Enhanced Receiver Techniques for Galileo E5 AltBOC Signal ...

Enhanced Receiver Techniques for Galileo E5AltBOC Signal Processing

By

Nagaraj Channarayapatna Shivaramaiah

A thesis submitted for the degree of

Doctor of Philosophy

Surveying & Spatial Information Systems,

The University of New South Wales.

June 2011

Abstract

In Global Navigation Satellite Systems (GNSS) the structure of the signal predom-

inately determines the system performance and wideband signals, in general, o�er good

performance. Consequently signal design and the receiver processing for wideband signals

have attracted signi�cant research attention in recent years. The L5/E5 frequency band

has been exploited for such high performance wideband signals as a part of the process of

GNSS modernisation. Due to their appealing performance, the wideband signals are likely

to be used in applications that demand high ranging accuracy.

The wideband Alternate Binary-O�set-Carrier (AltBOC) modulation is the most soph-

isticated among the GNSS signals. The receiver baseband signal processing has to overcome

several challenges before maximising the bene�ts o�ered by AltBOC modulation in terms

of computational complexity, resource utilisation, and power consumption. This disser-

tation proposes e�cient acquisition, tracking and multipath mitigation techniques for the

L5/E5 band signals, speci�cally the AltBOC(15,10). The signal detection probability and

mean acquisition time performance of di�erent acquisition strategies for AltBOC(15,10)

are analysed and a new acquisition method is proposed to increase the detection probabil-

ity, reduce the mean acquisition time, and reduce the computational resource requirement.

A generalised tracking architecture is described and a hybrid tracking architecture which

maximises the received signal energy, and hence the tracking performance is proposed. An

innovative Sideband-Carrier-Phase-Combination (SCPC) method is proposed to reduce

the pseudorange multipath to less than one metre. Results from simulation as well as real

test signals are provided to verify the proposed algorithms. In addition, results from an

FPGA-based implementation are provided to validate the complexity reduction and power

consumption bene�t claims of these new methods.

Learning from the drawbacks of the AltBOC modulation as far as the reduction in

algorithm simplicity is concerned, this dissertation proposes a new modulation scheme

called Time-Multiplexed O�set-Carrier QPSK (TMOC-QPSK) that behaves exactly like

an AltBOC, yet is �receiver friendly�. A generalisation of the TMOC-QPSK, referred to

as Time-Multiplexed-Multi-Carrier (TMMC) modulation, is presented and its potential

in dealing with radio frequency interference and frequency selective propagation delay

distortion is discussed.

i

Acknowledgements

I would like to express my deepest and most sincere gratitude to my supervisor,

Professor Andrew Graham Dempster for providing me an excellent opportunity to

work under his supervision. I will always remember his understanding, receptiveness

and professional approach that bene�ted this thesis. I am really thankful for all the

time he has spent guiding me and for what I have learnt from him during the course

of this research.

I am equally grateful to my co-supervisor and Head of School Professor Chris

Rizos for all the opportunities he has o�ered to me. His encouragement and support

towards ensuring a smooth �ow of the research activities helped me go that extra

mile. Thanks to both of my supervisors for all the trust they have put in my

work over the past years, providing me research assistantship, and providing me

opportunities to attend several conferences where I was able to discuss my research

and get fruitful feedback.

Sincere thanks to Associate Professor Dennis Akos for helping me collect the

data using the equipment available at the University of Colorado, Boulder. I will

never be able to forget my interaction with him though it was for a short duration.

Likewise, my sincere thanks to Dr. Sanjeev Gunawardena from Ohio University for

providing me with raw signal data sets that were useful for some of the experiments

in this thesis.

Sincere thanks to Prof. Letizia Lo Presti and Tung Hai Ta from Politecnico di

Torino, for their contribution to a collaborative work that supported a part of this

thesis. It was an excellent experience interacting with them and getting to know

their research methodology.

This research has been supported by funding from a variety of sources: The

Australian Research Council funding under the Discovery Project DP0556848, the

U.S. Institute of Navigation (ION) student paper award, the UNSW Dean's Excel-

lence award in Postgraduate Research via �rst prize in �The Digital Future� cat-

egory, and the UNSW Postgraduate Research Student Support travel sponsorship.

First prize awards at the 2009 European Satellite Navigation Competition (Baden-

Württemberg region) and at the 2008 GNSS summer school project added to my

motivation.

iii

iv ACKNOWLEDGEMENTS

I am also grateful to NewSouth Innovations for thoroughly scrutinising the out-

come of a part of my PhD research work for the suitability of a patent and sub-

sequently agreeing to fully sponsor an international patent application under Patent

Cooperation Treaty.

Warm thanks to my colleagues and friends: Peter Mumford, Kevin Parkinson,

Eamonn Glennon, Ravindra Swarna Babu, Fabrizio Tappero, Anthony Cole, Nonie

Politi, Sana Qaisar, Omer Mubarak, Faisal Khan, Jinghui Wu, Mazher Chowdhury

and all the other members of the Satellite Navigation and Positioning group for their

various kinds of support to overcome any di�culties during the PhD period.

I cannot omit to thank Prof. Eliathamby Ambikairajah, Head of School, Elec-

trical Engineering & Telecommunications UNSW and Dr. Oliver Diessel, School of

of Computer Science and Engineering UNSW for providing me teaching assistance

opportunities that helped �ll the �nancial gap that I needed to support my family

during the PhD period.

I would never have �nished the doctoral thesis without the support of my wife,

Ramya. I am especially grateful to her for her love and patience during all the past

years of hard work. I am equally thankful to my four year old son Praniil who used

to see me many days only at wake-ups. At times, I wondered I could have spent

more time playing with him if I had a similar cognitive learning pace as him during

the PhD period.

I feel a deep sense of gratitude for my mother, Savithri, for teaching me the

things that really matter in life, for being a constant source of encouragement in all

my endeavours, and for praying everyday for my success. I am deeply thankful for

my late father, Shivaramaiah for satisfying my thirst for knowledge always I wanted

to understand something, I feel the same warmth even after 10 years.

Finally, I feel proud to acknowledge the patience and continuous prayers of my

brother and his family, my in-laws and other family members for my success. I am

thankful for all their unconditional support, love and understanding.

Contents

Abstract i

Acknowledgements iii

Abbreviations and Symbols ix

List of Figures xiii

List of Tables xxi

Chapter 1. Introduction 1

1.1. History of the Global Positioning System 1

1.2. History of GLObal'naya NAvigatsionnaya Sputnikovaya Sistema 2

1.3. GPS and GLONASS Modernisation 2

1.4. Galileo 4

1.5. A Brief Overview of the Galileo E5 AltBOC Signal 4

1.6. Motivation and Objectives 6

1.7. Contributions 8

1.8. Structure of the Thesis 9

1.9. Publication Cross Reference Matrix 11

Chapter 2. Galileo E5 Signal and the Related Work 15

2.1. Introduction 15

2.2. GNSS Transmitted Signal structure 15

2.3. Galileo E5 AltBOC Signal Structure 16

2.4. The Correlation Function 22

2.5. GNSS Receiver Architecture 25

2.6. Signal Acquisition and Tracking: The Basics 26

2.7. Galileo E5 Signal Acquisition 33

2.8. Galileo E5 Signal Tracking 40

2.9. Multipath Mitigation in Galileo E5 43

2.10. Galileo E5 Baseband Hardware 46

2.11. Multiplexing in GNSS Modulations 48

2.12. Summary 50

Chapter 3. Experimental Setup 51

v

vi CONTENTS


3.2. Data Collection Apparatus 51

3.3. Summary 58

Chapter 4. Galileo E5 Signal Acquisition 61


4.2. Galileo E5 Acquisition Strategies 62

4.3. Acquisition Complexity and the Code Search Step Size 68

4.4. Considerations for the Cell Correlation E�ect 69

4.5. |V E2 + P 2| method for AltBOC 72

4.6. Envelope and Squared Envelope Detectors 80

4.7. Exploiting Secondary Codes to Increase Acquisition Performance 85

4.8. Summary 96

Chapter 5. Galileo E5 Signal Tracking 99


5.2. A Generalised Tracking Architecture 100

5.3. Candidate Local Reference Signals 104

5.4. Issues Related to the Di�erent Architectures 106

5.5. Hybrid Tracking Loop Architectures 108

5.6. An Extended Tracking Range DLL 123

5.7. Summary 131

Chapter 6. Galileo E5 Code Phase Multipath Mitigation 133


6.2. Performance of the Direct AltBOC Tracking Architecture 133

6.3. SCPC Method and an Architecture 138

6.4. Simulation and Test Results 149

6.5. A Group Delay Compensation Viewpoint for the SCPC Method 152

6.6. Summary 162

Chapter 7. Galileo E5 Baseband Hardware 165


7.2. GNSS Receiver Model and Search Dimensions 165

7.3. FFT Requirements for New GNSS Signals 168

7.4. The Proposed FFT Based Code Correlation Approach 171

7.5. Computational Complexity of the Proposed Approach 173

7.6. Implementation and Resource Utilisation on an FPGA 176

7.7. Case Studies and Discussion 178

7.8. E�cient Design of Core Correlator Blocks for Tracking 182

7.9. Summary 189

CONTENTS vii

Chapter 8. Time-Multiplexed O�set-Carrier QPSK for GNSS 193


8.2. Complexities with the AltBOC Modulation 194

8.3. Time-Multiplexed Modulations 196

8.4. Time-Multiplexed O�set-Carrier QPSK : The Signal Structure 200

8.5. Correlator Architecture for the TMOC-QPSK Signal 207

8.6. Resource Utilisation and Power Consumption 212

8.7. On E�cient Wideband GNSS Signal Design 218

8.8. Summary 232

Chapter 9. Conclusions and Recommendations 235

9.1. A Review of the Objectives 235

9.2. Acquisition, Tracking and Multipath Mitigation 236

9.3. Baseband Hardware Complexity 238

9.4. TMOC-QPSK and TMMC Modulation Schemes 240

9.5. Recommendations for Future Work 241

Bibliography 243

Appendix A. Fundamentals of AltLOC and AltBOC-NCE modulation 255

Appendix B. Signi�cance of the Product Signal in AltBOC(15,10) 261

Appendix C. Factorisation of the FFT Transform Lengths 265

Appendix D. Frequency Selective Propagation Delay Distortions 267

Appendix E. Envelope and Squared Envelope Detectors 269

Appendix F. Code Phase Jitter for the Generalised Tracking Architecture 271

Appendix G. EMLP Discriminator Function for AltBOC Signals in Multipath273

Appendix H. Carrier Phase Multipath Error for AltBOC Signals 277

Appendix I. Group Delay Error Caused by Multipath 279

Appendix J. Power Spectral Density of TMOC-QPSK 281

Appendix K. Output of the Correlator and Reference Signal Correlations 283

Appendix L. Approximation Tables 285

Abbreviations and Symbols

ACF Autocorrelation Function

ADC Analogue-to-Digital-Converter

AltBOC Alternate Binary O�set Carrier

AltLOC Alternate Linear O�set Carrier

ASIC Application Speci�c Integrated Circuit

AWGN Additive White Gaussian Noise

BOC Binary O�set Carrier

BPSK Binary PSK

C/A Coarse Acquisition

CC Cell Correlation

CC Central Carrier

CDMA Code Division Multiple Access

CDMA Multi-carrier CDMA

CEML Coherent EML

CW Continuous Wave

DME Distance Measuring Equipment

DS-SS Direct Sequence Spread Spectrum

DSB Dual (or Double) Sideband

EDA Electronic Design Automation

EML Early Minus Late

EMLP Non-coherent EML Power

ENC European Navigation Conference

FFT Fast Fourier Transform

FIC Full-band Independent Code

FLL Frequency-Locked Loop

FPGA Field Programmable Gate Array

GIOVE Galileo In-orbit Validation Element

GNSS Global Navigation Satellite System

GPS Global Positioning System

HDL Hardware Description Language

ID Integrate and Dump

IF Intermediate Frequency

ix

x Abbreviations and Symbols

ION Institute of Navigation

LE Logic Element

LNA Low Noise Ampli�er

LOS Line-Of-Sight

LUT Look-Up-Table

MF Matched Filter

NCE Non-constant envelope

NCO Numerically Controlled Oscillator

NLOS non-LOS

OC O�set Carrier

OFDM Orthogonal Frequency Division Multiplexing

OFDMA Orthogonal Frequency Division Multiple Access

OS Open Service

PC Pre-correlation

PLL Phase-Locked Loop

PNT Position Navigation and Timing

PRN Pseudo-Random Noise

PSD Power Spectral Density

PSK Phase Shift Keying

PVT Position-Velocity-Time

QPSK Quadrature Phase Shift Keying

RCPO Residual Code-Phase P�set

RF Radio Frequency

SBT Sideband Translation

SCPC Sideband Carrier-Phase Combination

SDR Software-De�ned Radio

SMR Singal-to-Multipath-Ratio

SNR Signal-to-Noise-Ratio

SPC Sub-carrier Phase Cancellation

SSB Single Sideband

TACAN Tactical Air Navigation

TMMC Time-mulriplexed multi-carrier

TMOC Time-Multiplexed O�set-Carrier

TRIGR Transform-domain Instrumentation GPS Receiver

USRP Universal Software Radio Peripheral

VCO Voltage Controlled Oscillator

DLL Delay-Locked Loop

DoD Department of Defense

XOR Exclusive-OR

Abbreviations and Symbols xi

δ Correlator chip spacing (chips)

δf Frequency step size (Hz)

δt Code delay step size (chips)

η Detection threshold

ωd Angular Doppler frequency estimate (rads/s)

ω0 Angular Intermediate frequency including Doppler frequency (rads/s)

ωc Angular carrier frequency (rads/s)

ωd Angular Doppler frequency (rads/s)

ωIF Angular Intermediate frequency (rads/s)

T acq Mean acquisition time

σ2φ Carrier phase error variance

σ2ε Code phase error variance

BL Loop noise bandwidth (Hz)

C Speed of light (m/s)

c(t) Primary spreading code sequence

C/N0 Carrier-to-Noise Density (dB-Hz)

cs(t) Secondary spreading code sequence

d(t) Navigation data sequence

Dε Code discriminator function (ideal)

e(t) Tiered spreading code sequence

fc Carrier frequency (Hz)

fco Code chipping rate (Hz)

fd Doppler frequency (Hz)

fIF Intermediate frequency (Hz)

fsc Subcarrier frequency (Hz)

fs Sampling frequency (Hz)

G(f) Power spectral density

L Primary spreading code repetition length (chips)

Ls Secondary spreading code repetition length (chips)

N0 Thermal noise density

Nc Coherent integration length (chips)

nW (t) White noise

Pd Probability of detection

Pfa Probability of false alarm

PT Transmitted signal power (W)

R(t) Autocorrelation function

rIF (t) Received IF signal

S(t) Transmitted signal

s(t) Baseband or Modulating signal

xii Abbreviations and Symbols

sc(t) Real part of the baseband signal

ss(t) Imaginary part of the baseband signal

Tc Code chip duration (s)

Td Data bit duration (s)

tg Group delay

tp Phase delay

Tcoh Coherent integration time (s)

Ts Sampling period (s)

List of Figures

1.1 E5 Signal Spectrum Representation . . . . . . . . . . . . . . . . . . . 5

1.2 Flow diagram showing the thesis structure . . . . . . . . . . . . . . . 10

1.3 Publication timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1 AltBOC multiplexer illustration . . . . . . . . . . . . . . . . . . . . . 17

2.2 AltBOC sub-carrier waveforms . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Constellation diagram of the constant envelope AltBOC signal. . . . 20

2.4 Tiered code generation . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 PSD of the constant envelope AltBOC(15,10). . . . . . . . . . . . . . 23

2.6 ACF of (a)Truly random sequence, (b)Maximal length sequence . . 24

2.7 Normalised autocorrelation value obtained using (2.19), (2.20) and

un�ltered GIOVE-A PRN 51 E5 codes. . . . . . . . . . . . . . . . . . 25

2.8 Typical architecture of a GNSS receiver . . . . . . . . . . . . . . . . . 26

2.9 Receiver search space . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.10 Correlator structure for acquisition - conventional scheme . . . . . . 28

2.11 Acquisition output illustration . . . . . . . . . . . . . . . . . . . . . . 29

2.12 FFT method of code acquisition in GNSS receivers . . . . . . . . . . 30

2.13 Typical tracking architecture . . . . . . . . . . . . . . . . . . . . . . . 32

2.14 Normalised autocorrelation value of the un�ltered GIOVE-A PRN

51 E5 code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.15 Autocorrelation of the GIOVE-A wideband E5 signal . . . . . . . . . 34

2.16 Correlation functions of di�erent components of the E5 signal . . . . 35

2.17 Normalised correlation value for E5a-Q code of GIOVE-A PRN 51 . 37

2.18 Categorisation of the code phase multipath mitigation methods . . . 44

2.19 Methods to reduce the FFT computational load . . . . . . . . . . . . 47

3.1 Overview of the simulation / experimental setup . . . . . . . . . . . . 52

3.2 GeNeRx1 receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 USRP2 data collection setup . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Averna setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 TRIGR front-end architecture. . . . . . . . . . . . . . . . . . . . . . . 57

xiii

xiv LIST OF FIGURES

3.6 TRIGR GNSS front-end unit . . . . . . . . . . . . . . . . . . . . . . . 57

4.1 Examples of search strategy based methods for acquisition . . . . . . 64

4.2 Normalised absolute correlation values for di�erent search strategies 66

4.3 |V E2 + P 2| method for AltBOC(15,10) . . . . . . . . . . . . . . . . . 67

4.4 E�ect of code search step size on the correlation value; worst case

and best case for AltBOC(15,10) and BPSK(n) ACFs . . . . . . . . . 69

4.5 E�ect of code search step size on the correlation values including

|V E2 + P 2| method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.6 Direct AltBOC acquisition architecture . . . . . . . . . . . . . . . . . 74

4.7 Direct AltBOC acquisition architecture with |V E2 + P 2| method;

speci�c sampling frequency . . . . . . . . . . . . . . . . . . . . . . . . 75

4.8 Direct AltBOC Acquisition Architecture with |V E2 + P 2|; Arbi-

trary (Valid) sampling frequency . . . . . . . . . . . . . . . . . . . . . 75

4.9 Error in correlation value computed using (2.19), (2.20) with respect

to that of GIOVE-A PRN 51 . . . . . . . . . . . . . . . . . . . . . . . 76

4.10 Worst case probability of detection for BPSK and AltBOC . . . . . . 78

4.11 Average probability of detection for BPSK and AltBOC . . . . . . . 78

4.12 Average Pd for di�erent acquisition approaches . . . . . . . . . . . . . 79

4.13 Average Pd for AltBOC and |V E2 + P 2| methods . . . . . . . . . . . 79

4.14 Worst case Pd for AltBOC and |V E2 + P 2| methods. . . . . . . . . . 80

4.15 T acq for the average Pd scenario. . . . . . . . . . . . . . . . . . . . . . 80

4.16 T acq for the worst case Pd scenario . . . . . . . . . . . . . . . . . . . . 81

4.17 Correlation waveforms, in�nite bandwidth . . . . . . . . . . . . . . . 82

4.18 Correlation loss, in�nite bandwidth . . . . . . . . . . . . . . . . . . . 83

4.19 Correlation waveforms, 50MHz bandwidth . . . . . . . . . . . . . . . 83

4.20 Correlation loss, 50MHz bandwidth . . . . . . . . . . . . . . . . . . . 84

4.21 Detector architecture, DA-envelope method . . . . . . . . . . . . . . . 84

4.22 Probability of detection, Nnc=1 (top), Nnc=4 (Middle), Nnc=8 (bot-

tom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.23 Pd for δt=0.85 (DA-squared envelope) and δt=1.0 (DA-envelope);

Nnc=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.24 T acq comparison; Nnc=1 . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.25 Autocorrelation plot of the secondary code CS251 . . . . . . . . . . . 89

4.26 Autocorrelation plot of the secondary code CS1001 . . . . . . . . . . 89

4.27 CLs for the Galileo E5 secondary codes . . . . . . . . . . . . . . . . . 91

4.28 Histogram of the CLs of E5 secondary codes . . . . . . . . . . . . . . 92

LIST OF FIGURES xv

4.29 Proposed system model for two cases; case 1: Nc � L, case 2:

Nc > L or Nc ≈ L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.30 Correlation value trend for increasing number of primary code pe-

riod integrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.31 Correlation values for all the secondary code hypotheses . . . . . . . 95

4.32 Absolute correlation value of the E5 signal; 1ms integration . . . . . 96

4.33 Absolute correlation value of the E5 signal; 4ms integration using

the secondary code chip position detection algorithm . . . . . . . . . 96

5.1 Generalised architecture for the E5 signal tracking . . . . . . . . . . . 101

5.2 Illustration of two types of data bit ambiguities . . . . . . . . . . . . 107

5.3 Coherent pilot signal tracking and aiding the data demodulation . . 110

5.4 A quasi-coherent (data wipe-o�) architecture . . . . . . . . . . . . . . 111

5.5 Correlation values with individual reference signals (top); with com-

bined reference signals (bottom) . . . . . . . . . . . . . . . . . . . . . 114

5.6 Cross correlation between the di�erent reference signal combina-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.7 Carrier phase error standard deviation for di�erent signal compo-

nents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.8 Code tracking error standard deviation for di�erent signal compo-

nents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.9 Tracking loop output parameters for 8-PSK-like tracking (no data

wipe-o�): (Data set-I) . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.10 Prompt correlation output for quasi-coherent E5, E5p and E5-PC

tracking methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.11 Tracking loop output parameters for quasi-coherent E5, E5p, and

E5-PC tracking methods; for Data set�I . . . . . . . . . . . . . . . . . 120

5.12 Tracking loop output parameters for quasi-coherent E5, E5p, and

E5-PC tracking methods; for Data set-II . . . . . . . . . . . . . . . . 120

5.13 Data bit demodulation with E5p tracking: (Data set-I) . . . . . . . . 121

5.14 Data bit demodulation with E5-PC tracking: (Data set-I) . . . . . . 121

5.15 Data bit demodulation with E5p tracking: (Data set-II) . . . . . . . 122

5.16 Data bit demodulation with E5-PC tracking: (Data set-II) . . . . . . 122

5.17 Galileo E5 correlation waveform; di�erent �lter bandwidths . . . . . 123

5.18 S-curve for the E5 8-PSK tracking (top) and BPSK(10) tracking . . 124

5.19 Receiver model with the proposed architecture . . . . . . . . . . . . . 125

5.20 Illustration of the proposed method; δc is the crossover point. . . . . 125

xvi LIST OF FIGURES

5.21 8-PSK AltBOC tracking without introducing any error . . . . . . . . 129

5.22 BPSK(10) E5ab tracking without introducing any error . . . . . . . . 129

5.23 8-PSK AltBOC tracking; error introduced from 60-105 ms . . . . . . 130

5.24 8-PSK AltBOC tracking with the hybrid DLL method; error intro-

duced from 60-105 ms . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.1 Generalised tracking loop architecture for the Galileo E5 signal . . . 135

6.2 Code multipath error envelope of E5a and E5 correlators with CEML

discriminator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.3 Carrier phase multipath error comparison . . . . . . . . . . . . . . . . 137

6.4 Envelope of the attenuation for E5a, E5b, and E5 signals under

multipath conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.5 Carrier phase error for E5, E5a and E5b under multipath conditions 139

6.6 Code pseudorange error for E5, E5a and E5b under multipath con-

ditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.7 Di�erence in carrier phase errors: E5-E5a, E5b-E5 and E5b-E5a . . 141

6.8 Di�erence of carrier phase error and code pseudorange error under

multipath conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.9 Multipath a�ected signal strength for E5, E5a and E5b signals . . . 142

6.10 Di�erence in SNRs of the received signals, E5-E5a, E5b-E5 and

E5b-E5a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.11 Architecture for the SCPC method . . . . . . . . . . . . . . . . . . . . 144

6.12 Illustrating the formation of the 8PSK / AltBOC-like correlation

function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.13 Illustrating the e�ect of phase shift while multiplying a sine wave

and a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.14 Illustration of the e�ect of di�erent phase shifts of the sine wave

due to the multiplication. . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.15 Test setup (a) simulation (b) with real satellite signal . . . . . . . . . 149

6.16 Code phase error (top) and di�erence in carrier phase errors of E5a

and E5b (bottom), for di�erent multipath delays; from simulation. . 150

6.17 Code phase error (top) and di�erence in carrier phase errors of E5a

and E5b (bottom), for di�erent multipath delays; with real signal . . 150

6.18 Error in corrected code phase estimate for di�erent multipath de-

lays; from simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.19 Error in corrected code phase estimate for di�erent multipath de-

lays; with real signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

LIST OF FIGURES xvii

6.20 Code multipath comparison; Standard, Narrow and SCPC . . . . . . 152

6.21 Phase delay and group delay vs frequency around the E5 band for

di�erent multipath delays; no ionospheric errors . . . . . . . . . . . . 154

6.22 Di�erence in the phase delays in E5a and E5b with respect to E5

(top); di�erence of the two curves in the top �gure (bottom) . . . . . 156

6.23 Di�erence in the correlation values in E5a and E5b signal compo-

nents with respect to Ionosphere free situation (top); di�erence of

the two curves in the top �gure (bottom) . . . . . . . . . . . . . . . . 157

6.24 Di�erence of E5a and E5b phase and group delays for di�erent mul-

tipath delays (analytical); single re�ected signal case; A=0.5; . . . . 157

6.25 Envelope of the group delay error due to multipath . . . . . . . . . . 158

6.26 Di�erence of E5a and E5b phase and group delays for di�erent mul-

tipath delays (analytical); single re�ected signal; A=0.5 . . . . . . . . 159

6.27 Phase delay and group delay for E5a, E5b and E5 frequencies under

multipath condition for di�erent ionospheric delay . . . . . . . . . . . 160

6.28 Phase delay and group delay di�erences at di�erent ionospheric de-

lays and multipath delays . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.29 Multipath mitigation with the simulated signal with multipath at

5.4m and ionospheric delays of 50m and 100m at E5. . . . . . . . . . 163

6.30 Multipath mitigation with the pseudo-real signal with multipath at

5.4m and ionospheric delays of 50m and 100m at E5. . . . . . . . . . 163

7.1 Block diagram of a multi-band receiver . . . . . . . . . . . . . . . . . 166

7.2 Search dimensions in a (a) single-band GNSS receiver (b) multi-

band GNSS receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.3 GNSS signals in the Galileo and GPS bands (from (OSSISICD,

2010)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7.4 Number of real additions comparison for FFT of di�erent GNSS

signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.5 Number of real multiplications comparison for FFT of di�erent

GNSS signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

7.6 Example of Mixed-radix method for a 2048-point FFT . . . . . . . . 177

7.7 Example of the signal-time-sharing FFT architecture for Combination-

I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

7.8 Comparison of number of LEs for di�erent signal combinations . . . 180

7.9 Comparison of number of multipliers for di�erent signal combina-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

xviii LIST OF FIGURES

7.10 Acquisition results for the GPS L1 C/A signal; PRN 17; 2048-point

FFT; 1ms integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.11 Acquisition results for the GIOVE-A E1 C signal; 16364-point FFT

realised using standard approach and the proposed Mixed-radix

(2*8*1024) approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.12 A functional diagram of the baseband hardware . . . . . . . . . . . . 183

7.13 Realisation of the core correlator block for the GPS L1 C/A signal . 185

7.14 Local reference mixer for the complex modulation signals . . . . . . . 186

7.15 Ratio of the power estimate for new signals with respect to GPS L1

C/A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.16 Power consumption of the entire baseband circuit . . . . . . . . . . . 189

7.17 Power consumption for di�erent multi-signal con�gurations. . . . . . 190

8.1 A generalised tracking architecture for AltBOC signals . . . . . . . . 195

8.2 Time-multiplexing methods to construct the baseband signal (a)

spreading codes with optional data are time-multiplexed; (b) sub-

carriers are time-multiplexed; and (c) spreading codes with sub-

carriers are time-multiplexed . . . . . . . . . . . . . . . . . . . . . . . 196

8.3 Illustration of phase points in π4-QPSK modulation . . . . . . . . . . 198

8.4 Code-multiplexing in GPS L2C signal . . . . . . . . . . . . . . . . . . 199

8.5 The proposed L1C pilot code generation scheme using the TMBOC

technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

8.6 Time-multiplexing methods and the corresponding sub-carrier wave-

form: (a) TMOC-QPSK-ab multiplexing method; (b) TMOC-QPSK-

IQ; (c) one cycle of sub-carrier waveform . . . . . . . . . . . . . . . . 201

8.7 TMOC-QPSK and TMOC-π4-QPSK transmitted signal . . . . . . . . 203

8.8 PSD of the AltBOC, AltBOC-NCE, TMOC-QPSK and TMOC-π4-

QPSK; analytical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

8.9 PSD of the AltBOC, AltBOC-NCE, TMOC-QPSK and TMOC-π4-

QPSK; simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

8.10 Correlator architecture to process the wideband TMOC-QPSK sig-

nal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

8.11 Incoming and reference sub-carrier correlations for AltBOC, TMOC-

QPSK, TMOC-π4-QPSK and AltLOC modulations. . . . . . . . . . . 210

8.12 Normalised auto-correlation waveforms of AltBOC,TMOC-QPSK,TMOC-π4-QPSK and AltBOC-NCE; in�nite bandwidth; simulation . . . . . 210

LIST OF FIGURES xix

8.13 Normalised auto-correlation waveforms of AltBOC,TMOC-QPSK,TMOC-π4-QPSK and AltBOC-NCE; 50 MHz bandwidth; simulation . . . . . 211

8.14 Correlator architecture to process the individual signal components

(a and b) in TMOC-QPSK modulation . . . . . . . . . . . . . . . . . 212

8.15 Output of the correlator for independent sideband processing; sim-

ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

8.16 Direct computation method of AltBOC reference signal generation . 214

8.17 LUT method of AltBOC reference signal generation . . . . . . . . . . 214

8.18 Direct computation method of TMOC-QPSK reference signal gen-

eration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

8.19 LUT method of TMOC-QPSK reference signal generation . . . . . . 215

8.20 Direct computation and LUT method of code mixer implementation

in AltBOC modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

8.21 Code mixing operation in TMOC-QPSK modulation . . . . . . . . . 216

8.22 Possible AltBOC signals within a 20.46MHz band . . . . . . . . . . . 221

8.23 Correlation functions for BPSK(10), AltBOC(5,5) and AltBOC(9,1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

8.24 A 20.46MHz band used with AltBOC variants; odd m . . . . . . . . 222

8.25 Combined correlation waveform of AltBOC variants; odd m . . . . . 223

8.26 A 20.46MHz band used with AltBOC variants; even m . . . . . . . . 223

8.27 Combined correlation waveform of AltBOC variants; even m . . . . . 224

8.28 A 20.46MHz band used with AltBOC variants; both odd and even m224

8.29 Combined correlation waveform of all AltBOC variants; both odd

and even m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

8.30 AltBOC(5,1) covering a 20.46MHz band in the �scan type� time-

multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

8.31 An illustration of time-multiplexing with multiple sub-carriers: �spread

type� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

8.32 An illustration of time-multiplexing with multiple sub-carriers: �scan

type� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

8.33 Code and sub-carrier for a partial sequence of TMMC(10,1); only

the real component of the complex sub-carrier is shown . . . . . . . . 228

8.34 Comparison of code multipath error-envelope for TMMC(10,1) and

BPSK(10) signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

8.35 Illustration of CW interference a�ecting only one sub-band. . . . . . 230

8.36 DME/TACAN interference at the gaps between sub-bands . . . . . . 231

xx LIST OF FIGURES

8.37 Illustration of forming a phase delay pro�le with the aim of aiding

the compensation of group delay at the centre of the band. . . . . . . 232

A.1 spectrum of the cosine-AltLOC . . . . . . . . . . . . . . . . . . . . . . 256

A.2 spectrum of the sine-AltLOC . . . . . . . . . . . . . . . . . . . . . . . 257

A.3 subcarrier in AltBOC-NCE . . . . . . . . . . . . . . . . . . . . . . . . 258

A.4 AltBOC-NCE modulation : constellation diagram . . . . . . . . . . . 259

A.5 PSD of the constant envelope AltBOC(15,10) . . . . . . . . . . . . . 259

B.1 PSD of the AltBOC-NCE(15,10) in a wider frequency range . . . . . 261

B.2 ACF of the product signal . . . . . . . . . . . . . . . . . . . . . . . . . 262

B.3 ACF of the AltBOC(15,10) signal with and without the product

signal with in�nite bandwidth. . . . . . . . . . . . . . . . . . . . . . . 263

B.4 ACF of the AltBOC(15,10) signal with and without the product

signal with 70 MHz bandwidth . . . . . . . . . . . . . . . . . . . . . . 263

C.1 Prime factor FFT approach . . . . . . . . . . . . . . . . . . . . . . . 265

C.2 Mixed radix FFT approach . . . . . . . . . . . . . . . . . . . . . . . . 266

E.1 Typical quadrature detector . . . . . . . . . . . . . . . . . . . . . . . . 269

List of Tables

1.1 E5 Signal Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Publication vs. Chapter cross reference matrix . . . . . . . . . . . . . 13

1.2 Publication vs. Chapter cross reference matrix (contd...) . . . . . . . 14

2.1 AltBOC sub-carrier coe�cients . . . . . . . . . . . . . . . . . . . . . 19

2.2 Galileo E5 OS signal code structure . . . . . . . . . . . . . . . . . . . 21

2.3 Galileo E1 Open Service signal code structure (from OSSISICD

(2010)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1 Summary of the resource usage in search strategy based schemes . . 67

4.2 E�ect of the cell correlation phenomenon on the performance . . . . 72

4.3 CLs for Galileo secondary codes . . . . . . . . . . . . . . . . . . . . . 91

5.1 Possible reference signals with the SBT method . . . . . . . . . . . . 104

5.2 Possible reference signals with the FIC method. . . . . . . . . . . . . 105

5.3 Possible reference signals with the 8-PSK-like method . . . . . . . . . 106

5.4 Indicative performance of di�erent tracking architectures . . . . . . . 109

5.5 Summary of the hybrid tracking architectures . . . . . . . . . . . . . 115

5.6 Performance comparison of the hybrid tracking architectures . . . . 118

7.1 GPS and Galileo signal parameters of interest . . . . . . . . . . . . . 169

7.2 Transform length requirements Case 1 � 0.5 chip step . . . . . . . . . 169

7.3 Transform length requirements Case 2 � other chip steps . . . . . . . 170

7.4 Transform length requirement summary . . . . . . . . . . . . . . . . . 170

7.5 1023 point FFT factorisation . . . . . . . . . . . . . . . . . . . . . . . 171

7.6 Transform length factorisation . . . . . . . . . . . . . . . . . . . . . . 171

7.7 FFT blocks required for GNSS signals in consideration . . . . . . . . 172

7.8 Complexity of small-point blocks . . . . . . . . . . . . . . . . . . . . . 172

7.9 Operation count for 1023 and 1024 point FFTs . . . . . . . . . . . . . 173

7.10 Revised transform lengths for di�erent signals . . . . . . . . . . . . . 173

7.11 FFT blocks required for GNSS signals in consideration � revised. . . 173

7.12 Computational complexity comparison. . . . . . . . . . . . . . . . . . 174

xxi

xxii LIST OF TABLES

7.13 Operations count for the correlator employing time-based (2046-

tap) and FFT-based (2048-points) methods . . . . . . . . . . . . . . . 176

7.14 FPGA resource utilisation for the basic building blocks . . . . . . . . 177

7.15 FPGA resource utilisation for 1024-point FFT . . . . . . . . . . . . . 177

7.16 FPGA resource utilisation for di�erent transform lengths . . . . . . . 178

7.17 Some new GNSS signals and their parameters of interest . . . . . . . 182

7.18 Resource utilisation and power consumption estimates of the core

correlator for di�erent signals . . . . . . . . . . . . . . . . . . . . . . . 187

8.1 Transmitted sub-carrier signal phases in the proposed modulation . . 204

8.2 Comparison of relative transmit signal power levels . . . . . . . . . . 206

8.3 Correlator complexity comparison summary for AltBOC and TMOC-

QPSK modulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

8.4 Logic resource and estimated power consumption for AltBOC and

TMOC-QPSK correlators . . . . . . . . . . . . . . . . . . . . . . . . . 217

8.5 Comparison summary of AltBOC, TMOC-QPSK and TMMC mod-

ulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

L.1 Scaled integer approximations for {±1.2071,±0.5} . . . . . . . . . . 285

L.2 Scaled integer approximations for {±1,±0.7071} . . . . . . . . . . . 285

CHAPTER 1

Introduction

Global Navigation Satellite System (GNSS) technology is rapidly penetrating

into day-to-day activities of our lives. Market researchers forecast 1.1 billion GNSS

system shipments by 2014 (ABIResearch, 2009) and revenue growing to US$330 bil-

lion by 2020 (Jacobson, 2007; GSA, 2009). The Global Positioning System (GPS)

modernisation e�orts, combined with the development of both supplementary and

complementary satellite navigation systems over the last few years and in the com-

ing decade, has brought a radical transformation in the potential of GNSS. GNSS

technology is like a highly reactive building block in chemistry (Arthur, 2009) that

drives, in unison with the evolution of other technologies, the creation of myriad

applications. Arguably, GNSS platforms, products and services that enable precise

position and seamless navigation are set to create a global technological revolution

during this decade, similar to the impact of the Internet and mobile phones in the

recent past.

1.1. History of the Global Positioning System

The Global Positioning System, originally designated as NAVSTAR (Navigation

System with Timing And Ranging) is a space-based radionavigation system devel-

oped by the United States Department of Defense (DoD)(USCG, Internet). GPS

provides all-weather, round-the-clock, worldwide reliable positioning, navigation and

timing (PNT) services free of cost to users. GPS is divided into three segments, the

space segment consisting of satellites and signals, the control segment and the user

segment consisting of GPS user receivers. The satellites broadcast navigation and

timing signals on di�erent frequencies in the L-band of the frequency spectrum that

are continuously monitored and controlled by the ground stations in the control seg-

ment. GPS receivers that receive and process the signals from at least four satellites

provide three dimensional position, velocity and time information to users. GPS

was o�cially declared to have met the requirements of Full Operational Capability

(FOC) in 1995 (TychoUSNO, Internetb).

GPS provides two types of services, the Precise Positioning Service (PPS), avail-

able to the U.S. military and other authorised users, and the Standard Positioning

1

2 1. INTRODUCTION

Service (SPS), originally designed to provide civil users with a less accurate posi-

tioning capability than PPS through the use of a technique known as Selective Avail-

ability (SA). SA was discontinued e�ective midnight May 1, 2000 (TheWhiteHouse,

2000), thus enabling the improved SPS performance that all users experience today.

At that time, the SPS was available only using the coarse acquisition (C/A) code

signal on the L1 frequency band centred at 1575.42 MHz with the Root-Mean-Square

(RMS) User Range Error (URE) performance of ≤6 metres across the constellation

(GPSSPS, 2001). In terms of positioning performance, this URE supported a global

average 95% error better than 13 metres (horizontal) and 22 metres (vertical) under

the speci�ed conditions detailed in GPSSPS (2001). The accuracy of the position

and timing solution is attributed to various error sources (Parkinson and Spilker,

1995; Kaplan and Hegarty, 2006).

1.2. History of GLObal'naya NAvigatsionnaya Sputnikovaya Sistema

GLObal'naya NAvigatsionnaya Sputnikovaya Sistema (English: Global Naviga-

tion Satellite System, GLONASS) is a space-based radionavigation system managed

by the Russian Space Forces(Ho�mann-Wellenhof et al., 2008). The system concept

and the purpose of use is similar to that of GPS. Though the constellation was

complete by 1995, the system rapidly fell into decay. However, over the last few

years new satellites have been progressively launched and FOC is expected some

time in 2011. GLONASS provides two types of services, an encrypted High Preci-

sion (HP) signal for military use and a Standard Precision (SP) signal for civil use.

The SP service was available only on the L1-frequency band with a signal struc-

ture di�erent to that of GPS L1 C/A. The GLONASS Interface Control Document

(GLOANSSICD, n.d.) initially speci�ed a position accuracy of 50-70 m horizontal

(99.7%) and 70 m vertical (99.7%) (Eissfeller et al., 2007), though several speci�ca-

tions exist (Ho�mann-Wellenhof et al., 2008). Due to the lack of global coverage,

the worldwide use of GLONASS was limited until very recently.

1.3. GPS and GLONASS Modernisation

One of the key objectives of the modernisation program was to support the user

segment with more reliable and better system performance metrics viz. accuracy,

availability, continuity and integrity (GPSJPO, 2000). Modernisation e�orts started

in the late 1990s, marked by several milestones in the following years that included

the use of the generic term GNSS. Some of the performance metrics were addressed

by the Satellite Based Augmentation Systems (SBAS) such as the U.S. based Wide

Area Augmentation System (WAAS) (WAASSPS, 2008). As with any system design

problem, there were di�erent aspects to the modernisation, such as requirements,

facilitators, constraints and innovations.

1.3. GPS AND GLONASS MODERNISATION 3

The requirements comprised of user demands, research thrusts, Government

policies and directives among others. The advancement of hardware/software and

atomic clock technologies and the opportunity to replace the old satellites were

among the facilitators. Some of the constraints were due to �nancial reasons,

while others were technical, e.g. backward compatibility. At around the same

time, the European Commission (EC) and the European Space Agency (ESA)

had initiated the development of a new system under the GNSS umbrella called

�Galileo�(EuropeanCommission, Internet). Starting with this, there were added con-

straints to the system design to address compatibility and interoperability among

GNSSs. As a result of the modernisation process, GNSS programmes bene�tted

from several technical innovations which harnessed e�cient signal structures and

additional signals to mitigate the errors, thus enabling better performance.

1.3.1. New GPS Civil Signals. The �rst result of the modernisation e�ort

for civilian users came with the launch of Block IIR-M (Replenishment-Modernised)

GPS satellites in late 2005 that transmitted a new civilian signal called L2C in the

L2 frequency band at 1227.6 MHz. Around the same time, new monitoring stations

were incorporated into the ground control segment. As of November 2010, there are

eight satellites transmitting the L2C signal. The most recent was the Block IIF-1

satellite, launched in May 2010 which also transmits the third civil signal in the L5

frequency band at 1176.45 MHz(TychoUSNO, Interneta). The latest performance

standard document (GPSSPS, 2008) speci�es the improved accuracy parameters for

a receiver using only the L1 C/A signal as ≤9 metres (horizontal) and ≤15 metres

(vertical). This improvement is a result of the GPS system's commitment to a

URE of ≤4 metres (RMS). A fourth civil signal called L1C which co-exists with

the L1 C/A signal in the same frequency band planned for the next generation GPS

satellites will further enhance system performance. A recent study (GPSGALPERF,

2010) predicts a 95% open sky accuracy of 1.22 m (horizontal), 2.11 m (vertical)

with the combination of current and future GPS civil signals in L1 and L5 bands.

1.3.2. New GLONASS Civil Signals. In 2001, the Russian government de-

cided to restore its system with a plan to complete the constellation by 2011. With

the launch of new longer life satellites, GLONASS is now able to achieve an accuracy

of 5-7 m (1σ, horizontal) (Oleynik, 2010). The new GLONASS K series satellites

with the plan of additional civil signals in the GLONASS L1, L2 and L3 frequency

band, are expected to o�er competitive performance to that of GPS (Cameron,

2010).

4 1. INTRODUCTION

1.4. Galileo

Currently under development, Galileo is Europe's GNSS. One unique feature of

Galileo is that it is under civilian control. The Galileo programme has two phases:

the In-Orbit Validation (IOV) phase and the Full Operational Capability (FOC)

phase. The de�nition, development and IOV phase of the Galileo programme were

carried out by ESA and the FOC phase is managed by the EC with ESA acting as a

design and procurement agent on behalf of the EC (EuropeanCommission, Internet).

The IOV phase, consisting of four satellites, is expected to be complete by the end

of 2011 and the last of the 18 satellites that provide intermediate initial operational

capability is expected to be launched in March 2014. The full system will consist of

30 satellites.

Galileo has �ve types of services planned. The Open Service (OS) consists of the

basic signals provided free-of-charge; the Safety-of-Life Service (SoLS) o�ered to the

safety-critical transport community; the Commercial Service (CS), providing higher

accuracy authenticated data; the Public Regulated Service (PRS), with controlled

access for speci�c users such as governmental agencies; and the Search And Rescue

Service (SARS). Two experimental satellites were launched in 2005 and 2008 known

as GIOVE-A (Galileo IOV Element) and GIOVE-B respectively. Galileo will have

passive hydrogen maser satellite clocks which will provide an order of magnitude

higher accuracy than the rubidium clocks used in other satellite systems.

The Galileo OS comprises of signals in the E1 frequency band (centred at the

GPS L1 frequency of 1575.42 MHz) and the E5 frequency band (centred at 1191.795

MHz). A recent user receiver test with simulated signals and scenarios (van den

Berg et al., 2010) demonstrated a 95% accuracy of 0.8 m (horizontal) and 1.02 m

(vertical) for the combination of E1 and E5 signals under speci�ed conditions. With

the compatibility and interoperability agreements among several systems, a Galileo

receiver will be able to take advantage of an increased number of satellites to improve

the performance.

1.5. A Brief Overview of the Galileo E5 AltBOC Signal

The Galileo E5 signal is by far the most sophisticated signal among all the sig-

nals used for GNSS. Like most of the GNSS signals, the Galileo E5 is a Direct

Sequence Spread Spectrum (DS-SS) with Code Division Multiple Access (CDMA)

signal. However, with four codes modulated onto the two phases of orthogonal com-

plex sub-carriers, the �rst two main lobes of the signal occupy 51.15 MHz bandwidth

centred at 1191.795 MHz. Galileo E5 signal employs a special modulation known

as Alternate Binary O�set Carrier (AltBOC) modulation to achieve this. The sub-

carrier waveforms are chosen so as to obtain a constant envelope at the transmitter.

1.5. A BRIEF OVERVIEW OF THE GALILEO E5 ALTBOC SIGNAL 5

1191

.795 M

Hz

1176

.45 M

Hz

1207

.140 M

Hz

E5a E5b

E5a

-I

E5b

-I

E5a-QE5b-Q

20.46 MHz20.46 MHz

51.15 MHz

f

Figure 1.1. E5 Signal Spectrum Representation

The result of this AltBOC modulation is a split spectrum around the centre fre-

quency as shown in Fig.1.1; the lower side band referred to as E5a and the upper

sideband referred to as E5b. In other words, each sideband comprises two di�er-

ent pseudorandom codes modulated onto the orthogonal components. The in-phase

components E5aI and E5bI carry the navigation data modulation. The quadrature

components E5aQ and E5bQ are pilot signals, i.e. they carry no data. Alternatively,

the complete modulation can be seen as an 8-PSK (Phase Shift Keying) modulation

(OSSISICD, 2010; GIOVEABICD, 2008; Issler et al., 2003).

The code chipping rate is 10.23 MHz and the sub-carrier frequency is 15.345

MHz, and hence the corresponding modulation used in Galileo E5 is denoted as

AltBOC(15,10). The Galileo E5 signal includes two independent navigation data

streams with a rate 12and constraint length 7 convolution encoding scheme. E5a

includes navigation data at 25 bits per second or bps (i.e. 50 symbols per second

or sps), and E5b includes navigation data at the 125 bps (i.e. 250 sps). The E5

AltBOC(15,10) has the largest bandwidth of any GNSS signal. The salient features

of the Galileo E5 signal are provided in Table 1.1.

AltBOC modulation belongs to the family of Binary O�set Carrier (BOC) mod-

ulations. The basic principle of BOC modulation is to reduce the width of the main

peak of the code correlation function without an unreasonable increase in the band-

width of the signal (which could be achieved simply by increasing the code chipping

rate). One of the attractive features of AltBOC is that the two sidebands can be

independently demodulated as BPSK (Binary PSK) signals. The Galileo E5 signal

o�ers unprecedented performance with a theoretical code tracking error less than 5

cm at signal strength of 35 dB-Hz (Sleewaegen et al., 2004). This performance is

6 1. INTRODUCTION

Table 1.1. E5 Signal Parameters

Parameter(Notation) Value / Function Units

Carrier frequency (fc) 1191.795 MHzCode frequency(fco) 10.23 MHz

Sub-carrier frequency(fsc) 15.345 MHzSub-carrier waveform Special

Code length 10230 chipsModulation AltBOC(15,10)

Lower sideband centre frequency(fca) 1176.45 MHzUpper sideband centre frequency(fcb) 1207.14 MHz

Bandwidth (E5) 51.15 MHzBandwidth (E5a, E5b) 20.46 MHz

Minimum Received Power Level -155 dBW

way beyond that o�ered by any other existing or planned GNSS signal.

1.6. Motivation and Objectives

The Galileo E5 AltBOC(15,10) signal plays a major role in achieving the ex-

tremely good 0.8 metres horizontal and 1.02 metre vertical positioning performance

predictions mentioned earlier in this chapter. Therefore it is of interest for the re-

ceiver designers to develop e�cient signal processing methods that help actualising

these performance �gures.

Devising a high performance signal is one of the many challenges of GNSS system

design. As far as the GNSS community is concerned, Galileo E5 AltBOC(15,10)

quali�es as an �invention� in terms of the signal design, next only in innovation to

the introduction of BOC modulation. The processing complexity required at the

receiver to reap the bene�ts of an E5 AltBOC(15,10) signal is challenging due to its

sophistication. The sophistication of E5 AltBOC(15,10) can be mainly attributed

to:

• the high signal bandwidth (51.15 MHz for the �rst two main lobes itself)

• the presence of four spreading codes at 10.23 MHz chipping rate and the

corresponding secondary codes, and

• the presence of a special four-level complex sub-carrier waveform.

Scaling algorithms used for other signals may su�ce but often turn out to be in-

e�cient in terms of hardware and software resource requirements and hence result

in increased power consumption and cost. Overcoming these hurdles can accelerate

the process of bringing E5 AltBOC(15,10) to mass market receivers. Therefore one

of the objectives of this thesis is to explore e�cient methods to process the E5 Alt-

BOC(15,10) signal, speci�cally, the signal acquisition, signal tracking and hardware

realisation.

1.6. MOTIVATION AND OBJECTIVES 7

The E5 AltBOC(15,10) signal encompasses certain unique features that are not

available in other GNSS signals. A non-coherent combination of the individually

processed E5a and E5b correlation outputs produces a BPSK(10)-like correlation

triangle in contrast to the narrow correlation peak obtained by treating the whole

signal as a wideband signal. While a wide correlation peak favours signal acquisition,

the signal tracking errors are generally low for a signal with a narrow correlation

peak. Therefore it is of interest, and a topic in this thesis, to explore signal acquisi-

tion and signal tracking algorithms that utilise the favourable features at each stage

of the signal processing.

The E5a and E5b sidebands can be considered as carrying the same ranging

information (at least the `pilot only' channels) and hence possess the property of

frequency diversity. The e�ect of multipath fading is frequency dependent. One

of the earliest works related to multipath fading and frequency diversity noted the

presence of negative correlation among individual frequency components for di�erent

multipath delays (Haber and Noorchashm 1974). Therefore another topic of interest

in this thesis is to exploit the frequency diversity in order to reduce the e�ect of

multipath on the signal measurements.

A GNSS receiver using the Galileo E5 signal may fall into one of the two cate-

gories, either a standalone Galileo E5 receiver or a multi-band (or multi-frequency)

GNSS receiver where E5 is one of the processed signals. In its simplest form, a

typical civilian multi-band receiver is expected to accommodate the signals in the

L1/E1, L2 and the L5/E5 bands. The Galileo E5a sideband shares frequency spec-

trum with the GPS L5 signal. Moreover, there are several advantages that a receiver

can exploit by using signals from both the E5/L5 and E1/L1 frequency bands, such

as the ionospheric delay estimation and mitigation of RF interference. Therefore,

another objective of this thesis is to study receiver complexity and explore signal

processing algorithms for the E5 AltBOC(15,10) signal in a multi-band receiver, in

addition to those for a standalone E5 receiver.

One of the purposes of having shorter codes in a GNSS is to aid the acquisition

of longer codes in the system, an example being the GPS L1 C/A code aiding the P

code acquisition (Parkinson and Spilker, 1995). However, there is another issue when

the shorter and longer codes are at di�erent frequency bands. If the signal carrying

the shorter code is a�ected due to interference or jamming, the receiver has to spend

its resources searching the entire code delay and carrier frequency ambiguity for the

longer code. Moreover, a receiver has to be designed to handle such worst case

scenarios, which may be an overkill from the system design perspective. To explore

the issue of e�ciently designing core signal acquisition blocks is another aim of this

thesis.

Having discussed the complexity of processing the Galileo E5 AltBOC(15,10)

8 1. INTRODUCTION

signal, it is of interest to examine the signal structure in detail to analyse the un-

derlying contributions to this complexity. Therefore another objective of this thesis

is to explore the possibility of devising a signal that possesses AltBOC-like features,

but reduces the receiver signal processing complexity.

Due to the attractiveness of the AltBOC(15,10) signal, the Chinese naviga-

tion system, COMPASS (a.k.a. Beidou) Phase-III plans to transmit signals called

B2a and B2b that are generated using the AltBOC(15,10) modulation centred at

1191.795 MHz. COMPASS uses the same code chipping rate as the Galileo E5 and

same data rate on B2a as on Galileo E5a (the data rate on B2b is 50 bps/100 sps

instead of the 125 bps/250 sps on Galileo E5b)(Lu, 2010). This supports the view

that most of the receiver algorithms developed for AltBOC(15,10) will eventually

be useful in processing signals from more than one GNSS.

1.7. Contributions

The following are contributions of this thesis:

• A detailed analysis of the Galileo E5 AltBOC(15,10) signal acquisition

strategies, focusing on the acquisition performance and categorisation of

acquisition strategies; studying the signi�cance of cell correlations on the

matched �lter acquisition performance.

• Proposal for a sequential detection acquisition algorithm to acquire sec-

ondary code phase in Galileo receivers.

• Designing a hybrid tracking architecture for the Galileo E5 AltBOC (15,10)

signal that uses a pre-correlation combination method to combine individual

components of the E5 signal with the aim of maximising the received signal

energy.

• Proposal for a novel extended tracking range Delay Locked Loop (DLL)

that combines the bene�ts of BPSK(10) correlation output and wideband

AltBOC(15,10) correlation output, especially for low signal strength and/or

high dynamics applications.

• Design of a mixed-radix Fast Fourier Transform (FFT) architecture that

can be adapted in real-time to perform code acquisition of di�erent GNSS

signals, to eliminate the need for dedicated large-point FFT blocks.

• Evaluation of baseband hardware complexity of multi-band GPS and Galileo

correlators; and estimating the power consumption of modernised GPS and

Galileo receivers.

• A novel (PCT/AU2010/000268) code phase multipath mitigation algorithm

called the Sideband Carrier-Phase Combination (SCPC) method that utilises

1.8. STRUCTURE OF THE THESIS 9

the frequency diversity property of Galileo E5 AltBOC(15,10) to compen-

sate for the group delay at the centre of the E5 band by estimating the

phase delays at E5a and E5b centre frequencies.

• Proposal for a Time-Multiplexed O�set-Carrier Quadrature PSK (TMOC-

QPSK) modulation that resembles a non-constant envelope AltBOC mod-

ulation and TMOC-π4-QPSK modulation scheme that resembles constant

envelope AltBOC modulation, with the proposed modulation schemes re-

quiring reduced complexity at the receiver.

1.8. Structure of the Thesis

The structure of the thesis is depicted in Fig. 1.2.

Chapter 2 provides a detailed description of the Galileo E5 signal structure and

relevant work in the literature concerning the topics dealt within this thesis. The

basics of the Galileo E5 signal structure, acquisition, tracking and multipath are

described here. Context and scope of the topics in this thesis is established and

justi�ed by reviewing the recent work carried out by other researchers.

Chapter 3 discusses Galileo E5 signal acquisition strategies, their performance

and secondary code acquisition. To start with, the probability of detection and

mean acquisition time performances are analysed for the primary code acquisition.

Next, the signi�cance of cell correlations on the acquisition performance is discussed.

Finally, an algorithm to acquire the secondary code phase is proposed.

The �rst part of Chapter 4 discusses hybrid tracking architectures that combine

the correlation outputs of di�erent components of the E5 signal. The second part

proposes a novel way of combining BPSK-like and AltBOC delay locked loops to

obtain an extended tracking range performance for the code tracking.

Chapter 5 explains SCPC, the code phase multipath mitigation algorithm, in de-

tail. A group delay compensation perspective for the proposed multipath mitigation

algorithm is also provided.

The Galileo E5 baseband hardware architecture in the context of a multi-band

(or multi-frequency) and multi-GNSS receiver is described in Chapter 6. First, the

multi-band and multi-GNSS signal processing techniques are categorised. Next, a

novel resource-sharing FFT algorithm is presented and the hardware complexity

results are provided. Finally, the baseband hardware complexity and power con-

sumption of modernised GNSS receivers is analysed.

Chapter 7 introduces the TMOC-QPSK modulation and discusses the bene�ts

that make it attractive as an alternative to AltBOC modulation. Hardware complex-

ity and power consumption estimates of the TMOC-QPSK correlator are compared

to those of an AltBOC correlator.

10 1. INTRODUCTION

Chapter 2Galileo E5 Signal & Related Work

Chapter 3Experimental Setup

Chapter 4Acquisition

Chapter 5Tracking

Chapter 6Multipath Mitigation

Chapter 7Galileo E5 Baseband Hardware

Chapter 8Time-Multiplexed Offset-Carrier

Modulation

Chapter 9Conclusions & Future Work

Chapter 1Introduction (this chapter)

Figure 1.2. Flow diagram showing the thesis structure

1.9. PUBLICATION CROSS REFERENCE MATRIX 11

Chapter 8 concludes the thesis with a summary of recommendations and outlines

some future work.

Appendices A to E provide basic information related to some of the topics dis-

cussed in this thesis. Appendices F to L provide the supporting information required

during the course of development of the methods proposed in this thesis.

1.9. Publication Cross Reference Matrix

The research related to this thesis was conducted between September 2007 and

January 2011. The publication timeline is shown in Fig. 1.3. Publication vs. chapter

cross reference matrix is shown in Table 1.2.

12 1. INTRODUCTION

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1.9. PUBLICATION CROSS REFERENCE MATRIX 13

Table 1.2. Publication vs. Chapter cross reference matrix

Publication Chapter/ Section

N. C. Shivaramaiah and A. G. Dempster, An Analysis of GalileoE5 Signal Acquisition Strategies, ENC-GNSS, Toulouse, France,

Apr 2008.

Chapter 4

N. C. Shivaramaiah and A. G. Dempster, Galileo E5 SignalAcquisition Strategies, Coordinates Magazine, vol. IV(8), no. 8,

Aug 2008, pp. 1216,

Chapter 4

N. C. Shivaramaiah, A. G. Dempster, and C. Rizos, Exploiting theSecondary Codes to Improve Signal Acquisition Performance inGalileo Receivers, ION-GNSS, Savannah, GA, Sep. 2008, pp.

1497-1506.

Chapter 4

N. C. Shivaramaiah and A. G. Dempster, An UnambiguousDetector Architecture for Galileo E5 Signal Acquisition, in Signals,

Systems and Computers, Asilomar Conference on, 2008, pp.2076-2080.

Chapter 4

N. C. Shivaramaiah and A. G. Dempster, ProcessingComplex-modulated Signals Involving Spreading Code and

Subcarrier in Ranging Systems, PCT/AU2010/000268, Prioritydate 11 Mar 2009

Chapter6

N. C. Shivaramaiah, A. G. Dempster, and C. Rizos, A HybridTracking Loop Architecture for Galileo E5 Signal, ENC-GNSS,

Naples, Italy, May 2009.

Chapter 5

N. C. Shivaramaiah and A. G. Dempster, A Novel ExtendedTracking Range DLL for AltBOC Signals, in IEEE VTC-FALL,

Anchorage, AK, Sep. 2009.

Chapter 5

N. C. Shivaramaiah, Code Phase Multipath Mitigation byExploiting the Frequency Diversity in Galileo E5 AltBOC, ION

GNSS, Savannah, GA, September 2009.

Chapter6

N. C. Shivaramaiah and A. G. Dempster, Group DelayCompensation in AltBOC Receivers to Mitigate the E�ect ofFrequency Selective Propagation Delay Distortions, IEEE/ION

PLANS, May 2010, pp. 227 235.

Chapter6

N. C. Shivaramaiah, A. G. Dempster, and C. Rizos, Application ofPrime-factor and Mixed-radix FFT Algorithms in Multi-bandGNSS Receivers, Journal of GPS, vol. 8, pp. 174-186, 2009.

Chapter7

N. C. Shivaramaiah and A. G. Dempster, The Galileo E5 AltBOC:Understanding the Signal Structure, IGNSS Symp, Gold Coast,

Australia, Dec 2009.

Chapter2

N. C. Shivaramaiah and A. G. Dempster, Design Challenges of aGalileo E1 Correlator on the Namuru Platform, in IGNSS Symp,

Gold Coast, Australia, Dec 2009.

Chapter7

N. C. Shivaramaiah and A. G. Dempster,Time-Multiplexed-O�set-Carrier Modulations for GNSS, IEEE

Trans. AES (Manuscript Under Review)

Chapter8

14 1. INTRODUCTION

Table 1.2. Publication vs. Chapter cross reference matrix (contd...)

N. C. Shivaramaiah, A. G. Dempster, and C. Rizos, On E�cientWideband GNSS Signal Design, Proceedings of the ION ITM,

24-26 Jan 2011, San Diego CA

Chapter8

T. H. Ta, N. C. Shivaramaiah, A. G. Dempster and Letizia LoPresti, Signi�cance of Cell Correlations in Matched Filter GPS

Acquisition Engines, in IGNSS Symp, Gold Coast, Australia, Dec2009. Contribution : Ideating the topic, Literature survey,Suggestions to state diagram, Mean acquisition time andPerformance analysis sections. % Contribution : 50

Chapter 4

T. H. Ta, N. C. Shivaramaiah, and A. G. Dempster, Signi�cance ofCell Correlations in GNSS Matched Filter Acquisition Engines,

IEEE Trans. AES, (Revised Manuscript Under Review).Contribution : Ideating the topic, Literature survey, Suggestions tostate diagram, Mean acquisition time and Performance analysis

sections, State optimisation, Generalisation section. %Contribution : 50

Chapter 4

S. U. Qaisar, N. C. Shivaramaiah, and A. G. Dempster, Exploitingthe Spectrum Envelope for GPS-L2C Signal Acquisition,

ENC-GNSS, Toulouse, France, Apr 2008. Contribution : Proposingthe resampling architecture, Developing the FPGA based resampling

correlator hardware and testing with the real signals. %Contribution : 50

Discussionsin

Chapter 7

S. U. Qaisar, N. C. Shivaramaiah, A. G. Dempster, and C. Rizos,Filtering IF Samples to Reduce Computational Load of FrequencyDomain Acquisition in GNSS Receivers, ION-GNSS, Savannah,

GA, Sep 2008, pp. 236-243. Contribution : Proposing theresampling architecture, Designing the FPGA based �lter,

Developing the correlator hardware, Various FFT size computationon the hardware, Experiments on real data. % Contribution : 50

Discussionsin

Chapter 7

P. Mumford, N. C. Shivaramaiah and E. Glennon, An Investigationof Correlator Design Architecture to Support QZSS L1 Signals,Proceedings of the ION ITM, 24-26 Jan 2011, San Diego CA. %

Contribution : 15

Discussionsin

Chapter 9

CHAPTER 2

Galileo E5 Signal and the Related Work

2.1. Introduction

This chapter provides the background information required for the rest of the

thesis, including a discussion on relevant previous work. To start with, Global

Navigation Satellite System (GNSS) signal structures are introduced and the Galileo

E5 signal is explained in detail. Next, a brief overview of receiver signal processing is

provided, followed by a detailed discussion of Galileo E5 AltBOC signal processing.

The challenges associated with signal acquisition, tracking, multipath mitigation

and the receiver hardware realisation are discussed in the context of previous work.

The scope of the research work related to this thesis is then established to support

the later chapters.

2.2. GNSS Transmitted Signal structure

Most GNSSs employ the Direct Sequence Spread Spectrum (DS-SS) technique

with all the satellites in the constellation synchronously transmitting navigation sig-

nals. Each satellite is assigned a Pseudo-Random Noise (PRN) spreading sequence

orthogonal (or quasi-orthogonal) to all other PRNs in the system. This technique,

which allows all the satellites to share the same carrier frequency, is the principal

feature of the Code Division Multiple Access (CDMA) technique. The signal energy

of the carrier is spread across a band of frequencies whose bandwidth is determined

by the rate of the PRN sequence, known as the �chipping rate�. In some systems, a

secondary PRN sequence is combined with the assigned PRN spreading sequence (or

primary PRN sequence) to form a tiered spreading sequence. A relatively low-rate

navigation data signal that contains the necessary information to estimate the range

to the satellite, such as time of transmit, satellite orbital parameters and corrections,

is modulated on to the PRN sequence. In general, a GNSS broadcast signal consists

of three components: a radio frequency carrier, a PRN spreading sequence (a.k.a.

ranging code) and the navigation data.

A generic DS-SS CDMAGNSS signal at the transmitter of any particular satellite

at a designated link (or frequency) X can be expressed as:

SX (t) =√

2PT,X < [sX (t) · exp (jωc,X t)] (2.1)

15

16 2. GALILEO E5 SIGNAL AND THE RELATED WORK

where PT,X is the transmitted signal power (W), ωc,X represents the angular carrier

frequency (rads/s),

sX (t) = sX I(t) + jsXQ(t) (2.2)

is the complex baseband signal and < is the real value function operator. Without

loss of generality, the carrier phase at the time of transmission can be assumed to

be zero and hence not included in (2.1). X corresponds to one of the many links

of the system usually associated with a number (and another signi�er identifying

the type of the signal, if there are more than one at the same frequency from the

same satellite). For example, in the Global Positioning System (GPS), the Standard

Positioning Service (SPS) service is provided through three links and X takes on

values L1, L2 or L5, whereas in Galileo, Open Service (OS) signals are available

though the links E1 and E5, and in Compass the links are designated with the letter

B. The complex baseband signal sX (t) consists of a PRN spreading sequence and

the navigation data. The components of sX (t) depend on the type of signal. For

the GPS L1 C/A with X=L1C/A,

sX (t) =+∞∑i=−∞

[cX ,|i|LX

· dX ,⌊ iDX

⌋ · gTc,X (t− i · Tc,X )

](2.3)

where

• cX ∈ {−1, 1} is the spreading code with repetition period LX (in chips)

and chip duration Tc,X (seconds); LL1C/A = 1023 and Tc,L1C/A = 11.023×106

≈977.5× 10−9

• dX ∈ {−1, 1} is the navigation data with DX =Td,XTc,X

code chips per symbol,

Td,X being the symbol duration (seconds); DL1C/A=20460 and Td,L1C/A =150

= 0.02.

• gTc,X (t) = uTc,X (t) is the rectangular pulse shaping function of width Tc,X

and unit amplitude

Observe that the L1C/A baseband signal does not possess quadrature component

and hence the transmitted signal can be written as

SL1C/A(t) =√

2PL1C/A · sL1C/A(t) · cos (2πfc,L1t) (2.4)

where 12πωc,L1 = fc,L1 = 1575.42× 106 is the carrier frequency (Hz). Since all signal

types on a particular link use the same carrier frequency, the signal type signi�er

C/A is omitted from the carrier frequency subscript.

2.3. Galileo E5 AltBOC Signal Structure

This section describes the Galileo E5 signal structure.

2.3. GALILEO E5 ALTBOC SIGNAL STRUCTURE 17

2.3.1. The transmitted signal. The Galileo E5 transmitted signal can be

represented as

SE5(t) =√

2PT,E5 < [sE5(t) · exp (jωc,E5t)] (2.5)

Since this thesis deals extensively with the E5 signal, the su�x `E5' is omitted in

future equations and no su�x refers to the E5 signal. Expanding (2.5) and using

(2.2), the transmitted signal can also be written as

S(t) =√

2PT [sI(t) cos(ωct)− sQ(t) sin(ωct)] (2.6)

where ωc= 1191.795 MHz is the centre (carrier) frequency of the E5 signal band

and s(t) = sI(t) + jsQ(t) is the baseband signal.

2.3.2. The baseband signal - constant envelope AltBOC modulation.

The Galileo E5 signal consists of four spreading codes that are modulated onto in-

phase and quadrature-phase components of orthogonal complex sub-carriers. These

phase-points are combined to form a single baseband signal that phase modulates

the carrier according to (2.6). A straightforward way of combining the complex

phase-points is by adding them together. However, such a method results in a non-

constant-envelope baseband signal, i.e. the normalised magnitudes of the resulting

signal do not lie on the unit circle in the complex plane. A detailed description of

the Alternate Linear O�set Carrier (AltLOC) and non-constant-envelope AltBOC

referred to in this thesis as AltBOC-NCE is provided in Appendix A.

In order to avoid the issue of a non-constant-envelope, Ries et al. (2003) proposed

a method wherein the sub-carrier waveforms are modi�ed to obtain a constant en-

velope. (Note: In this thesis, the constant envelope AltBOC is simply referred to as

AltBOC). In this case the sub-carriers are chosen such that the sum and di�erence

of complex values always lie on the unit circle in the complex plane.

The generation of the baseband signal is illustrated in Fig. 2.1.

AltBOC Mux

caI

daI

caQ

cbI

cbQ

dbI

saI

saQ

sbI

sbQ

sE5

Figure 2.1. AltBOC multiplexer illustration


The baseband signal can be represented as (CAB = CosAltBOC):

sCAB(t) =1

2√

2

[(saI(t) + j · saQ(t))

(scs(t)− j · scs(t− Tsc

4))

+

(sbI(t) + j · sbQ(t))(scs(t) + j · scs(t− Tsc

4))

+

(saI(t) + j · saQ(t))(scp(t)− j · scp(t− Tsc

4))

+

(sbI(t) + j · sbQ(t))(scp(t) + j · scp(t− Tsc

4))]

(2.7)

where Tsc = 1fsc

is the period of the sub-carrier. This is the type of modulation used

in the Galileo E5 signal, i.e. s(t) in (2.6) is the same as sCAB(t) (the other type is

SineAltBOC whose linear counterpart is explained in Appendix A).

The de�nitions of the individual components of the baseband signal are provided

in equation form below followed by a brief description:

saI(t) =+∞∑i=−∞

[caI,|i|LaI

· cs,aI,|i|Ls,aI · daI,⌊

iDaI

⌋ · uTc(t− i · Tc)]

(2.8)

saQ(t) =+∞∑i=−∞

[caQ,|i|LaQ

· cs,aQ,|i|Ls,aQ · uTc(t− i · Tc)]

sbI(t) =+∞∑i=−∞

[cbI,|i|LbI

· cs,bI,|i|Ls,bI · dbI,⌊

iDbI

⌋ · uTc(t− i · Tc)]

sbQ(t) =+∞∑i=−∞

[cbQ,|i|LbQ

· cs,bQ,|i|Ls,bQ · uTc(t− i · Tc)]

• sbI(t) is the E5bI signal component (i.e. E5b data signal), saI(t) is the

E5aI signal component (i.e. E5a data signal), sbQ(t) is the E5bQ signal

component (i.e. E5b pilot signal), saI(t) is the E5aQ signal component (i.e.

E5a pilot signal),

• dbI(t) and daI(t) are the data bit modulations of E5b and E5a respectively,

• ex(t) = cx(t) · cs,x(t) is the tiered spreading code, with cx(t) denoting the

primary spreading code of length 10230 with a repetition rate of one mil-

lisecond, cs,x(t) denoting the secondary code of varying length (OSSISICD,

2010) and x denoting the corresponding signal

• LaI , LaQ, LbI , LbQ denote the primary code length (repetition period) in

chips, Ls,aI , Ls,aQ, Ls,bI , Ls,bQ denote the secondary code lengths.


The secondary codes and the tiered code generation method are explained in sec.

2.3.4. The dashed (over-strike) signal components are the product signals:

saI(t) = saQ(t) · sbI(t) · sbQ(t) (2.9)

saQ(t) = saI(t) · sbI(t) · sbQ(t) (2.10)

sbI(t) = saI(t) · saQ(t) · sbQ(t) (2.11)

sbQ(t) = saI(t) · saQ(t) · sbI(t) (2.12)

The parameters scs(t) and scp(t) are the �single� and �product� components of

the four-level sub-carrier which is used to generate the constant envelope. They are

represented as in (2.13) and (2.14).

scs(t) =+∞∑i=−∞

AS|i|8 uTsc8

(t− i · Tsc8

) (2.13)

scp(t) =+∞∑i=−∞

AP|i|8 uTsc8

(t− i · Tsc8

) (2.14)

where the coe�cients are de�ned according to Table 2.1.

Table 2.1. AltBOC sub-carrier coe�cients

i 0 1 2 3 4 5 6 7

ASi

√2 + 1

2

1

2

−1

2

−√

2− 1

2

−√

2− 1

2

−1

2

1

2

√2 + 1

2

APi−√

2 + 1

2

1

2

−1

2

√2− 1

2

√2− 1

2

−1

2

1

2

−√

2 + 1

2

The sub-carrier waveform for a duration of one cycle is shown in Fig. 2.2.

The product sub-carrier has a very interesting role to play in the total signal,

apart from that of helping to produce a constant envelope modulation (Appendix

B). Its role can be summarised as follows.

• The inter-modulation product carries around 15% of the total power.

• The inter-modulation product helps in sharpening the autocorrelation func-

tion of the wideband E5 signal, but only if the front-end �lter covers the

spectrum of the product sub-carrier, i.e. the product signal is of signi�cance

only for �lter bandwidths >90MHz.

• For typical receiver bandwidths, the product sub-carrier can be safely ne-

glected (from the local replica generation).

The constellation diagram of the constant envelope AltBOC signal is shown in

Fig.2.3. One advantage of the 8-PSK type of representation is that the modulation

(and demodulation) can be realised using a look-up-table (OSSISICD, 2010). The


0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−1.5

−1

−0.5

0

0.5

1

1.5

tTsc

Am

plitu

de

scs(t)

scp(t)

−√2−12

−√2+12

Figure 2.2. AltBOC sub-carrier waveforms

Galileo E5 signal uses this constant envelope AltBOC with a sub-carrier frequency

of 15.345 MHz and code chipping rate of 10.23 MHz.

0 1

-j

j

-1

Figure 2.3. Constellation diagram of the constant envelope AltBOCsignal

2.3.3. Nomenclature. The BPSK modulation in GNSS is primarily identi�ed

by associating a number that is related to the primary spreading code chipping

frequency, as in �BPSK(n)�. This number n is simply multiplication factor required

to obtain the chipping frequency relative to 1.023 MHz, i.e. n = fco1.023×106

, fco being

the chipping rate of the signal in question. Hence, the modulation used for GPS

L1 C/A is written as BPSK(1). Another notation PSK-R has also been used in the

literature instead of BPSK, where R stands for �Rectangular� pulse shaping function

whose width is the same as the chip duration. For example, the chipping frequency

used in BPSK(10) modulation (used by the GPS L5 signal) is 10.23 MHz.


Table 2.2. Galileo E5 OS signal code structure (from (OSSISICD, 2010))

Signal Component Tiered Code Secondary Code Secondary CodePeriod (ms) Length (Chips) Mnemonic

E5aI 20 20 CS201

E5aQ 100 100 CS1001−50

E5bI 4 4 CS41

E5bQ 100 100 CS10051−100

1 2 Lc

1 2 Ls 1

1 2 Lcs

Primary Code

Secondary Code

Tiered Code

1 2 Lc 1 2 Lc 1 2 Lc

1 2

XOR

Figure 2.4. Tiered code generation

The family of BOC modulations have a sub-carrier on top of the spreading code

and are hence identi�ed with two numbers, as in �BOC(m,n)�. In BOC(m,n), n

has the same meaning as given in the previous paragraph and m = fsc1.023×106

, i.e.

it identi�es the sub-carrier frequency as a multiple of 1.023 MHz. For example,

Galileo E1B signal uses a primary code chipping rate of 1.023 MHz and a sub-carrier

frequency of 1.023 MHz, and hence is denoted as BOC(1,1). Another notation

BOC(pn,n) has also been used in the literature instead of BOC(m,n), in which

p = mn

= fscfco

is the ratio of sub-carrier frequency to the chipping rate. AltBOC

modulation also follows this BOC nomenclature and hence the modulation used

for Galileo E5 is AltBOC(15,10). Yet another notation used in the literature is

BOC(fsc, fco) where the arguments are only the ratios with respect to 1.023 MHz,

though they represent actual frequencies. In this thesis, the �rst notation BOC(m,n)

is used unless otherwise speci�ed.

2.3.4. Code length and the generation of tiered codes. All the four pri-

mary codes in the Galileo E5 signal have a length of 10230 chips, i.e. LaI = LaQ =

LbI = LbQ =10230. Secondary codes are multiplied with the primary codes to obtain

the tiered codes. Secondary codes are much shorter in length (i.e. fewer chips) and

much slower (each secondary code chip multiplies one code period of the primary

code).

Table 2.2 details the code structure. Fig. 2.4 illustrates the tiered code gener-

ation. Each chip of the secondary code spans one complete primary code period.

In other words, the chip transition of the secondary code is aligned with the `zero'


point of the primary code. The two codes are XORed to generate the tiered code.

If Lc is the number of chips in the primary code (the length of the primary code)

and Ls is the number of chips in the secondary code, then the tiered code will have

Lcs = Lc · Ls chips.The secondary codes used in Galileo are �memory� codes. Unlike the register

codes that can be generated on-the-�y using shift registers (or a combination of

shift registers), memory codes have to be stored and retrieved for use. The primary

purpose of the secondary codes is to provide better correlation properties for the

�nal pseudorandom sequences without demanding an unreasonably long code delay

search in a receiver. Each secondary code has a code identi�er mnemonic as given

in Table 2.2. For E5aI and E5bI signals, all the satellites use the single secondary

code sequence CS201 and CS41 respectively. For the E5aQ and E5bQ signals, each

satellite has di�erent secondary code sequences (of the same 100-bit length) with

the su�x distinguishing the sequences. Tiered code generation method is explained

in OSSISICD (2010).

2.3.5. The power spectral density (PSD). In order to represent the signal

in the frequency domain, it is required to obtain the power spectral density function

of the signal. The PSD of a constant envelope AltBOC signal is not straightforward

to derive since it involves computing the Fourier Transform of individual pieces of

the sub-carrier waveform. From Rebeyrol et al. (2005); Rebeyrol (2007):

GAltBOC(f) =4

Tcπ2f 2

cos2(πfTc)

cos2(πf Tsc

2

)[cos2

(πf

Tsc2

)− cos

(πf

Tsc2

)− 2cos

(πf

Tsc2

)cos

(πf

Tsc4

)+ 2

],

2fscfco

odd

(2.15)

Equation (2.15) is plotted in Fig. 2.5.

2.4. The Correlation Function

A basic property which enables a receiver to estimate the signal travel time from a

transmitter is the �autocorrelation� property of the spreading code, and the property

that enables CDMA technique to operate in the presence of other signals is the

�cross-correlation� property. Correlation indicates the closeness of match between

two entities, in this context, between two spreading codes. The autocorrelation

function (ACF) of a signal c(t) is de�ned by Braasch and van Dierendonck (1999):

Rc(τ) = limA→∞

1

2A

∫ A

−Ac(t)c(t− τ)dt. (2.16)

2.4. THE CORRELATION FUNCTION 23

−6 −4 −2 0 2 4 6

x 107

−110

−105

−100

−95

−90

−85

−80

−75

−70

−65

−60

frequency (Hz)

Am

plitu

de (

dBW

)

AltBOC − Const. Env.

Figure 2.5. PSD of the constant envelope AltBOC(15,10); zero fre-quency refers to the centre of the band

The autocorrelation of an in�nite random sequence is:

R(τ) = 1− |τ |Tc, for|τ | ≤ Tc

= 0, otherwise (2.17)

where τ is the lag value (in seconds). A truly random sequence is not practical as

it requires in�nite length. In practice, maximal-length sequences (m-sequences or

pseudo-noise sequences) provide a close approximation to random sequences. The

periodic autocorrelation function of an m-sequence of length L is given by:

R(τ) = 1− |τ |Tc

(1

L+ 1

), for|τ | ≤ Tc

= − 1

LforTc ≤ |τ | ≤ (L− 1)Tc (2.18)

The autocorrelation functions of in�nite length and �nite length codes (for the noise-

less case) are shown in Fig. 2.6(Braasch and van Dierendonck, 1999).

GPS L1 C/A signals employ spreading codes known as Gold codes. These codes

are formed by combining two m-sequences. The result is a family of pseudo-noise

codes with low cross-correlation between codes.

A spreading code is said to possesses a good autocorrelation property if the

correlation value for non-zero time lags is a minimum (as close to zero as possible).

Identifying the zero time lag point (the �correlation peak�) is the primary means of

estimating the time delay between the transmitter and the receiver. Since a GNSS

receiver receives combined signals from several visible satellites, the spreading codes

used for di�erent satellites should be distinguishable from each other. In other


1

- T c T c

(a)

1 1

(b)

L -1/L

0

R( )

0

- T c T c

R( )

Figure 2.6. ACF of (a)Truly random sequence, (b)Maximal lengthsequence

words, a set of spreading codes is said to be good if the correlation between any

code sequence in the set with any other code sequence is minimum (as close to zero

as possible) for all time lags.

For BOC(pn, n) signals, the autocorrelation function of the un�ltered signal (p =

1, 2, ...) is given by (Borre et al., 2007; Nunes et al., 2007)

R (τ) =

(−1)k+1[

1p

(−k2 + 2kp+ k − p)− (4p− 2k + 1) |τ |Tc

]|τ | < Tc

0 otherwise(2.19)

where k =⌈2p |τ |

Tc

⌉. Little has been reported in the literature concerning the exact

expression for the autocorrelation function for AltBOC(m,n). However, a very close

approximation is provided in Lohan et al. (2006) as a general expression for Complex

Double-Binary-O�set Carrier (CDBOC) modulations. The equation is repeated here

for AltBOC signals.

R1(τ) =

N1−1∑i=0

N2−1∑k=0

N1−1∑i1=0

N2−1∑k1=0

(−1)i+i1+k+k1ΛTB (τ − (i− i1)TB1 − (k − k1)TB12)

(2.20a)

R2(τ) =

N3−1∑l=0

N4−1∑m=0

N3−1∑l1=0

N4−1∑m1=0

Nres−1∑p=0

Nres−1∑p1=0

(−1)l+l1+m+m1

ΛTB (τ − (l − l1)TB3 − (m−m1)TB34 − (p− p1)TB12) (2.20b)

R(τ) = R1(τ) +R2(τ) (2.20)

where N1 = 3, N2 = 2, N3 = 3, N4 = 1, TBi = Tc/Ni, TBij = Tc/NiNj, Nres =

2.5. GNSS RECEIVER ARCHITECTURE 25

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Code Delay (chips)

Nor

mal

ized

Aut

o−co

rrel

atio

n V

alue

BOC(15,10)CDBOC EquationPRN 51(real)

Figure 2.7. Normalised autocorrelation value obtained using (2.19),(2.20) and un�ltered GIOVE-A PRN 51 E5 codes

N1N2/N3N4, ΛTB(t) is a triangular pulse of support 2TB12 and Tc is the chip period of

AltBOC(15,10).

Fig. 2.7 shows the autocorrelation function calculated using these two methods

along with that obtained from the ACF of a simulated GIOVE-A PRN 51 signal. It

is interesting to note that despite the similarities in shape, subtle di�erences exist

between the BOC(15,10) and AltBOC(15,10) autocorrelation functions.

2.4.1. Received signal power and the pre-correlation SNR. Since the

minimum received signal power for E5a and E5b is -155 dBW, the minimum received

signal power for the wideband E5 will be -152 dBm. Typical wideband receiver front-

end bandwidth can be considered to be 51.15 MHz. Hence, the noise within this

51.15 MHz will be −201.5 − (−77) ≈ −124.5 dBW. In addition, a front-end �lter

of about 50 MHz bandwidth will introduce ≈1.5 dB loss (see Fig.10 in Sleewaegen

et al. (2004)) compared to the signal power in an in�nite bandwidth. Therefore the

pre-correlation SNR for the wideband E5 signal is1

SNRpre ≈ −153.5− (−124.5) = −29 dB (2.21)

2.5. GNSS Receiver Architecture

Fig. 2.8 shows the typical architecture of a GNSS receiver. The received signal

is �ltered, down-converted and passed through an Analogue-to-Digital-Converter

(ADC) to obtain the Intermediate Frequency (IF) samples. The baseband signal

1The actual signal loss due to the receiver front-end �lter will be less than 1.5 dB since the Galileosatellite payload will only broadcast the signal over a �nite bandwidth and the minimum requiredpower level of -155 dBW per E5a and E5b will likely be referenced to the transmit, or some other�nite, bandwidth. Hence the actual pre-correlation SNR will be slightly higher than -29 dB.


RF Front end

(Down-

converter +

ADC)

Digital

Baseband

(Correlator)

Processing

(Software)

Antenna

PVT

Solution

Figure 2.8. Typical architecture of a GNSS receiver

processing (widely known as the correlator) is implemented in either hardware or

software, each having its own advantages and disadvantages.

In order to compute the receiver-satellite range and to demodulate the data

streams, a GNSS receiver must �rst synchronise its reference code sequence and

carrier frequency with the satellite signal. To achieve this synchronisation, the

receiver must search for the Doppler frequency and the PRN code chip delay. Signal

synchronisation is generally carried out in two steps: the coarse synchronisation

(referred to as Acquisition), and �ne synchronisation (called Tracking). During

signal acquisition, the receiver searches for available satellites and estimates the

approximate chip delay and Doppler frequency.

The baseband module is responsible for the initial estimation of the time delay

and the Doppler frequency o�set of the received signal with respect to the trans-

mitted signal. With the help of feedback control algorithms (implemented either as

a part of the digital hardware or as a part of the software processing), the base-

band module provides accurate and continuous estimates of the delay, phase and

frequency of the carrier and spreading code in the received signal. The process-

ing, usually implemented in software, computes the position-velocity-time (PVT)

solution corresponding to the phase centre of the receiver antenna. Since the sig-

nal acquisition module is required to search over a range of code delay and carrier

frequencies, both massively parallel time domain correlators and frequency domain

acquisition methods have been used to reduce the search time (Holmes, 2007).

2.6. Signal Acquisition and Tracking: The Basics

The received signal at the IF can be represented as (considering any one satellite)

(Falcone et al., 2006):

rIF (t) =√

2P · < [s(t− τ) · exp (ωIF t+ ωdt+ θ)] + nW (t) (2.22)

where is P the received power (W), ωIF is the intermediate frequency (rads/s),

fd = 12πωd is the Doppler frequency (Hz), θ is the phase of the received signal (rads),

s(t−τ) is the complex baseband signal with a time delay τ (seconds) with respect to

2.6. SIGNAL ACQUISITION AND TRACKING: THE BASICS 27

the transmitted signal, and nW (t) is additive white Gaussian noise. (Note: For sim-

plicity, the time variation of the time delay, the Doppler frequency and the received

carrier phase are not shown). Due to various factors such as the relative dynamics

between the satellite and the user, the receiver clock-frequency instability and the

environment, the uncertainty of τ and ωd spans a range of time and frequencies

respectively. This is referred to as the receiver �search space�.

2.6.1. Receiver search space. The acquisition process is a two-dimensional

search: one search is in time uncertainty (∆t) and the other in the frequency uncer-

tainty (∆f ) as shown in Fig. 2.9. In Fig. 2.9, δt and δf represent the search step

size in time and frequency respectively. δt is expressed in fraction of a chip, and δf

is in Hertz. The total number of search �cells� is thus:

Figure 2.9. Receiver search space

N =

⌈∆f

δf

⌉·⌈

∆t

δt

⌉(2.23)

where the total initial frequency uncertainty region ( both positive and negative

deviations from the carrier at f0) is given by

∆f = 2 ·(|∆fosc|+

∣∣∣∣∆v · f0

c

∣∣∣∣) (2.24)

where ∆v is the magnitude of the receiver's initial velocity uncertainty, and ∆fosc

is the magnitude of the receiver's initial oscillator frequency uncertainty. Similarly,

the total initial time uncertainty that the receiver must search through during the

acquisition is given by:

∆t = 2 ·(|∆tosc|+

∣∣∣∣∆x

c

∣∣∣∣) (2.25)


where ±∆tosc is the receiver's total initial oscillator time uncertainty, and ±∆x is

the receiver's total position uncertainty. Because the code repeats, the maximum

value that ∆t can take is the code length.

2.6.2. The acquisition process. The conventional correlator structure is shown

in Fig. 2.10. The received signal is multiplied by both in-phase and quadrature com-

ponents of the locally generated carrier. The resultant is multiplied with the locally

generated replica signal referenced to a particular time lag. An Integrate and Dump

�lter follows the code mixing for a prede�ned duration MTc. Both in-phase and

quadrature correlation values for a chip shift are then used for envelope detection.

The acquisition decision variable is then compared against a threshold value.

Reference

Generator

Complex

Carrier

c

t

t MT

2

2

To

Tracking( )IFr t

t ct T

[ ]Z n

c

t

t MT

[ ]IX n

[ ]QX n

t ct T

Figure 2.10. Correlator structure for acquisition - conventional scheme

The acquisition decision variable Z[n] is computed as XI [n]2 + XQ[n]2. For the

detection process, the Neyman-Pearson likelihood criterion is used where the Z[n]

is compared with a predetermined threshold η to decide which hypothesis between

H0(Z[n] < η) and H1(Z[n] > η) is true, where

H0: means absence of the desired signal, and

H1: means presence of the desired signal

If Z[n] passes the threshold η (in this situation the receiver time and Doppler fre-

quency estimates are said to be in the H1 region) then a �hit� is declared and the

tracking process is initiated with the approximate time delay and frequency es-

timates. If the acquisition decision variable Z[n] fails to pass the threshold (H0

region), then the code phase of the locally generated code is advanced by δt and

the process is continued until all the cells are evaluated. Several time cells can be

evaluated simultaneously using parallel time domain search techniques or by using

frequency domain correlation methods.

The Integrate and Dump �lter determines the allowable Doppler ambiguity for

detection. For example, an integrate and dump duration of 15.625µs implies a

frequency window of 64 KHz assuming a brick-wall �lter at the zero crossing. The

useful range decreases further because of the sinc nature of the �lter. Depending

on the a priori information about the satellite orbit parameters and the local clock


Code Delay (chips)

Do

pp

ler

Sh

ift

(kH

z)

Starting cell

x x x

1

|R(t)|2

t (chip)0xx

-1 1-0.5 -0.5

Correct Cell

H0 region

H1 region

-5-4

.5-4

-3.5

-3-2

.5-2

-1.5

-1-0

.50

0.5

11.5

22.5

33.5

44.5

5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

f (

kH

z)

|sinc(fT

coh)|

2

Threshold

Thre

shold

Figure 2.11. Acquisition output illustration

characteristics, the Doppler ambiguity can range from few kHz (moderately stable

clock) to tens of kHz (for lower stability clock or high relative dynamics scenarios).

Parallel frequency search methods that employ an FFT increase the search speed.

The two-dimensional acquisition output is illustrated in Fig. 2.11 for the case of a

GPS L1 C/A signal, with a random cell as the starting point2.

2.6.3. FFT approach for code correlation. PRN code acquisition in a GNSS

receiver involves correlating the received signal at baseband with all possible time-

delayed versions of the local replica code and searching for the maximum value of

correlation. The correlation value indicates whether proper alignment between the

codes has been achieved. It is well known that the autocorrelation and power spec-

tral density are Fourier Transform pairs and hence time delay searches for the PRN

code alignment can be performed simultaneously for all time delay values using con-

volution according to the Wiener-Kinchine theorem (Proakis and Manolakis, 1995).

The process is to multiply the Fourier Transform of the received signal with the com-

plex conjugate of the Fourier Transform of the local replica code, and then perform

the Inverse Fourier Transform of the product to obtain the result - which is nothing

but a vector of correlation values for all possible time delays. As the sequences are

periodic this can be achieved by making use of FFTs, as has been proposed for GPS

with BPSK signal modulation (Van Nee and Coenen, 1991). Considering a sequence

2Observe that there could be multiple H1 cells during the search as shown in Fig. 2.11.


FFT FFT -1

FFT*

( | |2 )

N-1

∑ ( . )

0

`s(n+m)

Choose

max

Base-band

signal

s(n)

Figure 2.12. FFT method of code acquisition in GNSS receivers

with period L, the autocorrelation in the time domain and via the frequency domain

are given by (Proakis and Manolakis, 1995)

R(m) =L−1∑n=0

s(n) s(n+m) (2.26a)

R = F−1(C(k)C

∗(k))

(2.26b)

where s and s are the received and local code plus sub-carrier, C and C are the

corresponding Discrete Fourier Transforms, and * is the complex conjugate operator.

Fig. 2.12 depicts the code acquisition process using the FFT method. Note that R

is a vector representing the correlation values for each time cell. To decide whether

the proper alignment is achieved between the local code and the code present in

the incoming signal, max(R) is tested against a threshold value. If the local FFT

is stored in memory, only one forward FFT and one inverse FFT operations are

required. It can be seen that the parallel search using FFTs reduces the time required

for code acquisition (Holmes, 2007; Sajabi et al., 2006) by computing the vector R

at once.

2.6.4. Acquisition performance measures. Generally the acquisition per-

formance is measured by two related parameters, the Probability of Detection Pd

(or Detection Probability) and Mean Acquisition Time T acq for a speci�ed Proba-

bility of False Alarm (Pfa), η and the Carrier-to-Noise density C/N0 (Kaplan and

Hegarty, 2006).

The decision statistic for the Direct AltBOC architecture is given by (in the

absence of non-coherent integration)

Z[n] = XI [n]2 +XQ[n]2 (2.27)

Along with the signal components, the output of both the I and the Q channelsXI [n]

and XQ[n] contain noise components nI and nQ and are assumed to be statistically

independent, zero mean, and Gaussian distributed. Also the signal components in

the I and Q channels are assumed to be independent Gaussian distributed random

variables with mean mI and mQ respectively and equal variance of σ2. It has been


shown (Peterson et al., 1995) that the sum of squares of M independent Gaussian

random variables of the same variance is non-centrally distributed with M degrees

of freedom and with non-centrality parameter3 λ. The non-centrality parameter is

given by

λ2 =M∑i=1

m2i (2.28)

For the Direct AltBOC case, under the hypothesisH0 when there is no signal present,

the decision statistic has a central chi-square distribution with 2 degrees of freedom4

with PDF pn(x) and the Pfa is then given by

Pfa =

∞∫η

pn(x) dx = exp

(−η2σ2

)(2.29)

Under the hypothesis H1 when the signal is present the decision statistic has a

non-central chi-square distribution with 2 degrees of freedom with PDF ps(x) and

non-centrality parameter λ2 = m2I +m2

Q. The Pd is then given by

Pd =

∞∫η

pn(x) dx (2.30)

In order to compute the probabilities, �rst the threshold η for a chosen Pfa is

computed by numerically evaluating the inverse chi-square distribution. Then the

Pd is computed using (2.30) and with the help of the cumulative distribution function

(CDF) of the chi-square distribution (Shnidman, 1989).

Using the probabilities of detection for di�erent received signal strengths and

the probability of false alarm, the mean acquisition time is evaluated. Assuming a

single dwell search the mean acquisition time is given by (Holmes, 1982)

T acq =2 + (2− Pd)(∆t − 1)(1 + kpPfa)

2PdTcohNnc (2.31)

where ∆t is the size of the uncertainty region, kp is the penalty due to false alarm

and Tcoh is the pre-detection coherent integration duration.

2.6.5. The tracking process. For a particular PRN as soon as the presence of

the signal is established and the approximate Doppler frequency and code delay are

estimated, the very next step is to lock onto the signal by more accurately estimating

the Doppler frequency and the code delay. Then the receiver must continuously

3With non-coherent integration, the number of degrees of freedom increases to 2Nnc4The �no signal present� hypothesis modelled using a central chi-squared neglects the possiblepresence of appreciable decision metric amplitude due to cross-correlation with the other PRNs.A more accurate approach to assess acquisition performance uses non-central chi-squared distri-butions for both the �signal present� and �no signal present� hypotheses (see e.g. Appendix B toDierendonck (1996)).


Code Generator

Carrier

Generator

( )IFr t

Code

Loop

Filter

Code

NCO

( )x t

Integrate &

dump

Carrier

Loop

Filter

Carrier

discrimin

ator

Carrier

NCO

Code

discrimin

atorIntegrate &

dump

Integrate &

dump

Early

Late

Prompt

Figure 2.13. Typical tracking architecture; bold lines carry complexvalues

lock onto the signal, or in other words �track�, it in order to generate continuous

carrier Doppler, carrier phase and code phase estimates. Accurate estimation of

these parameters results in accurate estimation of the range measurement. Tracking

the signal is also a requirement to demodulate the navigation data bits transmitted

from the satellites. Signal tracking in GNSS is a vast topic whose fundamentals

are explained in detail in such texts as Parkinson and Spilker (1995); Kaplan and

Hegarty (2006).

As with the acquisition process, the tracking process is also two-dimensional.

Both the carrier frequency and the code delay peaks (the red points in Fig. 2.11)

must be tracked.

A typical tracking architecture is shown in Fig. 2.13. Carrier lock is typically

maintained by employing a Phase-Locked Loop (PLL) or a Frequency-Locked Loop

(FLL) (or a combination of both) whose input is the �Prompt� correlation value

generated by correlating the input signal with the local replica code aligned at the

estimated code delay. Code (delay) lock is maintained by employing a Delay-Locked

Loop (DLL) whose inputs are the correlation values of time shifted, i.e. the �Early�

and the �Late� versions of the local replica code. The code discriminator estimates

the error in the time delay by determining the di�erence in the early and late correla-

tion values, which is then �ltered and provided as a feedback to the code Numerically

Controlled Oscillator (NCO). The code generator then produces the replica codes at

the time delays instructed by the code NCO. The carrier discriminator establishes

the frequency/phase correction required (in terms of frequency or phase depending

on whether it is PLL or FLL) that is �ltered and fed back to the carrier NCO.

When a satellite signal is being continuously tracked, the navigation data bits are

demodulated and measurements are generated. The navigation data bits are used

to extract the time information and compute the satellite position. The satellite

position, time and the phase measurements from the tracking loop are used to

2.7. GALILEO E5 SIGNAL ACQUISITION 33

compute the user position and velocity solution and to estimate any drift in the

receiver clock.

The process of correcting the code delay and the carrier frequency continues as

long as the strength of the received signal is above a point that can be handled by

the tracking loops, and below which the loops cannot generate proper corrections

and the loops lose lock. The loops can also lose lock in other situations such as when

the dynamics of the receiver is beyond the loops' control, or in situations where the

receiver is a�ected by Radio Frequency Interference (RFI), and so on. The ability of

the tracking loops to produce proper corrections is also challenged under multipath

scenarios which distort the shape of the correlation function seen by the receiver.

The performance of the tracking loops depends on several design parameters such

as the type of discriminator used, �lter order, �lter bandwidth and the integration

duration. In general, the performance of the tracking loops is measured in terms

of the standard deviation of the carrier phase error, standard deviation of the code

delay error, the dynamics (velocity, acceleration and jerk) that the loops can handle,

the drift of the clock that the loops can cope up with, and the ability to track weak

signals.

2.7. Galileo E5 Signal Acquisition

The topic of AltBOC(15,10) signal acquisition is interesting due to three reasons.

First, the unique features of the signal, such as independent processing of either the

E5a or E5b side bands or the direct wideband processing, o�er several paths to

estimate the time delay and Doppler frequency of the received signal. Second, the

correlation function of the AltBOC signal resembles those of the BOC modulations,

and therefore the methods developed for the BOC modulation can be explored for

their applicability to AltBOC. Finally, the demand in resources, such as computa-

tional load and the time for acquisition, that basically stem from the fact that the

receiver has to accommodate high bandwidth and four signal components, should

be addressed.

A direct method to process the E5 signal at the receiver is to receive the signal

with at least 51.15 MHz front-end bandwidth and perform the correlation with the

locally generated replica of the modulating signal. This results in a correlation

function as shown in Fig. 2.14.

2.7.1. Autocorrelation waveform of the wideband E5 signal. Fig. 2.15

shows the autocorrelation plot of the wideband E5 signal for di�erent front-end

�lter bandwidths. Observe that even for a 50 MHz �lter, the peak is only slightly

degraded due to the fact that the energy in the two main lobes is still captured with

this bandwidth.


4900 5000 5100 5200 5300−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Sample Number

Nor

mal

ized

Aut

ocor

rela

tion

Val

ue

Real ComponentImaginary Component

Figure 2.14. Normalised autocorrelation value of the un�lteredGIOVE-A PRN 51 E5 code with 120 samples per chip and arbitrarychip shift

−1.5 −1 −0.5 0 0.5 1 1.5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time delay (chips)

Nor

mal

ized

aut

ocor

rela

tion

valu

e

Infinite BW90 MHz70 MHz50 MHz

Figure 2.15. Autocorrelation of the GIOVE-A wideband E5 signal

2.7.2. Correlation function of the components of the E5 signal. In the

receiver signal processing, there may be situations when only a component of the

signal is used. For example, a typical tracking algorithm uses only the pilot signal

or a receiver employing a front-end for GPS L5 (which shares the same main-lobe

frequency range as the E5a signal) may use only E5a component. Fig. 2.16 shows

the correlation waveforms of some components (or combinations of components) of

the E5 signal. Observe that the E5 pilot (combination of E5a and E5b pilot signals)

and the E5 data (combination of E5a and E5b data signals) signals result in similar

correlation waveforms as for the wideband signal. The separate E5a and E5b signals,

being free from the e�ect of sub-carriers, result in single-peak correlation waveforms.


−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

0

0.5

1E5 Pilot

−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

0

0.5

1E5 Data

−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

0

0.5

1E5a

Nor

mal

ized

cor

rela

tion

valu

e

Time delay (chips)

−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

0

0.5

1E5b

Figure 2.16. Correlation functions of di�erent components of theE5 signal; each waveform is normalised to its own component

2.7.3. Issues with AltBOC(15,10) signal acquisition and the existing

methods. As noted already, the autocorrelation function has side peaks which re-

sult in ambiguous signal acquisition. However, this is not the only issue with Direct

AltBOC acquisition.

First, the required receiver bandwidth to accommodate the two main lobes of

51.15 MHz imposes a limitation on the minimum sampling frequency and is much

higher than that required by other GNSS signals. Typical sampling frequencies

greater than 100 MHz have been used, for example, 122.76 MHz in Dovis et al.

(2007) and 112 MHz in the Septentrio GeNeRx1 receiver, to generate the local

replica and to perform the correlation process. Second, the sharp main peak in the

autocorrelation function restricts increasing the code search step size as is the case

for BOC signals (De Wilde et al., 2006). Third, the side peaks of the autocorrelation

function cause the threat of false transition to the tracking process, unless this is

resolved during the tracking process. Reducing the code search step size increases

the number of cells to be searched during the acquisition.

Di�erent acquisition approaches that exploit independent processing of sidebands

are explained in Martin et al. (2003); Dovis et al. (2007). In the Single Sideband

(SSB) approach, one of the main lobes, either E5a or E5b, is �ltered with a 20.46

MHz �lter and then correlated with respective codes (no sub-carrier is required).

This results in a BPSK(10)-like correlation triangle. In the Dual (or Double) Side-

band (DSB) approach, both the E5a and E5b lobes are �ltered, correlated with

respective codes and combined. Again, this results in a �BPSK(10)-like� correlation

triangle.

The Direct AltBOC acquisition method makes use of the 8-PSK principle and

the local replica can be generated using the look-up table method (OSSISICD, 2010).


In Dovis et al. (2007) a multi-resolution approach to �nd the code delay is demon-

strated. First, coarse acquisition uses the SSB strategy with E5aQ pilot code. With

this the acquisition engine can search at 0.5 chips with a total of 20460 cells for

a one millisecond pre-detection integration. As a second step, �ne estimate of the

code delay is performed with code search step size of 112over 2 chips covering only 24

cells. This reduces the total number of cells searched and also avoids the side peak

ambiguity. However note that in the method proposed in Dovis et al. (2007), the

�rst code delay estimate is going to incur a loss of about 6 dB in Pd (because of only

21% power) compared to the Direct AltBOC processing. In order to compensate for

this loss a minimum of four millisecond pre-detection integration time is required,

which increases the time spent in each cell to 4 ms.

Another class of acquisition technique proposed in the literature addresses the

problem of side peak ambiguity in BOC signals (Heiries et al., 2004; Burian et al.,

2006). These techniques concentrate on the correlation function and try to synthesise

a correlation waveform devoid of any strong side peaks. These techniques hardly

address the correlation loss for larger code search step size scenarios. Some of the

related techniques are:

(1) `BPSK-like' method proposed in Martin et al. (2003) and modi�ed in Burian

et al. (2006)

(2) Sub-carrier Phase Cancellation Method (SPC) proposed in Heiries et al.

(2004)

(3) Very Early + Prompt method mentioned in Heiries et al. (2004)

The `BPSK-like' method essentially falls into the category of the SSB/DSB ap-

proach.

The SPC method is based on the idea of removing the sub-carrier from the

received signal (after carrier removal). In this method, the complex local replica is

generated as in (2.32) where s(t) is the local replica, c(t) is the local code, sc(t) is

the local sub-carrier τ is the delay estimate, and Tsc is the sub-carrier period:

s∗(t− τ) = c(t− τ)

(sc(t− τ)− i · sc

(t− τ − Tsc

4

))(2.32)

As seen from (2.32), the sub-carrier in the quadrature arm of the local replica is

phase-shifted by one quarter of the sub-carrier period. It turns out that when the

BOC signal is multiplied with this local replica the shape of the correlation function

is similar to the BPSK triangle.Fig. 2.17 shows the E5a-Q correlation waveform.

The Very Early + Prompt method, referred here as |V E2 + P 2| works on the

basis that if magnitudes of two correlation values of the BOC signal separated by

an appropriate delay are combined, then it results in a correlation waveform whose

shape is similar to the BPSK triangle. In the |V E2 + P 2| method the local replica


300 350 400 450 500 550 600 650 700−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Sample number

Nor

mal

ized

Cor

rela

tion

Val

ue

Real

Imaginary

Magnitude

Figure 2.17. Normalised correlation value for E5a-Q code ofGIOVE-A PRN 51

is generated as follows (Heiries et al., 2004):

s∗P (t− τ) = c(t− τ) · sc(t− τ) (2.33a)

s∗V E(t− τ) = c

(t− τ − Tsc

4

)· sc(t− τ − Tsc

4

)(2.33b)

However, the performance of the SPC and the |V E2 + P 2| methods for the Galileo

E5 AltBOC(15,10) signal were not studied in Heiries et al. (2004).

With these discussions it can be observed that there is a need for proper cat-

egorisation, in a broader sense, of the several existing methods. In addition, the

applicability of some of the methods that are not discussed explicitly targeting

AltBOC have to be examined for their feasibility, and the performance of the appli-

cable methods has to be evaluated. For example, in the multi-resolution acquisition

method the feasibility of using the DSB method instead of the SSB method (for the

�rst/coarse search) has to be analysed. A part of Chapter 4 addresses this topic.

2.7.4. The cell-correlation phenomenon. GNSS signal acquisition engines

can be broadly classi�ed into two categories: active correlator and passive matched

�lter (Polydoros and Weber, 1984a,b; Shivaramaiah, 2004). In active correlators,

the local replica code phase is actively self-adjusted to make it compliant with the

incoming code. The dwell time needed to produce a correlation output is equal to

the product between the coherent and noncoherent integration times. However, in

matched �lter (MF) correlators, the local code with a predetermined phase is loaded

into the code bu�er and remains unchanged throughout the acquisition. The MF

just passively waits until the code in the received signal obtains the predetermined

phase. A correlator output is produced whenever a new signal sample arrives in the


Table 2.3. Galileo E1 Open Service signal code structure (from OS-SISICD (2010))

Signal Component Tiered Code Secondary Code Secondary CodePeriod (ms) Length (Chips) Mnemonic

E1B 4 1 -N/A-E1C 100 25 CS251

shift register. This means that the dwell time is just one sample duration, which is

much faster than that of the active correlators.

Due to their attractive acquisition time performance MF correlators promise to

be a good choice for GNSS signal acquisition engines (Holmes, 2007; Lyusin, 1998).

However, consecutive MF correlator outputs are made by the signal segments which

are just di�erent in some samples. Therefore, depending on the sample spacing,

there might exist strong correlations among the consecutive MF correlator outputs.

This e�ect has been studied to some extent for other spread spectrum systems (Sheen

et al., 1999; Giunta and Benedetto, 2007). Existing GNSS signal acquisition methods

that target the design (e.g. threshold determination) and performance evaluation

(detection probability and mean acquisition time), do not consider the e�ect of

cell-correlation. In Chapter 4, a brief description of the e�ect of cell-correlation is

provided (Ta et al., accepted for publication).

2.7.5. Secondary code acquisition. In GNSS, longer integrations are re-

quired to obtain better signal-to-noise ratio during the signal synchronisation pro-

cess. However the presence of secondary codes on the top of primary codes puts a

constraint on the coherent integration duration for pilot channels in a similar way

to the e�ect of data bits in data-carrying channels.

Sec. 2.3.4 described the secondary code structure in Galileo E5. For completeness

Table 2.3 also lists the secondary code characteristics of the Galileo E1 signal. Note

that except for the E1B signal component, which carries only the navigation data,

all the other Galileo OS signals have a two-tiered code structure.

A well known method of weak signal acquisition is to integrate the correlation

values for a longer period in order to achieve a good post-correlation signal-to-noise

ratio, and hence allow a su�cient margin for the decision statistic to pass the acqui-

sition threshold test. Pre-detection integration over one primary code duration is

often not su�cient to acquire weak signals and the presence of a secondary code of

unknown phase prevents the receiver from performing a longer integration. Knowl-

edge of the secondary code phase is required to perform longer coherent integrations.

Extended integration can be achieved by a suitable combination of coherent and

non-coherent integration of the correlation values. Even though the non-coherent

integration performance is inferior to that of coherent integration (Diggelen, 2009),


it is a preferred choice in traditional receivers so as to integrate across the data bit

boundaries. Secondary codes have the same e�ect on the correlation values as data

bit transitions, but code values are known a priori (but not code phase) by the

receiver, unlike the data.

For a smooth transition from acquisition to tracking it is also necessary to ac-

quire the secondary code chip position. If a receiver employs any of the existing

acquisition methods, then the receiver must follow a two-step procedure: acquire

the primary code �rst and then acquire the secondary code by trying out all the

possible secondary code delays. For longer integrations in weak signal environments

this is a time-consuming task and increases the acquisition time, especially with

long secondary code lengths (such as 100 chips). The other option is to consider

the secondary codes as data bits, in which case the maximum coherent integration

duration during the tracking will be restricted to one primary code period (until the

secondary code phase is determined), which in turn a�ects the tracking performance.

Previous work related to Galileo signal acquisition considers only the primary

code period for coherent integration, and then the result is integrated non-coherently

for longer integration periods, e.g. De Wilde et al. (2006). A closely related work

(Corazza et al., 2007) which uses a multi-hypothesis technique demands larger and

larger memory as the integration time is increased, and in addition the secondary

code phase has to be acquired in a separate process, or one should wait until the

maximum length of the secondary code is reached.

2.7.5.1. Challenges in the presence of secondary codes. In order to reduce the

noise in the acquisition process, a typical method is to integrate the correlation

samples over longer durations so as to reduce the noise bandwidth and hence the

noise at the decision point. Integration for a single period of the primary code may

not yield the required noise performance. For example, the primary codes in E5

signal have a period of one millisecond which o�ers only 30 dB gain in the C/N0,

whereas each doubling this integration time further increases the gain by 3 dB.

Hence in principle, one can keep on increasing the integration time, and for the pilot

signals this process is often only limited by the receiver dynamics and reference

clock frequency drift. In the absence of a secondary code, the integration longer

than one primary code period is performed by coherently combining the successive

correlation samples of one primary code length. This is possible because the primary

code period is only moderately large. On the other hand, the length of the tiered

code is very long, perhaps too long, to be acquired in a single step. Hence the initial

task of the acquisition engine is to align the local primary code replica with the

primary code boundary of the received signal. Once the primary code chip shift

is found, the secondary code chip position is then acquired, thus completing the

acquisition process.


The presence of secondary codes basically imposes two challenges to this process.

The �rst challenge is for longer integration. Because the acquisition engine will not

have knowledge of the secondary code chip position (and hence the chip value),

the secondary code chip transition may result in loss of the coherently combined

correlation value. This problem is similar to the data bit transition problem for

the GPS L1 C/A signal. The simplest solution is to non-coherently combine the

correlation values of one primary code length to obtain the �nal correlation value.

However, the non-coherent combining results in a lower integration gain (mainly

due to the squaring loss). A more sophisticated approach to the longer integration

problem is to analyse the coherent combination of all the secondary code transition

hypotheses and then select the maximum among them. This approach was proposed

earlier in Hegarty (2006) for GPS L5 and later in Corazza et al. (2007). This results

in an evolutionary tree with Ls leaves (where Ls is the number of secondary code

chips) whose size doubles for every additional primary code period, i.e. 2Ls−1, with

the �-1� entering because the receiver does not need to determine the absolute phase

at this stage. For example, in order to integrate for 4 primary code lengths, one

needs to perform and analyse the correlations for 8 di�erent combinations. For 8

primary code lengths integration there will be 128 combinations. It is very di�cult

to handle such high integration durations in practice. Also note that the standard

evolutionary tree approach for the longer integration does not consider the receiver's

knowledge of the secondary code.

The second challenge is to acquire the chip position of the secondary code. It

is necessary to know the secondary chip position so as to pass the information to

the signal tracking stage. Without this information the tracking process can not

�wipe-o�� the e�ect of the secondary code. In the two-step acquisition process, to

acquire the secondary code chip position in E5a-Q or E5b-Q signal, 100 combinations

of 100 consecutive one millisecond correlation values have to be examined. These

two challenges are the main motivations to explore the properties of the Galileo

secondary codes - which is the topic of the secondary code acquisition section in

Chapter 3.

2.8. Galileo E5 Signal Tracking

As with the acquisition, it is possible to track the Galileo E5 AltBOC(15,10)

signal by employing di�erent local reference signals, each focusing on di�erent com-

ponents (or the combination of components) of the signal. The �rst of its kind,

a brief discussion on Galileo E5 AltBOC tracking is provided in Sleewaegen et al.

(2004). The simulation results of tracking with the conventional type architecture

are also provided in Sleewaegen et al. (2004), which is useful in showing the kind of

performance achievable with the AltBOC(15,10) signal.

2.8. GALILEO E5 SIGNAL TRACKING 41

In Soellner and Erhard (2003), the concepts of Central Carrier Single-Sideband-

Tracking (CC-SSB) and O�set Carrier Single-Sideband-Tracking (OC-SSB) are in-

troduced. In CC-SSB the receiver demodulates E5a and E5b signals either by cen-

tring the local carrier frequency at the centre of the respective bands or by translating

the sidebands to the baseband before processing them. In OC-SSB the local carrier

frequency corresponds to the centre of the E5 band, i.e. 1191.795 MHz, and the

sidebands are processed as though they are at an IF of ±15.345 MHz. A tracking

architecture that assists in comparing the performance of Alternate Linear O�set

Carrier (AltLOC) and AltBOC modulation with AltBOC and AltBOC-NCE local

reference signals is presented in Soellner and Erhard (2003). However, Soellner and

Erhard (2003) does not discuss the issue of combining the di�erent components of

the AltBOC(15,10) signal, and also does not discuss increasing the integration du-

ration beyond the data bit boundaries. Depending on whether a receiver transits

to tracking with only the primary code phase estimate or both the primary and

secondary code phase estimates, constraints on the integration duration is imposed,

which in turn a�ects the tracking accuracy. Hence this is an interesting problem to

investigate further.

The general problem of dealing with integration beyond data bit boundaries in

the presence of a pilot and a data signal is not unique to E5 AltBOC. The GPS

L5 signal also consists of a pilot and a data signal component that are in phase

quadrature with each other. In principle, the structure of the Galileo E5a signal,

when considered independently, is similar to the GPS L5 signal structure. Methods

which tackle the problem of data bits can be broadly categorised into post-correlation

combination methods and pre-correlation combination methods.

In post-correlation combination methods, the signal components are combined

after the correlation operation, at the discriminator stage. These methods have

been explored in Hegarty (1999), Tran (2004), Tran (2002) for the GPS L5 and

GPS L2C signals. Basically, a weighted combination of the discriminator outputs

is used to estimate the carrier phase and the code delay corrections within the

tracking loops. These methods can be applied separately to the E5a pilot and

data components denoted here as L(E5aI , E5aQ) and to the E5b pilot and data

components denoted here as L(E5bI , E5bQ). A coherent combination method has

been discussed to improve the signal-to-noise ratio during the acquisition, in the case

of GPS L5 signal acquisition (Yang et al., 2004). The process of linear combination

becomes more di�cult in the presence of two data carrying signals for coherent

integrations more than the symbol duration, which is the situation with the E5

signal.

Some of the architectures in the literature employ the pre-correlation method.

Mattos (2006) proposed a method for combination when there is only one data


component for the Galileo E1 signal. Gerein (2007) proposed a method to demod-

ulate the data by combining reference signals of the data components for the E5

signal. Unlike the post-correlation method, where the correlation for the di�erent

components are obtained independently, pre-correlation methods require a new local

reference signal to be generated.

The challenge of devising an architecture that allows coherent integration longer

than the data bit duration (or the secondary code bit duration) in the presence of

two pilot and two data-carrying components still remains, and is the topic addressed

in Chapter 5. With di�erent architecture modi�cations in context, the e�ect on the

complexity of the pre-correlation combination process in the presence of two data

components is also covered in Chapter 5.

2.8.1. Code lock loop tracking range. The discriminator in the case of a

DLL converts the di�erence in the Early and Late correlation values into the time

delay error (from the correlation main peak) of the Prompt reference signal. The

spacing between the Early and the Late reference signals and the sharpness of the

correlation peak determine the range of time delay errors over which the DLL can

produce corrections so as to �pull� the Prompt to correspond to the peak. Code

tracking linear range and the code phase tracking jitter have always been opposing

design criteria for a code lock loop in GNSS. BOC-type signals o�er better code

tracking jitter and multipath performance by sharpening the correlation triangle.

However the linear tracking range is reduced, which directly results in poor dynamics

and low signal strength performance of the tracking loop. The problem is most

severe in AltBOC modulation, which has the sharpest correlation peak among all

the existing GNSS signals.

The basic principle of the BOC modulations is to reduce the width of the main

peak of the code correlation triangle without an unreasonable increase in the band-

width of the signal (which could be achieved simply by increasing the code chipping

rate). A correlation triangle with a sharp main peak increases the slope of the DLL

discriminator S-curve which in turn results in reduced code phase jitter at the output

of the DLL (Kaplan and Hegarty, 2006). However, the sharpening of the correlation

triangle puts a constraint on the Early-Late correlator chip spacing, and hence the

linear tracking range of the DLL is reduced. While higher dynamics can be handled

by employing a FLL for carrier tracking (Kaplan and Hegarty, 2006), the linear

tracking range of the code lock loop directly a�ects the degree of dynamics that can

be handled and its low signal strength performance. While the e�ect of dynamics

stress on a DLL can be reduced to some extent by carrier Doppler aiding (Kaplan

and Hegarty, 2006), the degraded robustness of the DLL at low signal strength still

remains an issue.

2.9. MULTIPATH MITIGATION IN GALILEO E5 43

Extending the range has been studied in the past (Wilde, 1995, 1998a,b) by

increasing the number of correlator arms and combining their correlation outputs.

These methods are not directly applicable to the BOC family of signals because the

standard Early-Late chip spacing used for BOC signals will already be at the edge of

the linear region, and any additional correlator arm may drive the loop into a false

locking region due to the multiple correlation peaks within one chip. BPSK tracking

can use up to 1 chip as the Early-Late spacing, but till now (Margaria and Dovis,

2008), separate E5a/E5b loops have only been used for the data demodulation with

carrier tracking loops and not for the code phase measurements, so the chip spacing

was of little signi�cance in AltBOC(15,10). The AltBOC modulation technique

can address this problem because it allows the Galileo E5 signal, for example, to

independently process its E5a and E5b components as BPSK signals in addition to

the high performance 8-PSK like main signal. This means that the same satellite

signal can be demodulated to obtain the AltBOC(15,10) correlation function and a

BPSK(10)-like correlation function. Combination of these two types of code tracking

loops is the key point to achieve the extended tracking region discussed in the second

half of Chapter 4.

2.9. Multipath Mitigation in Galileo E5

It is well known that multipath a�ects the performance GNSS. As ways to deal

with other errors progress, the impact of multipath is becoming a more signi�cant

issue. Mitigation of multipath has been a research focus for several decades and the

problem is still persistent (Ward et al., 2006). The research community is trying

to combat multipath fading in two major ways. First, by a signal structure design

approach. Reduction of the e�ects of multipath has been one of the main design

criteria for the newly proposed signals in the GNSS modernisation process (Falcone

et al., 2006). Second, by a receiver design approach. A number of techniques focus

on di�erent stages of the receiver signal processing chain to resolve the e�ect of

multipath on the measurements (Irsigler and Eissfeller, 2003).

Due to the code chipping rate and higher signal bandwidth, AltBOC (15,10) also

helps in eliminating the long-range multipath e�ects on code phase measurements.

However with existing DLL and code discriminator architectures, the short-range

multipath e�ects still remain.

Multipath mitigation via enhanced receiver design methodologies can be broadly

categorised as follows. First, using the Receive Time Diversity (or tapped delay)

techniques and the maximum likelihood principle, the multipath parameters viz. the

attenuation, the multipath delay and multipath phase are estimated. Often, the re-

ceive time diversity has been achieved by employing multiple correlator arms spaced

in time. The Multipath Estimating DLL (MEDLL) (Townsend and Fenton, 1995),


Frequency Diversity

Figure 2.18. Categorisation of the code phase multipath mitigationmethods

Pulse Aperture Correlator (PAC), and the Vision Correlator belong to this category.

The second category is the Receive Spatial Diversity schemes. A familiar method

is to use multiple antennas in a prede�ned pattern with associated processing tech-

niques (Moelker, 1997; Ray et al., 1999). The third category is the Receive Antenna

Design itself (Brown and Mathews, 2005; Counselman, 1999; Kunysz, 2003). This

can include antennas with low gain below the horizontal and the use of choke rings.

The fourth category is Receiver Signal Processing by virtue of tracking loop

design, S-curve shaping, �ltering, and so on. A number of techniques exist in this

category such as the {Narrow, High Resolution, Strobe, Gated, Shaping} correlators

(McGraw and Braasch, 1999; Garin and Rousseau, 1997; Garin, 2005; Veitsel et al.,

1998; Nuñes et al., 2004; Pany et al., 2005; Bhuiyan et al., 2007). Peak tracking,

Adaptive and Particle �ltering, are also popular. A detailed comparison of most of

these techniques is provided in Irsigler and Eissfeller (2003); Braasch (2001). Each

of these techniques has its own advantages, disadvantages and pertinence. The cate-

gorisation of some of the techniques belonging to Receive Time Diversity or Receiver

Signal Processing are arguable. Nevertheless, the aforementioned techniques do not

exploit all of the properties of the transmitted signal.

Chapter 6 introduces a method which exploits the Frequency Diversity feature

of the Galileo E5 AltBOC signal. The aforementioned categorisation is depicted

in Fig. 2.18. In wireless communication systems other than ranging systems, Fre-

quency Diversity has been used to address the problem of multipath fading, but in

a di�erent context. In these systems, transmitting and receiving multiple frequen-

cies e�ectively carrying the same information are used to combine the energies in

multipath channels via some special techniques such as Maximal Ratio Combining

(MRC) and Equal Gain Combining (EGC). The main aim of these techniques is

to increase the channel capacity by reducing the Inter Symbol Interference (ISI)

(Goldsmith, 2005; Tse and Viswanath, 2005; Proakis, 2000). However the focus of

a ranging system is to estimate the code delay and carrier phase of the direct signal

2.9. MULTIPATH MITIGATION IN GALILEO E5 45

excluding all of the superimposed multipath components at the receiving antenna.

As mentioned earlier, the Galileo E5 signal encompasses a number of features.

The two sideband components of the Galileo E5 signal can be considered as car-

rying the same ranging information (at least the `pilot only' channels). The two

sidebands when considered as BPSK(10) di�er in their modulation type to that of

the main signal which is 8-PSK AltBOC (this feature can be argued as being Signal

Diversity - Barnes et al. (2006)). Since the e�ect of multipath fading is frequency

dependent and also depends on the type of modulation, it is of interest to exploit

these features in reducing the multipath e�ect. In addition, one of the earliest works

related to multipath fading and frequency diversity analyses the negative correlation

among the individual frequency components for di�erent multipath delays (Haber

and Noorchashm, 1974). The AltBOC signal exhibits a basic frequency diversity

feature in terms of having a signal transmitted from the same satellite in two di�er-

ent bands, E5a and E5b. The exploitation of these features for multipath mitigation

is the topic of Chapter 5.

Ionospheric e�ects on the wideband signals are discussed in Gao et al. (2007). It

is well known that the ionospheric path is characterised by the direction of arrival,

the wave polarisation, the carrier frequency and the group delay. Only the last two

parameters vary for the E5a and E5b signals. The dispersion within the 51 MHz

band on either side of the E5 centre frequency of 1191.795MHz is nearly symmetric

(Sleewaegen et al., 2004). Even under severe conditions (or storms) of about 100m

ionospheric delay, the di�erence due to the dispersion between E5a and E5b signals

is only 0.33ns. There remains the e�ect of ionosphere dispersion within the useful

bandwidth of 51MHz of the Galileo E5 AltBOC(15,10) signal, but the e�ect due to

the dispersion can be neglected for all practical purposes, as detailed in Sleewaegen

et al. (2004). The second part of Chapter 5 discusses the e�ect of ionosphere on the

multipath mitigation technique that utilises Frequency Diversity.

The e�ect of multipath on group delay has been studied in Otoshi (1993b);

Bishop et al. (1985). Unlike the ionosphere, which directly delays the Line-Of-Sight

(LOS) signal, under multipath conditions the non-LOS (NLOS) (re�ected) signals

are superimposed onto the LOS signal at the receiver antenna. It is of interest to

explore whether the phase at which the NLOS signal is superimposed is di�erent for

E5a and E5b signals, and whether this di�erence follows a pattern, and by combining

the multipath-a�ected carrier phases of the E5a and E5b signals, whether the code

phase multipath error at E5 can be mitigated. Chapter 5 also discusses this group

delay compensation viewpoint.


2.10. Galileo E5 Baseband Hardware

There has been a tremendous increase in interest in satellite-based radio navi-

gation technologies with the announcement of designs for multiple satellite systems

and new ranging signals that will complement those of the venerable GPS. The

greatest interest has been directed to Galileo, as well as the modernisation plans for

GPS. The development of GNSS and Regional Navigation Satellite Systems (Rizos

2007) have posed new challenges to the receiver development community, especially

with regard to the baseband signal processing of multiple GNSS signals (Dempster

2007, Dempster and Hewitson 2007).

As discussed in sec. 2.6.3, due to their fast operation, FFT-based code acquisi-

tion methods are an attractive option for GNSS receiver baseband signal processing

(Van Nee and Coenen, 1991). The requirement for FFT-based code correlation

is more attractive for longer codes, whose acquisition would consume considerable

time if serial code search is employed. However, there are several reasons why the

utility of FFT-based methods is dependent on understanding the trade-o� between

acquisition speed and the required processing power. First, the new signals of the

GNSS family employ longer period PRN codes and higher signal bandwidths, which

demand FFTs of large transform lengths. Secondly, to gain an advantage in posi-

tioning performance, next generation receivers target multiple GNSS signals, and

since each signal has its own code length, the receiver should accommodate FFT

blocks of varying lengths. In a multi-band, multi-system receiver, to handle all the

scenarios including the worst case scenario, i.e. of being able to independently ac-

quire longer codes, the acquisition engine has to be designed to accommodate the

largest FFT required in the system.

An acquisition engine is more useful initially when the receiver is powered on,

rather than during normal operation. Hence the design goal of the acquisition engine

in a multi-band GNSS receiver is to have the �exibility to search any of the desired

signals using as little hardware resource as possible. Though FFT methods speed-up

the acquisition process, this speed-up comes at a cost of increased computational

burden due to the increased FFT length.

There are several schemes to reduce the computational burden by reducing the

number of points required to perform the FFT (see Fig. 2.19).

Acquisition architecture can be modi�ed to realise the correlation with smaller

subsets of the signal and the local code (Yang, 2001; Sajabi et al., 2006), or to

reduce the e�ective transform length (Sajabi et al., 2006; Starzyk and Zhu, 2001;

Qaisar, Shivaramaiah and Dempster, 2008; Qaisar, Shivaramaiah, Dempster and

Rizos, 2008). Another method is to use the assistance data (Diggelen, 2009) from

an external source to narrow down the search space and hence to reduce the size of

2.10. GALILEO E5 BASEBAND HARDWARE 47

Acquisition

Architecture

Modification

Assistance

Information

Reducing the FFT

Computational

Burden

Efficient FFT

Implementation

· External (e.g. AGNSS)

· Internal (e.g. L1 to L5)

e.g.

· Partial Correlation

· Averaging

Figure 2.19. Methods to reduce the FFT computational load

the FFT. This assistance can be from an internal source such as in the case when

the receiver is already tracking a signal at a di�erent frequency band (from the same

satellite) that has shorter code.

In a multi-band GNSS receiver, the receiver has to compute several FFTs of

di�erent sizes due to the varying code lengths of di�erent signals. Hence the receiver

has to accommodate dedicated FFT blocks of varying sizes. Moreover, depending

on the code length and the required chip step, the FFT block requirement may not

always be a power-of-two. For example, a triple-band receiver designed to acquire

GPS L1, Galileo E1, GPS L5 and GPS L2C has to have 2046, 8184, 20460 and

40920 point FFT blocks (half chip step assumed). The transform lengths that are

closer to a power-of-two number can be made power-of-two, by padding zeros (Yang,

2001) and then the power-of-two FFT can be e�ciently computed. In our example

this operation gives 2046 � 2048 and 8184 � 8192. However these techniques do

not solve either the problem of having dedicated FFT blocks of di�erent sizes for

di�erent signals or the problem of having a programmable FFT engine which caters

to all the required transform lengths in the receiver.

Since there are several GNSS modernisation signals with signal structures that

employ di�erent code lengths (apart from the GPS and Galileo civilian signals dis-

cussed in the previous paragraph), having dedicated or individual FFT blocks for

all the signals becomes extremely demanding in terms of the hardware resource and

the computational burden. Chapter 6 proposes methods that fall into a third cat-

egory shown in Fig. 2.19 in which the FFT for a larger set of code lengths (which

includes shorter, moderate and larger codes) is computed e�ciently by a method of

combining the FFT blocks used for shorter code signals.

2.10.1. Resource and power consumption estimate for tracking chan-

nels. Almost all the published work related to the baseband signal processing for

new GNSS signals concentrate on the algorithms, architectures and performance but

do not explore the resource and power requirements from a hardware standpoint.


The large signal bandwidth, multiple longer spreading codes and the split-spectrum

modulations demand wider registers and wider accumulators at higher operating

frequencies compared to the baseband hardware of the existing GPS L1 C/A sig-

nal. Some of the signals with the memory spreading codes have a completely new

requirement of up to 0.5 M memory bits.

Even with the advancement of software receivers, the hardware realisation of the

core correlator channels may become inevitable due to the large number signals that

need to be accommodated (because of multiple signals and multiple systems). A

receiver designed to be capable of processing a multitude of signals faces interest-

ing challenges when it comes to the resource and power consumption requirements

(Dempster, 2007; Dempster and Rizos, 2009). A recent publication McGraw (2010)

also draws attention to the concerns about modernised GNSS receiver baseband sig-

nal processing complexity. In Dempster (2007), it was estimated that a baseband

signal processing module that processes �all� the current and new GNSS signals will

roughly consume 200 times more power than that of a GPS L1 C/A! The contri-

bution of AltBOC(15,10) is the highest among all the other signal types due to its

complexity. In this context, the second part of Chapter 6 analyses the resource

requirement and power consumption of a baseband hardware designed to process

GPS and Galileo civil signals.

2.11. Multiplexing in GNSS Modulations

The GNSS community has paid great attention to signal design during the last

decade, mainly focusing on e�cient spectral utilisation and resistance to propagation

and environmental degradation (e.g. Barker et al., 2000; Betz, Winter 2001-2002;

Betz and Goldstein, 2002; Ries et al., 2002, 2003; Hegarty et al., 2004; Cahn et al.,

2007; Avila-Rodriguez et al., 2008; Stansell, Online).

The major contributors to the AltBOC(15,10) receiver baseband signal process-

ing complexity are the following. First, the receiver must generate four primary

spreading codes and combine them with the secondary codes. An alternative option

is to store the codes in memory. This may not be feasible in all situations (e.g.

if frequency domain techniques are used and FFTs of the local replica are stored)

due to the code length and the number of PRN codes (50 PRNs, four codes each

of 10230 chips length). Second, in the wideband 8-PSK-like processing, the phase

points should be generated using the special sub-carriers that demand accurate rep-

resentation of the sub-carrier amplitude (larger bit-width). Non-constant envelope

AltBOC could be used as an alternative reference signal instead of the constant

envelope AltBOC, but this incurs a loss of up to 1.38 dB (Soellner and Erhard,

2003).

2.11. MULTIPLEXING IN GNSS MODULATIONS 49

The aforementioned points indicate that there exists scope for improvement in

some of the aspects of the AltBOC modulation technique. The tracking and multi-

path performance depend heavily on the RMS bandwidth (or the Gabor bandwidth)

of the signal and hence complexity reduction by reducing the signal bandwidth harms

the performance. However, the signal structure itself can be explored for a speci�ed

signal bandwidth. One such parameter is the way the individual components of the

signal are multiplexed.

AltBOC modulation can be viewed as a method of phase-multiplexing two QPSK

signals. A direct way of phase-multiplexing two QPSK signals disturbs the constant

envelope properties of the signal (Ries et al., 2003). In the constant envelope AltBOC

this issue is resolved by modifying the shape of the sub-carrier waveform and adding

product-signal components. Several signal combining methods have been introduced

in the literature to combine more than one signal component to form a single trans-

mitted signal. These include the Interplex modulation (Butman and Timor, 1972;

Rebeyrol et al., 2006; Fan et al., 2008), Majority Vote combination (Spilker, 1977;

Pratt and Owen, 2005; Rodríguez, 2008) and Phase-Optimized Constant-Envelope

(POCET) combination (Dafesh and Cahn, 2009), which concentrate on the signal

components in a single frequency band. The AltBOC modulation still remains the

best of all the constant envelope phase-multiplexing schemes in the context of four

signal components at two frequency bands (complex sub-carrier), except for the

penalty of receiver complexity required for its demodulation. Time-multiplexing is

another method of combining the individual signal components. Time-multiplexing

has been successfully implemented for two BPSK and BOC signal components in

GPS L2C (ISGPS200E, 2010) and L1C (ISGPS800, 2010). Ries et al. (2002) de-

scribes a time multiplexing method for real sub-carrier BOC signals, but does not

discuss the detailed signal structure and properties, nor the complex sub-carrier

modulation.

Since any improvement in the signal structure is fundamental to the perfor-

mance of the entire system, it is of interest to further explore the time multiplexing

techniques - a topic of Chapter 7.

Apart from the receiver complexity issues, the AltBOC modulation su�ers from

two more drawbacks. First, the AltBOC, with the �rst two main lobes occupying

approximately 20MHz each (for the E5 AltBOC(15,10) example), is more prone to

interference. Second, frequency-selective channel impairments, such as phase delays

due to ionospheric dispersion and multipath, a�ect the signal's performance due to

the higher signal bandwidth. The search for a new signal especially for a wideband

GNSS signal is a very new topic (Mateu et al., 2010). Therefore Chapter 8 also

looks into this topic.


2.12. Summary

This chapter introduced discussions and the earlier work on �ve topics related

to Galileo E5 AltBOC signal viz. signal acquisition, signal tracking, code phase

multipath mitigation, baseband hardware realisation and multiplexing during signal

modulations. During the discussions, important gaps were identi�ed and the scope

for more research work was established. These �ve topics broadly re�ect the contents

of Chapters 4-8.

CHAPTER 3

Experimental Setup

3.1. Introduction

This chapter describes the data acquisition apparatus used to collect Global Nav-

igation Satellite System (GNSS) data required for various experiments conducted

for this thesis. Since this thesis deals with baseband signal processing algorithms,

the type of GNSS data collected is raw Intermediate Frequency (IF) signal samples.

The basic criterion for the experimental setup was to collect the IF samples of the

Galileo E5 and other GNSS signals at least using the same apparatus (except for the

front-end �lters and associated circuitry), if not simultaneously. The data collected

by the data acquisition apparatus was processed in one of the following ways:

• With the help of Matlab software developed by the author; the program

running on a host computer.

• With the help of a Hardware Description Language (HDL) program (Verilog

& VHDL) developed by the author; the program running in simulation

mode in an Electronic Design Automation (EDA) environment (Altera), on

a host computer.

• With the help of a HDL program (as above); the program running on an

Altera Field Programmable Gate Array (FPGA) device, on a hardware

platform.

For simulation studies, signal samples were generated with the help of a Matlab soft-

ware program developed by the author. An overview of the experimental method-

ology is provided in Fig. 3.1. With the data collection apparatus, IF samples were

collected from satellites for GPS L1 C/A, GPS L5 (SVN49), GIOVE-A & GIOVE-B

E1, and GIOVE-A & GIOVE-B E5 signals.

3.2. Data Collection Apparatus

Initially, �nding a data collection apparatus or research platform that can pro-

vide IF signal samples (or at least allow tapping the IF output in a receiver) was a

challenge, especially due to the wideband (>50 MHz) requirement of the Galileo E5

AltBOC(15,10) signal. Fortunately, in the course of this research, four di�erent data

collection apparatus were identi�ed and used as required by the experiments. It is

also worth mentioning here that the reason for using di�erent apparatus was mainly

51

52 3. EXPERIMENTAL SETUP

Signal Generation (MATLAB)

Data Collection Apparatus

Post processing (MATLAB)

IF Samples

Results

Host Computer Host Computer

Post processing (EDA Environment)

Host Computer

Storage (file)

Results

Post processing (FPGA Device)

FPGA Board Results

Host Computer

Hardware Setup

Antenna

OR

OR

OR

FPGA output capture

Host Computer

Signal Generation / Collection

Baseband Algorithms Implementation

Software Components

Hardware / HDL / Firmware Components

Figure 3.1. Overview of the simulation / experimental setup

because each setup had its own limitations. This section provides an overview of

these data collection apparatuses. The following items are discussed for each setup:

block diagram or picture, GPS / Galileo civil signals accommodated, bandwidth,

sampling frequency, number of Analogue-to-Digital-Converter (ADC) bits per sam-

ple (or number of processed bits per sample), limitations of the setup, antennas

used, and usage instances in this thesis.

3.2.1. Septentrio GeNeRx1. The GeNeRx1 is a multi-frequency GPS/Galileo

receiver developed by Septentrio (Septentrio, 2006). GeNeRx1 allows logging of IF

signal samples to a host computer via an Ethernet link. Though GeNeRx1 is capable

of simultaneously processing GPS and Galileo signals at di�erent frequency bands

as a receiver, IF logging is possible for only one frequency band at a time. The GPS

Galileo L1/E1 band is down-converted to an IF of 140.12 MHz with a bandwidth

of 40 MHz. The GPS L5 / Galileo E5 has an IF of 139.905 MHz with a bandwidth

of 55 MHz. The sampling frequency for all the signals is 112 MHz. Quantisation is

8-bits / sample. The front-end used for the GPS L2 signal has a bandwidth of 25

MHz, but IF sample logging is not allowed. The GeNeRx1 receiver is shown in Fig.

3.2. A Leica AR25 antenna (Bedford et al., 2009) was used to collect the signals.

Advantages

• Wide bandwidth (55 MHz) to capture the Galileo E5 AltBOC(15,10) signal.

• Accessibility - this setup was located at the UNSW Satellite Positioning and

Navigation (SNAP) Lab and hence it was available for data collection most

of the time (GeNeRx1 was also being used by the German Aerospace (DLR)

for GIOVE-A/B orbit monitoring), from Feb 2008 through Dec 2010.

Limitations

3.2. DATA COLLECTION APPARATUS 53

Figure 3.2. GeNeRx1 receiver

• Maximum data collection size is 16MBytes and hence the maximum dura-

tion of continuous IF data that can be captured is 149.79 ms.

� This is not an issue for signal acquisition since one primary code period

of Galileo E5 signal is one millisecond and the maximum length of the

secondary code is 100 (which results in a 100 ms tiered code period).

Therefore the signal of length one tiered code period can be easily

observed. However, when the acquisition module passes the estimates

to the tracking module, tracking loops may not settle within 149 ms and

hence it is di�cult to measure the tracking performance. This problem

was addressed to some extent by improving the estimates provided

to the tracking module (�ne synchronisation) to help the loops settle

within the �rst 50 ms (say) or less.

• Either the signal from the L1/E1 band or the L5/E5 band can be captured

in one stretch (but not both simultaneously), and hence no dual-frequency

algorithms can use the data collected with this setup.

• The GeNeRx1 receiver operation halts during the IF sample download op-

eration. Hence permission from the DLR was required every time logging

of the IF samples was attempted.

Due to the third limitation no performance comparison was possible on the same

dataset with respect to the baseband algorithms running inside the GeNeRx1 re-

ceiver. However, the carrier Doppler and the signal strength output of the acquisition

algorithms/tracking developed by the author were found to agree with those pro-

vided by the GeNeRx1 receiver (observed a few seconds before and after the data


USRP2

Antenna

GNU Radio capturing the

samples

Host Computer

Figure 3.3. USRP2 data collection setup

collection).

3.2.2. Universal software radio peripheral (USRP / USRP2) with

GNU radio. The USRP, developed by Ettus Research LLC, is a family of gen-

eral purpose platforms for software-de�ned radios (SDR)(Ettus, 2010). GNU Radio

is an open-source SDR platform. The USRP product family consists of the mother-

boards, which contain an FPGA for high speed signal processing, and interchange-

able daughterboards that cover di�erent frequency ranges. Together they provide a

�bridge� between the host computer and one or more antennas.

USRP is the �rst generation motherboard, and USRP2 is the second generation

motherboard. The maximum possible bandwidth in USRP is 16 MHz, and for

USRP2 it is 50 MHz; IF being zero in both cases. However the USRP2 �rmware and

the GNU radio software currently do not support more than 25 MHz bandwidth and,

in addition, the data collection at bandwidths 20 MHz and beyond are unreliable

(frequent loss of samples).

The daughterboards that cover the GNSS bands are1 DBSRX(800 MHz to 2.4

GHz receiver), RFX1200 (1150-1450 MHz transceiver) and RFX1800 (1.5-2.1 GHz

transceiver). Initial evaluation showed that the RFX1200 and RFX1800 were slightly

better (about 1.5 dB) in signal strength than the DBSRX boards, and also in terms

of having fewer spurious frequency components in the collected signal spectrum.

The USRP2 setup is shown in Fig. 3.3.

Two antennas were used for the data collection: the Leica AR25 which was

on the roof-top of the Electrical Engineering building at UNSW and the Antcom

2GNSSA-XMNS-1(Antcom, 2010).

Advantages

• Portability - extremely easy to carry around to collect IF samples, especially

USRP2 and the Antcom 2GNSSA antenna1WBX (50 MHz to 2.2 GHz transceiver) is a recent addition and was not available when theexperiments were carried out


Averna record & playback

systemLNA

40 dB pre-amp

Roof-top antenna

Figure 3.4. Averna setup

• Accessibility - always available, UNSW SNAP Lab has several USRP2

motherboards and daughterboards

Limitations

• 50 MHz wideband signal could not be captured with a single USRP2.

� USRP2 boards are designed to be used in a Multiple-Input-Multiple-

Output (MIMO) setup, in a master-slave arrangement, and one should

be able to coherently sample and capture two 20 MHz wide E5a and

E5b bands. However this arrangement had other issues, such as using

the external clock in the slave and synchronising the bits transferred

from two di�erent boards.

• IF signal capture at 20 MHz for longer durations (longer than about 2

seconds) is unreliable (discontinuities in the sample collection).

The data collected from the USRP2 was useful in observing the GPS L5 and Galileo

E5a signals together for a duration longer than 150 ms (which was not possible with

GeNeRx1).

3.2.3. Averna / National Instruments (NI) record & playback system

at University of Colorado, Boulder (UCB). During the �nal stages of the

research, the wideband Galileo E5 signal samples of longer duration were collected

with the aid of an Averna / NI record and playback system available in the GNSS

Lab at the UCB Aerospace Engineering department. A/Prof. Dennis Akos provided

data �les and also accommodated the author in the GNSS Lab for a short duration

to collect additional data.

The Averna record and playback system is built on NI's PXI Express architecture.

The captured data is streamed in real time to a high capacity RAID system. The

Averna system at UCB can capture data up to a bandwidth of 50 MHz, and can

cover all the GNSS frequency bands. LabView-based software in the host computer

helps control di�erent parameters such as the bandwidth/sampling frequency, gain,

etc. The Averna setup is shown in Fig. 3.4. Several datasets were collected using

two types of antennas: a NovAtel 704-X(NovAtel, 2010) and an Antcom 42GNSSA-

XT-1(Antcom, 2010). Datasets collected using the NovAtel antenna were found to


be better (in signal strength at the output of the correlator, tested with the help of

an SBAS satellite) than the Antcom antenna. The IF was zero and the sampling

frequency was 62.5 MHz. The 16-bit ADC output was re-quantised to 8-bit / sample.

A dataset was also collected by Dr. Akos, using an 18 m parabolic dish antenna

situated about 40 miles from UCB.

Advantages

• This setup allowed capture of wideband (50 MHz) IF signal samples for

longer durations (a few minutes)

• One of the datasets collected using the 18 m dish was helpful in obtaining

the GIOVE-B signal with a high Carrier to Noise density (C/N0)

Limitations

• High DME/TACAN interference was experienced within the E5 band in

Boulder. The area surrounding Boulder has a high density of VOR-DME /

TACAN stations and, as a result, signal strength degradation of 10-14 dB

was observed at the output of the correlator. Therefore only some of the

datasets collected using this setup could be used2.

• A di�erence in signal strengths of about 2-3 dB between the E5a and E5b

bands were observed, even with the 18 m parabolic antenna and, the source

of this error could not be established (the processing software used to de-

termine this di�erence was the same as that used with the GeNeRx1 setup,

except for the modi�cations in sampling frequency and IF values).

• Not being available at the UNSW SNAP Lab, opportunities to use it were

limited.

3.2.4. Transform-domain instrumentation GPS Receiver (TRIGR) plat-

form from Ohio University (OU). The information related to TRIGR and the

pictures included here were provided by Dr. Sanjeev Gunawardena of the Ohio Uni-

versity Avionics Engineering Center. Dr. Gunawardena also collected the IF samples

when the GIOVE-A or GIOVE-B satellites were visible over Athens, as requested

by the author. This setup was used during the �nal stages of the research3.

TRIGR is a GNSS receiver platform developed by the Ohio University Avionics

Engineering Center, which can be used to collect data from up to 8 front-end modules

with a maximum continuous data streaming rate (to disk) of about 250 MB/sec.

The maximum bandwidth possible for each front-end channel is 24 MHz. The raw

2The signals could be used with the pulse blanking technique, but the experiments in this thesisfocused on the signals without DME/TACAN interference for most of the tests, to match theprocess followed with other setups.3TRIGR was initially capable of collecting data in the L1, L2 and L5/E5a bands. Upon an enquiryby the author, Dr. Gunawardena agreed to include the E5b capability.


Front-End Architecture

Active

Splitter

Networks

or

Direct

Connect

to

Antennas

-11.5 to 20 dB

DVGA2

PLOfLO1 =

fc-70 MHz

LPFRF-BPF IF-BPF

DVGA1

-11.5 to 20 dB

GNSS Front-End Module

PLO

High Speed

Serial LVDS

Interface

to

Data

Collection

Server

ADC4 or 8 Ch

12-bit

65 MSPS

max

CH_A

Enc

Ant. Feeds

OCXO

SPI

Interface

ADC

Control

10 MHz Reference Osc.

(Internal or External)

CH_BGNSS Front-End Module fc: any, BW: 20-24 MHz

fs = 56.32 MHz

RF/IF Digital

Gain ControlSPI Interface

CH_CGNSS Front-End Module fc: any, BW: 20-24 MHz

CH_DGNSS Front-End Module fc: any, BW: 20-24 MHz

Figure 3.5. TRIGR front-end architecture

GPS L1, L2, L5or custom GNSSRF Front-EndModules

4-Channel or 8-Channel

Coherent Sampling Module

Active Signal SplittersIncluding Antenna Power

High-SpeedLVDS Datalink toData Collection Computer

GNSS Front-End Unit (top)

Figure 3.6. TRIGR GNSS front-end unit

data from the ADC is sent to an FPGA processor inside the server unit via a high-

speed serial interface. The FPGA packs the channels into a 32-bit format which is

then DMA (Direct Memory Access) transferred (via 8x PCI-Express) to the server

and written to an 8-disk RAID array. Depending on the packing format selected,

the 250 MB/sec data rate may contain up to any two channels at 16-bits/sample or

8 channels at 4-bits/sample. Fig. 3.5 shows the TRIGR front-end architecture and

Fig. 3.6 shows the TRIGR unit itself .

The IF samples were collected using a NovAtel GPS-740x antenna with a ~40 dB

gain <1 dB noise �gure Low Noise Ampli�er (LNA). The frequency bands covered

were GPS L1 / Galileo E1 (20 MHz), GPS L2 (20 MHz), GPS L5 / Galileo E5a

(24 MHz), and Galileo E5b (24 MHz). The sampling frequency was 56.32 MHz,


with IF centred at 13.68 MHz for L1 and L2 signals, 14.13 MHz for L5/E5a signals

and 13.82 MHz for the E5b signal. The number of quantised bits were modi�ed to

2-bits/sample to reduce the �le size (since a lot of data �les were transferred via

FTP).

Advantages

• This setup allowed capture of the civil signals with 20 MHz bandwidth in

L1/E1, L2, L5/E5a and E5b bands simultaneously for longer durations (a

few minutes).

Limitations

• Initial tests showed a slight di�erence in the signal strengths received by

the E5a channel and the E5b channel for both GIOVE-A and GIOVE-B

signals. Since the transmission correctness of the GIOVE-A and GIOVE-B

test satellites could not be established, the source of the problem could not

be identi�ed. Hence this data was not used for the tests on algorithms that

required E5a and E5b signal strength comparison.

• Not being available at the UNSW SNAP Lab, opportunities to use it were

limited.

3.3. Summary

Matlab-based software was developed by the author to acquire and track the

Galileo E5 signal, along with GPS L1 C/A, GPS L2, GPS L5 and Galileo E1 signals.

The Matlab-based software was veri�ed with the aid of IF signals captured using

the GeNeRx1 receiver.

The Verilog-based baseband hardware components (the core correlator modules)

were developed by the author for the Galileo E5 signal. The functionality of the

correlator modules were veri�ed with the aid of EDA simulation tools and the data

captured using the GeNeRx1 receiver.

Due to the 150 ms signal capture duration limitation of the GeNeRx1 setup,

other data collection apparatus viz., the USRP setup, the Averna setup and the

TRIGR setup were identi�ed and examined for their usability, though the last two

setups were used only during the �nal stages of the research. The advantages of each

experimental setup were applied in order to test some of the algorithms developed

during this research.

This exercise not only helped in obtaining some useful datasets for the experi-

ments conducted during the research, but also helped identify some limitations in

these setups (and any other similar experimental setup). Some of the datasets that

could not be used for the experiments in this thesis - and in fact pose di�erent sets

3.3. SUMMARY 59

of challenges, that would be experienced by other researchers while processing E5

signals - is an interesting topic to address in future work.

CHAPTER 4

Galileo E5 Signal Acquisition

4.1. Introduction

This chapter covers the topics related to the Galileo E5 primary code and sec-

ondary code acquisition identi�ed in Chapter 2. Sections 4.2, 4.3 and 4.5 contain

the work published in ENC-GNSS 2008 (Shivaramaiah and Dempster, 2008a); sec.

4.4 contains a part of the work accepted for publication in IEEE TAES (Ta et al.,

accepted for publication); sec. 4.6 contains the work published in ASILOMAR 2008

(Shivaramaiah and Dempster, 2008b), and sec. 4.7 contains the work published in

ION GNSS 2008 (Shivaramaiah et al., 2008).

The contributions of this chapter are:

• categorisation of the existing Galileo E5 AltBOC acquisition methods,

• analysis of the e�ect of code search step size on AltBOC(15,10) signal ac-

quisition,

• examining the e�ect of cell correlation in Global Navigation Satellite System

(GNSS) matched �lter (MF) receivers and discussing the importance of its

inclusion in the acquisition performance analysis,

• analysing the performance of the |V E2 + P 2| method for AltBOC(15,10),

• analysing the correlation properties of secondary codes used in GPS L5,

Galileo E1 and Galileo E5,

• de�ning and describing the �characteristic length� property of secondary

codes used in GNSS, and

• proposing a sequential method (based on the principles of convolutional

decoding) to acquire the secondary code phase

This chapter is organised as follows: Sec. 4.2 details the di�erent signal acquisition

strategies; sec. 4.3 discusses the acquisition complexity followed by a description of

the e�ect of cell correlation phenomenon in sec. 4.4; sec. 4.5 analyse the performance

of the |V E2 + P 2| method. Sec. 4.6 revisits the |V E2 + P 2| method in the presence

of �ltering and the type of decision methodology. Sec. 4.7 describes the properties

of secondary codes and proposes a method to acquire the secondary code phase.

Sec. 4.8 summarises the �ndings of this chapter.

61

62 4. GALILEO E5 SIGNAL ACQUISITION

4.2. Galileo E5 Acquisition Strategies

4.2.1. Categorisation of the acquisition methods. The acquisition perfor-

mance parameters Pd and T acq are directly and inversely proportional respectively,

to the signal-to-noise-ratio (SNR) of the desired signal. The code length and the au-

tocorrelation properties of the ranging code, granularity of time and frequency steps

and the search strategy all in�uence the Pd and T acq for a speci�ed Pfa. BPSK

modulated ranging codes o�er triangular autocorrelation function (ACF) with a

single main peak which, in most situations provides a good Pd and T acq for half-chip

search steps. However, as seen in Chapter 2, BOC and AltBOC modulations pro-

duce multiple peaks in the ACF which create ambiguities during acquisition, and

require better processing to achieve Pd and T acq comparable to that of GPS L1 C/A

under similar conditions.

The principal aim of any acquisition method in GNSS is to address the issue of

e�ciently achieving the required Pd and T acq. Though Pd and T acq are inter-related,

the methods discussed in Chapter 2 can be broadly classi�ed into two groups, one

which primarily targets Pd and the other that primarily addresses T acq.

�Search strategy� based methods. The methods which target T acq basically try

to avoid acquiring the narrow correlation peak since such a narrow peak demands

smaller search steps1. For example, by correlating the two sidebands independently

acquisition engines obtain a correlation waveform without multiple peaks. This al-

lows the acquisition engine to increase the code search step size (or equivalently

reduce the number of search cells).In other words, these methods look for a di�er-

ent search strategy and can be appropriately grouped into �search strategy� based

methods. An example application of the search strategy based method is the multi-

resolution acquisition method described in Dovis et al. (2007).

�Correlation scheme� based methods. The methods that target Pd basically try

to widen the correlation function so as to have higher probability of �hitting� the

correct cell. In other words these methods modify the correlation function and hence

can be appropriately grouped into �correlation scheme� based methods. The sub-

carrier phase cancellation (SPC) method and the |V E2 + P 2| method (Heiries et al.,

2004; Burian et al., 2006) are two examples in this category. It should be noted that,

because Pd and Tacq are related to each other, the correlation scheme based methods

indirectly result in a better T acq.

1Another way of targeting T acq is to reduce the number of search cells by reducing the ambiguityin the code delay domain. For example, acquisition or tracking estimates from the Galileo E1signal, or from an external aiding source such as A-GNSS, if available in the receiver, can be usedto drastically narrow down the code delay search space for the E5 signal. These methods are notdiscussed in this chapter.

4.2. GALILEO E5 ACQUISITION STRATEGIES 63

4.2.2. Search strategy based methods for acquisition. The list below con-

solidates di�erent methods into the search strategy category. The list assumes that

the secondary code phase is unknown and the aim of the acquisition engine is to

acquire only the primary code. Fig. 4.1 shows one example block diagram in each

type of search strategy based methods for acquisition.

4.2.2.1. Single sideband acquisition (SSB). This method is equivalent to the CC-

SSB method where the input IF signal is mixed with the local carrier centred at one

of the sidebands and the result is then mixed with the local reference signal. The

reference signal here is void of any subcarrier. The variants of this method are:

(1) Any one of the E5aQ, E5bQ, E5aI and E5bI: Pilot or data channels with

the integration limit of one millisecond (one secondary code chip duration)

(2) Non-coherent combination of {E5aQ, E5aI} or non-coherent combination of

{E5bQ, E5bI}: Integration duration not constrained by the spreading codes

or the data (but constrained by the receiver clock and the user dynamics).

Note that a coherent combination of the pilot and data channels is not directly pos-

sible because the secondary codes used in E5aI, E5bI, E5aQ and E5bQ are di�erent

to each other.

4.2.2.2. Double sideband acquisition (DSB). Here, the correlation results from

both the sidebands are used together. The variants of this method are:

(1) Non-coherent combination of {E5aQ, E5bQ} or non-coherent combination

of {E5aI, E5bI}: Combination of pilot channels or data channels.

(2) Non-coherent combination of {E5a, E5b}.

4.2.2.3. Full-band independent code acquisition (FIC). This method is equivalent

to the OC-SSB method where the local carrier is centred at the centre of E5 band.

The reference signal contains the spreading code and the subcarrier corresponding

to the signal component of interest. The variants of this method are:

(1) Any one of the E5aQ, E5bQ, E5aI and E5bI: Pilot or data channels with

the integration limit of one millisecond (one secondary code chip duration).

(2) Non-coherent combination of {E5aQ, E5aI} or non-coherent combination of

{E5bQ, E5bI}: Integration duration not constrained by the spreading codes

or the data (but constrained by the receiver clock and the user dynamics).

(3) Non-coherent combination of {E5aQ, E5bQ} or non-coherent combination

of {E5aI, E5bI}: Combination of pilot channels or data channels.

(4) Non-coherent combination of {E5a, E5b}.

In the FIC acquisition method a locally generated individual code with the corre-

sponding sub-carrier is multiplied with the received signal without �ltering (i.e. no

�lter apart from the RF front-end �lter). This is possible because each of the codes


20 MHz Filter Correlator

E5bQ Code

E5bQ Correlation Value

E5bQ

E5bI

E5bE5a

E5aI

E5aQ E5bQ

E5bI


E5bQ Code

E5b Correlation Value

E5bQ

E5bI

E5bE5a

E5aI

E5aQ E5bQ

E5bI

Correlator

E5bI Code

+

( )2

( )2


E5bQ Code

E5 pilot components Correlation Value

E5bQ

E5bI

E5bE5a

E5aI

E5aQ E5bQ

E5bI

Correlator

E5aQ Code

+

( )2

( )220 MHz Filter

E5aQ

E5aI


E5bQ Code x exp(jwsct)

E5 pilot components Correlation Value

E5bE5a

E5aI

E5aQ E5bQ

E5bI

Correlator

+

( )2

( )2

E5aQ Code x exp(-jwsct)


AltBOC LUT output

E5 Correlation Value

E5bE5a

E5aI

E5aQ E5bQ

E5bI

SSB – E5bQ Single Component example

SSB – E5bQ and E5bI non-coherent combination example

DSB – E5bQ and E5aQ non-coherent combination example

FIC – E5bQ and E5aQ non-coherent combination example

Direct AltBOC (data-bit wipe off not shown)

Figure 4.1. Examples of search strategy based methods for acquisition


used in E5 is quasi-orthogonal to the other. Even though the magnitude of individ-

ual correlation values is a BPSK(10)-like correlation triangle, coherent combination

(which su�ers from the secondary code / data bit issue) of pilot-only or data-only

channels yields a correlation waveform similar to AltBOC(15,10). However, the com-

bination of E5a and E5b channels results in a BPSK(10)-like correlation triangle as

in the case of DSB. Another di�erence between SSB or DSB and the FIC method is

the frequency of operation of the correlation circuit. Although the sideband �ltering

overhead is removed in FIC, there is no scope for down-sampling and the correlator

circuit should operate at the original sampling frequency.

4.2.2.4. Direct AltBOC method (8-PSK-like processing). In the direct AltBOC

method, the reference signal generator employs a look-up-table (LUT) so as to com-

bine all four signal components. However, this method su�ers from the secondary

code and data bit sign issues when only the primary codes are considered for the lo-

cal code replica. Even though the correlator circuit operates at the original sampling

frequency, it is not required to generate the individual sub-carriers to combine with

the individual codes because the LUT essentially maps the sub-carrier phase-points.

One method to resolve the data bit ambiguity is to form four LUT outputs corre-

sponding to all the four possible data bit combinations (due to two data components

E5aI and E5bI), correlate the input signal with these four LUT outputs to obtain

four correlation values and select the maximum correlation value among these four

correlation values to use in the acquisition decision process. This brute-force data

wipe-o� method is assumed in the rest of this chapter for the Direct AltBOC ac-

quisition method. However, for the Galileo E5 signal, one data bit period is longer

than one secondary code chip period. Moreover, during the primary code acquisi-

tion, the receiver will generally not have the knowledge of the secondary code phase

(and hence the secondary code chip value). Therefore, the problem of coherent in-

tegration while combining di�erent signal components is constrained more by the

unknown secondary code phase rather than the data bit ambiguity. The topic of

longer coherent integrations including the unknown secondary code phase is dealt

in sec. 4.7.

Fig. 4.2 shows the correlation waveforms for some of the approaches mentioned

above. In Fig. 4.2 a sampling frequency of 122.76 MHz was used and the values

were normalised with respect to the correlation values obtained with the direct

AltBOC method for an un�ltered signal. The sidebands are �ltered with a �lter of

bandwidth 20.46MHz. Observe that a single component results in a maximum value

close to ~21% of the Direct AltBOC processing. Table 4.1 summarises the search

strategy based acquisition approaches discussed so far. The term �extra �lter� is

used here since, to take advantage of the AltBOC modulation, the tracking process

requires at least a 51 MHz front-end �lter, and whatever is used in the acquisition


0 50 100 150 200 250 300 350 400 4500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sample number

Nor

mal

ized

Abs

olut

e A

utoc

orre

latio

n V

alue

E5aQ−SSB

E5a−SSB

E5a and E5b−DSB

E5aQ−FIC

E5 Coherent Pilots−FIC

E5 Pilots and Data−FIC

Direct AltBOC(15,10)

Figure 4.2. Normalised absolute correlation values for di�erentsearch strategies using GIOVE-A PRN 51

apart from this �lter is an extra �lter for the receiver. For example, if only one

sideband is processed during the acquisition (i.e. SSB method), but the 51.15 MHz

wideband AltBOC is used for the tracking, then the 20 MHz �lter required for the

SSB processing is considered to be an extra �lter.

4.2.3. Correlation scheme based methods. In this category, the two impor-

tant methods are the SPC method and the|V E2 + P 2| method whose applicability

to AltBOC is discussed below.

4.2.3.1. Applicability of the SPC method to AltBOC. It is interesting to note

that this is the principle used to combine four codes in AltBOC modulation scheme

(see the AltBOC modulation equations in (2.7)). The correlation process in the

FIC approach works on the basis of the SPC method. This is also evident from the

similarities in shape of the correlation waveforms of E5aQ-FIC in Fig. 4.2 and the

result of the SPC method in 2.17. Therefore, it is concluded that the SPC method

is equivalent to the FIC method.

4.2.3.2. Applicability of the |V E2 + P 2| method to AltBOC. The |V E2 + P 2|method combines correlation values from two time-shifted versions of the local ref-

erence signal as given in (2.33a) and (2.33b). The time delay is one quarter of a

subcarrier cycle. For the AltBOC(15,10) signal, this delay is 0.167 chips. The re-

sulting correlation waveform with this method is shown in Fig. 4.3. Observe that

the shape somewhat resembles a BPSK triangle and also that the peak is �at across

0.167 chips. The bias in the centre of the resulting correlation waveform can be eas-

ily compensated as it is one half of the time delay (0.5*0.167 chips). This method

is discussed in more detail in Sec. 4.5.


Table 4.1. Summary of the resource usage in search strategy basedschemes

Extra

Filter

Re-

quired?

Down

sam-

pling

Possi-

ble?

Code /

Subcar-

rier

Genera-

tors

Shape of

the Cor-

relation

Wave-

form

Correlation

Power(%

of

Direct-

AltBOC)

SSB Any One

Code

Yes

(one)

Yes 1 / 0 BPSK(10) 21.34

One

sideband

Yes

(one)

Yes 2 / 0 BPSK(10) 42.68

DSB, Both

sidebands

Yes

(two)

Yes 4 / 0 BPSK(10) 85.36

Any One

Code

No No 1 / 1 BPSK(10) 21.34

FIC Pilots or

Data

No No 2 /2 AltBOC

(15,10)

42.68

Pilots &

Data*

No No 4 / 2 AltBOC

(15,10)

85.36

Direct AltBOC No No 4 Code,

one 16x8

LUT

AltBOC

(15,10)

100

*Coherent combination of the pilot channels combined non-coherently with the coherent combination of

the data channels.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Code Delay (chips)

Nor

mal

ized

Abs

olut

e C

orre

latio

n V

alue

|P||VE|

|VE2+P2|

Figure 4.3. |V E2 + P 2| method for AltBOC(15,10)


4.3. Acquisition Complexity and the Code Search Step Size

For the AltBOC(15,10) signal the e�ect of code search step size on the correlation

value is shown in Fig. 4.4. The best case and the worst case are chosen to gain

an insight into the sharpness of the main peak and the e�ect of the side peaks.

The best case value is the highest maximum correlation that can be obtained for

any given code search step size. This value always corresponds to the peak of the

ACF. The worst case value is the lowest maximum correlation obtained by stepping

through the ACF with steps of a given size. As an example, for an ideal BPSK

autocorrelation triangle, when the search step is 0.5, the best case correlation value

is 1 and the worst case value is 0.75 (normalised). For the BPSK case the worst case

correlation value follows a linear degradation with increasing step size, as expected

with a symmetrical triangular correlation function. For the AltBOC(15,10) case, not

only is the degradation more steep, but also there are nulls produced by the regularly

spaced autocorrelation nulls between side peaks2. The worst case correlation loss

has smaller �sub-peaks� at step sizes of about 0.5 and 0.83. A typical code search

step size of 0.5 experiences a loss of up to 8.8 dB compared to the best case and up

to 6.3 dB loss compared to BPSK correlation waveform with the same search step.

The actual loss depends on a parameter called �residual-code-phase-o�set� (RCPO)3

(Yoon et al., 2000; Shivaramaiah, 2004), which in practice is uniformly distributed

over (0, δt2

] and hence the average case is theoretically mid-way between best and

the worst cases.

As an example of calculating the number of search cells, consider a one mil-

lisecond pre-detection integration period which is the length of a primary code of

E5. For the same worst case correlation loss as the BPSK case of 2.5 dB with a

code search step size of 0.5, the step size should be set to about 0.083 chips for

AltBOC(15,10). This results in 10230 · (1/0.083) ≈ 122760 search cells, which is the

same as the number of samples in one millisecond assuming a sampling frequency

of 122.76 MHz. Alternatively, one can decide to use a search step of 0.5 and 6 dB

loss can be regained by increasing the pre-detection coherent integration time by

four times, which results in a total of 10230 · (1/0.5) · 4 = 81840 cell searches. With

the latter approach one requires a �ner search of the code delay for a smooth and

unambiguous transition to the tracking process. The �ner search involves cells cor-

responding to only 2 chips ambiguity (24 cells). However, increasing the coherent

integration duration decreases the Doppler bin size, and this in turn increases the

2The minor irregularities in the shape are due to the usage of GIOVE PRN51 spreading codeinstead of an analytical AltBOC ACF3In other words, each time the receiver starts the acquisition process, the amount by which the�rst sample considered for the correlation is away from the chip boundary (or chip boundary +δtTc), is the residual-code-phase.

4.4. CONSIDERATIONS FOR THE CELL CORRELATION EFFECT 69

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

Chip Spacing (chips)

Nor

mal

ized

Abs

olut

e C

orre

latio

n V

alue

BPSK Worst CaseAltBOC(15,10) Worst CaseBest Case

Figure 4.4. E�ect of code search step size on the correlation value;worst case and best case for AltBOC(15,10) and BPSK(n) ACFs

number of frequency cells to search. In e�ect the total search time increases sixteen

fold. In addition, the coherent integration su�ers from the secondary code sign issue,

unless otherwise compensated for.

Note that this approach of �nding a step size for the sub-peaks of the worst case

correlation loss curve holds good for any BOC signal, and each BOC signal would

have a particular value of code search step size away from zero that has an acceptable

correlation loss (such as the 0.5 step size and 6.3 dB for AltBOC(15,10)). However,

at low received signal strengths, setting the code search step corresponding to the

sub-peaks might not always perform well due to noise. Thus the code search step

size plays an important role in determining Pd and T acq. Throughout this chapter

only the search dimension along the time axis will be considered. The analyses do

not consider Doppler because the methodologies discussed in this chapter do not

directly relate to Doppler search, but it has to be borne in mind that increasing the

integration time increases the number of frequency cells to search.

4.4. Considerations for the Cell Correlation E�ect

Before moving onto a detailed analysis of the |V E2 + P 2| method, it is important

to understand the other implications of the code search step size on the acquisition

performance. One such implication, often neglected in the literature dealing with

the acquisition performance of matched �lters (MF) is the e�ect of �cell correlation�.

This section summarises the important �ndings of the cell correlation analyses car-

ried out in collaboration with Tung Hai Ta, whose details are available in Ta et al.

(accepted for publication). Tung helped in verifying quantitatively the e�ect of cell


correlation in BPSK(1) and BOC(1,1) signals, through extensive software simula-

tions.

The correlation function of the BPSK(10) signal has only one main peak, whilst

that of the AltBOC(15,10) has two strong side-peaks which are located one third

chip away from the main peak (see Fig. 2.14). As discussed in sec. 4.3, the code

search step size is a acquisition algorithm design parameter. Let λP and λA be

the number of samples per chip for BPSK(10) and AltBOC(15,10) respectively, i.e.

λP = 1δBPSKt

, λA = 1δAltBOCt

.

4.4.1. Cell correlation phenomenon. Considering two correlator outputs

Y [k − l0] and Y [k] at two di�erent time instances, where Y [k] = XI [k] + XQ[k]

the correlation coe�cient between two outputs is

ρYk,Yk−l0 =cov(Yk, Yk−l0)

σYkσYk−l0(4.1)

where

σZkσZk−l = σ2Z = Mσ2

n (4.2)

σ2n is the variance of the additive white Gaussian noise (AWGN) nW (t) present at

the input of the correlator (see (2.22)) and M is the number of samples in the

integration period. Now,

cov(Yk, Yk−l0) = E [(Yk − E[Yk|φ,τ,fd ])(Yk−l − E[Yk−l0|φ,τ,fd ])∗]

= E

[∑p

∑q

nW [k + p]nW [k − l0 + q]ci[p]ci[q] (4.3)

· exp(j2π(fIF + fdk)(k + p)TS

)exp

(j2π(fIF + fdk−l0 )(k − l0 + q)Ts

)]where Ts = 1

fsis the sampling period. With AWGN only, the terms with

k + p = k − l0 + q (4.4)

p = q − l0survive. Let the Doppler search be performed sequentially in the search space. If

Yk, Yk−l0 belong to two di�erent Doppler trials (i.e. fd 6= fdk−l0) then they are

computed based on two di�erent incoming signal segments. Hence there are no

terms in (4.3) that satisfy the condition in (4.4). Therefore, (4.3) becomes

cov(Yk, Yk−l0) = σ2n

M−|l0|−1∑p=0

ci[p]ci[p+ l0] exp(−j2π(fIF + fdk − fdk−l0 )Ts

)= (M − |l0|)σ2

nR[l0]. (4.5)

4.4. CONSIDERATIONS FOR THE CELL CORRELATION EFFECT 71

Note that because fs is chosen such that there is no aliasing, (4.5) depends only

on the signal type through the correlation function R[l0]. With the help of the

equations for the correlation functions for BPSK(10) and AltBOC(15,10), and using

(4.5), ρY k,Y k−l0 in (4.1) can be obtained for λP = 2 (δt = 0.5) andλA = 6 (δt = 0.167)

as:

For BPSK(10)

ρYk,Y k−l0 ≈

1− |l0|λP

, 0 ≤ |l0| ≤ (λP − 1)

0 , |l0| ≥ λP

(4.6)

For AltBOC(15,10)

ρYk,Y k−l0 ≈

1− |l0|

λA, 0 ≤ |l0| ≤ (λA

2− 1)

−23

(1− |l0|

λA

), λA

2≤ |l0| ≤ (λA − 1)

0 , |l0| ≥ λA

(4.7)

Equations (4.6) and (4.7) show that the correlation or the dependency between

two correlator outputs depends on the distance of the two cells in the search space

where the outputs are calculated. If the distance is less than one chip length, then

the two outputs are correlated. This is the so-called cell correlation phenomenon. It

should be stressed that this phenomenon is always visible in MF correlators due to

the correlation of the local code stored in the MFs under the in�uence of the noise

component induced in the received signal at di�erent time instances within a chip

period. Moreover, from (4.6), ρYk,Yk−l0 is always positive, therefore the correlated

cells have an increasing linear relationship for BPSK(10). For the AltBOC(15,10)

signal the correlation between some of the consecutive outputs is linearly increasing,

the others are linear decreasing (Ta et al., accepted for publication). This fact is

apparent when considering the shape of the AltBOC(15,10) correlation waveform

(see Fig. 2.14).

4.4.2. Detection and false alarm probabilities and mean acquisition

time considering cell correlation. Without considering the cell correlation phe-

nomenon Pr(Zk > η|Zk−1 ≤ η, Zk−2 ≤ η...) = Pr(Zk > η), where Pr denotes the

probability function. Therefore all detection and false alarm probabilities belong

only to the uncorrelated group and are estimated at a single cell of the search space.

With the cell correlations in consideration, the detection and false alarm proba-

bilities can be divided into two groups: (i) correlated and (ii) uncorrelated probabil-

ities. The correlated group includes the conditional probabilities, which can be es-

timated if the associated distributions are determined. However, these distributions

(i.e. correlated central and non-central chi-square distributions) are very di�cult if


not impossible to derive as closed form representations. Therefore, the Monte Carlo

method is suggested to estimate the probabilities in this group (Ta et al., accepted

for publication). For the uncorrelated group, the conditional probabilities become

the marginal ones. These probabilities are estimated at a single cell of the search

space. Moreover it should be noted that for BPSK(10), because two consecutive

outputs have a linearly increasing correlation, the uncorrelated probabilities (i.e.

marginal probabilities) are always larger than the correlated ones (i.e. conditional

probabilities), Pr{Zk > η|Zk−1 ≤ η) < Pr{Zk > η). For AltBOC(15,10), because

the decision variable is Z[k] = |Y [k]|2 (i.e. not directly the MF output), the linearly

decreasing correlations between the correlator outputs do not a�ect the comparison

between conditional and marginal probabilities. Moreover, by taking the absolute

square of the output, the correlations between the decision variables again become

linearly decreasing in nature and, as a result, the marginal probabilities are larger

than the conditional ones.

As with the probability of detection, the mean acquisition time should also be

computed for the two groups of cells (the uncorrelated and the correlated), and then

the result has to be combined with the weighted averaging method, weighting each

group with the number of cells contained within that group.

In summary, the qualitative impacts of the cell correlation phenomenon on the

performance parameters are listed in Table 4.2.

Table 4.2. E�ect of the cell correlation phenomenon on the performance

C/N0 (dB-Hz) T acq η and Pd

low (/32) Twith CCacq < T

without CCacq ηwith CC < ηwithout CC and

Di�erence gets worse when η is high Pwith CCd > Pwithout CCd

high ('32) Twith CCacq < T

without CCacq Di�erence gets worse

Di�erence gets worse when η is low when C/N0 is low

Not taking into account the cell correlation phenomenon in the analysis of the

mean acquisition time results in a di�erence in the times of about 6-12% (Ta et al.,

accepted for publication) compared to that obtained with accounting for the cell

correlation, depending on the signal type and signal strength. This di�erence is

then propagate to the threshold setting and eventually the detection probability.

4.5. |V E2 + P 2| method for AltBOC

The problem of reduced Pd with increasing code search step size has been studied

earlier for CDMA systems. The problem which exists for BOC modulated signals is

not totally di�erent from this. Even for the BPSK signals, an increase in the step

size reduces Pd. In addition there is an issue of RCPO. As mentioned earlier, the

RCPO arises from the fact that the point where the receiver starts the correlation

4.5.∣∣V E2 + P 2

∣∣ METHOD FOR ALTBOC 73

operation may not be aligned with the correlation peak and there can be an initial

ambiguity of up to half of the step size. The issue of RCPO brings in the best

and worst case scenarios to the analysis and without loss of generality, the start of

the receiver correlation operation can be assumed to be a uniform random variable

within an interval of (0, δt2

]. In (Yoon et al., 2000) this problem was addressed in

detail and as a solution a method of addition of successive correlation samples was

proposed. For BPSK this method �attens the correlation function around the peak

and hence increases the Pd, and also makes the correlation function less sensitive to

RCPO(Shivaramaiah, 2004). The |V E2 + P 2| method used for BOC signals is a spe-

cial case of this successive correlation samples addition method. In the |V E2 + P 2|method the delay between the samples that are combined is �controlled� by design

so as to obtain a BPSK-like correlation triangle.

In what follows, the e�ect of code search step size on the |V E2 + P 2| method

is analysed, see Fig. 4.54. The worst case correlation values for the |V E2 + P 2|method are close to those of the BPSK worst case values and swings around it. For

example, in order to obtain the losses similar to that for 0.5 step sizes of BPSK,

one should use a 0.4 step size for the |V E2 + P 2| method. For a 0.5 step size only

about 1 dB loss is incurred compared to the BPSK worst case. An observation of the

|V E2 + P 2| worst case loss curve indicates an interesting phenomenon. The curve

has a �attened response at three places. The middle one is worth closely observing.

For step size of 0.5 to 0.85 (and slightly beyond that) step size, the correlation loss

remains constant at 0.67. This means that even at a 0.85 step size one will incur

only a loss of 3.5 dB and this loss is less than even the BPSK worst case at 0.85

step size.

To understand the advantage in terms of the number of cell searches consider

again a one millisecond pre-detection integration period. Even though the �at region

extends a little bit beyond 0.85 chips, the chip step chosen for the analysis is 0.85

thus allowing a margin for the other factors like �ltering that may a�ect the shape of

the correlation waveform. With a 0.85 step size, only 10230 · (1/0.85) ≈ 12036 cells

are needed in the �rst (or coarse) acquisition step and around 36 cells (assuming 3

chip ambiguity and 1/12 chip step) in the second (or �ne) search step. This is a

huge reduction in the number of cells to search for acquisition (which requires a 0.1

chip step for the same loss with the Direct AltBOC case). When compared to the

0.5 chip stepping case which requires 20460 cell searches, an improvement of about

≈41% is obtained.

The above description does not consider the e�ect of noise and the e�ect of

RF front-end �ltering. In practice whenever two signal components are added, noise

4Fig. 4.5 is the same as Fig. 4.4 but with the∣∣V E2 + P 2

∣∣ method included


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

Code Search Step Size (chips)

Nor

mal

ized

Abs

olut

e C

orre

latio

n V

alue

Best CaseBPSK Worst CaseAltBOC(15,10) Worst Case

|VE2+P2|

Figure 4.5. E�ect of code search step size on the correlation valuesincluding |V E2 + P 2| method

AltBOC

Look-up

Table

Complex

CarrierDecision

{aI, bI, aQ,

bQ} Codes

c

t

t MT

2

2

To

Tracking( )IFr t

c

t

t MT

t ct T

t ct T

)(nu

)(nuI

)(nuQ

Figure 4.6. Direct AltBOC acquisition architecture

components are also added and the results degrade at lower received signal strengths.

Nevertheless, the advantage of this method is su�cient to overcome the degradation

due to noise as will be seen when Pd and T acq are evaluated. When the received

signal is �ltered the correlation functions will no longer be sharp and hence the delay

value of 0.167 chips may not be valid. However, without loss of generality, it can

be stated that an optimum delay can be found for the particular �lter used in a

receiver to make use of the |V E2 + P 2| method.

4.5.1. System description for the |V E2 + P 2| method. Figure 4.6 shows

the Direct AltBOC acquisition architecture. δt is the code search step size used

for stepping the energy search. As discussed earlier, this value is typically 0.083

chips. Once the decision is made, the control is handed over directly to the tracking

process. Fig. 4.7 shows the architecture with the addition of VE and P correlation

values when the sampling frequency is such that it enables us to provide the required

code delay D between the VE and P samples used for the addition. This is the case

with a sampling frequency of 122.76 MHz which can be used to realise the required

D = 0.167 chips (every alternate sample). Observe that the architecture does not

use any additional correlators compared to that in Fig. 4.6. In practice it may not

4.5.∣∣V E2 + P 2


AltBOC

Look-up

Table

Complex

Carrier

Decision

{aI, bI, aQ,

bQ} Codes

c

t

t MT

2

2

To

Tracking

( )IFr t

)(nu

cDT

2

t ct TcDT

2)(1 nu

)(2 nu

)(1 nuI

)(1 nuQ

)(2 nuI

)(2 nu Q

c

t

t MT

t ct T

Figure 4.7. Direct AltBOC acquisition architecture with|V E2 + P 2| method; speci�c sampling frequency

AltBOC

Look-up

Table

Complex

Carrier

Decision{aI, bI, aQ,

bQ} Codes

c

t

t MT

2

To

Tracking

( )IFr t

)(nu

)(1 nu

)(2 nu

cDT

2

2

2

)(1 nuI

)(1 nuQ

)(2 nuI

)(2 nu Q

t ct T

t ct T

c

t

t MT

c

t

t MT

c

t

t MT

Figure 4.8. Direct AltBOC Acquisition Architecture with|V E2 + P 2|; Arbitrary (Valid) sampling frequency

be possible for the designer to realise this delay from the sampling interval due to

RF �ltering e�ects or other hardware limitations. Fig. 4.8 shows the architecture of

the |V E2 + P 2| method for the case where the required delay D is di�cult to realise

using the sampling frequency. Here the output of the AltBOC LUT is delayed by a

value DTc and separate channels are used for correlation. This increases the number

of correlators required but the advantage is that it can be used for any arbitrary

(but valid) sampling frequency. A couple of points should be noted regarding these

architectures. All the lines after the carrier mixing stage carry complex values. In

Fig. 4.7 and Fig. 4.8, δt can be as large as 0.85, as discussed earlier. Furthermore,

with these two architectures, the control may need to be transferred to a �ner code

delay search instead of tracking, if the tracking architecture demands code phase

estimates better than half of the code search step.

Decision block. A simple hypothesis test with a threshold η is performed by

comparing the output of the correlator u(n) with η . By making use of the individual

correlation outputs in the case of the |V E2 + P 2| method, one can estimate the code

phase more accurately and select this better estimate instead of randomly selecting

either V E or P .

Case 1. u(n) ≥ η: decision is made to examine u1(n) and u2(n). If u1(n) ≥ u2(n)

then the delay corresponding to the VE correlation sample is considered the


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Code Delay (chips)

Err

or in

Cor

rela

tion

Val

ue

With BOC EquationWith CDBOC Equation

Figure 4.9. Error in correlation value computed using (2.19), (2.20)with respect to that of GIOVE-A PRN 51

estimate, otherwise the delay corresponding to the P correlation sample is

considered the estimate and the �ner search process is initiated.

Case 2. u(n) < η: decision is made to test the next set of cells, until all the

search cells are exhausted.

4.5.2. Pd and T acq for the |V E2 + P 2| method. For the purpose of perfor-mance evaluation, without loss of generality, both the architectures in Figs. 4.7 and

4.8 can be considered to be the same.

In order to analyse theoretically the performance of E5 signal acquisition it

is necessary to have a closed-form expression to compute the ACF. However, as

mentioned in sec. 2.4, there is only an approximate equation and hence it is neces-

sary to understand its error behaviour. Another option is to use the equation for

BOC(15,10). Fig. 4.9 plots the error in correlation values of these two options with

respect to the correlation waveform obtained from the GIOVE-A PRN 51 primary

code. Note that the ACF generated with the CDBOC expression (2.20) has less

than 5% error, whereas the ACF expression of BOC(15,10) 2.19 has more than 15%

deviation compared to that obtained with GIOVE-A PRN 51. Since the theoretical

evaluation of acquisition performance parameters largely depends on ACF, 2.20 is

used for all analyses unless otherwise stated.

The decision statistic for the direct AltBOC architecture is similar to the decision

variable of the conventional architecture of Fig. 2.10 and hence the equations in sec.

2.6.4 can be used for the performance evaluation.

The decision statistic for the |V E2 + P 2| architecture is given by

u(n) = u21(n) + u2

2(n) = u21I(n) + u2

1Q(n) + u22I(n) + u2

2Q(n) (4.8)

4.5.∣∣V E2 + P 2


Again, assuming the Gaussian distribution and statistical independence of the

individual correlation outputs, one can compute the detection and false alarm prob-

abilities.

Under the hypothesis H0 when there is no signal present, the decision statistic

has a central chi-square distribution with four degrees of freedom (2.29).

Under the hypothesis H1 when the signal is present the decision statistic has a

non-central chi-square distribution with four degrees of freedom with PDF ps(x) and

non-centrality parameter λ2 = m21I +m2

1Q +m22I +m2

2Q and Pd is given by (2.30).

In both the Direct AltBOC and the |V E2 + P 2| methods, one can perform a

non-coherent integration of the decision statistic to improve the sensitivity, and

correspondingly the degree of freedom parameter of the chi-square distributions will

have to be multiplied by the number of non-coherent summations. Therefore if

Nnc is the number of non-coherent summations, then the decision statistic in the

|V E2 + P 2| method will have 4Nnc degrees of freedom.

In these analyses and simulations a Pfa = 10−3 was considered and a pre-

detection coherent integration time of one millisecond was assumed. Note that,

in practice, the penalty due to false alarm will be comparatively less in the case of

the |V E2 + P 2| method, as one will be entering the �ner code delay search process

and not the tracking process. For the approaches with a two-step search process,

only the coarse (initial) step is considered since it is the dominant part.

Fig. 4.10 shows the worst case probability of detection for BPSK and AltBOC.

To show the step size required to achieve BPSK-like correlation losses and to show

the e�ect of larger step size, δt= 0.083 as well as δt=0.5 are chosen for AltBOC. Note

that a loss of 6.3 dB between AltBOC with δt=0.5 and BPSK δt=0.5 (or AltBOC

=0.083) can be noted from the plot.

Fig. 4.11 shows the average probability of detection for BPSK and AltBOC with

δt as in Fig. 4.10. The average correlation loss scenario stems from the fact that one

will not always encounter the worst case and RCPO will have a uniform distribution

in [0, δt). Note that the di�erence between AltBOC δt=0.5 and the BPSK δt=0.5

reduces to around 2.2 dB in such a case.

Fig. 4.12 shows the average probability of detection for the acquisition ap-

proaches considered in Fig. 4.2. Note that the di�erence in the power distribution

among di�erent approaches is evident with the probability of detection curve.

Figs. 4.13 and 4.14 provide the theoretical and simulated average and worst

case probability of detection for the |V E2 + P 2| method with both δt=0.5 and 0.85

scenarios respectively.

In Fig. 4.13 one can see that the average probability of detection for the

|V E2 + P 2| method is 0.4 dB worse than the BPSK case, and that the |V E2 + P 2|method outperforms Direct AltBOC approach by about 2.2 dB. Fig. 4.14 shows


20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Carrier to Noise Ratio C/N0 (dBHz)

Pro

babi

lity

of D

etec

tion

Pd

Best CaseBPSK Delta=0.5AltBOC Delta=0.083AltBOC Delta=0.5

Figure 4.10. Worst case probability of detection for BPSK and AltBOC

20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Carrier to Noise Ratio C/N0 (dBHz)

Pro

babi

lity

of D

etec

tion

Pd

Best CaseBPSK Delta=0.5AltBOC Delta=0.083AltBOC Delta=0.5

Figure 4.11. Average probability of detection for BPSK and AltBOC

4.5.∣∣V E2 + P 2


20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Carrier to Noise Density C/N0 (dBHz)

Pro

babi

lity

of D

etec

tion

Pd

Best CaseE5aQ−SSBE5a−SSBE5a & E5b – DSBE5aQ−FICE5 Pilots – FICE5 Pilots & Data FICDirect AltBOC

Figure 4.12. Average Pd for di�erent acquisition approaches

20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

babi

lity

of D

etec

tion

Pd

BPSK theor.AltBOC δ

t=0.5 theor.

VEP δt=0.5 theor.

VEP δt=0.85 theor.

BPSK sim.AltBOC δ

t=0.5 sim.

VEP δt=0.5 sim.

VEP δt=0.85 sim.

Figure 4.13. Average Pd for AltBOC and |V E2 + P 2| methods (�:theory, •: simulation)

that the worst case loss for the |V E2 + P 2| method is only 1 dB worse than that of

the BPSK method, and there is an improvement of 5.3 dB compared to the Direct

AltBOC approach.

Figures. 4.15 and 4.16 compare the mean acquisition time for the probability of

detection scenarios considered in Fig. 4.13 and 4.145. Observe that the |V E2 + P 2|method for the average case with δt=0.85 chip step performs better than the BPSK

case with δt=0.5. This is also true for the worst case scenario, but only at higher

5Since Pd in�uences T acq and Pd is less than 0.1 for C/N0 values below 30 dBHz, the Monte-Carlo

simulations do not yield proper results. Hence, T acq for C/N0 values below 30 dBHz are plotted.


20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

babi

lity

of D

etec

tion

Pd


t=0.5 theor.

VEP δt=0.5 theor.

VEP δt=0.85 theor.

BPSK sim.AltBOC δ

t=0.5 sim.

VEP δt=0.5 sim.

VEP δt=0.85 sim.

Figure 4.14. Worst case Pd for AltBOC and |V E2 + P 2| methods(�:theory, •: simulation)

30 35 40 45 500

50

100

150

200

250

300


Mea

n A

cqui

sitio

n T

ime

/Tac

q (s

)


t=0.5 theor.

VEP δt=0.5 theor.

VEP δt=0.85 theor.

BPSK sim.AltBOC δ

t=0.5 sim.

VEP δt=0.5 sim.

VEP δt=0.85 sim.

Figure 4.15. T acq for the average Pd scenario (- - :theory, • : simulation)

signal strengths. At lower signal strengths for the worst case scenario, the advan-

tage of higher chip step is nulli�ed by the increased contribution from the noise.

Even with this disadvantage due to higher noise, the performance of the |V E2 + P 2|method is close to that of the BPSK case. In both the scenarios, the |V E2 + P 2|method outperforms the Direct AltBOC method.

4.6. Envelope and Squared Envelope Detectors

It is clear from the previous sections that the |V E2 + P 2| method requires the

correlation waveforms to be of a certain shape for the combination to provide the

4.6. ENVELOPE AND SQUARED ENVELOPE DETECTORS 81

30 32 34 36 38 40 42 44 46 48 500

50

100

150

200

250

300


Mea

n A

cqui

sitio

n T

ime

/Tac

q (s

)


t=0.5 theor.

VEP δt=0.5 theor.

VEP δt=0.85 theor.

BPSK sim.AltBOC δ

t=0.5 sim.

VEP δt=0.5 sim.

VEP δt=0.85 sim.

Figure 4.16. T acq for the worst case Pd scenario (- - :theory, • :simulation)

expected result. However, the shape of the correlation function depends on the front-

end �lter used in the receiver. In addition, the combination could be either of the

�squared envelope� or �envelope� type. This section considers these two parameters

to analyse their e�ect on the |V E2 + P 2|method. In a general sense, the |V E2 + P 2|method can be considered as a �Delayed Addition� (DA) method, and this is the

generic term used in this section.

4.6.1. The DA method and the e�ect of pre-correlation �ltering. It

was shown in the previous section that the nulls in the AltBOC correlation wave-

form severely a�ect acquisition, and this can be mitigated by using the |V E2 + P 2|method, a special case of the DA method.

Let Ik1 , Qk1 be the correlation values corresponding to the local code C(t − τ)

and Ik2 , Qk2 be the correlation values corresponding to the local code C(t− τ−DTc).The parameter D ∈ (0, 1] is chosen to obtain the BPSK-like correlation triangle. In

the DA-squared envelope method, the decision statistic is formulated as:

da_zs =Nnc∑k=1

sk, sk = I2k1

+Q2k1

+ I2k2

+Q2k2

(4.9)

Now, observe the shape of the correlation waveform by combining the envelope

outputs in the DA method instead of squared envelope outputs. To do this, let the

DA-envelope decision statistic be formulated as:

da_ze =Nnc∑k=1

ek, ek =√I2k1

+Q2k1

+√I2k2

+Q2k2

(4.10)


−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

Code Delay (chips)

Nor

mal

ized

Abs

olut

e C

orre

latio

n V

alue

Direct AltBOCDA−Sq.EnvelopeDA−Envelope

Figure 4.17. Correlation waveforms, in�nite bandwidth

For both the methods D=0.167 chip is used, which gives a BPSK-like triangle. Figs.

4.17 and 4.18 show the correlation waveform and the e�ect of code search step,

without pre-correlation �ltering. Figs. 4.19 and 4.20 have a 5th order Butterworth

�lter with a 3 dB bandwidth of 50 MHz. The outputs are normalised with respect

to the Direct AltBOC case. Observe that the correlation output in the DA-envelope

case (with �ltering) is slightly better than that of the DA-squared envelope case and

has smaller triangles sitting on top of it. In addition, the �at region from about 0.5

to 0.85 chip search steps in the case of the DA-squared envelope is extended to about

1.0 chip in the case of the DA-envelope method. In addition, the �ltering alters the

position of the sub-peaks. For example the second sub-peak is moved close to 0.3

chips from the initial 0.33 of the in�nite bandwidth case. The implication of these

di�erences on the acquisition performances are analysed in the following section.

4.6.2. The DA-envelope detector architecture. The correlator architec-

ture for the DA-envelope method can be realised in two ways. In the �rst approach

the output of the correlators separated in time by DTc can be added together to

obtain the in-phase and quadrature phase correlator outputs. The parameter D

then depends on the sampling frequency. Figure 4.21 illustrates this architecture.

On the other hand, if the second set of in-phase and quadrature phase correlation

values can be generated in a similar way to Fig. 4.8 with a separate set of local code

replicas, then D can be controlled by appropriately programming the phase of the

NCO that is used to generate the code.

4.6. ENVELOPE AND SQUARED ENVELOPE DETECTORS 83

0 0.5 1 1.5 20

0.5

1

1.5


Nor

mal

ized

Cor

rela

tion

Loss


Figure 4.18. Correlation loss, in�nite bandwidth

−2 −1 0 1 20

0.2

0.4

0.6

0.8

1

1.2

1.4

Code Delay (chips)

Nor

mal

ized

Abs

olut

e C

orre

latio

n V

alue


Figure 4.19. Correlation waveforms, 50MHz bandwidth


0 0.5 1 1.5 20

0.5

1

1.5


Nor

mal

ized

Cor

rela

tion

Loss


Figure 4.20. Correlation loss, 50MHz bandwidth

cDT

cDT

Code

Generator

Complex

Carrier

2

2

)()(

Envelope

t Tt c

_da ze

2

2

( )IFr t

t Tt c

Figure 4.21. Detector architecture, DA-envelope method

4.7. EXPLOITING SECONDARY CODES TO INCREASE ACQ. PERFORMANCE 85

4.6.3. Performance analysis. As explained earlier, the acquisition perfor-

mance of the DA-squared envelope is straightforward to calculate, but for the DA-

envelope method it is not.

The closed form approximations of the in�nite series expressions derived in Hu

and Beaulieu (2005b,a) can be used in this context. Note that for the DA-squared

envelope method there are 4Nnc degrees of freedom because each decision statistic

comprises two sets of correlator outputs. Coherent integration duration Tcoh of

one millisecond is used with probability of false alarm Pfa set to 10−3. For the

DA methods a chip search step of 0.85 is used. As a reference, the probability of

detection curve for the Direct AltBOC acquisition is also considered. Fig. 4.22 shows

the probability of detection curves for Nnc=1, Nnc=4 and Nnc=8. Observe that the

DA methods perform better for higher signal strengths at smaller values of Nnc. As

the signal strength decreases or as Nnc increases, the Direct AltBOC behaves well.

This can be explained as follows. When Nnc=1, the decision statistic will have two

signal components and two noise components. At high SNR values, the e�ect of

an increase in noise is masked. At low SNR the increase in noise because of the

two noise components reduces the advantage of the DA method. As Nnc increases,

the Direct AltBOC method will experience more SNR gain compared to the DA

methods, again because of the additional noise components in the DA method.

At smaller values of SNR the DA-envelope method provides about 1 dB improve-

ment over the DA-squared envelope method. A small part of this improvement is

due to the envelope detector itself (as discussed earlier) and the rest is due to the

reduced loss in the correlation output in the case of the DA-envelope method com-

pared to the DA-squared Envelope method. As Nnc increases the advantage of

the DA-envelope method reduces because both the decision statistics tend towards

Gaussian distributions according to the central limit theorem.

Fig. 4.23 shows the probability of detection curves for the DA-squared envelope

method and the DA-envelope method at code search steps of 0.85 and 1.0 respec-

tively. Note that both the methods behave similarly as expected from the correlation

waveform. Fig. 4.24 compares the T acq for the two methods. Here the search step

used for the DA-squared envelope method is 0.85 and the search step used for DA-

envelope method is 1.0 since both the methods have similar probability of detection

performances. The Direct AltBOC method is also shown for reference. Note that an

improvement of about 10% is obtained with the DA-envelope method with respect

to the DA-squared envelope method.

4.7. Exploiting Secondary Codes to Increase Acquisition Performance

The architecture presented in this section takes a di�erent approach to perform

longer integrations to that of non-coherently combining one primary code period


20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C/N0 (dBHz) at the detector input

Pro

babi

lity

of D

etec

tion

Pd


20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

babi

lity

of D

etec

tion

Pd


20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

babi

lity

of D

etec

tion

Pd


Figure 4.22. Probability of detection, Nnc=1 (top), Nnc=4 (Mid-dle), Nnc=8 (bottom)


20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

babi

lity

of D

etec

tion

Pd

DA−EnvelopeDA−Sq.Envelope

Figure 4.23. Pd for δt=0.85 (DA-squared envelope) and δt=1.0 (DA-envelope); Nnc=1

20 25 30 35 40 45 5010

1

102

103

104

105

106


Mea

n A

cqui

sitio

n T

ime

/Tac

q (s

)


Figure 4.24. T acq comparison; Nnc=1


correlation values. Correlation values obtained by integrating over the primary

code period Tp = L · Tc are coherently accumulated with succeeding values. This

accumulation is performed by using knowledge of the secondary code, i.e. an output

is produced for all the Ls delays of the secondary code. This coherent integration is

continued for the desired duration and then the decision statistic is found by taking

the maximum value among a maximum of Ls correlation values. Note that when

correlations are performed with the pilot signals, the integration can be extended to

any desired length as long as the receiver dynamics does not alter the code phase

delay.

As the secondary codes are memory codes, the second problem of �nding the

secondary code chip position leads to a question: �does one need to search the

entire length of the secondary code to �nd the chip position?� This is important

because the computational resources and time taken for completion of secondary

code acquisition can be reduced if one can �nd the secondary code delay within the

�rst few accumulations. It will be shown that out of all Ls correlation values that

are accumulated in each Tp seconds, one accumulated correlation value which is a

potential �winner� clearly distinguishes itself from the others by producing a higher

and constantly increasing correlation value. This trend is seen at very early stages

of the accumulation process (e.g. around chip 15 for the E5aQ whose secondary

code length is 100). This shows that one need not integrate for the entire secondary

code length in order to identify the phase of the secondary code.

4.7.1. Some properties of Galileo secondary codes.

4.7.1.1. Full-sequence autocorrelation. Figs. 4.25 and 4.26 show the autocorre-

lation plots of two selected secondary codes. As can be seen from these �gures, the

codes have very good autocorrelation properties. The autocorrelation sub-peak is

about 18 dB below the main peak in CS251 and about 21dB below the main peak

in CS1001.

4.7.1.2. Minimum sequence length required to identify the chip position, the �Char-

acteristic Length�. For pseudorandom sequences which are m-sequences (Sarwate

and Pursley, 1980) generated by a Linear Feedback Shift Register (LFSR) of length

n, it is known that the chip position of any chip in the sequence can be uniquely

identi�ed by just examining n chips (including the current one). This is possible

because while generating the maximal length sequence the shift register traverses

through all possible binary combinations (except `all zeros'). If k is the current chip

position, the chip value depends only on the previous n values. Mathematically this

is expressed as:

ck+1 = f (ck−n−1, ck−n−2, . . . , ck) (4.11)


−10 −5 0 5 10−30

−25

−20

−15

−10

−5

0

Chip Shift (Chips)

Nor

mal

ized

Cor

rela

tion

Val

ue (

dB)

Figure 4.25. Autocorrelation plot of the secondary code CS251

−40 −20 0 20 40−40

−30

−20

−10

0

Chip Shift (Chips)

Nor

mal

ized

Cor

rela

tion

Val

ue (

dB)

Figure 4.26. Autocorrelation plot of the secondary code CS1001


The length which is just su�cient to identify the chip shift is called the linear span

(LS) L of that code sequence(Cherubini and Benvenuto, 2002; Chan and Games,

1990). Note that the Berlekamp-Massey algorithm (Cherubini and Benvenuto, 2002)

can be used to reconstruct the entire sequence if one considers 2L chips (without the

knowledge of feedback taps). However, because the memory codes are not directly

generated using LFSR, use is made of the general concept of LS and not the sequence

reconstruction.

For the Gold codes (as in the case of those used in GPS L1 C/A) generated with

two n-bit shift registers, the LS is 2n chips. This implies that if the sequence is

broken into smaller sequences, each of length equal to the LS, then no two of these

smaller sequence bit patterns will be identical to each other.

This concept can be extended to the memory codes. Even though the memory

codes are not generated using a LFSR, their spans can be used. This span can be

referred to as the characteristic length (CL) (and use the same notation L) of thesequence.

4.7.1.3. Procedure for evaluating the characteristic length. In order to do this,

the following these steps are used:

(1) Let k be the number of contiguous `zeros' or `ones' (whichever is maximum)

in the sequence whose length is Ls.

(2) Form a matrixM with the partial sequences of length k as the rows where

each row is shifted by one bit with respect to the previous row. Hence the

size of the matrix will be Ls · k.

M =

c1 c2 . . ck

c2 c3 . . ck+1

. . . . .

. . . . .

cNs−k cNs−k−1 . . cNs. . . . .

cNs c1 . . ck−1

(4.12)

(3) Examine the matrix for identical rows. If any two rows are identical then

increment k and repeat Step 2 until no two rows of the matrix are iden-

tical. The uniqueness of the rows can be found by computing the linear

rank correlation coe�cient matrix X = CORR(M) of the rows of M and

examining whether any entry of X is unity.

(4) The smallest value of k which satis�es the condition in Step 3 is the CL Lof the sequence.

The CL obtained using the aforementioned procedure for di�erent secondary code

sequences of Galileo is given in Table 4.3. For the E5 secondary code, which is


Table 4.3. CLs for Galileo secondary codes

Secondary Code Characteristic Length (L)CS4 3CS201 8CS251 7

CS100b(GIOVE-A) 15CS100d(GIOVE-A) 13

0 10 20 30 40 50 60 70 80 90 1009

10

11

12

13

14

15

16

17

18

Index i in CS100i

Cha

ract

eris

tic L

engt

h

Figure 4.27. CLs for the Galileo E5 secondary codes

di�erent for each satellite, Fig. 4.27 shows the CLs (note the vertical scale does not

start at zero) and Fig. 4.28 shows the histogram of CLs.

As can be seen from the Table 4.3 and Figs. 4.27 and 4.28, the CLs are much

smaller than the sequence lengths, and that the E5 secondary codes have CLs be-

tween 9 and 18. In addition, most of the E5 sequences have a CL of 11.

4.7.2. System model. This section describes the proposed method for longer

integration and for �nding the secondary code chip position. Coherent integration

must be extended for a duration more than one code period to achieve the required

integration gain. If Nc is the number of primary code periods (or secondary code

chips) used for the coherent integration, then the total coherent integration duration

is Tcoh = Nc · Tp, where Tp = LcTc is the primary code period. Assume that Nc is

designed such that, in the absence of the secondary code, only one trial (i.e. one

coherent integration over a duration of Tcoh) is su�cient to detect the primary code

chip shift. Depending on the CL L of the code sequence under consideration, Nc

may be smaller or larger than L and the problem of detecting the signal and �nding

the secondary code chip position has to be addressed appropriately.

Fig. 4.29 shows the system model for the proposed approach.


9 10 11 12 13 14 15 16 17 180

5

10

15

20

25

30

35

40

45

Characteristic Lengths in CS100i

Occ

uran

ce o

f the

Cha

ract

eris

tic L

engt

h

Figure 4.28. Histogram of the CLs of E5 secondary codes

Primary

Code

Correlation

Secondary

Code

Hypothesis

Bi branches

Branch

Elimination

Logic

Case 1:

Iterations

≥ (L/Nc) ?

OR

Case 2:

max (S)

≥ ξ ?

Advance to

next K

chips

No

Yes

Input signal

(after carrier

removal)Retrieve

sec. code

chip

position

To

Tracking

Case 1=> Nc<< L

Case 2: => Nc > L or Nc ≈ L

Figure 4.29. Proposed system model for two cases; case 1: Nc � L,case 2: Nc > L or Nc ≈ L

4.7.2.1. Primary code correlation. The primary code correlator performs the cor-

relation of the input signal with the local primary code replica. The primary code

correlation can use any of the three correlation approaches viz.the active serial cor-

relator, the passive matched �lter and the parallel FFT-based approach. The active

serial correlator and the FFT-based approaches su�er from the problem of cor-

relation loss due to the possible secondary code sign change across the primary

code boundary. However the correlation output in a passive matched �lter will not

have losses when there is a match between the input code sequence and the local

code sequence (since the integration occurs from code boundary to code boundary).

Therefore the matched �lter type of acquisition is more suitable for the primary

code correlation block in Fig. 4.29.

4.7.2.2. Secondary code hypothesis block. The secondary code hypotheses block

evaluates all the required secondary code combinations using the evolutionary tree

approach described in Corazza et al. (2007). This block evaluates Bi branches at a


time.

Case 1. If Nc � L, the value of B will be 2Nc−1 to start with. Hence

Bi =

2Nc−1 i = 0

Bi−1 − Ei i 6= 0(4.13)

where Ei is the number of branches eliminated in the ith iteration.

Case 2. If Nc > L or Nc ≈ L, then the value of B at each iteration will depend

on the strength of the received signal. This means that as one extends

the integration time, the branch elimination logic can better decide on the

branches to be eliminated.

4.7.2.3. Branch elimination logic. The branch elimination logic examines all the

hypotheses output by the secondary code hypothesis block. The criterion for any

branch elimination is the lower correlation value relative to other branches. Let

S = (s1, s2, . . . , sBi)Tbe the vector containing the entire secondary code hypotheses,

where si is the ith secondary code hypothesis. A vector D containing the di�erence

of S with respect to the maximum is formed. Thus:

D = max(S)− S (4.14)

At each iteration, Ei number of branches whose correlation value exceeds a prede-

�ned threshold ξ are eliminated.

4.7.2.4. Decision to end the iteration.

Case 1. When Nc � L (which is the case when the signal strength is moderately

high) one would have detected the primary code (and its chip shift) before

determining the secondary code chip position. In this case the iteration

should be continued to determine the secondary code chip position and hence

the emphasis is on the crossing point of the CL. Thus the iteration ends when

the total number of primary code periods used in all the iterations is greater

than L. At each stage, the secondary code is advanced by K chips (typically

K = Nc).

Case 2. When Nc > L or Nc ≈ L, (which is the case when the signal strength is

low) one needs to integrate long enough so that the decision statistic

Λ = max (λ(si)) i = [0, Bi − 1] (4.15)

is greater than the threshold ξ, to detect the primary code. Here λ(si) is

the detector output of the ith secondary code hypothesis. Hence in this

case the emphasis is on the signal (i.e. primary code) detection and the

hypothesis which causes the decision statistic to cross the threshold is used


5 10 15 200

1

2

3

4x 10

6

Number of primary code periods

Max

imum

cor

rela

tion

valu

e

Figure 4.30. Correlation value trend for increasing number of pri-mary code period integrations (di�erent colours show all the 100 hy-potheses)

to determine the secondary code chip position. At each stage, the secondary

code is advanced by K chips, which is su�cient to eliminate some of the

branches.

4.7.2.5. Secondary code chip position retrieval. The secondary code hypothesis

that wins corresponds to a sub-sequence within the complete secondary code. The

process of obtaining this sub-sequence is equivalent to the �traceback� technique used

in Viterbi decoding(Viterbi and Omura, 1979). Thus retrieving the chip position is

performed by searching for this sub-sequence in the larger sequence and determining

the index of the shift.

4.7.3. Results. In order to evaluate the performance of the proposed method,

150ms of real data collected from Septentrio GeNeRX1 receiver for the E5 signal

from the GIOVE-A satellite was used. Determining the secondary code chip position

is di�cult in the case of E5 since the primary code period is only one millisecond

and the secondary code 100 chips for the pilot signals.

The procedure followed to acquire the E5 signal is:

(1) acquire E5aQ pilot signal whilst �nding the secondary code chip position,

and then

(2) use this information of the secondary code chip position and acquire E5

signal.

Fig. 4.30 shows the trend in correlation values for increasing integration time. To

show the applicability of the proposed method all 100 hypotheses were used. Note


0 50 1000

1

2

3

4x 10

6 2ms Integration

0 50 1000

1

2

3

4x 10

6 4ms Integration

0 50 1000

1

2

3

4x 10

6 8ms Integration

0 50 1000

1

2

3

4x 10

6 12ms Integration

0 50 1000

1

2

3

4x 10

6 16ms Integration

0 50 1000

1

2

3

4x 10

6

Secondary code sub−sequence index

Cor

rela

tion

valu

e

20ms Integration

Figure 4.31. Correlation values for all the secondary code hypothe-ses (sub-sequence indices)

that there is only one potential �winner�, which can be clearly distinguished from

other hypotheses as the integration time increases. The deviation point of other

sequences compared to the potential winner depends on:

i. the Hamming distance of the potential winner with respect to the other sub

sequences (in the X matrix), and ii. the position of the chip di�erences that result

in this Hamming distance.

As an example, consider a sequence with CL of 15. If the minimum Hamming

distance of the potential winner with respect to other sub-sequences is 4 (say) and the

chip di�erences appear after 11, then the closest incorrect contender grows with the

winner before deviating at an integration time interval of 11 ms. For the same case

of Hamming distance of 4, if there are some bit di�erences early on in the sequence

then the closest contender will grow but in a parallel track below the potential

winner eventually guaranteed to deviate at 11ms. Fig. 4.30 used GIOVE-A CS100b

code (whose CL is 15) and in the received signal, the winning sub-sequence was

located starting at the 46th chip of the secondary code sequence. Without loss of

generality, it can be concluded that in all cases this deviation point will occur ahead

of the CL.

Fig. 4.31 shows the correlation plots for di�erent integration durations. Note

that the correlation values of di�erent hypotheses are close to each other when the


0 1 2 3 4 5

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

Sample number

Cor

rela

tion

valu

e

3.268 3.27 3.272 3.274 3.276 3.278 3.28

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

Sample number

Cor

rela

tion

valu

e

Figure 4.32. Absolute correlation value of the E5 signal (right pic-ture is the zoomed version around the peak);1 ms integration

0 2 4 6 8 10

x 104

0

5

10

15x 10

5

Sample number

Cor

rela

tion

valu

e

3.268 3.27 3.272 3.274 3.276 3.278 3.28

x 104

0

5

10

15x 10

5

Sample number

Cor

rela

tion

valu

e

Figure 4.33. Absolute correlation value of the E5 signal (right pic-ture is the zoom version around the peak); 4ms integration using thesecondary code chip position detection algorithm; data collected forGIOVE-A satellite on 5th Feb 2008

integration time is less than the CL (because the primary code period is 1 millisecond

for E5, the terms `integration time' and `number of primary code periods can be used

interchangeably). As soon as the integration time reaches the CL, a clear peak pops

up.

The secondary code chip position obtained in this case was 46 which corresponds

to the sub-sequence [1 0 0 0 1 0 1 1 0 0 0 1 1 1 0]. Using this information the E5

signal was acquired with 1 millisecond and 4 millisecond integrations. The plots

(and the zoomed versions around the peak) are shown in Figs. 4.32 and 4.33.

4.8. Summary

The �rst part of this chapter discussed the complexity and problems with the

Galileo E5 signal acquisition and revisited di�erent strategies which address these

problems. The probability of detection and the mean acquisition time for these

strategies were studied, especially concentrating on the |V E2 + P 2| method along

with the acquisition engine architecture. For the same probability of detection,

compared to the Direct AltBOC approach, the |V E2 + P 2| method results in an

4.8. SUMMARY 97

improvement in C/N0 of about 2.2 dB for the average scenario and about 5.3 dB

for the worst case scenario. In addition an interesting observation shows that the

correlation loss in the |V E2 + P 2| method remains constant for chip step sizes from

0.5 to 0.85 which can be exploited to reduce the mean acquisition time by 41%. It

was concluded that the |V E2 + P 2| method with this step size, is a good candidate

for implementation in Galileo E5 receivers.

Next, the performance of squared-envelope and envelope detectors applied to

the |V E2 + P 2| method, which is a speci�c case of the DA method were analysed.

It was shown that for a �xed code search step size of 0.6 chips, the DA-envelope

method improves the probability of detection by about 1dB. For a �xed probability

of detection, the code search step in the case of the DA-envelope method can be

increased, thus reducing the mean acquisition time by about 10% compared to the

DA-squared envelope method.

The problem of coherent integration over periods longer than one primary code

length and the acquisition of secondary code chip position was also examined. An

acquisition engine architecture which can handle both these problems simultaneously

was proposed. The acquisition architecture presented in this chapter is unique in the

sense of achieving longer integration by exerting secondary codes, and also acquiring

the secondary code chip position as a by-product of the acquisition process. It was

shown that the secondary code chip shift can be uniquely identi�ed by shorter length

sequences than the code itself. Most of the E5 secondary codes of length 100 can be

identi�ed by shorter sequences of length of around 15.

CHAPTER 5

Galileo E5 Signal Tracking

5.1. Introduction

Due to their structure, Galileo E5 signals o�er a number of ways to synchronise

the signal and to demodulate the data. The presence of several parameters in the

AltBOC modulation from its four primary codes, four secondary codes, four phases

of a complex sub-carrier and two data components, all of which are appropriately

mapped onto four signal components, makes the e�cient synchronisation of the

AltBOC signal an interesting and challenging task. Several strategies for signal

acquisition have been discussed in sec. 4.2. A sequential search method to identify

the phase of the secondary code has been discussed in sec. 4.7. The subsequent step

of tracking the signal is the topic of this chapter.

This chapter describes the architectures required, and discusses the pros and cons

of several methods, for tracking the E5 signal. Sections 5.2 to 5.5 contain the work

published in ENC GNSS 2009 (Shivaramaiah et al., 2009a) and sec. 5.6 contains the

work published in IEEE VTC FALL 2009 (Shivaramaiah and Dempster, 2009a).

The contributions of this chapter are:

• Development of a generic tracking architecture for the Galileo E5 AltBOC

signal and deriving an equation for the code tracking error

• Proposal of a hybrid tracking architecture with pre-correlation combinationmethod to address the issue of coherent integration beyond one data bit

duration

• Proposal of an extended tracking range DLL that combines the bene�ts

of the wide tracking range DLL discriminator characteristics of the E5a

/ E5b signal and the more accurate DLL discriminator characteristics of

AltBOC(15,10)

This chapter is organised as follows. In sec. 5.2, a generic tracking architecture that

can describe several ways of tracking a Galileo E5 signal, i.e. an architecture that

generalises the use of several local reference signals, is described and an equation

for the code tracking error is derived. This is followed by a list of candidate local

reference signals and a brief discussion of the advantages and disadvantages of using

di�erent reference signals in sec. 5.3. The issues with the combination of several

components of the E5 signal are discussed in sec. 5.4. In sec. 5.5, hybrid tracking

99

100 5. GALILEO E5 SIGNAL TRACKING

architectures that address the issues mentioned in sec. 5.4 are studied and a pre-

correlation combination architecture is proposed. Results of real satellite signal

tracking with the proposed hybrid architecture are provided in sec. 5.5. In sec.5.6 a

unique way of combining the Delay Locked-Loop (DLL) of an architecture employing

a reference signal that results in a BPSK-like correlation waveform and the DLL of

the Direct AltBOC signal tracking architecture is presented.

5.2. A Generalised Tracking Architecture

The received signal at an intermediate frequency (IF) in the case of Galileo E5

AltBOC can be represented as (considering any one satellite):1

rIF (t) =√

2P · < [s(t− τ) · exp (ωIF t+ ωdt+ θ)] + nW (t) (2.22)

where P is the received power, ωIF is the intermediate frequency, ωd is the Doppler

frequency, θ is the phase of the received signal, s(t − τ) is the complex baseband

signal with a time delay τ with respect to the transmitted signal, and nW (t) is

additive white Gaussian noise. The complex baseband signal can be represented as

s(t) = sc(t) + jss(t). Hence (2.22) can be written as:

rIF (t) =√

2P (sc(t− τ) · cos (ω0t+ θ)− ss(t− τ) · sin (ω0t+ θ)) + nW (t) (5.1)

where ω0 = ωIF + ωd.

The cosine and sine components are generated according to the AltBOC modula-

tion scheme described in sec. 2.3. The receiver front-end typically uses a bandwidth

of at least 51.15 MHz so as to pass the �rst two main lobes of the signal spectrum.

As with the processing methods for the acquisition explained in sec. 4.2, the sig-

nal tracking can also use similar techniques to track the complete E5 AltBOC or

any component of the signal. The �rst method in which the E5a and E5b signals

are translated from their centre frequencies to the baseband is the Single Sideband

(SSB) method or the Double Sideband (DSB) method when components from both

the sidebands are combined. While referring to either SSB or DSB method where

the local reference signal is generated free of sub-carriers, generally the term Side-

band Translation (SBT) method is used (Margaria and Dovis, 2008). The second

method is the Full-band Independent Correlation (FIC) in which the local signal is

generated for the required signal component mapped onto the appropriate phase of

the sub-carrier. The third method is the 8-PSK-like processing which allows only

the complete E5 signal correlation by the use of a look-up-table (LUT). With the

�rst two methods, several combinations of signal components are possible with the

combination variables being coherent and non-coherent, data and pilot, E5a and

E5b.1This equation is same as (2.22)

5.2. A GENERALISED TRACKING ARCHITECTURE 101

Reference

Baseband Signal

Generator

Complex

Carrier

( )IFr t

Code

Loop

Filter

Code

NCO

1,my1( )y t

2 ( )y t( )x t

( )y t

1

1( 1)

nT

n T

dt

1T

1T

2,my *

1ˆs t

*

2ˆs t

*

0ˆs t

Carrier

Loop

Filter

Carrier

discrimin

ator

Carrier

NCO

0 ( )y t

2

2( 1)

nT

n T

dt

0,ly

2T

1

1( 1)

nT

n T

dt

Code

discrimin

ator

Figure 5.1. Generalised architecture for the E5 signal tracking

Fig. 5.1 shows a generalised tracking architecture for the E5 signal. This ar-

chitecture holds good for all three methods of processing mentioned in the previous

paragraph. The received signal is multiplied with the locally generated carrier to

obtain the baseband signal. For the SBT method:

x(t) = xSBT (t) = exp(−j(ω0t+ θ ± ωsct

))(5.2)

where ωsc = 2πfsc is the sub-carrier angular frequency and the preceding sign de-

pends on whether one is interested in E5a (−) or E5b (+). For the FIC and 8-PSK

methods:

x(t) = xFIC(t) = x8−PSK(t) = exp(−j(ω0t+ θ

))(5.3)

Without loss of generality, the analogue representation of the signals at di�erent

points in the tracking loop up to the integrate and dump stage can be used. The

received signal is multiplied with the complex carrier generated at IF plus the esti-

mated Doppler (ω0 = ωIF + ωd ) and the estimated phase (θ ) to obtain y(t). The

reference baseband signal generator produces three signals s∗0(t − τ), s∗1(t − τ) and

s∗2(t− τ) where τ is the estimate of the code delay. The output of the carrier mixer

is then multiplied by the reference baseband signal to obtain the individual sample

correlations y0(t), y1(t) and y2(t) corresponding to s∗0(t− τ), s∗1(t− τ) and s∗2(t− τ)

respectively. The output obtained by multiplying with the di�erent versions of the

reference signal are integrated over a speci�ed duration T1 seconds to use in the

code tracking loop and T2 seconds to use in the carrier tracking. The carrier dis-

criminator uses the prompt reference signal (e.g. arctan). The two integration times

are given two di�erent notations to show that two loops can operate with di�erent

integration durations when required (which is more generic). The code tracking in

this architecture uses the delay-locked loop (DLL), whose inputs are the correlation

values of the received signal with at least two time-displaced versions of the local

reference signal. Note that more than three �taps� can also be used, although this


case is not discussed here.

Below are some examples of interpreting the reference signals under speci�c

cases:

Case 1. Direct AltBOC / 8-PSK-like code tracking: The reference base-

band signal in the case of the 8-PSK method comprises both the code and

the sub-carrier. In its simplest form, the reference signals are early, late

and prompt versions of the AltBOC signal with a spacing of 2δ = d chips

between early and late samples:

s∗1(t− τ) = s∗(t− τ + δTc) (5.4a)

s∗2(t− τ) = s∗(t− τ − δTc) (5.4b)

s∗0(t− τ) = s∗(t− τ) (5.4c)

where Tc is the chip duration.

Case 2. FIC code tracking: In the case of BPSK(10) tracking, say with the

E5a component, assuming that both the pilot and data channels are used

for code tracking and only the pilot signal is used for carrier tracking, the

reference signals are:

s∗1(t− τ) =1

2√

2c∗a (t− τ + δTc) · sc(t− τ + δTc) (5.5a)

s∗2(t− τ) =1

2√

2c∗a (t− τ − δTc) · sc(t− τ − δTc) (5.5b)

s∗0(t− τ) =−j

2√

2caQ (t− τ) · sc(t− τ) (5.5c)

where ca = caI +jcaQ is the E5a code (including secondary codes). A similar

approach can be used for the BPSK(10) tracking of the E5b component.

Case 3. Carrier tracking: Since the signal is comprised of pilot components

that can be used for e�cient carrier tracking (and hence no necessity of

having a Costas loop), the prompt arm can carry the combined E5a and

E5b pilot channels:

s∗0(t− τ) = − j

2√

2(caQ (t− τ) · sc(t− τ) + cbQ (t− τ) · sc∗(t− τ)) (5.6)

(5) where sc(t) = scs(t) + j · scs(t − Tsc4

), scs is the sum-sub-carrier, Tsc

is the sub-carrier period and caQ and cbQ are the E5a and E5b quadrature

spreading codes (including secondary codes) respectively.

The choice of the baseband reference signal is the di�erentiating parameter by which

additional tracking architectures are possible in addition to the SBT, FIC and 8-PSK

methods. In this architecture, a delay-lock tracking is assumed for the code with

5.2. A GENERALISED TRACKING ARCHITECTURE 103

two time-delayed reference signals s1(t− τ) and s2(t− τ). The generalisation of this

architecture stems from the fact that the two reference signals can be generated in

many possible ways and also that they need not be from the same signal component.

It is interesting to note that this generic architecture �nds a relation to an existing

patent De Wilde et al. (2007) that uses s1(t − τ) = sE5a(t − τ) and s1(t − τ) =

sE5b(t − τ) as the local code arms to produce an output that resembles a scaled

version of the correlation value produced by the Direct AltBOC early-late arms

(case #1).

5.2.1. Code tracking error with noise in the absence of multipath. Due

to the use of complex signals and the special sub-carrier waveforms, the derivation

of the code tracking jitter is quite challenging. Moreover, the derivation has to

consider the generic nature of the reference signals. For the non-coherent early-

minus-late power code discriminator, the equation for the code tracking error is (for

the complete derivation see Appendix F):

σ2ε =

4N0BL

K2P

[(|R1|2 + |R2|2 −R1 ·R∗2 ·R∗r −R∗1 ·R2 ·Rr

)+N0

(1− |Rr|2

)PT1

](5.7)

where

Rz =

∫ T1

0

s(t− τ) s∗z(t− τ) dt, z = 1, 2 (5.8)

represents the correlation of the input signal and the reference signal,

Rr =

∫ T1

0

∫ T1

0

δ(t− u) s1(t− τ) s∗2(u− τ) dtdu (5.9)

represents the auto-correlation of the reference signal, and

K =d

dε(D (ε))|ε=0 (5.10)

is the slope of the S-curve with the discriminator function given by:

D (ε) = |R1 (ε)|2 − |R2 (ε)|2 (5.11)

σ2ε denotes the error variance in chips, N0 is the one-sided noise spectral density, BL

is the one-sided closed loop noise bandwidth of the code lock loop and ε is the error

in code delay estimate: ε = τ − τ .For the Direct AltBOC / 8-PSK tracking, the reference signals sx(·) are as given

in (5.4a),(5.4b) and (5.4c). Therefore, R1 = R(δTc), R2 = R(−δTc), R1(ε) =

R(ε + δTc), R2(ε) = R(ε − δTc), Rr1 = Rr2 = Rr(2δTc) which when substituted in


Table 5.1. Possible reference signals with the SBT method

Signalcomponentof interest

Reference baseband signalsr(t− τ)

E5aI (data) eaI(t− τ)E5aQ (pilot) eaQ(t− τ)E5bI (data) ebI(t− τ)E5bQ (pilot) ebQ(t− τ)

E5a ±eaI(t− τ)+j · eaQ(t− τ)E5b ±ebI(t− τ)+j · ebQ(t− τ)

(5.7) becomes

σ2ε,DirectAltBOC =

4N0BL

K2P

|R (δTc)|2 + |R (−δTc)|2−

R (δTc) ·R∗ (−δTc) ·R∗r (2δTc)−R∗ (δTc) ·R (−δTc) ·Rr (2δTc)

+N0

(1− |Rr (2δTc)|2

)PT1

](5.12)

and the discriminator function is now given by:

D (ε) = |R (ε+ δTc)|2 − |R (ε− δTc)|2 (5.13)

Equation (5.12) matches with the result given by Soellner and Erhard (2003).

Note that Soellner and Erhard (2003) provides only the result and the derivation

is not available in public domain (con�rmed via a private communication with the

author).

At this point, it is worth mentioning that (5.12) also accommodates real reference

signals. With the real reference signal, (5.12) reduces to

σ2ε =

4N0BL

K2P

[(2R2 (δTc) (1−R (2δTc))

)+N0 (1−R2 (2δTc))

PT1

](5.14)

which is exactly the same as (7.2-86) of Holmes (2007), derived for real BPSK and

BOC signals. Hence, (5.12) and more generally (5.7) are very powerful in the sense of

accommodating all the currently used GNSS signals. Note that the channel �ltering

e�ects are not accounted for in (5.7).

5.3. Candidate Local Reference Signals

5.3.1. Reference signals with the SBT method. Table 5.1 shows several

possibilities of the reference signals with the SBT method. The reference signal

could be applied to any of the cases r = 0, 1, 2, typically within the same signal

component of interest. The parameter e•(•) is the primary code plus secondary

5.3. CANDIDATE LOCAL REFERENCE SIGNALS 105

Table 5.2. Possible reference signals with the FIC method

Signal component of interest Reference baseband signal sr(t− τ)

E5aI (data) 12√

2· eaI(t− τ) · scsum(t− τ)

E5aQ (pilot) j

2√

2· eaQ(t− τ) · scsum(t− τ)

E5bI (data) 12√

2· ebI(t− τ) · sc∗sum(t− τ)

E5bQ (pilot) j

2√

2· ebQ(t− τ) · sc∗sum(t− τ)

E5a 12√

2·(±eaI(t− τ)

+j · eaQ(t− τ)

)· scsum(t− τ)

E5b 12√

2·(±ebI(t− τ)

+j · ebQ(t− τ)

)· sc∗sum(t− τ)

E5p*(E5aQ and E5bQ)j

2√

2· eaQ(t− τ) · scsum(t− τ)+

j

2√

2· ebQ(t− τ) · sc∗sum(t− τ)

E5d* (E5aI and E5bI)± 1

2√

2· eaI(t− τ) · scsum(t− τ)±

12√


E5ab

12√

2·(±eaI(t− τ)

+j · eaQ(t− τ)

)· scsum(t− τ)

+ 12√

2·(±ebI(t− τ)

+j · ebQ(t− τ)

)· sc∗sum(t− τ)

*In this chapter, the combination of the E5a pilot and the E5b pilot signal isdenoted as E5p. Similarly, the combination of the E5a data signal and the E5b

data signal is denoted as E5d.

code plus navigation data (for the I components) as de�ned in OSSISICD (2010);

GIOVEABICD (2008). When correlated with y(t), all the reference signals result

in a BPSK(10)-like correlation triangle due to the absence of the sub-carrier. Note

that the last two reference signals are coherent summation of the data and pilot

components. Since the reference signal is generated without the data, there is an

ambiguity in deciding the sign of the summation.

5.3.2. Reference signals with the FIC method. Table 5.2 shows the pos-

sible reference signals using the FIC method. As explained in sec. 2.3, in order

to achieve the constant envelope modulation at the transmitter, the sub-carrier is

separated into two parts: the sum-sub-carrier and the product-sub-carrier. The

sum-sub-carrier scsum(t− τ) = scs(t)− j · scs(t− Tsc4

) is the major part whose phase

is used to modulate the four components of the E5 signal (Tsc is the sub-carrier

period). The product-sub-carrier scprod(t − τ) = scp(t) − j · scp(t − Tsc4

) modulates

the product codes. The waveforms scs and scp are shown in Fig. 2.2. It is possible

to incorporate the product codes also into the reference signal. As shown in Appen-

dix B, the product signals are of no advantage for receiver bandwidths less than 90

MHz. Hence we can safely neglect the product signal from inclusion in the reference

signal.


Table 5.3. Possible reference signals with the 8-PSK-like method

Signalcomponent of

interest

Reference baseband signal E5 with sr(t− τ)

E5 exp(j π

4k(t− τ)

)with

k(t) =LUT (eaI(t), eaQ(t), ebI(t), ebQ(t), iTsc)

When correlated with y(t), the �rst six reference signals in Table 5.2 result in a

BPSK(10)-like correlation waveform, while the last three reference signals result in

a AltBOC(15,10)-like correlation waveform. Note that as the data and pilot signals

are being combined without considering the data bit in the reference signal, E5a,

E5b, E5d and E5ab will have ambiguities for the sign of the coherent summation.

5.3.3. Reference signals with the 8-PSK-like method. Table 5.3 shows

the reference signal with the 8-PSK-like tracking method. The parameter iTsc is the

quantised sub-carrier phase k(t) and is the output of the LUT (called here as the

function LUT ) de�ned in OSSISICD (2010). Again the data ambiguity exists due to

the E5a and E5b data components. It should be noted that unlike the FIC method,

the product signal cannot be separated from the sum signal. This reference signal

when correlated with y(t), produces an AltBOC(15,10) correlation triangle.

5.4. Issues Related to the Di�erent Architectures

Each of the reference signals yields an architecture to track the signal or a com-

bination of signals of interest listed in the left column of the Tables 5.1, 5.2 and 5.3.

As mentioned in sec. 2.8, the tracking performance measured in terms of carrier

phase jitter and code phase jitter is directly dependent on the signal strength of the

received signal; in general, the higher the signal strength the better the performance.

Hence it is wise to combine the di�erent components of the signal in order to extract

as much power from the received signal as possible.

5.4.1. The data bit ambiguity. In GNSS signals that have only the data

component (i.e. no pilot component, such as the GPS L1 C/A signal) the data bit

ambiguity occurs when the correlation values across two or more data bit periods

needs to be combined. This requirement arises during coherent integrations beyond

one data bit duration. However, for GNSS signals that have more than one signal

component with at least one data component, there is an additional problem of data

bit ambiguity across the signal components. This problem arises when the receiver

needs to coherently combine two or more signal components. These two types of

data bit ambiguities are illustrated in Fig.

5.4. ISSUES RELATED TO THE DIFFERENT ARCHITECTURES 107

Data-bit

Duration

Component 1

Component 2

Across data bit

periods

Across

components

Time

Figure 5.2. Illustration of two types of data bit ambiguities

A receiver processing the E5 signal experiences both the above-mentioned types

of data bit ambiguities, since there are two data and two pilot components. The need

for coherent integration beyond one data bit period arises because of the requirement

to suppress the in�uence of noise at the output of the tracking loops. The need for

coherent combination across signal components arises because such a combination

maximises the received signal power and at the same time provides an AltBOC

correlation waveform at the output of the correlator. Because of data bit ambiguity,

coherent combination is not possible in many cases, as indicated in (blue) colour in

Tables 5.1, 5.2 and 5.3. Note that the only coherent combination possible is the E5a

and E5b pilot signal combination which results in the E5 pilot signal combination.

In the SBT and FIC methods, the alternate combination (for the data bit �ip-

over) can be represented by changing the sign of the summation between E5a and

E5b components. However the correlation operation has to be performed to test

both the direct and the alternate combinations, and the most probable combination

can be estimated by comparing both the correlation outputs (the combination with

a sign opposite to that present in the input will produce a very low correlation

value, the other one will produce a high correlation value). On the other hand, for

the 8-PSK-like tracking, the inputs to the AltBOC LUT needs to be modi�ed and

four correlation values need to be tested, as described in the next section.

The non-coherent combination during the correlation process, which is a classical

solution to data bit ambiguity, does not directly serve the purpose of achieving better

tracking since the phase of the signal is lost during the combination process.

5.4.2. Carrier tracking, pilot and the data signals. The 8-PSK-like track-

ing method makes the best use of received signal power, but requires a Costas-loop

for the carrier tracking. If the E5p component is tracked, then it is possible to use a

pure phase-locked loop (PLL) to gain a 3dB advantage (Kaplan and Hegarty, 2006)

over the E5 signal. The E5 pilot tracking (E5p) will still be short of 3dB compared


to the maximum achievable gain that can be obtained when a combination of the

E5 signal and a pure PLL is used (under the same integration duration criteria).

On the other hand, the pilot signal tracking can cater for greater signal dynamics.

5.4.3. Code tracking linear range. The code tracking linear range directly

depends on the sharpness of the underlying correlation function. The E5 signal

tracking with the AltBOC(15,10) correlation output has a much sharper main peak

compared to the architectures which produce BPSK(10)-like correlation output. On

the other hand, as discussed in previous paragraphs, the 8-PSK-like tracking method

gives the best received signal-to-noise ratio among all the architectures. The carrier

aiding of the code loop lessens the e�ect of the code loop dependency, but does not

completely eliminate it, especially at lower sampling frequencies. Sec. 5.6 proposes

a novel method of combining the BPSK(10) and AltBOC discriminator outputs to

overcome this problem.

Table 5.4 summarises the important discussion issues so far. Later in this chap-

ter, performance evaluation results of some of these architectures will be provided.

5.5. Hybrid Tracking Loop Architectures

Based on the discussions in the previous section, the requirements for a hybrid

architecture can be stated as follows:

• An important requirement of a hybrid architecture is to utilise the received

signal power to the maximum extent possible (in an ideal situation, the

complete E5 signal should be used for tracking) and at the same time ob-

taining a narrow correlation main peak (in an ideal situation, this would be

the correlation waveform obtained with the Direct AltBOC / 8-PSK-like

tracking)

• The data bit ambiguity problem should be avoided, both across the com-

ponents and across the data bit durations, and

• The resources required to achieve the �rst two requirements along with the

resources required to demodulate the data bits (which is the other major

task of the tracking loop) should be as low as possible.

The discussions of the three methods presented below will be focused on the above

three requirements.

5.5.1. Coherent pilot signal tracking and aiding the data demodula-

tion (denoted here as �E5p�). This method is already described in the litera-

ture(Gerein, 2007; Margaria and Dovis, 2008). Therefore the task of this section is

5.5. HYBRID TRACKING LOOP ARCHITECTURES 109

Table 5.4. Indicative performance of di�erent tracking architectures

Signal

Component

Performance (relative to each other)

Code

phase

jitter

Carrier

phase

jitter

Code

tracking

linear

range

Power

sharing

(in�nite

band-

width)

E5 8-PSK

AltBOC

Very

good

Good Poor 100%

E5a pilot / E5b

pilot (with

SBT)

Poor Good Good 21.34%

E5a data / E5b

data (with

SBT)

Poor Poor Good 21.34%

E5a pilot / E5b

pilot (with FIC)

Poor Good Very

good

21.34%

E5a data / E5b

data (with FIC)

Poor Poor Very

good

21.34%

E5 pilot (with

FIC)

Good Very

Good

Very

good

42.68%

E5 data (with

FIC)

Good Good Very

good

42.68%

E5ab (with

FIC)

Good Good Poor 85.36%

to study the suitability of this method keeping in mind the requirements mentioned

above. The intention of this method is to

• obtain the measurements by tracking a coherent combination of the pilot

components (E5aQ and E5bQ) (Gerein, 2007; Margaria and Dovis, 2008),

and

• demodulate the data

� together from a combination of the data components (E5aI and E5bI)

(Gerein, 2007) OR

� separately from the individual data components E5aI and E5bI, which

is nothing but the FIC equivalent of the SBT method proposed in

Margaria and Dovis (2008)

Fig. 5.3 shows the architecture of the coherent pilot tracking method.

Carrier and code tracking: The reference signal for the carrier and code tracking

correspond to the E5p signal component mentioned in Table 5.2. The X in Fig. 5.3

which indicates the reference signal type should be replaced by E5p (i.e. coherent


Reference

Baseband Signal

Generator

Complex

Carrier

( )IFr t

Code

Loop

Filter

Code

NCO

1 ,X my1,Xy

2,Xy 0

ˆˆ

( )j t

x t e

( )y t

1

1( 1)

nT

n T

dt

1T

1T

2 ,X my

*

1,Xs

*

2,Xs

*

0,Xs

Carrier

Loop

Filter

Carrier

discrimin

ator

Carrier

NCO

0,Xy

2

2( 1)

nT

n T

dt

0 ,X ly

2T

1

1( 1)

nT

n T

dt

Code

discrimin

ator

E5a, E5b

data bits

0 5 ,E a my

0, 5E by

2

2( 1)

a

a

nT

n T

dt

2aT

2bT

0 5 ,E b my

2

2( 1)

b

b

nT

n T

dt

angle(.)

angle(.)

LUT /

Mapp

ing

0, 5E ay

*

0, 2Ds

*

0, 1Ds

Figure 5.3. Coherent pilot signal tracking and aiding the data de-modulation

combination of E5aQ and E5bQ) to obtain the speci�c architecture. Since this

component is free of data, the integration time is only limited by the loop time

constant and the signal dynamics that need to be catered for. Observe that both

the carrier tracking and the code tracking use the combined pilot components. The

carrier and code tracking require three sets of code mixers and accumulators.

Data bit demodulation: To demodulate the data, observe that the reference sig-

nals are generated from the reference signal generator which is driven by the code

NCO used to track the combined coherent pilot signals. Demodulating the data bits

separately from the individual data components (Margaria and Dovis, 2008) requires

two complex code mixers and accumulators (the integration block) and uses only a

quarter of the signal power to detect the data bit transitions. However, the method

of (Gerein, 2007) forms the sum and di�erence of E5aI and E5bI (data components):

s∗0,D1 = 12√

2· eaI(t− τ) · scsum(t− τ) + 1

2√

2· ebI(t− τ) · sc∗sum(t− τ) (5.15)

s∗0,D2 = 12√

2· eaI(t− τ) · scsum(t− τ)− 1

2√


= s0,D1 (5.16)

Observe that even the data components combination method uses two code mix-

ers and two accumulators. The angles of the complex correlation vales y0,E5a,m and

y0,E5b,m have to be mapped to E5a and E5b data bits correspondingly, and is the only

additional processing requirement compared to that of the single data component

method. This mapping can be easily implemented with the help of a LUT(Gerein,

2007). Using the combined data components is more suitable (and hence used to

represent this architecture in the following discussions) in addressing the require-


Reference

Baseband Signal

Generator

Complex

Carrier

( )IFr t

Code

Loop

Filter

Code

NCO

1 ,i my1iy

2iy 0

ˆˆ

( )j t

x t e

( )y t

1

1( 1)

nT

n T

dt

1T

1T

2 ,i my

*

1is

*

2is

*

0is

Carrier

Loop

Filter

Carrier

discrimin

ator

Carrier

NCO

0iy

2

2( 1)

nT

n T

dt

0 ,i ly

2T

1

1( 1)

nT

n T

dt

Code

discrimin

ator

Selector

1 2,z z 0z

E5a and E5b

data bits

Figure 5.4. A quasi-coherent (data wipe-o�) architecture

ments laid at the beginning of this section due to the 3dB signal power advantage

compared to using the individual data components.

5.5.2. A quasi-coherent architecture (denoted here as �E5�). This ar-

chitecture is similar to the coherent and semi-coherent integration proposed in Yang

et al. (2004) for the acquisition of GPS L5 signals. This method is also a data wipe-

o� method across the components. Though the data wipe-o� tracking architecture

for the Galileo E5 presented in this subsection is a logical extension (from two signal

components to four signal components) of the data wipe-o� method that already

exists for other GNSS signals, such as the method in Yang et al. (2004), the actual

description of the architecture and the method for demodulating the data is original.

Fig. 5.4 shows the architecture of the data wipe-o� method.

Carrier and code tracking: The data bits in E5a and E5b can be added con-

structively or destructively. Because of the two data bits, there are four cases and

one can combine the correlation outputs for these cases. Basically, four correlation

outputs, y0i,l, i = 1, 2, 3, 4 are obtained from the AltBOC LUT2 from the reference

signals:

s0i(t− τ) = exp(jπ

4ki(t− τ)

)(5.17)

where

k1(t) = LUT (caI(t), caQ(t), cbI(t), cbQ(t), iTsc)

k2(t) = LUT (caI(t), caQ(t),−cbI(t), cbQ(t), iTsc)

k3(t) = LUT (−caI(t), caQ(t), cbI(t), cbQ(t), iTsc)

k4(t) = LUT (−caI(t), caQ(t),−cbI(t), cbQ(t), iTsc) (5.18)

Note in (5.18), the negative sign used for the data signal components E5a-I and

E5b-I. Also note that y0i,l are the complex correlation outputs. Instead of the non-

coherent combination, one can use the magnitudes of these complex correlation

2AltBOC LUT is the look-up-table provided in (OSSISICD, 2010) for the AltBOC modulation


outputs to form the quasi-coherent combination:

z0,l = maxi

(∣∣y0i,l

∣∣) (5.19)

The index l represents the integration period: 4 ms, if the secondary code delay

is available at this stage in the receiver, in which case the code sequences c•(t) in

(5.18) includes the secondary codes OR 1 ms if the secondary code delay is unknown

at this stage in the receiver.

The output z0 = y0imax,l obtained with the �prompt� reference signal is used for

the carrier tracking, imax being the index corresponding to the sign combination that

satis�ed the condition in 5.19. The corresponding early and late correlator outputs

are selected as z1 = y1imax,m and z2 = y2imax,m, index m representing the integration

period. Assuming that z1,m and z2,m are required at the same instant (real-time3)

as that of z0,l, the correlation outputs for all the combinations in (5.18) must be

generated simultaneously corresponding to all the three reference signals s∗0, s∗1 and

s∗2. The selector block then passes z1,m and z2,m to the code discriminator. In this

case, 12 sets (three for each sign combination) of code mixers and accumulators are

required for the tracking.

Data bit demodulation: Data bit demodulation is very simple in this architecture.

It is done by examining the E5aI and E5bI sign combination that results in the

maximum value during each integration duration (within a data bit period).

Coherent integration beyond one data bit period: Increasing the integration dura-

tion for the carrier and code tracking beyond the symbol duration (4 ms) is possible

in this architecture. This is done by coherently adding the complex correlation values

to get z0 =Lcarcoh∑l=1

y0imaxl ,l, z1 =Lcodecoh∑m=1

y1imaxm ,m and z2 =Lcodecoh∑m=1

y2imaxm ,m, where Lcoh = TcohTd

, Td

being the data bit duration (s). However, there is a drawback of this combination.

Though the correlation values correspond to the longer integration duration, the

correctness of the selection of individual correlation values within a data bit period

is the weak link because the �max� operation is performed over only one data bit

period (and will be more noisy compared to the total coherent integration). The

other methods for data wipe-o�, such as examining all the branches in the data bit

binary tree, are not considered in this thesis.

5.5.3. Pre-correlation combination architecture (denoted here as �E5-

PC�). In this sub-section a new architecture called the pre-correlation (PC) combi-

3The other option is to wait for the selector block to decide the best sign combination and then usethe code mixer and accumulator for s1 and s2 to generate only that particular correlation output.However, to do this, (i) the input signal should be stored for use in the resource sharing and (ii)the receiver design should be able to synchronise the usage of prompt correlator output and theearly/late correlator outputs by some other means. Such an architecture is not considered in thisthesis.


nation method is proposed. The key idea here is to move the process of combining

the signal components from a post correlation operation to a pre-correlation opera-

tion. In other words, in the pre-correlation combination method, the local reference

signals are combined together before performing the correlation.

Carrier and code tracking: There are four possible reference signals s01,s02,s03

and s04 as per (5.17) and (5.18). s01 and s04 correspond to the cases where the

two data bits match, (+,+) and (-,-) respectively; and s02 and s03 correspond to

the cases where the two data bits di�er from each other, (+,-) or (-,+) respectively.

Now there are two cases:

Case 1. if the data bits in the input signal match (+,+) or (-,-) : one of the

reference signals s01 or s04 will produce a maximum value and the other will

result in noise AND both s02 and s03 will result in 50% correlation loss, and

Case 2. if the data bits in the input signal are di�erent to each other (+,-) or

(-,+) : one of the reference signals s02 or s03 will produce a maximum value

and the other will result in noise AND both s01 and s04 will result in 50%

correlation loss

Since there are only two possibilities after addition (data bits can be the same or data

bits can be di�erent) it is su�cient to add two reference signals, one corresponding

to the `same sign' data bit case and the other corresponding to the `di�erent sign'

data bit case. In other words, if the reference signal is s∗01 + s∗04 or s∗02 + s∗03 then

the output will always be maximum irrespective of the data bit pattern in the input

signal - Fig. 5.5 is an example with real data.

The PC architecture uses the same architecture depicted in Fig. 5.3 but the

reference signals have to be modi�ed. The identi�er X has been replaced by PC

(pre-correlation) and the reference signal for the prompt arm is

s0,PC = s01(t− τ) + s04(t− τ) OR s02(t− τ) + s03(t− τ) (5.20)

and similar equations hold for s1,PC and s2,PC reference signals. Before adding indi-

vidual reference signals, one needs to ensure that they are su�ciently uncorrelated.

Fortunately, the codes used for Galileo and GIOVE E5 signals exhibit such a prop-

erty. For example, Fig. 5.6 shows the cross-correlation between s01(t−τ) & s04(t−τ)

and s02(t− τ) & s03(t− τ) for the GIOVE-A code. Observe that the cross-correlation

is around 19 dB below zero for both cases, which is su�cient to separate the two

reference signals and hence either combination can be used to obtain s0,PC(t). It

should be noted that the entire received signal power is used in this method by

virtue of using all four signal components. However, during the addition, there will

be an additional noise component and hence the signal-to-noise ratio will be slightly


0 50 100 1500

2

4

6x 10

5

Cor

rela

tion

valu

e

Individual reference signals

(+,+)(+, −)(−, +)(−, −)

0 50 100 1500

2

4

6x 10

5 Combined reference signals

Time (ms)

Cor

rela

tion

valu

e

(+,+) + (−, −)(+,−) + (−, +)

Figure 5.5. Correlation values with individual reference signals(top); with combined reference signals (bottom)

0 2 4 6 8 10 12

x 105

−60

−50

−40

−30

−20

−10

0

Chip shift (in samples)

Cro

ss c

orre

latio

n va

lue

(dB

)

s01

with s04

s02

with s03

Figure 5.6. Cross correlation between the di�erent reference sig-nal combinations (55 MHz front-end bandwidth; 112 MHz sampling);GIOVE-A spreading code


Table 5.5. Summary of the hybrid tracking architectures

E5p(Coherentpilot)

E5 (Quasi-coherent)

E5-PC(Pre-

correlationcombina-tion)

% of received signal powerused for tracking

50 100 100

Coherent integrationbeyond one data bit period

Possible Possible, buthas a weaklink due tothe �max�operation

Possible

Resource utilisation (# ofcomplex code mixers and

accumulators)

5 12 5

less than that of the data-wipe-o� case. This will be considered in the next section

while analysing the performance.

Data bit demodulation: The data bit demodulation in this architecture is similar

to that used in the pilot signal tracking architecture in sec. 5.5.1.

Coherent integration beyond one data bit period: Since the correlation process

always generates the maximum correlation value, coherent integration beyond one

data bit period is possible by simple addition of the consecutive correlation values.

Hardware considerations: This architecture requires only three sets of code mix-

ers and accumulators. The only additional requirement is an adder to add the two

reference signals, but the width of this adder is much less than the width of the

accumulators (4-5 bits as against the 20+ bits for the accumulator) and hence not

a signi�cant overhead.

5.5.4. Summary of the three hybrid architectures. Table 5.5 summarises

the three hybrid architectures with respect to the requirements laid out at the be-

ginning this section.

5.5.5. Performance analysis. For the performance analysis, the following re-

ceiver design parameters were used. The sampling frequency and the front-end

bandwidth are chosen to be 112 MHz and 55 MHz respectively, to match the Septen-

trio GeNeRx1 receiver which was used to collect real signal samples for the bench

tests. For the tracking, an early-late chip spacing d=0.3, one-sided closed code loop

bandwidth BL of 1 Hz and one-sided closed phase-locked loop (PLL) bandwidth of

10 Hz was chosen (stationary receiver tests).


For the carrier tracking, a pure PLL has been used for the pilot component

channels and a Costas PLL for the data carrying components. The carrier tracking

noise variance is given by (Kaplan and Hegarty, 2006):

σ2φ,data =

BPLL

αdataC/N0

(1 +

1

2 · C/N0 · T2

)(5.21)

σ2φ,pilot ≈

BPLL

αdataC/N0

(5.22)

where αdata and αpilot are the power sharing factors referenced to the complete E5

signal, αdata=αpilot= 0.214 in the case of single component and αdata=αpilot= 0.428

in the case of two component combination.

The code tracking noise variance is also analysed for two di�erent discriminator

types. The pilot component uses the coherent dot-product type discriminator and

the data component uses the non-coherent dot-product discriminator. The code

tracking jitter for the BPSK and the AltBOC modulation is given by (Dierendonck,

1996) (in chips2):

σ2ε,pilot =

BL [1−R(dTc)]

2αdata · C/N0 ·K2(5.23)

σ2ε,data =

BL [1−R(dTc)]

2αdata · C/N0 ·K2

(1 +

1

αdata · C/N0 · T1

)(5.24)

where Tc is the chip duration and R is the underlying correlation function. The

slope K is unity for the signal components that produce a BPSK(10)-like correlation

waveform. For the signal components that produce the AltBOC(15,10) correlation

waveform, K ≈ 8.5.

Fig. 5.7 shows the carrier phase error for di�erent signal components. Leg-

end description: The quasi-coherent architecture which uses the Direct AltBOC

/ 8-PSK-like tracking with data wipe-o� is denoted in the �gure as �E5�. �E5p�

denotes the coherent pilot signal architecture that combines both the pilot signal

components. �E5-PC� denotes the proposed pre-correlation combination method.

�L(E5aI,E5aQ)� denotes the post-correlation linear combination architecture applied

to the E5a signal, originally proposed in Hegarty (1999). �E5a� denotes the tracking

with data and pilot components. �E5a-I� denotes the E5a data channel tracking,

and �nally �E5a-Q� denotes the E5a pilot channel tracking. Observe that the Direct

AltBOC quasi-coherent tracking outperforms all the other signal components. The

performance of the E5-PC method is slightly worse than the quasi-coherent track-

ing. This is expected because when the two local reference signals are added, an

additional noise component is incorporated. The interesting part is that the PC

architecture performs better than the coherent pilot tracking architecture and can

be attributed to the increased signal strength due to the four signal components


20 25 30 35 40 45 500

10

20

30

40

50

60

70

80

90

C / N0 (dB−Hz)

Car

rier

phas

e er

ror

stan

dard

dev

iatio

n (d

eg)

E5E5pE5aE5a−IE5a−QL(E5aI, E5aQ)E5−PC

Figure 5.7. Carrier phase error standard deviation for di�erent sig-nal components

20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

C/N0 (dB−Hz)

Cod

e tr

acki

ng e

rror

sta

ndar

d de

vitio

n (m

)

E5E5pE5aE5aIE5aQL(E5aI,E5aQ)E5−PC

Figure 5.8. Code tracking error standard deviation for di�erent sig-nal components

instead of two. The E5 coherent pilot signal tracking o�ers better performance than

the linear combination of E5 data and pilot components. The E5a-Q pilot tracking

performs better than the E5a pilot and data combination especially at lower sig-

nal strengths because of the absence of data bit ambiguity. The E5a-I data signal

has the worst performance among all the considered methods, because of the power

sharing as well as the data bit ambiguity. Fig. 5.8 shows the code tracking error

standard deviation for the signal components in Fig. 5.7. Again observe that the


Table 5.6. Performance comparison of the hybrid tracking architectures

E5 E5p E5-PC E5pcomparedto E5

E5-PCcomparedto E5

σ2ε given 25 8.19 11.58 9.73 -41.39% -18.80%

Code C/N0 35 2.59 3.66 3.08 -41.31% -18.92%tracking C/N0 0.1 23.27 26.28 24.77 3.01 dB 1.5 dB

@ σ2ε 0.05 29.30 32.30 30.80 3.00 dB 1.5 dB

σ2φ given 25 10.19 14.41 12.12 -41.41% -18.94%

Carrier C/N0 35 3.22 4.58 3.83 -41.61% -18.94%tracking C/N0 @ σ2

φ 150 21.65 24.66 23.15 3.01 dB 1.5 dB

* C/N0 is in dB-Hz, σ2ε is in centimetres, σ2

φ is in degrees

8-PSK E5 method o�ers the best performance among all the other components.

The errors in the E5 and E5p components are much less than other components

because the underlying correlation waveform is of the AltBOC(15,10) instead of the

BPSK(10)-like correlation waveform.

From the above discussion, it is clear that the three hybrid architectures outper-

form all the other architectures. Table 5.6 compares the performance of these three

hybrid tracking architectures with the values obtained through simulation. Since

the E5p architectrue does not use the data signal components, the e�ective signal

strength input to the tracking loops is less than that in the E5 architecture and hence

the tracking performance of E5p architecture is inferior to that of E5 architecture.

The performance of E5-PC architecture is in between that of E5 and E5p architec-

tures. In addition, E5-PC o�ers advantages in terms of complexity compared to the

E5 architecture as discussed earlier in this chapter.

5.5.6. Tracking results with the real signal. Using the Septentrio GeN-

eRx1 receiver, the GIOVE-A satellite signal was collected as digitised intermediate

frequency (IF) signal samples during the E5 signal transmission. These data sets

were tracked using a Matlab-based acquisition and tracking module (with all three

data wipe-o� 8-PSK tracking, E5-pilot tracking and E5 pre-correlation combining

methods), and data bits were demodulated.

Fig. 5.9 shows the output of tracking loops for the coherent 8-PSK tracking of the

E5 signal without data bit wipe-o�. Observe that the correlation value drops due

to the destructive pattern of the data bits. Fig. 5.10 shows the prompt correlation

outputs for the three types of hybrid tracking methods quasi-coherent 8-PSK, E5

pilot and pre-correlation combination. Observe that the pre-correlation method

produces slightly nosier output as expected because of two noise components.

Figs. 5.11 and 5.12 show the tracking loop output parameters for the three

types of hybrid tracking methods quasi-coherent 8-PSK, E5 pilot and Pre-correlation


0 50 100 150740

750

760Carrier Doppler (Hz)

0 50 100 150−0.1

0

0.1Carrier phase error (cycles)

0 50 100 150−0.5

0

0.5Code phase error (chips)

0 50 100 1505

6

7

8Code doppler (Hz)

0 50 100 1500

2

4

6x 10

5

Time (ms)

Early, Prompt and Late Correlation Values

Figure 5.9. Tracking loop output parameters for 8-PSK-like track-ing (no data wipe-o�): (Data set-I); colour legend: Prompt (-, Blue),Early (�, Green), Late (.., red)

0 50 100 1501

2

3

4

5x 10

5

Time (ms)

Pro

mpt

cor

rela

tion

valu

es

Dataset − II

0 50 100 1501

2

3

4

5

6x 10

5

Time (ms)

Dataset − I

Figure 5.10. Prompt correlation output for quasi-coherent E5 (�,green), E5p (..,blue), and E5-PC (-, purple) tracking methods


0 50 100 150735

740

745

750

755Carrier doppler (Hz)

0 50 100 150−0.1

−0.05

0

0.05


0 50 100 150−0.2

−0.1

0

0.1

0.2

0.3


Time (ms)0 50 100 150

5.5

6

6.5

7

7.5Code doppler (Hz)

Figure 5.11. Tracking loop output parameters for quasi-coherent E5(�,green); E5p (..,blue); and E5-PC (-,purple) tracking methods; forData set�I

0 50 100 150−0.2

−0.1

0

0.1

0.2

0.3


Time (ms)

0 50 100 150590

600

610

620

630

640Carrier doppler (Hz)

0 50 100 150−0.2

−0.1

0

0.1


0 50 100 1504

4.5

5

5.5

6

6.5Code doppler (Hz)

Figure 5.12. Tracking loop output parameters for quasi-coherent E5(�,green); E5p (..,blue); and E5-PC (-,purple) tracking methods; forData set-II

combination. Again observe that the E5-PC method produces slightly nosier output

compared to the data bit wipe-o� method, but comparable to the E5 pilot tracking

method.

Fig. 5.13-5.16 show the correlation values are obtained without any dedicated

tracking loops for the E5a and E5b components. Legend for Fig. 5.13-5.16 (only for

Correlation values): Prompt (-, blue), Early (�,green), Late (.., red). The phase


0 50 100 1500.5

1

1.5

2

2.5x 10

5 Correlation values (E5a)

0 50 100 150−2.2

−2

−1.8

−1.6

−1.4

−1.2angle (Prompt), E5a

0 50 100 1501

1.5

2

2.5

3x 10

5

Time (ms)

Correlation values (E5b)

0 50 100 150−2

−1

0

1

2angle (Prompt), E5b

Figure 5.13. Data bit demodulation, showing the magnitude andthe phase of the correlation values; E5p tracking: (Data set-I)

0 50 100 1500.5

1

1.5

2

2.5x 10

5Correlation value (E5a)

0 50 100 150−2

−1.8

−1.6

−1.4

−1.2angle(Prompt),E5a

0 50 100 1501

1.5

2

2.5

3x 10

5Correlation value (E5b)

0 50 100 150−2

−1

0

1

2angle(Prompt),E5b

Time (ms)

Figure 5.14. Data bit demodulation, showing the magnitude andthe phase of the correlation values; E5-PC tracking: (Data set-I)


0 50 100 1500.5

1

1.5

2x 10


0 50 100 150−4

−2

0

2

4angle (Prompt), E5a

0 50 100 1501

1.5

2

2.5

3x 10

5

Time (ms)

Correlation Value (E5b)

0 50 100 150−4

−2

0

2

4angle (Prompt), E5b

Figure 5.15. Data bit demodulation, showing the magnitude andthe phase of the correlation values; E5p tracking: (Data set-II)

0 50 100 1500.5

1

1.5

2x 10


0 50 100 150−3

−2

−1

0

1

2angle(Prompt),E5a

0 50 100 1501

1.5

2

2.5

3x 10

5Correlation value (E5b)

0 50 100 150−3

−2

−1

0

1

2angle(Prompt),E5b

Time (ms)

Figure 5.16. Data bit demodulation, showing the magnitude andthe phase of the correlation values; E5-PC tracking: (Data set-II)

5.6. AN EXTENDED TRACKING RANGE DLL 123

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Code delay (chips)

Nor

mal

ized

Cor

rela

tion

Val

ue

E5 − UnfilteredE5a/b − UnfilteredE5 −70MHz filterE5a − 20MHz filter

Figure 5.17. Galileo E5 correlation waveform; di�erent �lter bandwidths

of the prompt correlation value shown decodes the data bit as explained earlier.

The data sets were chosen such that the data changes are only in E5b (Data set-I)

and data changes are in both E5a and E5b (Data set-II). The data bit �ip-overs are

indicated by the 180 deg phase jumps in Figs. 5.13-5.16.

5.6. An Extended Tracking Range DLL

In the previous sections, several ways of tracking the AltBOC(15,10) signal were

discussed. Combining the independently processed E5a and E5b correlation re-

sults (through the DSB method or the FIC method) is the best way to maximise

the received signal energy in order to obtain a BPSK(10)-like triangle. Using 8-

PSK-like tracking is the best way to maximise the energy in order to obtain an

AltBOC(15,10) correlation waveform. Fig. 5.17 revisits the correlation plots for

the methods required in this section. In this section, a novel method to combine

the DLL discriminators for the BPSK(10)-like and the AltBOC(15,10) correlation

waveforms is discussed. For the discussion throughout this section, a coherent early

minus late (EML) discriminator has been used due to its simplicity.

5.6.1. DLL considerations. The maximum possible chip spacing assuming

an ideal correlation triangle for 8-PSK tracking is 0.33 chip (this depends on the

bandwidth of the RF front-end �lter). Therefore, an early-late chip spacing of 0.3

is a typical value used in receivers for the 8-PSK tracking, as mentioned in sec. 5.5.

Beyond a 0.3 chip spacing the discriminator has the potential to drive the loop into

a false lock region. Fig. 5.18 shows the S-curve along with the variation of the

S-curve linear region against the chip spacing d for both 8-PSK and BPSK tracking

loops.


−1.5 −1 −0.5 0 0.5 1 1.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Code deay error (chips)

Dis

crim

inat

or o

utpu

t

d = 0.1d = 0.2d = 0.3d = 0.4d = 0.5d = 0.6

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Code delay error (chips)

Dis

crim

inat

or o

utpu

t

d = 0.1

d = 0.2

d = 0.3

d = 0.4

d = 0.5

d = 0.6

Figure 5.18. S-curve for the E5 8-PSK tracking (top) and BPSK(10)tracking for di�erent chip early-late spacing (in�nite bandwidth)

5.6.2. Extended tracking range DLL.

5.6.2.1. The receiver model. In the proposed method, the correlator outputs of

8-PSK and BPSK tracking are combined and are provided as the input to the loop

�lter. Fig. 5.19 shows the receiver model for the proposed architecture. The cor-

relator outputs from the two loops are passed onto a combiner and selector block.

There is only one VCO which updates the timing estimate for both the loops. Note

that the data demodulation is not shown for simpli�cation of the diagram.

5.6.2.2. Discriminator combination. Fig. 5.20 illustrates the regions of interest

of the two discriminators. Consider the 8-PSK tracking loop with 0.3 early-late chip

spacing. For delay errors up to ±0.15 chips, the slope of the S-Curve is positive

and the DLL produces an output to lock to the zero error point. Now consider the

BPSK tracking case. Assume that the chip spacing used for BPSK tracking is 0.6

chips. For delay errors up to ±0.15 chips the slope of the S-curve is positive and


AltBOC

Code

Generator

Complex

Carrier

)(trsLPF

Early

Late

LPF

E5a/b

band

translator

( )ey t

( )ly t

E5ab

Code

Generator

LPF

Early

Late

LPF

1( )ey t

1( )ly t

Selector and

DiscriminatorLoop

FilterNCO

( )e t

1( )r t

( )r t ( )ey t

( )ly t

1( )ey t

1( )ly t

Figure 5.19. Receiver model with the proposed architecture

Discriminator

Output

Delay

Error

8-PSK

AltBOC

BPSK

Combined

δc

Figure 5.20. Illustration of the proposed method; δc is the crossoverpoint


the DLL produces an output to lock to the zero error point. Beyond this chip error,

and up to +/- 0.3 chip error, the DLL for the 8-PSK has negative slope whereas

the BPSK triangle will still have positive slope. The 8-PSK DLL in e�ect may push

the VCO voltage to lock to the secondary peak whereas the BPSK DLL will still

be aiming for the correct central point. Due to the opposite slopes, the two error

outputs cross each other, the 8-PSK discriminator error starts diminishing, but the

BPSK discriminator error will still be growing. Hence the selector block will perform

the following operation:

e(t) =

e0(t) if |e0(t)| < ε

e0(t) if |e0(t)| > Kc |e1(t)|e1(t) if |e0(t)| < Kc |e1(t)|

(5.25)

where

e0(t) = ye(t)− yl(t) (5.26)

e1(t) = ye1(t)− yl1(t) (5.27)

are the individual discriminator outputs of the 8-PSK and BPSK tracking loops

respectively and Kc is the scaling factor which depends on the chip spacing used

for the BPSK discriminator. The parameter ε is added to avoid any unwanted

triggering of the selector at very small discriminator errors, say e0(t) < 0.05. Note

that the delay error corresponding to the crossing point δc depends on the scaling

factor Kc. The error range, for which this algorithm is valid, depends on the chip

spacing used for the BPSK tracking loop (assuming that one always sets the 8-PSK

loop at its typical spacing of 0.3 chips). In addition this algorithm can be extended

for early-late spacing of up to 1.0 chip for the BPSK-triangle.

The advantage in the code tracking range of this method is obvious. Even though

the overall tracking range is controlled by both the DLLs, the output of the BPSK

DLL will be used only rarely, for large displacements. For small displacements,

the 8-PSK loop will take over and the tracking jitter is not a�ected. Others have

addressed a related problem of coming out of the false lock point, for example using

the bump-jumping correlator (Fine and Wilson, 1999) which often requires more

processing by way of having more correlator arms and computing the appropriate

weighting coe�cients for combining them (Fante, 2003).

5.6.3. Performance analysis.

5.6.3.1. Equation for the code tracking jitter of the proposed method. In order to

analyse the proposed algorithm, the code phase tracking jitter is evaluated in terms

of signal and noise components that enter the tracking loop. The performance

analysis of the BPSK and BOC coherent DLLs is given in (Holmes, 2007). Here the


analysis for the extended range method is provided. This linear analysis is valid for

Gaussian input noise. The received spreading waveform (including the sub-carrier)

is denoted here by c(t − τ) where is the τ timing o�set between transmitter and

the receiver. The local spreading waveform is denoted by c(t − τ) where τ is the

receiver estimate of the timing o�set τ . The received signal after the carrier mixing

(assuming a perfect carrier match) in Fig. 5.19 is given by

r(t) =√Pc(t− τ) + nW (t) (5.28)

where P is the signal power and nW (t) is the additive white Gaussian noise with two

sided power spectral density of N0

2Watts/ Hz. Because the signal power in both the

components of the E5a and E5b sidebands is combined, the power loss compared

to the main 8-PSK AltBOC signal is negligible. Hence the same received power

levels at the input of E5a and E5b combined discriminator are assumed. Also the

combined spreading code is represented as c1(t− τ). The input at the E5a and E5b

combined loop is given by

r1(t) =√Pc1(t− τ) + nW (t) (5.29)

The received signal is correlated with the early and late reference waveforms

c(t − τ ± δTc) and c1(t − τ ± δ1Tc) where 2δ = d is the spacing between the early

and late correlator arms for the 8-PSK DLL and 2δ1 = d1 for the BPSK DLL and

Tc denotes the chip period. The code phase error is de�ned as

ε = (τ − τ) (5.30)

Note that a single VCO to adjust the local reference code phase. The correlator

outputs are given by:

ye(t) =√Pc(t− τ)c(t− τ + δTc) + nW (t)c(t− τ + δTc) (5.31)

yl(t) =√Pc(t− τ)c(t− τ − δTc) + nW (t)c(t− τ − δTc) (5.32)

ye1(t) =√Pc(t− τ)c1(t− τ + δ1Tc) + nW (t)c1(t− τ + δ1Tc) (5.33)

yl1(t) =√Pc(t− τ)c1(t− τ − δ1Tc) + nW (t)c1(t− τ − δ1Tc) (5.34)

Assuming that the autocorrelation noise and the correlation noise between indi-

vidual codes are negligible, the output of the LPF yields:

ye/l(t) =√PR(τ − τ ± δTc) + ne/l(t)

=√PR(ε± δTc) + ne/l(t) (5.35)

ye1/l1(t) =√PR1(τ − τ ± δ1Tc) + ne1/l1(t)

=√PR1(ε± δ1Tc) + ne1/l1(t) (5.36)

where R(•) and R1(•) are the correlation functions of 8-PSK AltBOC and BPSK


respectively. The selection algorithm now chooses the error value according to (5.25).

Hence

e(t) =

√PD(ε) + n′(t) 0 6 |ε| < δc√PD1(ε) + n′1(t) δc 6 |ε| < δ1

(5.37)

where D(ε) is the discriminator error function. Note that because no additional

noise components are added in the selection process, the noise in the two cases will

comprise only two noise components. The two-sided power spectral density of the

resultant noise at f = 0 is given by (Holmes, 2007)

Sn′1(0) = N0 [1−R1(2δ1Tc)] = N0d1 (5.38)

Sn′(0) = N0 [1−R(2δTc)] = N0 [1−R(dTc)] (5.39)

The code tracking jitter is given by (Holmes, 2007)

σ2ε =

N0 · 2BL [1−R(dTc)]

PK2(5.40)

where BL is the one-sided closed-loop noise bandwidth of the code-tracking loop

and K is the slope if the discriminator curve. Since a closed form expression for the

AltBOC correlation function does not exist to the authors' knowledge, as mentioned

in sec. 2.4, the correlation function is retained as it is and computed during the

simulation process. The code tracking jitter is then given by

σ2ε =

N0BL1d12P

forBPSK

N0·BL[1−R(dTc)]PK2 for 8-PSK AltBOC

where BL and BL1 are the corresponding code lock loop �lter bandwidths. For the

AltBOC(15,10) signal with 70 MHz front-end �lter bandwidth and d= 0.3, the value

of K is found to be 9 and [1−R(dTc)] ≈ 0.4.

Only outside the region {−δc,+δc}, is the code tracking jitter of the BPSK valid.

As soon as the loop has errors less than ±δc, one obtains a smaller jitter dictated

by the 8-PSK loop. Note that for delay errors from ±δ to ±δc the jitter will be

slightly more than that in the linear region because the slope of the discriminator

curve is slightly less. Since the scaling factor Kc controls δc, Kc can be adjusted (for

a given δ and δ1 ) for a given design and will not be discussed in this chapter. A

coherent EML discriminator with δ=0.15 and a value of Kc=3 gives δc≈0.2. It doesnot depend on δ1 as the slope of the BPSK coherent EML discriminator does not

depend on the chip spacing.

5.6.3.2. Results. The proposed algorithm has been tested on the Galileo In-Earth

Orbit Validation Equipment (GIOVE) satellite E5 signal collected using the Septen-

trio GeNeRx1 receiver. Since the existing setup is bench test equipment, errors in

the estimated code delay have been added during the simulations to make the loop


0 50 100 150−0.5

0

0.5

Cod

e ph

ase

erro

r (c

hips

)

0 50 100 1505

6

7

8

Cod

e D

oppl

er (

Hz)

0 50 100 1500.5

1

1.5

2

2.5x 10

5

Cor

rela

tion

valu

e

Time (ms)

Figure 5.21. 8-PSK AltBOC tracking without introducing any error

0 50 100 150−0.5

0

0.5

Cod

e ph

ase

erro

r (c

hips

)

0 50 100 1505

6

7

8C

ode

Dop

pler

(H

z)

0 50 100 1500

0.5

1

1.5

2

2.5x 10

5

Cor

rela

tion

valu

e

Time (ms)

Figure 5.22. BPSK(10) E5ab tracking without introducing any error

deviate from normal behaviour.

Figs. 5.21 and 5.22 show the code tracking without any errors induced during

testing. The bottom portion of the �gure shows the early (green, -*-), prompt (blue,

�) and late (red, -�-) correlation values. When an error is introduced from 60-105

ms and 8-PSK tracking is used, then the result is as shown in Fig. 5.23. Note that

the code lock loop is diverging. With the proposed method shown in Fig. 5.24, the

code lock loop caters for this change and pulls the loop back to the proper code

Doppler and hence the code phase error remains small.


0 50 100 150−1

−0.5

0

0.5

1

Cod

e ph

ase

erro

r (c

hips

)

0 50 100 1504

6

8

10

Cod

e D

oppl

er (

Hz)

0 50 100 1500

0.5

1

1.5

2

2.5x 10

5

Cor

rela

tion

valu

e

Time (ms)

Figure 5.23. 8-PSK AltBOC tracking; error introduced from 60-105 ms

0 50 100 150−1

−0.5

0

0.5

1

Cod

e ph

ase

erro

r (c

hips

)

0 50 100 1504

6

8

10

Cod

e D

oppl

er (

Hz)

0 50 100 1500.5

1

1.5

2

2.5x 10

5

Cor

rela

tion

valu

e

Time (ms)

Figure 5.24. 8-PSK AltBOC tracking with the hybrid DLL method;error introduced from 60-105 ms

5.7. SUMMARY 131

5.7. Summary

This chapter discussed several tracking architectures that are possible for track-

ing the E5 signal and its components by making use of di�erent local reference

signals. The advantages and disadvantages of using di�erent reference signals are

discussed and a pre-correlation combination type of hybrid tracking architecture is

described. Performance of the proposed architecture was analysed, and real sig-

nal tracking results for GIOVE-A were obtained. In summary, the two preferred

tracking methods are the quasi-coherent E5 and the proposed pre-correlation com-

bination method. The quasi-coherent E5 tracking tops the performance but requires

12 sets of code mixers and accumulators. In addition, longer coherent integration in

the quasi-coherent E5 architecture has a weak link due to the �max� operation that

helps perform the data wipe-o� operation. The performance of the pre-correlation

combination architecture is 18% inferior to the quasi-coherent E5 architecture. How-

ever, the pre-correlation combination method requires only 5 sets of code mixers and

accumulators (less than half compared to the quasi-coherent E5 architecture), and

also allows longer coherent integration beyond beyond one data bit duration. Hence

the pre-correlation combination is useful in situations where the correlator resources

are limited (or if there is a requirement of low power consumption).

This chapter also proposed a new method of increasing the e�ective tracking

range for code tracking of AltBOC signals without a�ecting the jitter performance

in the existing linear range of the code tracking loop. By selecting the appropriate

discriminator output between the BPSK tracking loop and the 8-PSK tracking loop,

it is shown that errors larger than the linear tracking range of the 8-PSK tracking

loop can be pulled back to the linear region without any degradation in the tracking

jitter. The proposed algorithm can be used in systems where there is a possibility

of obtaining a wider and narrower correlation triangle for a given signal.

CHAPTER 6

Galileo E5 Code Phase Multipath Mitigation

6.1. Introduction

This chapter proposes a method to mitigate the code phase multipath by ex-

ploiting the frequency diversity which is inherent to the AltBOC modulation used

in Galileo E5 satellite navigation signals. The �rst part of this chapter deals with the

multipath mitigation technique in detail and the second part introduces the group

delay compensation viewpoint for the proposed frequency diversity based multi-

path mitigation scheme. Sec. 6.2 to sec. 6.4 contain the work published in ION

GNSS 2009 (Shivaramaiah, 2009)(student paper sponsorship award winner and best

presentation award winner in the session). This work is also a part of the patent

(Shivaramaiah and Dempster, 2009b) handled by NewSouth Innovations Pty Lim-

ited (NSi), a wholly owned subsidiary and controlled entity of University of New

South Wales (UNSW). Sec. 6.5 contains the work published in IEEE/ION PLANS

2010 (Shivaramaiah and Dempster, 2010a).

In sec. 6.2, the multipath performance of the Galileo E5 Direct AltBOC sig-

nal tracking architecture (presented in chapter 5) is discussed. In sec. 6.3, the

proposed multipath mitigation scheme, called the Sideband Carrier Phase Combi-

nation (SCPC) method, is described with the associated tracking loop architecture.

Sec. 6.4 presents the simulation and test results for the SCPC. Sec. 6.5 discusses

the group delay compensation viewpoint for the frequency diversity based multipath

mitigation scheme. Section 6.6 summarises the contents of this chapter.

6.2. Performance of the Direct AltBOC Tracking Architecture

6.2.1. Model of the received signal. The received E5 AltBOC signal of any

one satellite in the presence of a direct signal and N re�ected components can be

represented as:

rIF (t) =N∑i=0

√2aiP · <

[s (t− ti) · ej(ωIF t+ωdi t+θi)

]+ nW (t) (6.1)

where s(t) = sc(t)+jss(t) is the complex baseband signal, ωIF and ωd are the carrier

and Doppler frequencies respectively, θi is the phase, ti is the delay and ai is the

attenuation of the ith signal, P is the received signal power, nW (t) is the additive

white Gaussian noise, and < is the real argument function. For most applications,

133

134 6. GALILEO E5 CODE PHASE MULTIPATH MITIGATION

it can be assumed that the di�erence in Doppler between the direct signal and the

re�ected signals is negligible (especially for delays less than one chip of the Galileo

E5 spreading code). The attenuation, time delay and phase of each signal can be

written relative to that of the direct component as:

αi =aia0

, τi = ti − t0 , φi = |θi − θ0|2π (6.2)

Letting ω0 = ωIF + ωd, (G.1) can then be written as:

rIF (t) =N∑i=0

√2aiP ·

(sc (t− t0 − τi) · cos (ω0t+ θ0 + φi)−ss (t− t0 − τi) · sin (ω0t+ θ0 + φi)

)+ nW (t) (6.3)

6.2.2. Signal tracking with the Direct AltBOC architecture. Signal track-

ing in Galileo E5 receivers can be achieved in several ways as discussed in Chapter

5. Two major techniques are important in the context of this chapter. One is the

wideband signal tracking, where the received signal is passed through a wideband

�lter (at least 51.15 MHz bandwidth so as to pass the two dominant lobes) centred

around 1191.795 MHz. This type of tracking allows full utilisation of the shape of

the AltBOC correlation function and the received power and has been termed Direct

AltBOC tracking in Chapter 5 (also referred to as 8-PSK AltBOC tracking). The

second is side-band tracking, where the E5a and E5b sidebands are extracted from

the received signal by multiplying it with an appropriate complex sub-carrier. Since

this operation results in a BPSK(10)-like correlation function, it is also referred to

as BPSK(10) tracking. The same terminologies are retained in this chapter.

The generalised tracking architecture presented in Chapter 5 for the Galileo E5

signal is shown again in Fig. 6.1. In Fig. 6.1, all the lines with x, s and y as the label

carry complex signals. The reference signals are the early, late and prompt versions

of the AltBOC signal with a spacing of 2δ chips between early and late samples:

s∗1(t− τ) = s∗(t− τ + δTc) (6.4a)

s∗2(t− τ) = s∗(t− τ − δTc) (6.4b)

s∗0(t− τ) = s∗(t− τ) (6.4c)

where Tc is the chip duration.

6.2.3. Code tracking error in the presence of multipath, without noise.

In accordance with the customary method used in Holmes (2007) and Braasch (1996)

a single re�ected signal without noise is assumed in order to derive an equation for

the discriminator function, and hence to analyse the e�ect of multipath on the

code tracking error. The discriminator function in the presence of a single re�ected

signal for the coherent early-minus-late (CEML) discriminator is given by (derived

6.2. PERFORMANCE OF THE DIRECT ALTBOC TRACKING ARCHITECTURE 135

Reference

Baseband Signal

Generator

Complex

Carrier

( )IFr t

Code

Loop

Filter

Code

NCO

1my

1( )y t

2 ( )y t 0

ˆˆ

( )j t

x t e

( )y t

1

1( 1)

nT

n T

dt

1T

1T

2my

*

1ˆs t

*

2ˆs t

*

0ˆs t

Carrier

Loop

Filter

Carrier

discrimin

ator

Carrier

NCO

0 ( )y t

2

2( 1)

nT

n T

dt

0ly

2T

1

1( 1)

nT

n T

dt

Code

discrimin

ator

Figure 6.1. Generalised tracking loop architecture for the GalileoE5 signal

in Appendix G):

Dceml (ε) =√P

((|R (ε+ δTc)| −|R (ε− δTc)|

)+√α1 cos(φ1)

(|R (ε+ δTc + τ1)| −|R (ε− δTc + τ1)|

))(6.5)

For the non coherent early-minus-late-power (EMLP) discriminator it is (also in

Appendix G):

Demlp (ε) = P

(|R (ε+ δTc)|2−|R (ε− δTc)|2

)+ α1

(|R (ε+ δTc + τ1)|2−|R (ε− δTc + τ1)|2

)+2√α1 cos(φ1)

(R′((ε+ δTc) , τ1)−R′ ((ε− δTc) , τ1)

) (6.6)

where

R′((ε+ δTc) , τ1) =

Rc (ε+ δTc)Rc (ε+ δTc + τ1) +

Rs (ε+ δTc)Rs (ε+ δTc + τ1) +

Rc (ε+ δTc)Rs (ε+ δTc + τ1) +

Rs (ε+ δTc)Rc (ε+ δTc + τ1)

(6.7)

Rc and Rs being the normalised correlation functions of cosine and sine terms respec-

tively. Interestingly, (6.5) and (6.6) are similar to those derived by Holmes (2007)

and Braasch (1996) for the GPS L1 C/A case, except for the use of complex corre-

lation functions. The function R′can be thought of as a `correlation of correlation

functions indicator'. It represents in a sense the similarity between the correlation of

the direct signal with the reference signal and the correlation of the re�ected signal

with the reference signal. In this case, the code tracking loop estimates the code

delay as τ = tc+ ε where tc is the composite code delay determined by (6.5) or (6.6)

and the error with respect to the direct signal delay is τc = tc − t0. ε is the errorassociated in estimating τ .

Fig. 6.2 shows the multipath error envelope of the 8-PSK AltBOC and BPSK(10)

tracking architectures of the E5 signal with the CEML discriminator. Note that


0 10 20 30 40 50 60−3

−2

−1

0

1

2

3

Multipath delay (m)

Cod

e m

ultip

ath

erro

r (m

)

Code multipath error comparison, SMR = 6dB

E5a−20MHz; δ=0.15E5−50MHz; δ=0.15

Figure 6.2. Code multipath error envelope of E5a and E5 correlatorswith CEML discriminator

Direct AltBOC tracking with at least 50 MHz bandwidth performs better than the

E5a tracking with 20 MHz bandwidth (5th order Butterworth �lters have been used).

However, there is still scope for improvement for multipath delays less than 20 m,

if the goal (say) is to bring the multipath error down to 0.5 m or less.

6.2.4. Carrier phase error in the presence of multipath and code phase

multipath error, without noise. The concepts developed for carrier phase error

for the other signals also valid for the E5 AltBOC tracking architectures. This is

because as far as the carrier is concerned, there is no change in the signal structure

except that they are at di�erent frequencies. In the absence of multipath, the carrier

lock loop estimates the carrier phase φ = θ0 + εφ where εφ is the error associated

with the estimate. In the presence of multipath, the carrier lock loop estimates the

carrier phase θc of the composite signal. The carrier phase multipath error for the

AltBOC signal consisting of direct signal component and a single re�ected signal

component is given by (derived in Appendix (H)):

φc = arctan

( √α1 |R (ε+ τ1)| sin (φ1)

|R (ε)|+√α1 |R (ε+ τ1)| cos (φ1)

)(6.8)

where φc = θc−θ0 is the phase error (which is the same as the phase of the composite

signal when θ0 = 0), and again R is the complex autocorrelation function between

the input signal and the reference signal. Note that the carrier phase error depends

on the error in estimating the code phase .

The carrier phase multipath error envelope is shown in Fig. 6.3. Observe that

unlike the case for BPSK tracking both the code and carrier phase errors have nulls

6.2. PERFORMANCE OF THE DIRECT ALTBOC TRACKING ARCHITECTURE 137

0 10 20 30 40 50 60−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

Multipath delay (m)

Car

rier

phas

e m

ultip

ath

erro

r (m

)

Carrier multipath error comparison, SMR = 6dB

E5a−20MHzE5−50MHz

Figure 6.3. Carrier phase multipath error comparison

within a single chip. This is similar to any BOC(m,n) signal, and is due to the

phenomenon of negative correlation between frequency components of the signal.

In other words, the presence of the sub-carrier alters the shape of the correlation

function of the spreading code which is the major in�uencing factor for the shape

of the carrier phase multipath error envelope.

6.2.5. Signal-to-noise-ratio (SNR) in the presence of multipath. If the

composite signal received at the IF (single re�ector case) is expressed as:

rc =√

2acP · <[(scc(t− tc) + jssc(t− tc)) · ej(ω0t+θc)

]+ nW (t) (6.9)

=√

2acP ·(scc (t− t0 − τc) · cos (ω0t+ θ0 + φc)−ssc (t− t0 − τc) · sin (ω0t+ θ0 + φc)

)+ nW (t) (6.10)

where the subscript c denotes the `composite' signal, then the strength of the com-

posite signal at the output of the correlator can be derived equivalently as:

bc = |R (ε)|2 + α1 |R (ε+ τ1)|2 + 2√α1 |R (ε)R∗ (ε+ τ1)| cos(φ1) (6.11)

If one assumes that the correlator output strength for the direct signal is b0 = |R (ε)|2then the equivalent attenuation of the composite signal with respect to the direct

signal from (6.2) is:

βc =bcb0

= 1 + α1β1 + 2√α1β1 cos(φ1) (6.12)

where β1 is the attenuation of the re�ected signal with respect to the direct signal

at the correlator output. Fig. 6.4 shows the minimum and maximum attenuation

in the case of the E5a, E5b and E5 signals. Again note the shape of the attenuation


0 50 100 150 200 250 300 3500

0.5

1

1.5

E5a

Minimum attenuation and maximum amplification, SMR = 6dB

0 50 100 150 200 250 300 3500

0.5

1

1.5

E5b

0 50 100 150 200 250 300 3500

0.5

1

1.5

E5

Multipath delay (decimeter)

Figure 6.4. Envelope of the attenuation (normalised ratio) for E5a,E5b, and E5 signals under multipath conditions

for E5, which is due to the shape of the correlation function.

6.3. SCPC Method and an Architecture

6.3.1. E5a and E5b carrier phases under multipath. The E5a, E5 and

E5b signals have wavelengths of λE5a=25.48 cm, λE5=25.15 cm and λE5b=24.83

cm respectively, due to their corresponding carrier frequencies. When the signal

travels from the satellite to the receiver, a certain number of cycles plus a fraction

of a cycle elapse. This fraction of a cycle converted to radians is the phase of the

received signal i.e. θ = 2πdλ

where d is the distance between the transmitter and the

receiver. In the absence of re�ected signals, the phase of the received signal depends

on the distance between the satellite and the receiver plus any associated errors in

estimating the pseudorange. Denote the phase of the received signals in this case

for E5, E5a and E5b as θ0 = 2πdλE5

, θ0a = 2πdλE5a

and θ0b = 2πdλE5b

respectively. In the

presence of multipath, there can be any number of re�ected signals. In this thesis

only the single re�ection case is considered. However, the method described in this

chapter applies equally well to the case where there are more than one re�ected

signals arriving at the receiver antenna.

The re�ected signal always arrives after the direct signal. This means that the

re�ected signal always travels a longer distance than the direct signal. If the distance

travelled by the re�ected signal is d1(> d) then the phases of the re�ected signal

will be θ1 = 2πd1λE5

, θ1a = 2πd1λE5a

and θ1b = 2πd1λE5b

. The multipath delay, which is nothing

6.3. SCPC METHOD AND AN ARCHITECTURE 139

50 100 150 200 250 300−0.02

0

0.02

E5a

Error in carrier phases (m), SMR = 6dB

0 50 100 150 200 250 300−0.02

0

0.02

E5b

0 50 100 150 200 250 300−0.02

0

0.02

E5


Figure 6.5. Carrier phase error for E5, E5a and E5b under multi-path conditions

but the di�erence in time between the re�ected signal and direct signal, is t1 = d1−dC

where C is the velocity of light.

In the case of E5a and E5b processing the carrier phase multipath error phase

is given by:

φca = arctan

( √α1a |RP (εa + τ1)| sin (φ1a)

|RP (εa)|+√α1a |RP (εa + τ1)| cos (φ1a)

)(6.13)

φcb = arctan

( √α1b |RP (εb + τ1)| sin (φ1b)

|RP (εb)|+√α1b |RP (εb + τ1)| cos (φ1b)

)(6.14)

where RP (·) is the autocorrelation function of the BPSK(10) (or PSK-R) baseband

signal and ε• is the corresponding code estimate error.

Suppose the time estimates for the E5a and E5b code lock loops are provided by

the E5 code tracking loop, then εa = εb = ε. In addition, the re�ector attenuation

can be considered to be frequency independent within the 50 MHz band around

1191.795 MHz (Seybold, 2005). Hence a1aa0a

= a1ba0b

= a1a0

= α1. Therefore the carrier

phase error for the E5a and E5b components of the signal can be written as:

φca = arctan

( √α1 |RP (ε+ τ1)| sin (φ1a)

|RP (ε)|+√α1 |RP (ε+ τ1)| cos (φ1a)

)(6.15)

φcb = arctan

( √α1 |RP (ε+ τ1)| sin (φ1b)

|RP (ε)|+√α1 |RP (ε+ τ1)| cos (φ1b)

)(6.16)

These carrier phase errors are plotted in Fig. 6.5 for di�erent multipath delays.

The E5 code tracking loop estimates the E5 code delay as τE5 = tcc + ε where

tcc is the composite code delay experienced by the E5 signal and is determined by


50 100 150 200 250 300−2

0

2

E5a

Error in pseudorange (m), SMR = 6dB

50 100 150 200 250 300−2

0

2

E5b

50 100 150 200 250 300−2

0

2

E5


Figure 6.6. Code pseudorange error for E5, E5a and E5b undermultipath conditions

(6.5). The error with respect to the E5 direct signal delay is τcc = tcc − t0. The

E5a code tracking loop estimates the E5a code delay as τE5a = tca + εa where tca is

the composite code delay experienced by the E5a signal and is determined by the

real component of (6.5). The error with respect to the E5a direct signal delay is

τca = tca − t0. Similarly, for the E5b signal, τE5b = tcb + εb and τcb = tcb − t0. With

the SCPC architecture, as mentioned earlier, ε = εa = εb. Hence τcc = τE5−(t0 +ε),

τca = τE5a − (t0 + ε) and τcb = τE5b − (t0 + ε). The code phase error thus calculated

due to multipath is plotted in Fig. 6.6 for E5, E5a and E5b for di�erent multipath

delays.

In these simulations, without loss of generality, it is assumed that the phase

change of the re�ected signal is only due to the path delay (usually, the phase is

inverted upon re�ection). In addition, the phase change is considered to be constant

over the entire E5 band. This is a reasonable assumption since the re�ectors are

negligibly dispersive within 50 MHz at around 1.2 GHz (Seybold, 2005). Other error

sources are discussed later in the chapter. If the carrier phase errors in (6.8), (6.15)

and (6.16) are subtracted from each other, then the results obtained are as shown

in Fig. 6.7.

The shape of the di�erence in carrier phase errors resembles the shape of the

code phase multipath error of the E5 signal. If one again subtracts the code phase

error from this di�erence in carrier phases, then the results shown in Fig. 6.8 are

obtained. The subtraction is formulated as follows:


0 50 100 150 200 250 300 350−0.5

0

0.5

E5−

E5a

Difference in carrier phase errors(m), SMR = 6dB

0 50 100 150 200 250 300 350−0.5

0

0.5

E5b

−E

5

0 50 100 150 200 250 300 350−1

0

1

E5b

−E

5a


Figure 6.7. Di�erence in carrier phase errors: E5-E5a, E5b-E5 andE5b-E5a

0 50 100 150 200 250 300 350−1

0

1

K2(E

5−E

5a)−

ε

Difference of carrier phase difference and pseudorange(m), SMR = 6dB

0 50 100 150 200 250 300 350−1

0

1

K2(E

5b−

E5)

−ε

0 50 100 150 200 250 300 350−1

0

1

K1(E

5b−

E5a

)−ε


Figure 6.8. Di�erence of carrier phase error and code pseudorangeerror under multipath conditions


0 50 100 150 200 250 300 3500.5

1

1.5

E5a

Composite Signal Strength (normalized ratio), SMR = 6dB

0 50 100 150 200 250 300 3500.5

1

1.5

E5b

0 50 100 150 200 250 300 3500

0.5

1

1.5

E5


Figure 6.9. Multipath a�ected signal strength for E5, E5a and E5bsignals

εab = τc −K1 (φcb − φca)εbc = τc −K2 (φbc − φc) (6.17)

εca = τc −K2 (φc − φca)

The resulting values εab, εbc and εca plotted in Fig. 6.8 show that the code phase

multipath errors are reduced to a signi�cant extent if these for the measurements

are used. The optimum value of the constants K1 and K2 depends on the receiver

bandwidth. In addition K2 = 2K1. Typically, the value of is found to be around 2.0

(experimentally determined).

6.3.2. E5a and E5b SNRs under multipath. Consider the received signal

consisting of a single re�ected signal as described in (6.9). The receiver processing

the signal a�ected by multipath estimates the SNR as the composite SNR instead

of the SNR of the direct signal. In the case of E5, E5a and E5b processing, this

SNR is given by:

bca = R2P (ε) + α1R

2P (ε+ τ1) + 2

√α1RP (ε)RP (ε+ τ1) cos(φ1a) (6.18)

bcb = R2P (ε) + α1R

2P (ε+ τ1) + 2

√α1RP (ε)RP (ε+ τ1) cos(φ1b) (6.19)

where bca is the SNR of the composite signal as seen at the E5a band and bcb is

the SNR of the composite signal as seen at the E5a band. The equations (6.11),

(6.18) and (6.19) show that the composite signal strengths di�er among the three

signals and the di�erence is only related to the phase of the multipath signal. These

composite SNRs are plotted in Fig. 6.9 for di�erent multipath delays.


0 50 100 150 200 250 300 350−1

−0.5

0

0.5

E5−

E5a

Difference of Composite Signal Strengths (normalized ratio), SMR = 6dB

0 50 100 150 200 250 300 350−0.5

0

0.5

1

E5b

−E

5

0 50 100 150 200 250 300 350−1

0

1

E5b

−E

5a


Figure 6.10. Di�erence in SNRs of the received signals, E5-E5a,E5b-E5 and E5b-E5a

If the composite SNRs are subtracted from each other, then the result shown in

Fig. 6.10 is obtained. The shapes of the di�erence in composite SNRs somewhat

resembles the shape of the code phase multipath error of the E5 signal. Observe

the bottom plot in Fig 6.10 - the lobes are symmetric along the x-axis unlike the

code phase multipath error. However, close to the edges of these lobes the di�erence

vanishes as in the case of the code phase error. Since the di�erence in SNRs indicates

the presence of multipath, but not the actual code phase multipath error, this SNR

method can be used as additional information to detect the multipath. Hence, this

method can be used in conjunction with the carrier phase di�erence method.

The di�erence equation is formulated as follows (C-language tertiary operators

used for simplicity):

mba = (bcb − bca) > η?1 : 0

mbc = (bcb − bc) > η?1 : 0 (6.20)

mca = (bc − bca) > η?1 : 0

where mxy are the multipath indication �ags and η is the threshold optimised to

trigger the di�erence. Depending on the SNR estimation method used and the

tracking loop parameters, η is appropriately chosen. Typical values range from 0.5-

1.0 (in dB). One can see from Fig.6.10 that the check will be more robust with the

di�erence between E5b and E5a (i.e. using mba).


AltBOC Ref

Signal Generator

Complex

Carrier

( )IFr t

Code

Loop

Filter

Code

NCO

1my

1( )y t

2 ( )y t 0

ˆˆ

( )j t

x t e

( )y t

1

1( 1)

nT

n T

dt

1T

1T

2my

*

5ˆ

E cs t T

*

5ˆ

E cs t T

*

5ˆ

Es t

Carrier

Loop

Filter

Carrier

discrimin

ator

Carrier

NCO

0 ( )y t

2

2( 1)

nT

n T

dt

0ly

2T

1

1( 1)

nT

n T

dt

Code

discrimin

ator

E5a/b

band

translator

E5a and E5b

Code Generator

*

5ˆ

E as t *

5ˆ

E bs t

Carrier

Loop Filter

(E5b)

Carrier

discrimin

ator(E5b)

E5b

Carrier

NCO

0 ( )by t

2

2( 1)

nT

n T

dt

0by

2T

2

2( 1)

nT

n T

dt

0ay

2T

Carrier

Loop Filter

(E5a)

Carrier

discrimin

ator(E5a)

E5a

Carrier

NCO

0 ( )ay t Combiner

c

ca

cb

Code phase

measurement

cb cab cbb

Figure 6.11. Architecture for the SCPC method

6.3.3. Architecture for the SCPC method. The required architecture for

the SCPC method is illustrated in Fig. 6.11. Note that this is a special case of the

generalised architecture presented in chapter (5). Important points to observe are

that the code delay estimate from the output of the code NCO used for E5 tracking

is input to the E5a and E5b code generators (indicated by a bold blue line). No

additional code NCOs are required for the E5a and E5b signals because the timing

for both the 8-PSK AltBOC reference signal and the E5a/b code generator are the

same. In fact, the outputs of the E5a/b code generator shown explicitly in this

diagram are already available in the AltBOC reference signal generator and hence

additional hardware is not required except for the code mixers and loop modules.

The combiner block performs the scaled di�erence of the carrier phases according

to (6.17). In (6.17) it can be seen that this scaled di�erence of carrier phases

represents the code multipath delay error. Therefore when the scaled di�erence is

subtracted from the code delay estimate τ , one obtains the multipath mitigated

code delay measurement. The carrier phase errors are obtained via the loop �lters.

In addition, the measured SNRs at the output of the E5, E5a and E5b loops are

provided as input to the combiner block. This is used to gate the subtraction of

the scaled di�erence from the code delay estimate. A method to estimate the signal

strength parameter is to average the squared magnitude of the prompt correlator

output y0 of the corresponding signal component (not shown here).

6.3.4. Explaining the SCPC method. The �ltered correlation function of

the AltBOC(15,10) can be approximated as the product of a symmetrical triangle


−3 −2 −1 0 1 2 3−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time delay (primary code chips)

Pro

duct

of t

he tr

iang

ular

and

sin

e w

avef

orm

s

cos(2π fsc

t)

ACF of E5aACF of E5

Figure 6.12. Illustrating the formation of the 8PSK / AltBOC-likecorrelation function

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time delay (chips)Pro

duct

of t

he tr

iang

ular

and

sin

e w

avef

orm

s

cos(2π fsc

t)

cos(2π fsc

t−π/2)

cos(2π fsc

t+π/2)

Figure 6.13. Illustrating the e�ect of phase shift while multiplyinga sine wave and a triangle

of width two chips and a cosine wave at a frequency fsc Hz as shown in Fig. 6.12.

Of course the shape of the correlation function depends on the RF front-end �lter

parameters, nevertheless this is a good approximation.

Note that the sine wave is such that its zero phase corresponds to the zero delay

error. If the phase of the sine wave is advanced or retarded by π2then interestingly

the resulting waveform resembles the coherent early-late and late-early discriminator

S-curves with δ=0.167. This is shown in Fig. 6.13. The zero crossings are at zero

delay error and can be thought of as the case without multipath.

If the phase of the cosine wave is not zero, then the shape of the resultant

waveform is changed. This is shown in Fig. 6.14 for di�erent phase shifts from −π2


−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time delay (chips)

Pro

duct

of t

he tr

iang

ular

and

sin

e w

avef

orm

s

−π/2−3π/8−π/4−π/80+π/8+π/4+3π/2+π/2

Figure 6.14. Illustration of the e�ect of di�erent phase shifts of thesine wave due to the multiplication

to +π2. Observe that for phase shifts other than ±π

2, the zero crossing does not align

with the zero delay error. These zero crossings can be thought of as representing

the discriminator S-curve with multipath.

Now consider the equation for the carrier phase multipath error for E5a and

E5b given in 6.15, 6.16. The numerator contains the expression of a PSK-R(10)

correlation function multiplied by a sine function. The multipath phase relations

can be written as:

φ1a = φ1 − φscφ1b = φ1 + φsc

where φsc = 2πfscτ1 is the phase shift induced by the sub-carrier.

Note that the phase di�erences are less a�ected by a small change in the multi-

path delay because the beat frequencies have a wavelength of ~20 metres (E5-E5a,

E5b-E5) and ~10 metres (E5b-E5a). For E5a:

RP (ε+ τ1) sin (φ1a) =√α1RP (ε+ τ1) sin (φ1 − φsc)

=√α1RP (ε+ τ1) (sin (φ1) cos (φsc)− cos (φ1) sin (φsc)) (6.21)

For E5b:

RP (ε+ τ1) sin (φ1b) =√α1RP (ε+ τ1) sin (φ1 + φsc)

=√α1RP (ε+ τ1) (sin (φ1) cos (φsc) + cos (φ1) sin (φsc)) (6.22)

The denominators of (6.15) and (6.16) act as scaling factors for the discriminator

functions since they contain a dominant constant term RP (ε) which will be close to


the peak assuming ε to be small.

Now, consider the small argument approximation tan(x) ≈ x. Even though this

approximation can not be used for large arguments, it can be used here to analyse

the behaviour of the di�erence of the phases. Hence from (6.15),(6.16),(6.21) and

(6.22):

φcb − φca ≈√α1RP (ε+ τ1) (sin (φ1) cos (φsc) + cos (φ1) sin (φsc))

Cb

−√α1RP (ε+ τ1) (sin (φ1) cos (φsc)− cos (φ1) sin (φsc))

Ca(6.23)

Since the constants are dominated by the multipath independent factor, one can

assume that Cb ≈ Ca ≈ C1. Hence:

φcb − φca ≈2√α1 cos (φ1) (RP (ε+ δTc + τ1)−RP (ε− δTc + τ1))

C1

(6.24)

The term RP (ε+ τ1) · sin (φsc) has been replaced by the early�late term. The right

hand side of the equation resembles the term due to the re�ected signal component

in the discriminator equation (6.5) and hence that of the code multipath error. The

constant C1 is a product of four terms that are obtained while subtracting (6.15)

and (6.16). This value is close to 4 (found with the help of Monte Carlo simulations)

and hence K1 in (6.17) is ≈2.A similar approach can be followed for the di�erence between E5 and E5a, E5

and E5b. Even the E5 correlation function multiplied by a cosine wave at phase

shifts approximates the discriminator curve for E5, but this approximation is not as

good as the BPSK(10) case. Because of this the �nal result is also a less accurate

approximation to the true error.

6.3.5. Other considerations in the SCPC method. The multipath phase

di�erence formulation described above does not consider other sources of errors.

However, in practice, there would be other sources of errors due to satellite orbit,

satellite clock, ionosphere and troposphere. Of these errors only the ionospheric and

relative Doppler e�ects need be considered, as they are frequency dependent. The

other sources of errors can be assumed to be the same for the E5, E5a and E5b

components.

First consider the code delay due to the ionosphere. The process of providing

the code delay estimate of E5 to E5a and E5b reference signal generators (or the

assumption that εa ≈ εb ≈ ε) remains valid even in the presence of ionospheric delay.

This is because with respect to E5, the di�erence in ionospheric delay is given by:

∆IE5−E5a/b = τiono(E5)− τiono(E5a/b) = I0

(f 2E5a/b − f 2

E5

f 2E5a/b

)(6.25)


Even at extremely high ionosphere delays such as 100 metres (Sleewaegen et al.,

2004) where I0= 333.56 ns, the delay di�erences E5-E5a and E5-E5b are -8.42 ns

and 8.75 ns respectively. Note that it will not be on the peak of the E5a or E5b

triangle, but displaced by this amount. By tracking at this o�set, one removes the

ionosphere dispersion e�ect on E5a and E5b. The delay itself is small which in

this worst case scenario causes 0.8 dB loss in correlation value with respect to the

peak. The absolute delay di�erence is negligible and hence it will cause negligible

di�erence in the BPSK(10) correlation value between E5a and E5b.

It is shown in Sleewaegen et al. (2004) that the carrier phase advance due to the

ionosphere dispersion in the E5 band is negligible for most practical situations. In

addition, the phase of E5a retards by nearly the same amount as that of advance in

the phase of E5b:

∆φE5−E5a/b = φiono(E5)− φiono(E5a/b) = I0fE5

(1− fE5

fE5a/b

)(6.26)

(28) Considering similar extreme conditions of 100m ionospheric delay, the phase

di�erences E5-E5a and E5-E5b yield about 0.13 cycles di�erence between E5-E5a

and E5-E5b. The integer ambiguity which results as the wide-lane combination

of these phases can be resolved in a single epoch (e.g. Feng and Rizos, 2005).

There exist some methods to estimate accurately the ionospheric corrections for

the carrier phases (Watson, 2008). Alternatively, when the receiver is in navigation

mode, continuously providing the solution, the ionospheric e�ect can be modeled

and removed to some extent.

The problem of relative Doppler o�set between the direct signal and the re�ected

signal is discussed in Kelly et al. (2003) and can not be neglected for the carrier

phase. For multipath delays between 2-8 metres, which have the potential of errors

(see Fig. 6.2) higher than that of the subsequent lobes of the envelope, the Doppler

di�erence among the E5, E5a and E5b components can be considered as small.

Hence the SCPC method will have slightly degraded performance in high dynamics

scenarios. In addition, signal dynamics can be used to obtain more information on

the multipath (Kelly and Braasch, 1999). Nevertheless it still o�ers considerable

improvements in most practical situations.

The assumptions made during the derivation of the SCPC method derivation will

still be valid with the ionosphere and Doppler considered (although now they are

less accurate assumptions). The other sources of error are slowly varying and tend

to cancel each other out when one subtracts the E5a and E5b measurements. In

summary, the e�ect of other sources of errors will negligibly degrade the performance

of the SCPC method.

6.4. SIMULATION AND TEST RESULTS 149

SeptentrioGeNeRx1

Leica AR25

IF Samples Receiver

(Matlab)

Matlab based Signal

Simulator

IF Samples

Multipath Channel Model

(Matlab)

Receiver(Matlab)

Multipath Channel Model

(Matlab)

Measurement Output

(a)

(b)

Figure 6.15. Test setup (a) simulation (b) with real satellite signal

6.4. Simulation and Test Results

For both the simulation and real signal test purposes, the GIOVE-A (PRN 51) E5

AltBOC signal is considered. Fig. 6.15 shows the test setup. A multi-constellation

choke ring antenna AR25 from Leica was used with the Septentrio GeNeRx1 receiver

to collect the IF (Intermediate Frequency) signal samples. Multipath was simulated

by superimposing the delayed and phase shifted version of the direct signal onto the

direct signal. The composite signal was then acquired and tracked with the help

of a Matlab-based E5 receiver (developed by the author). The GeNeRx1's receiver

parameters were considered as the baseline: a front-end bandwidth of 55 MHz and

sampling frequency of 112 MHz with 8-bit sampling.

Multipath parameters were: SMR = 6 dB, multipath delay varied between 1 to

3 samples (2.68 m, 5.36 m and 8.02 m which lie within the maximum code error

envelope region) and multipath phase such that the code error is maximum. (For

the real signal the phase at the point of interest was estimated prior to adding the

delayed signal.) The re�ected signal was superimposed on the direct signal at the

60 ms point in the plots.

Fig. 6.16 shows the code phase error with and without multipath. When there

is multipath, the code phase delay estimate deviates from the true value. Observe

that the error is less for the 5.36 m delay than for the other two, thus verifying

the multipath error envelope. Also shown is the di�erence in carrier phase errors

between E5a and E5b. It can be seen that the di�erence in carrier phase error

follows that of the code phase error. Fig. 6.17 is the real signal case equivalent

of Fig. 6.16. Observe that the error curves are very noisy, but the deviation is

evident from the graph. In the simulated signal, the signal strength was very good

and there were no other errors (such as ephemeris, atmosphere-related or antenna

/ radio frequency front-end hardware related) except thermal noise. In case of the

real signal multipath is just one of the error sources and hence the error curves are

not smooth.


0 50 100 150−0.2

−0.1

0

0.1

0.2

Cod

e ph

ase

erro

r (c

hips

)

0 50 100 150−0.1

−0.05

0

0.05

0.1

Time (ms)Diff

eren

ce in

E5a

and

E5b

ca

rrie

r ph

ase

erro

rs (

rad)

No Multipathτ1 = 2.68m

τ1 = 5.36m

τ1 = 8.02m

No Multipathτ1 = 2.68m

τ1 = 5.36m

τ1 = 8.02m

Figure 6.16. Code phase error (top) and di�erence in carrier phaseerrors of E5a and E5b (bottom), for di�erent multipath delays; mul-tipath introduced from 60th ms; from simulation

0 50 100 150−1

−0.5

0

0.5

1

Cod

e ph

ase

erro

r (c

hips

)

0 50 100 150−0.3

−0.2

−0.1

0

0.1

Time (ms)

Diff

eren

ce o

f ca

rrie

r ph

ase

erro

rs (

rad)

No multipathτ

1=2.68 m

τ1=5.36 m

τ1=8.02 m

No multipathτ

1=2.68 m

τ1=5.36 m

τ1=8.02 m

Figure 6.17. Code phase error (top) and di�erence in carrier phaseerrors of E5a and E5b (bottom), for di�erent multipath delays; mul-tipath introduced from 60th ms; with real signal

6.4. SIMULATION AND TEST RESULTS 151

0 50 100 150−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time (ms)

Err

or in

cor

rect

ed c

ode

phas

e (c

hips

)

No Multipathτ

1=2.68 m

τ1=5.36 m

τ1=8.02 m

Figure 6.18. Error in corrected code phase estimate for di�erentmultipath delays; multipath introduced from 60th ms; from simulation

0 50 100 150−1

−0.5

0

0.5

1

Time (ms)

Err

or in

cor

rect

ed c

ode

phas

e (c

hips

)

No Multipathτ

1=2.68 m

τ1=5.36 m

τ1=8.02 m

Figure 6.19. Error in corrected code phase estimate for di�erentmultipath delays; multipath introduced from 60th ms; with real signal

Figs. 6.18 and 6.19 show the error in code phase estimate when the correction

for the multipath using the SCPC method is applied. Observe that the deviations

which appeared in the top portion of Fig. 6.16 and 6.17 are now greatly reduced

and multipath mitigated code phase estimates are obtained from the receiver.

Comparing SCPC with the Standard and Narrow Correlator for Galileo

E5 AltBOC(15,10). In a strict sense, the SCPC method of mitigating the code


0 10 20 30 40 50 60−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Multipath delay (m)

Cod

e m

ultip

ath

erro

r (m

)


E5−50MHz; Standard Correlator δ=0.3E5−50MHz; Narrow Correlator δ=0.05E5−50MHz; SCPC

Figure 6.20. A code multipath error envelope comparison of thestandard correlator, the narrow correlator and the SCPC method

phase multipath is di�erent to other receiver signal processing techniques available

for the code phase multipath mitigation such as the {Narrow, High Resolution,

Strobe, Gated, Shaping} correlators. For example, in the narrow correlator, the

magnitude of the multipath error is lessened by sampling the correlation function at

a higher rate than that of the standard correlator. In other words, in these �other

receiver signal processing� techniques, the correlation function which is input to the

code tracking loop is �altered� to �lessen� the e�ect of the disturbance caused by the

re�ected signal.

In the SCPC method, the input to the code tracking loop is still a distorted

correlation function as in the case of a standard (conventional wide chip-spacing)

correlator, but the erroneous code phase output is �corrected for� the contribution

from the re�ected signal using additional information from the sideband tracking.

The correctness of the correction depends on other factors as discussed in sec. 6.3.5.

For this reason, a direct comparison of the multipath error envelope between SCPC

and other methods is not recommended. However, one such comparison is shown in

Fig. 6.20. Note that the intermediate nulls (shown by the arrow marks) are due to

the absence of other errors mentioned in sec. 6.3.5. The error at these points (shown

by the arrow mark) starts to grow as the contribution from other error sources grows,

eventually inverting the �v� shaped valleys.

6.5. A Group Delay Compensation Viewpoint for the SCPC Method

It was discussed earlier in sec. 2.9 that E5 AltBOC modulation can be thought

of as a frequency-diverse transmission system. This section provides a di�erent

6.5. A GROUP DELAY COMPENSATION VIEWPOINT FOR THE SCPC METHOD 153

viewpoint for the SCPC method proposed in sec. 6.3 to mitigate the multipath e�ect.

Multipath is one of the two major sources of frequency selective radio frequency (RF)

propagation delay distortion in receivers demodulating an AltBOC signal the other

being the ionosphere. The viewpoint here is to observe the error from these two

sources as a combined e�ect on the group delay measured by the AltBOC tracking

loop and then compensate for this error with the aid of the phase delays at E5a and

E5b bands.

With this new viewpoint the underlying idea of the proposed SCPC method

turns out to be that the di�erence in the phase delays of the E5a and E5b signal

tracking is simply the slope of the phase response over the entire band and hence

represents the group delay at the E5 centre frequency of 1191.795 MHz. Since the

carrier phase measurements are representatives of the phase delays and the code

phase errors are representatives of the group delay, the group delay which appears

as the code phase error for the E5 signal tracking, can be compensated for except

for the occurrence of higher order errors.

This section makes use of the background information provided in Appendix D

to discuss the e�ect of phase and group delay for the Galileo E5 AltBOC(15,10)

signal. Appendix D reviews the well known frequency selective nature of the phase

delay and the group delay with respect to ionospheric errors.

Phase delay, group delay and multipath errors. The phase of the re�ected

signal that gets superimposed onto the direct signal at the receiver antenna depends

on the additional path length traversed by the re�ected signal. Equations for phase

and group delay errors in a multipath scenario for the single re�ected signal case are

derived in Otoshi (1993b). The combined phase delay is given by

tp = tp1 + tpmulti (6.27)

where tp1 is the total phase delay of the direct or the �rst signal (including the

ionosphere delay) and tpmulti is the error in phase delay due to the re�ected signals.

The phase delay multipath error is a function of the phase delays of the re�ected

signals tpmulti = F(tp2 , tp3 , . . .) and in the single re�ected signal case, it is given by

tpmulti = − 1

ωarctan

[A sin θ

1 + A cos θ

](6.28)

where θ is the composite phase given by θ = −(φ2 − φ1) , φ1 and φ2 being the

phases of the direct signal and the re�ected signal respectively, A is the ratio of the

amplitude of the re�ected signal to the direct signal.

The combined group delay of the signal is given by

tg = tg1 + tgmulti (6.29)


1.16 1.17 1.18 1.19 1.2 1.21 1.22 1.23

x 109

77.4603

77.4603

77.4603

77.4603

77.4603

Pha

se d

elay

(m

s)

tp & t

g vs f for different multipath delays, SMR = 6dB

1.16 1.17 1.18 1.19 1.2 1.21 1.22 1.23

x 109

−77.4603

−77.4602

−77.4602

−77.4601

−77.4601

Gro

up d

elay

(m

s)

Frequency (Hz)

Figure 6.21. Phase delay and group delay vs frequency around theE5 band for di�erent multipath delays; no ionospheric errors; arbitrarysatellite distance of 23222 km; single re�ected signal with SMR = 6dB;

where tg1 is the total group delay of the direct or the �rst signal (including the

ionosphere delay) and tgmulti is the error in group delay due to the re�ected signals.

The group delay multipath error is a function of the group delays of the re�ected

signals tgmulti = H(tg2 , tg3 , . . .) and in the single re�ected signal case, it is given by

Otoshi (1993b)

tgmulti = −A(tg2 − tg1)[

A+ cos θ

1 + 2A cos θ + A2

](6.30)

where tg1 is the total group delay of the direct, or �rst, signal and tg2 is the total

group delay of the second, or re�ected signal.

The phase of the re�ected signal at the receiving antenna depends on the type of

the re�ector, the frequency of the signal and the path length. Therefore both tpmultiand tgmulti are functions of ω. For a given type of re�ector, the frequency dependency

of the complex re�ection coe�cient within the Galileo E5 AltBOC(15,10) is negligi-

ble (Seybold, 2005). Hence the frequency dependency to the re�ector contribution

in the amplitude A and the phase θ can be neglected for all practical purposes. Un-

like the ionospheric errors tpiono in (D.4) and tgiono in (D.5), the relationship between

the phase delay and the group delay errors due to multipath at any two frequency

components is quite complicated to visualise, speci�cally tgmulti in (6.30). θ(ω) will

have a (modulo 2π) linear relationship with frequency, but tpmulti and tgmultiwill ex-

perience non-linearly damped oscillations according to (6.30) and (6.28). Fig. 6.21

shows the variations of the phase delay and the group delay versus frequency for

di�erent multipath delays. Observe that the variations follow a pattern which will


be further explored later in this chapter.

Relation between the phase/group delay and the phase measurement

errors in GNSS. From (D.4) and (6.28) it can be inferred that the phase delay

results in carrier phase measurement error. The relation is straightforward; if the

phase delay is tp, then the carrier phase measurement will incur a phase error of

the same amount. On the other hand, the relationship between the group delay in

(D.5) and (6.30), and the code phase measurement error is not easy to visualise.

The optimal tracking circuitry for synchronising the spreading code involves a delay

locked loop Parkinson and Spilker (1995). In an attempt to measure the group

delay in spread spectrum systems, Otoshi (1993a) shows that the timing error of an

early/late code correlator indicates the group delay experienced by the signal and

is explored further in the following sub sections for the AltBOC(15,10) signal.

6.5.1. The received signal and the tracking architecture. The received

signal along with N − 1 re�ected signals can be expressed as

r(t) = <{P0 · s(t− tgiono − tgoth) · exp (jωeff (t+ tpiono + tpoth)) +

N−1∑i=1

Pi · s(t− tgiono − tgimulti − tgoth) · exp (jωeff (t+ tpiono + tpimulti + tpoth))} (6.31)

where tpoth and tgoth represent the phase delay and group delay respectively, due

to all other sources than the ionosphere and the multipath, P denotes the received

signal power, ωeff denotes the e�ective frequency (including Doppler). These sources

mainly include the signal transit time, the troposphere error, antenna-induced errors,

and the errors in the receiver due to the RF down converter and �lter(s). Depending

on the method of down conversion, r(t) could be either complex or real. However,

bandpass sampling to a moderate IF is assumed in this work and as a result r(t) is

real.

The tracking architecture used in the following sections is the architecture used

for the SCPC method as shown in Fig. 6.11. Note that the code phase estimates

of the wideband E5 AltBOC(15,10) tracking are provided to the code generation

modules of the sideband tracking.

6.5.2. Mitigating the ionospheric and multipath errors.

6.5.2.1. Mitigating the e�ects of ionosphere (in the absence of multipath errors).

One of the main features of Galileo E5 AltBOC modulation is that the E5a and

E5b sidebands can be processed independently of each other. Hence it is possible to

obtain three code and carrier phase measurements from E5a , E5b and the wideband

E5 that can be used to obtain estimates of ionosphere-induced errors.


0 10 20 30 40 50 60 70 80 90 100−1

−0.5

0

0.5

1x 10

−8

∆ t p(s)

0 10 20 30 40 50 60 70 80 90 100−4

−3

−2

−1

0x 10

−10

∇∆t p(s

)

Ionospheric error at E5 (m)

E5−E5aE5−E5b

∆tpE5−E5a

− ∆tpE5−E5b

Figure 6.22. Di�erence in the phase delays in E5a and E5b withrespect to E5 (top); di�erence of the two curves in the top �gure(bottom)

Figures 6.22 and 6.23 show the di�erence in the phase delay and the correlation

values (note the di�erence in scales). The corresponding parameters at the centre

frequency are used as references to obtain the di�erences. An interesting point to

note is that within the E5 band, due to the near-symmetric nature of the ionospheric

group delay dispersion, the code phase estimates from the E5a and E5b tracking

loops are a�ected in an opposite manner when referred to the code phase estimates

of the wideband E5 signal tracking loop. Observe that the di�erence between E5-E5a

and E5-E5b is only up to 0.33 ns (0.33 ns is for a very high ionospheric delay of 100

m) which shows that one can use the combination of E5a and E5b measurements to

obtain a good estimate of the ionospheric delay at the centre frequency. Nevertheless,

the quality of such estimate depends on the contribution of the receiver noise.

6.5.2.2. Mitigating the e�ects of multipath (in the absence of ionospheric errors).

The e�ect of the phase delay and the group delay for di�erent multipath delays and

di�erent frequency components around the E5 band was shown in the previous

section. It is important to observe the behaviour of the phase and group delays at

the E5a, E5b and E5 centre frequencies. Fig. 6.24 shows the di�erence of phase

delay and group delay between E5a and E5b frequencies. This plot is generated

using (6.28) and (6.30). It should be noted that the equation only involves the

carrier frequencies and does not include the e�ects of the spreading code.

With the spreading code in place, the errors at larger multipath delays are at-

tenuated, following the shape of the correlation function. The composite phase

delay and the composite group delay for a single re�ection case at the output of the


0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

Cor

rela

tion

loss

(dB

)

0 10 20 30 40 50 60 70 80 90 1000

0.01

0.02

0.03

0.04∆

corr

elat

ion

loss

es (

dB)

Ionospheric error at E5 (m)

E5aE5b

E5a − E5b

Figure 6.23. Di�erence in the correlation values in E5a and E5bsignal components with respect to Ionosphere free situation (top);di�erence of the two curves in the top �gure (bottom)

0 5 10 15 20 25 30−2

−1

0

1

2x 10

−10

t p E5b

−t p E

5a

(s)

0 5 10 15 20 25 30−2

−1

0

1

2x 10

−7

t g E5b

−t g E

5a

(s)

Multipath delay (m)

Figure 6.24. Di�erence of E5a and E5b phase and group delaysfor di�erent multipath delays (analytical); single re�ected signal case;A=0.5;


0 5 10 15 20 25 30−6

−4

−2

0

2

4

6

Multipath Delay (m)

Gro

up D

elay

(m

)

E5 LBE5 UBE5a UBE5a LB

Figure 6.25. Envelope of the group delay error due to multipath

correlator is (see Appendix (I) for the derivation):

tpcomposite = tp1 −1

ωarctan

[AR(ε+ δ) sin θ

R(ε) + AR(ε+ δ) cos θ

](6.32)

tgcomposite = tg1 + AR(ε+ δ)(tg2 − tg1)[AR(ε+ δ) +R(ε) cos θ

R2(ε) + 2AR(ε)R(ε+ δ) cos θ + A2R2(ε+ δ)

](6.33)

where R(.) is the autocorrelation function of the underlying spreading code, ε is the

code phase error (in chips), δ is the path delay di�erence between the re�ected signal

and the direct signal (in chips). The envelope of the group delay error is shown in

Fig. 6.25. The e�ect of spreading code, via (I.1) and (6.33), is plotted in Fig. 6.26.

The e�ect of the correlation shape can be observed in both the phase delay

di�erence and the group delay di�erence responses. It was shown in sec. (6.3) that

a combination of phase delays in E5a and E5b can be used to mitigate the e�ect of

code phase multipath in E5 wideband tracking.

With the proposed architecture, the local baseband reference signals in the cases

of E5a and E5b components are generated not at the peak of the corresponding

correlation triangle, but at an o�set equal to the di�erence in group delay with

respect to the E5 (centre frequency) wideband tracking.

6.5.2.3. Group delay compensation when both ionospheric and multipath errors

are present. Previous sections discussed the ionosphere and multipath errors with-

out considering any dependency on each other. The individual analysis provides a

good insight into the errors. Situations with only ionosphere errors or only mul-

tipath error may occur in some applications. However, a more practical situation

is the case when both ionosphere and multipath errors co-exist. In addition, the


0 5 10 15 20 25 30−2

−1

0

1

2x 10

−10

t p E5b

−t p E

5a

(s)

0 5 10 15 20 25 30−2

−1

0

1

2x 10

−7t g E

5b

−t g E

5a

(s)

Multipath delay (m)

Without the spreading codeWith the spreading code

Without the spreading codeWith the spreading code

Figure 6.26. Di�erence of E5a and E5b phase and group delays fordi�erent multipath delays (analytical); single re�ected signal; A=0.5

receiver may experience range estimation errors from sources such as troposphere

errors, estimation of the clock error (both satellite and the receiver), satellite posi-

tion and velocity estimation (due to orbit parameters / ephemeris), previous epoch

pseudorange estimation error and other secondary e�ects. It is assumed in this work

that all errors from the other sources are frequency independent.

Fig. 6.27 shows the phase delay and group delay for E5a, E5b and E5 frequencies

vs. multipath delay for three ionospheric delays, viz. 0m, 50m and 100m, experi-

enced at the centre frequency. The error shape due to the multipath is not visible

because the magnitude of the multipath error is small compared to the scale of the

plots.

Fig. 6.28 shows the phase delay and group delay di�erences at di�erent iono-

spheric delays and multipath delays. Observe that the di�erence in phase delay

shows an o�set depending on the ionospheric delay. The error due to multipath

is around this o�set. The group delay di�erence also shows a similar behaviour.

However, the error due to multipath is of an order comparable to the ionosphere

errors for the delays shown in Fig. 6.28. The phase delay di�erences

tpbc = tpE5b− tpE5

tpca = tpE5− tpE5a

tpba = tpE5b− tpE5a

(6.34)

provide an easy computation of the ionosphere-a�ected phase at the E5 centre fre-

quency due to the similar di�erences with respect to both the side bands. This

is expected as the ionospheric delay is almost symmetrical around the E5 centre


0 5 10 15 20 25 30−4

−2

0

2

4

t p (ra

d)

0 5 10 15 20 25 304708

4709

4710

4711

4712

4713

t g (ch

ips)

Multipath delay (m)

E5a, 0mE5a, 50mE5a, 100mE5b, 0mE5b, 50mE5b, 100mE5, 0mE5, 50mE5, 100m

Figure 6.27. Phase delay and group delay for E5a, E5b and E5frequencies under multipath condition for di�erent ionospheric delay;nominal satellite distance of 23222km; single re�ected signal; A=0.5

frequency. It should be noted that the knowledge of integer number of cycles is not

required for two reasons:

(1) Because the phase di�erence method experiences the oscillations at the

frequency di�erence (30.690 MHz for E5b-E5a and 15.345 MHz for E5b-

E5c or E5b-E5) there can be a maximum of 3 cycles di�erence between E5a

and E5b for ionospheric delays of up to 100 m (1.5 cycles in the other two

cases).

(2) The availability of three phase measurements helps isolate the ionosphere

error at the centre frequency.

To mitigate the e�ects of multipath the, SCPC method discussed in sec. (6.3) is

used here. Referring back to Fig. 6.28, a two-step approach is followed here, �rst to

resolve the ionospheric error and second to apply the SCPC algorithm to mitigate

the multipath error. The di�erence in the phase delays of the E5a and E5b signal

component tracking is nothing but the slope of the phase response over the entire

band and hence represents the group delay at the centre frequency E5=1191.795

MHz.

E�ect of previous epoch pseudorange errors on the ionosphere and

multipath mitigation process. As mentioned earlier, apart from the two major

errors under consideration, the receiver may experience other frequency-independent

errors. As an example assume that the pseudorange is in error. The �rst consequence

of this erroneous pseudorange is that the phase and the chip shift (fractional) of the

incoming signal di�er from the actual values. However, this error will be common


0 5 10 15 20 25 30−3

−2

−1

0

1

2

t p diff

(ra

d)

0 5 10 15 20 25 30−0.4

−0.2

0

0.2

0.4

t g diff

(ch

ips)

Multipath delay (m)

E5b−E5a, 0mE5b−E5a, 50mE5b−E5a, 100mE5b−E5, 0mE5b−E5, 50mE5b−E5, 100mE5−E5a, 0mE5−E5a, 50mE5−E5a, 100m

Figure 6.28. Phase delay and group delay di�erences at di�erentionospheric delays and multipath delays

to all three components E5, E5a and E5b of the signal and the method of obtaining

the di�erence in phase delay and the di�erence in group delays nulli�es this common

error. The second consequence is the e�ect of this pseudorange error on the code

multipath error. In (I.1) and (6.33), the ε parameter which indicates the error in the

pseudorange alters the multipath error characteristics. Thanks to the SCPC method

the code delay estimates from the E5 AltBOC(15,10) tracking loop are provided to

the code delay estimates of the two sidebands. With this sort of aiding, all the three

components of the signal calculate the pseudorange estimates from a single source

(of the previous instant) and keep the noise characteristics undisturbed. Hence the

e�ect of pseudorange error on the multipath error is removed by the tracking loop

architecture.

E�ect of Doppler frequency on the ionosphere and multipath miti-

gation process. In moderate to high dynamics applications, each frequency com-

ponent of the wideband signal experiences di�erent Doppler shifts. In the case of

Galileo E5 AltBOC(15,10), the Doppler frequencies observed on the E5a and E5b

components di�er from those of the wideband AltBOC tracking that experiences a

Doppler corresponding to the centre frequency E5. However, aiding the E5a and

E5b carrier tracking loops with the frequency estimate of the E5 AltBOC tracking

loop eliminates the e�ect of any di�erence in Doppler frequency estimation in the

sideband tracking loops. Second, the e�ect of the rate of change of the user dynam-

ics on the multipath is discussed in Nedic (2009). In the case of E5a and E5b, these

errors will be opposing each other when referenced to the E5 centre frequency. Hence

the di�erence of E5a and E5b carrier phase measurements is devoid of range-rate

e�ects.


E�ect of antenna induced errors on the ionosphere and multipath miti-

gation process. Due to practical limitations in the antenna design, some properties

of the antenna depend on the frequency (Orban and Moernaut, 2009). Of interest

to this chapter are the phase centre and the axial ratio. The phase centre of the

antenna varies with frequency and boresight angle. The axial ratio of the antenna

which is an indicator of the amount of the rejection of a LHCP (re�ected) signal

varies with the frequency and the incidence angle (Yun et al., 2008; Zhuang and

Tranquilla, 1995). Knowing the frequency dependent variation will help in calibrat-

ing for that e�ect. However, the variation caused by the incidence angle cannot be

calibrated a priori because the angle of incidence of the re�ected signal will not

be known. A possible solution is to employ multiple closely spaced antennas (Ray,

2000). A detailed analysis of the antenna-induced errors is beyond the scope of this

thesis.

6.5.3. Simulation results and discussion. To test the multipath mitigation

technique in the presence of ionosphere, the GIOVE-A E5 AltBOC(15,10) signal

structure is used as a reference. Two types of veri�cation of the technique mentioned

in the previous section are performed. In the �rst case, the signal with the multipath

and the ionosphere e�ects is generated in a Matlab environment. In the second, an IF

signal is collected from the GIOVE-A satellite, ensuring that there are no re�ectors

within a distance of 30 metres that can cause multipath errors. Then, ionosphere

and multipath errors are added using Matlab (at the IF stage) to this signal. This

signal is termed a �pseudo-real� signal. The IF samples of the GIOVE-A satellite

signal were collected using the Septentrio GeNeRx1 receiver which has the capability

to output 150 ms of IF signal sampled at 112 MHz. The bandwidth of the receiver

is 55 MHz around the E5 centre frequency. The simulated delay distortions are

added to the signal from the 60th ms. Figs. 6.29 and 6.30 show the results of the

SCPC method with simulated ionosphere errors.Fig. 6.29 shows the performance

of the group delay compensation technique at a particular multipath delay for two

ionospheric delay errors of 50 m and 100 m. Again, the reduction in the code phase

error is clear from the simulated signal. With the real signal the error itself is small,

as shown in the top left portion of Fig. 6.30. However, the group delay compensation

brings down a signi�cant part of this error.

6.6. Summary

In this chapter, a code phase multipath mitigation method called SCPC (Side-

band Carrier Phase Combination) was presented. The SCPC method makes use of

the carrier phases of E5a and E5b sidebands to estimate the code phase multipath

at the centre of the E5 band that is experienced by a Direct AltBOC tracking. An

6.6. SUMMARY 163

0 50 100−0.04

−0.03

−0.02

−0.01

0

0.01

0.02Code phase

0 50 100−10

−8

−6

−4

−2

0

2

4x 10

−3 Difference in cp

0 20 40 60 80 100 120 140−0.01

−0.005

0

0.005

0.01

0.015Code minus scaled cp diff

Time (ms)

50m iono100m iono

Figure 6.29. Multipath mitigation with the simulated signal withmultipath at 5.4m and ionospheric delays of 50m and 100m at E5

0 50 100 150−0.2

0

0.2

0.4

0.6Code phase (chips)

0 50 100 150−0.3

−0.2

−0.1

0

0.1

0.2Code minus scaled cp diff

Time (ms)

0 50 100 150−0.15

−0.1

−0.05

0

0.05Difference in carrier phases

50m100m

Figure 6.30. Multipath mitigation with the pseudo-real signal withmultipath at 5.4m and ionospheric delays of 50m and 100m at E5


architecture to realise the SCPC method was described. The applicability of the

SCPC method was discussed and the results obtained with real satellite signal tests

were presented. It is shown that the method can reduce the code multipath error

to less than 0.5 metres for the E5 AltBOC.

The proposed SCPC technique compensates for the propagation delay distortion

without the need of estimating the absolute value of either the ionospheric or multi-

path delay. In summary, the proposed group delay compensation technique exploits

the multi-frequency, wideband feature of the AltBOC modulation to generate an

accurate range measurement.

CHAPTER 7

Galileo E5 Baseband Hardware

7.1. Introduction

This chapter mainly deals with two aspects of the Galileo E5 baseband hardware.

The �rst part addresses the problem of e�ciently computing the FFTs required

to acquire signals of varying code lengths. The requirements of FFT-based algo-

rithms for a multi-band receiver are discussed and the application of prime-factor

and mixed-radix FFT algorithms is analysed. A novel way of factorising di�erent

transform lengths into smaller transforms and then combining these smaller-point

FFTs to compute the larger required FFTs is described.

The second part of this chapter describes an e�cient way of realising the core cor-

relator functionality for the E5 signal, estimates the hardware resource requirement

and power consumption, and compares it with other signals.

Sec. 7.2 to sec. 7.7 of this chapter contain work published in Journal of GPS,

2009 (Shivaramaiah et al., 2009b), while sec. 7.8 contains the work published in

ISCAS 2010 (Shivaramaiah and Dempster, 2010b).

The chapter is organised as follows. Sec. 7.2 describes the receiver model and

search dimensions in the context of a multi-band GNSS receiver, sec. 7.3 describes

the FFT requirements of new GNSS signals, and sec. 7.4 describes the proposed

mixed-radix approach along with the result of transform length factorisation for the

di�erent signal types. Sec. 7.5 discusses computational complexity by comparing

the standard and proposed approaches. Sec. 7.6 describes the FPGA resource

utilisation of the proposed methods, followed by a comparison for some of the signal

combinations in sec. 7.7. Sec. 7.8 compares the resource utilisation and power

consumption of the core correlator used for signal tracking.

7.2. GNSS Receiver Model and Search Dimensions

7.2.1. The receiver model. The received signal in a multi-band GNSS re-

ceiver capable of receiving the open access signals (or components of these signals)

GPS L1 C/A, L2 and L5, Galileo E1 and E5 is down converted, sampled and digi-

tised to obtain an Intermediate Frequency (IF) equivalent. Assuming separate down

conversion paths for the L1, L2 and L5 bands, the received signal is transformed

165

166 7. GALILEO E5 BASEBAND HARDWARE

RF front-end

Digitized IF samples

Clock Source

Carrier Removal

Code Correlation Controller

Antenna

Commands

Decision

Correlation Values

Correlator

r(t) r1(n)

r2(n)

r5(n)

Figure 7.1. Block diagram of a multi-band receiver

into three equivalent IF signals:

r(t)down-conversion

=⇒ r1(t) + r2(t) + r5(t) (7.1)

with the individual signals being (for one satellite from each system)

r1(t) = AL1sL1(t− τL1) cos ((ωIF1 + ωdL1) t+ θL1)

+AE1sE1(t− τE1) cos ((ωIF1 + ωdE1) t+ θE1) (7.2)

r2(t) = AL2sL2(t− τL2) cos ((ωIF2 + ωdL2) t+ θL2) (7.2a)

r5(t) = AL5sL5(t− τL5) exp {j [(ωIF5 + ωdL5) t+ θL5]}

+AE5sE5(t− τE5) exp {j [(ωIF5 + ωdE5) t+ θE5]} (7.2b)

where AX is the amplitude, sX is the modulating baseband component, τX is the

delay and θX is the phase of the signal X, and ωIF1 ,ωIF2 ,wIF5 are the intermediate

frequencies. As the �rst stage within the receiver the nominal carrier frequency is

removed. The output of this carrier removal process then comprises only the base-

band component of the received signal plus any Doppler. It is this baseband version

that is considered for the proposed code acquisition approach in the discussions

throughout the remainder part of this chapter. Fig. 7.1 illustrates such a receiver.

Because of their signal structure, each of these signals has a di�erent requirement of

the minimum sampling frequency. As will be explained later, the minimum number

of cells to search within the code acquisition block depends on the length of the

spreading code and the modulation rather than on the sampling frequency. Note

that only the baseband signal is of interest and the RF down converter and antenna

are beyond the scope of this discussion. In addition, depending on the design, the

receiver may process any combination of the above signals.

7.2.2. Search dimensions for signal acquisition in a multi-band GNSS

receiver. In this section new parameters associated with the concept of search

engine dimensions are introduced. Acquisition has been discussed in the literature

as a two-dimensional search (Kaplan and Hegarty, 2006) when the receiver knows

7.2. GNSS RECEIVER MODEL AND SEARCH DIMENSIONS 167

PRN Code Number

Code Delay

Doppler Frequency

Code Length

Signal TypePRN Code Number

Doppler Frequency

(a) (b)

Figure 7.2. Search dimensions in a (a) single-band GNSS receiveremploying the time domain correlation approach (b) multi-band GNSSreceiver employing the FFT-based code acquisition approach

the PRN code which it is searching for, or as a three-dimensional search otherwise

(Djebouri et al., 2006). When the acquisition process is considered as a whole

(search engine plus the controller) instead of just the search engine, the search is

three-dimensional. The parameter along the code dimension in a GNSS receiver

employing a time domain correlation approach for acquisition is the code delay (see

Fig. 7.2a). In a single-band receiver employing an FFT-based acquisition approach

there is no parameter along the code dimension as the search is performed over the

entire code space at once. The size of the FFT block depends on the code length

and the desired resolution of the code search. However in a multi-band receiver the

code length is still a varying parameter and this is used along the code dimension.

With the new GNSS signals in context, because of the varying code length and

varying search step requirements, the acquisition engine needs to be re-arranged

whenever the same hardware resource needs to be used across di�erent signals.

Hence this parameter introduces another dimension in the search process which is

referred to here as the `signal' dimension. Fig. 7.2b depicts the four dimensions in

the context of a multi-band GNSS receiver. Note that the variables in each dimension

are not totally independent. For example the same PRN may have di�erent code

delay search requirements depending on the signal type.

During the signal acquisition process, the selection of the resolution of the coarse

estimates of chip delay and Doppler frequency depends on the requirements of the

succeeding tracking stage - typical values for GPS L1 C/A being 0.5 chips for the

code delay and 500 Hz (for one millisecond coherent integration) for the Doppler

frequency. The number of time cells to search depends on the code length. The

number of frequency cells to search depends on the total frequency ambiguity and

also on the coherent integration time. As an example, for an L5 code length of


ARNSRNSS

ARNSRNSS

960

1164

1176

.45

1191

.795

1207

.14

1215 12

3712

60

1278

.75 1300

156315

59

1575

.42 1587

1591

1610

Lower L-Band Upper L-Band

E5a E5b

L5 L2 L1

E6 E1

Galileo Navigation Bands GPS Navigation Bands

MHz

Figure 7.3. GNSS signals in the Galileo and GPS bands (from (OS-SISICD, 2010))

10230, Doppler frequency uncertainty region of ±5 KHz and an integration time of

1 ms, there will be 20460 time cells (at 0.5 chip steps) and 21 (at 500 Hz steps)

frequency cells to search.

For a multi-band multi-system receiver adapting the baseband for various num-

bers of time search cells is more important than the varying number of frequency

cells. This is because the variation in the number of frequency cells to search across

the signals is less compared to the variation in the number of time cells to search.

The shaded portion in Fig. 7.2b is the region that in�uences the size of the FFT in

the search engine. This region comprises the code length and the signal type there-

fore the focus of this chapter is to explore FFT-based acquisition methods which

span this region of interest. The aim is to search for a computationally e�cient FFT

method which can easily adapt to di�erent combinations of values along these two

dimensions.

7.3. FFT Requirements for New GNSS Signals

Fig. 7.3 shows the frequency bands for the Galileo and GPS signals. There are six

signals in the spectrum as shown, three each for Galileo and GPS. For the following

discussion only the �open� signals intended for civilian users are considered: GPS

L1C/A, L2C, L5, Galileo E1 and E5. The code length and bandwidth parameters for

these signals are listed in Table 7.1 (ISGPS705, 2010; ISGPS200E, 2010; OSSISICD,

2010).

The size of the FFT depends on the code length and the required chip step.

Whereas the chip step required for a signal with BPSK-like autocorrelation triangle is

0.5, the step size requirement for BOC signals (to ensure losses in SNR are restricted

to 1.15dB on average) depends on the BOC parameters. In the case of Galileo E1

a chip step of 0.167 is required in order to obtain a comparable correlation loss to

7.3. FFT REQUIREMENTS FOR NEW GNSS SIGNALS 169

Table 7.1. GPS and Galileo signal parameters of interest

Signal Name Code Length Chipping

Rate (MHz)

Receiver

Bandwidth

in MHz

(typical)

GPS L1 C/A 1023 1.023 2

GPS L2C-CM 20460 0.5115 2

GPS L2C-CL* 767250 0.5115 2

GPS L5 10230 10.23 20

Galileo E1B/ E1C 4092 1.023 4

Galileo E5 10230 10.23 50

Galileo E5a /E5b 10230 10.23 20

*L2C-CL is generally not targetted in the �rst stage of acquisition (Dempster, 2006) and

hence not included in the discussions in this chapter.

Table 7.2. Transform length requirements Case 1 � 0.5 chip step

Signal Name Chip StepSize

RequiredTransform Length

GPS L1 C/A 0.5 2046Galileo E1B/C� SA 0.5(Side-band

Acquisition)8184

GPS L2C 0.5 40920GPS L5 0.5 20460

Galileo E5a/E5b 0.5 20460

that for the BPSK 0.5 chip step (De Wilde et al., 2006). Hence for the Galileo

E1B/C signal, the number of time cells to search increases from 8184 to 24552

for one millisecond coherent integration duration, despite the signal bandwidth only

doubling. For the analysis of the FFT requirements and of the proposed approaches,

a coherent integration time of one primary code period is considered in this chapter.

As already mentioned, the transform length can be reduced to as much as twice

the code length for some of the signals. This is true for the GPS L1, L2, L5, Galileo

E5a and E5b signals where the shape of the autocorrelation function allows half chip

(or less) alignment between the received and local signals with an e�ective sample

size of twice the code length. For these signals the transform length requirements

are given in Table 7.2 (Case 1).

For the Galileo E1B/C and E5 signals, the e�ective sample sizes for one millisec-

ond, which is the transform length, are shown in Table 7.3 (Case 2). Note that for

the Galileo E1B/C, the signal can be acquired with 0.5 chip spacing with the Side-

band Acquisition (SA) method (but with 3dB correlation loss compared to BPSK

0.5 chip spacing), or the Direct Acquisition (DA) method with a 0.167 chip spacing


Table 7.3. Transform length requirements Case 2 � other chip steps

Signal Name Chip Step Size RequiredTransform Length

Galileo E1B/C� DA 0.167(Direct Acquisition) 24552Galileo E5a/E5b 0.083 122760

Table 7.4. Transform length requirement summary

Signal Name Required Transform Length

GPS L1 C/A 2046Galileo E1B/C� SA 8184

GPS L5, Galileo E5a/E5b 20460Galileo E1B/C � DA 24552

GPS L2C 40920Galileo E5a/E5b 122760

(no loss compared to BPSK 0.5 chip spacing).

It is clear from the discussions so far that there is a common transform length

requirement among the signals. Table 7.4 combines all the signals with respect

to the transform length and summarises the requirements for the signals under

consideration. The chip step size for each signal is the same as in Tables 7.2 and

7.3.

A typical implementation of FFT-based acquisition has two problems. The �rst

problem is due to the transform length. In order to simplify the FFT implementa-

tion, often a �next immediate of power-of-two� transform length is chosen instead

of the transform lengths listed in Table 7.4, by zero-padding the input sequence.

Even though this works well for the GPS L1 signal (1024 instead of 1023), for the

codes with longer lengths one might have to unnecessarily increase the transform

length by a signi�cant amount (e.g. 65536 instead of 40920 for the GPS L2C) which

also may reduce the SNR of the correlation output (Yang 2001). It should be noted

that there are di�erent contexts where the method of zero-padding is used. Yang

(2001) describes the method of zero-padding to perform circular correlation and

linear correlations at arbitrary lengths. On the other hand Dempster (2006) de-

scribes a method of zero-padding for the L2C signal acquisition. In the discussions

this zero-padding is considered as a consequence of making the transform length a

power-of-two (including any acquisition concept related zero-padding as in the case

of L2C). The second problem is the fact that di�erent signals require FFT blocks

of di�erent sizes. For example, assuming one millisecond coherent integration, a

receiver processing GPS L1 C/A and Galileo E1B/C will have to have both 2046

point, as well as 8184 point FFTs (in the case of the SA method for E1). This

results in allocating dedicated FFT blocks for each signal, which is a very expensive

7.4. THE PROPOSED FFT BASED CODE CORRELATION APPROACH 171

Table 7.5. 1023 point FFT factorisation

Transform length Factors

1023 3, 11, 31

Table 7.6. Transform length factorisation

Signal Name Transform Length Factors

GPS L1 C/A 2046 2, 1023Galileo E1B/C� SA 8184 8, 1023

GPS L5, Galileo E5a/E5b 20460 4, 5, 1023Galileo E1B/C � DA 24552 3, 8, 1023

GPS L2C 40920 5, 8, 1023Galileo E5a/E5b 122760 3, 5, 8, 1023

approach.

7.4. The Proposed FFT Based Code Correlation Approach

In this section, �rst the rationale behind the factorisation of large-point FFTs

is described. Next, the computational complexity of the small-point FFT blocks is

discussed and it is shown that a small modi�cation to the �brute-force� factorisa-

tion method can result in the e�cient computation of the small-point FFT blocks.

Finally, with the revised factorisation, a table of required small-point FFT blocks is

given.

7.4.1. Factoring of FFT transform lengths. The basic idea is to factor N

(for an N -Point FFT) into two or more smaller integers, implement the small-point

building blocks, and combine them to obtain the �nal result. Thus for the two-factor

case, i.e., if N can be factored into N = P · Q then, Q number of P -point FFTs

and P number of Q-point FFTs are combined to form the N -point FFT. Appendix

C describes two methods of factoring the FFT transform lengths, the prime-factor

method and the mixed-radix method.

All the transform length requirements listed in Table 7.4 are multiples of 1023.

It should be noted that 1023 is easily factored into three prime numbers: 3, 11 and

31 (Table 7.5). All other transform lengths can be factorised such that the factors

are relatively prime to 1023. This factorisation is shown in Table 7.6. It can be

seen that the transform of length 1023 is common across all the transform lengths,

hence it makes sense to have the 1023-point FFT as a single block, which can be

implemented using the prime-factor approach. The basic small-point building blocks

required for all transform lengths under consideration are listed in Table 7.7. This

list assumes a 1023-point block as a single entity (as mentioned above).


Table 7.7. FFT blocks required for GNSS signals in consideration

Basic Building Blocks

2, 3, 4, 5, 8, 11, 31, 1023

Table 7.8. Complexity of small-point blocks

Transform Length Additions Multiplications

2 4 03 12 44 16 05 34 108 52 411 168 4016 148 2031 776 160

7.4.2. Complexity of small-point blocks. Many algorithms are available

for computing the small-point FFTs, such as the Winograd, Rader, SWIFT, Prime-

length, etc (Smith, 1995). Each algorithm has its own complexity (number of ad-

ditions and multiplications). For the sake of commonality amongst di�erent combi-

nations of small-point blocks, Table 7.8 lists the number of real additions and mul-

tiplications that are required (Smith, 1995; Burrus and Selesnick, 1995). Note that

if there exists a method which can more e�ciently compute the small-point FFTs,

the improvement is directly observed in the proposed prime-factor and mixed-radix

approaches as well because the proposed method uses a combination of the basic

small-point blocks.

7.4.3. A note on 1023-point and 1024-point FFTs. As mentioned previ-

ously, 1023 is a common factor in the transform lengths of all the signals under

consideration. But because 1024 is the next immediate power-of-two number for

1023 it can be implemented using Radix-2, Radix-4 or other optimised algorithms.

Therefore it is necessary to compare the performance of the prime-factor approach

for a 1023-point FFT with a 1024-point FFT (with padding of one zero). It was

shown in Proakis and Manolakis (1995) that the Split-radix FFT algorithm requires

fewer multiplications and additions compared to the Radix-2 and Radix-4 algo-

rithms. Split-radix is a method in which at each stage the transform is divided into

Radix-2 and Radix-4 branches (not FFTs) and then blended in the next stage. It

should not be confused with the Mixed-radix which uses the factors (i.e. the smaller

FFTs) of the transform length at each stage. Table 7.9 lists the operation count

comparison for the 1023-point FFT using the prime-factor approach and the 1024-

point FFT using the Radix-2, Radix-4 and Split-radix approaches. Note that the

7.5. COMPUTATIONAL COMPLEXITY OF THE PROPOSED APPROACH 173

Table 7.9. Operation count for 1023 and 1024 point FFTs

Transform Length Algorithm Additions Multiplications

1023 Prime-factor 45324 103641024 Radix-2 46080 153601024 Radix-4 49920 115201024 Split-radix 27652 7172

Table 7.10. Revised transform lengths for di�erent signals

Signal Name Required Transform Length Factors

GPS L1 C/A 2048 2, 1024Galileo E1B/C� SA 8192 8, 1024

GPS L5, Galileo E5a/E5b 20480 4, 5, 1024Galileo E1B/C � DA 24576 3, 8, 1024

GPS L2C 40960 5, 8, 1024Galileo E5 122880 3, 5, 8, 1024

Table 7.11. FFT blocks required for GNSS signals in consideration� revised

Basic Building Blocks

2, 3, 4, 5, 8, 1024

Split-radix approach is the �cheapest� of all the considered approaches for the 1023

or 1024-point FFT. With this information it is therefore wise to choose 1024 as the

common factor instead of 1023.

7.4.4. Revised FFT transform lengths and their factors. The revised

transform length requirements are given in Table 7.10. The revised requirements of

the basic building blocks are given in Table 7.11. Since the factors are not relatively

prime, the method to be used to combine the small-point blocks is the Mixed-radix

method.

7.5. Computational Complexity of the Proposed Approach

In order to compare the complexity of the proposed approaches, the Split-radix

algorithms for the power-of-two approaches with the number of real additions and

real multiplications according to Sorensen et al. (1986) (the complex multiplications

are treated as 3 real multiplications and 3 real additions) are used. Table 7.12 gives a

comparison of the Split-radix approach and the Mixed-radix approach. To compute

the number of operations for the Mixed-radix algorithms, the factorisation according

to Tables 7.10 and 7.4 has been used. As an example, consider the transform length

2048 for which 2-point and 1024-point FFT blocks are needed. In the �rst stage,

all the 1024 two-point FFTs are computed. The outputs of these FFTs are then


Table 7.12. Computational complexity comparison

Split Radix Mixed Radix

Transform

Length

Additions Multipli-

cations

Transform

Length

Additions Multipli-

cations

2048 61444 16388 2048 62649 17413

4096 135172 36868 4096 136199 37895

8192 294916 81924 8192 295947 82955

16384 638980 180228 16384 640019 181267

32768 1376260 393220 20480 869399 279575

-NA- -NA- -NA- 24576 1035291 330779

65536 2949124 851972 40960 1856555 594987

131072 6291460 1835012 122880 6110331 1866875

0 1 2 3 4 5 6 7

x 106Number of Real Additions

Sig

nal T

ype

Split−radix FFT ApproachMixed−radix FFT Approach

L1 C/A

E1B/C−SA

L5, E5a/b

E1 B/C −DA

L2C

E5

Figure 7.4. Number of real additions comparison for FFT of dif-ferent GNSS signals (Split-radix method is used for the standard ap-proach)

multiplied with the (1024-1)*(2-1) = 1023 complex coe�cients (the other coe�cients

are unity). In the last stage two 1024-point FFTs are computed to obtain the �nal

output. Comparing the complexity of these two approaches suggests that the Mixed-

radix algorithm requires only a small amount of additional computations. Moreover,

because the Mixed-radix FFT approach makes use of small-point FFTs, the required

FFT can be built using the smaller-point FFTs. Hence Mixed-radix algorithms are

proposed to construct the di�erent sizes of FFTs that are required. Figs. 7.4 and

7.5 show the computational complexity for di�erent GNSS signals using the data

given in Table 7.12.

7.5.1. Comparing the complexity of the FFT-based correlator with the

time-based correlator. To obtain the same search time, the time-based correla-

7.5. COMPUTATIONAL COMPLEXITY OF THE PROPOSED APPROACH 175

0 0.5 1 1.5 2

x 106Number of Real Multiplications

Sig

nal T

ype

Split−radix FFT ApproachMixed−radix FFT Approach

L1 C/A

E1B/C−SA

E1 B/C −DA

L5, E5a/b

L2C

E5

Figure 7.5. Number of real multiplications comparison for FFT ofdi�erent GNSS signals (Split-radix method is used for the standardapproach)

tion process should provide the correlation values simultaneously for all code delays

since the FFT-based method does the same. The acquisition performance of such

parallel time-domain correlators was experimentally studied in Malik et al. (2009b)

and Malik et al. (2009a). A time-based correlator with 2-bit input and 2-bit local

carrier signal should typically process a 4-bit input for the code correlation (the

code correlation is a simple signum function) and the output of the code correlator

is then fed to the input port of an accumulator. For the GPS L1 C/A signal, compu-

tation of each correlation value involves 2046 additions/subtractions. Thus, with an

acquisition code phase resolution of half a chip, the time-based correlator requires

2046*2046 = 4186116 add/subtract operations. The accumulator width depends on

the integration duration, but for one millisecond integration a 16-bit accumulator is

su�cient (the details will be explained in sec. 7.8).

With FFT-based correlation, the number of additions and multiplications to

obtain all the correlation values can be computed using Table 7.12 and Fig. 2.12, and

considering that the FFT of the local code is pre-computed and stored in memory.

Table 7.13 illustrates the FFT-based and time-based correlator complexities for the

GPS L1 C/A signal with 2046 cell searches (2048-point FFT). Note that according to

Fig. 2.12, the FFT operations in Table 7.12 should be multiplied by two (there is an

FFT and an IFFT to execute in real-time), with a complex multiplication in between.

For the time-based correlator the number of accumulators has to be doubled because

of I and Q local carriers. The code correlation part of the time-based correlator when

implemented on a Altera Cyclone family FPGA device consumes 39 Logic Elements

(LEs) for each of the I and Q accumulators (i.e. an accumulator with a feature of


Table 7.13. Operations count for the entire correlator employingtime-based (2046-tap) and FFT-based (2048-point) methods

FFT-based Time-based

Additions 125301 8372232Logic Elements 11229 159588Multiplications 34829 0Multipliers 39 0

add/subtract based on the code bit input). The correlation values for all the delays

are obtained by cycling 2046 times through each of the 2046 correlators. It can be

observed from Table 7.13 that even though there are no multipliers required, the

time-based correlator consumes a huge number of LEs. Having saved 93% of LEs,

the multiplier requirement is easily addressed using the multiplier blocks available

in existing FPGAs. As a result, other signals are not considered due to the high

resource consumption of the time-based correlators. For the same reason, the time

domain correlators are not considered in the results section.

7.6. Implementation and Resource Utilisation on an FPGA

As discussed in the previous section, application of the proposed methods re-

duces the number of computations compared to directly computing the transforms

of power-of-two lengths. In addition, instead of having separate FFT blocks for each

signal in the receiver, the proposed algorithm uses basic building blocks in order to

construct the required larger FFTs. (`Constructing' here means the combination

of smaller blocks using the Mixed-radix approach.) The main task of the combi-

nation process is to con�gure the complex multiplication coe�cients between the

small-point FFT blocks as shown in Fig.C.2.

For hardware resource comparisons the number of LEs and multipliers of the

Altera Cyclone family FPGA devices are used. For the sake of commonality, a data

and twiddle precision of 16 is chosen across all the stages of the FFT. Note that in

the case of the Mixed-radix approach, by arranging the factors in an increasing order

it is possible to use smaller bit widths during the initial stages of the FFT. Also, it

is possible make use of the fully parallel architecture for the small-point FFT blocks

wherein each 16-bit addition consumes 16 LEs and each multiplication consumes

one multiplier block. The number of LEs and multiplier consumption was evaluated

for the small-point FFTs and are listed in Table 7.14. The reason for selecting a

fully parallel architecture is that the blocks can be operated at higher throughput

which helps when building larger length FFTs. Table 7.15 lists the FPGA resource

utilisation for the 1024-point FFT with streaming I/O architecture (Altera, 2007).

7.6. IMPLEMENTATION AND RESOURCE UTILISATION ON AN FPGA 177

Table 7.14. FPGA resource utilisation for the basic building blocks

Transform Length LEs Multipliers

2 9 03 192 44 256 05 544 108 832 4

Table 7.15. FPGA resource utilisation for 1024-point FFT

Transform Length Altera MegaCore IP (v7.2)LEs Multipliers

1024 5552 18

2-point FFT

1024

1 1024-point FFT

12

Multipliers

Complex Coefficients

Figure 7.6. Example of Mixed-radix method for a 2048-point FFT

For the other transform lengths the pipelined streaming input method is used

by combining the appropriate smaller-point FFTs. Hence the resource utilisation is

in�uenced by the resource consumed by the combination pattern of the basic build-

ing blocks. The requirements of the memory increases only by an amount of highest

value (P − 1) · (Q− 1) (which is required to hold the complex multiplication coe�-

cients, P and Q being the two factors in consideration) among all the desired signal

combinations. For example, as shown in Fig. 7.6, a 2048-point FFT will require one

2-point block which is serially operating on the input streaming data 1024 times and

one 1024-point FFT block which is operated twice, with 1023 complex multiplica-

tions in between. The depth [1...1024] and [1, 2] indicates the serial operation of the

corresponding FFT blocks. Table 7.16 lists the resource utilisation for longer length

FFTs. The * indicates that the Altera core does not exist for these sizes and has


Table 7.16. FPGA resource utilisation for di�erent transform lengths

Standard Approach (Altera

MegaCore IP )

Proposed Approach

(Time-shared architecture)

Transform

Length

LEs Multipliers Transform

Length

LEs Multipliers

2048 7610 36 2048 5556 18

4096 8011 36 4096 5808 18

8192 7760 36 8192 6384 22

32768 17337 80 20480 9008 58

-NA- -NA- -NA- 24576 9584 62

*65536 34864 160 40960 1464 118

*131072 69720 352 122880 38768 478

been computed by combining the N/2 point FFTs as explained in (Altera, 2004).

7.7. Case Studies and Discussion

The combinations considered here are:

• Combination-I: GPS L1 C/A and Galileo E1B/C-SA

• Combination-II: GPS L1 C/A and GPS L2C

• Combination-III: GPS L1 C/A, Galileo E1B/C-SA and Galileo E5a/E5b

/GPS L5

When using the proposed approach, depending on the acquisition engine design,

either the FFT blocks can be time-shared among di�erent signals, or each signal

can have its own (independent) FFT processing block. The time-sharing referred

to here is di�erent to the time-sharing referred to in the previous section. In the

previous section the computation of a particular length FFT is carried out by time-

sharing the smaller FFT blocks. In this section time-sharing indicates the re-use

of FFT blocks among di�erent signals that require FFTs of di�erent lengths. The

hardware resource required is the same as that required by the largest FFT among

the signals considered. This is referred to as the �signal-time-sharing� approach.

An example of the signal-time-sharing FFT architecture for Combination-I is

shown in Fig. 7.7. In a GPS L1 C/A + Galileo E1B/C receiver, �rst the GPS

satellites can be searched and then the Galileo satellites, or, for example, half the

number of channels can search for GPS satellites and the other half can search

for Galileo satellites or any other scheme. The basic FFT blocks required for this

combination are 2, 8 and 1024. For the GPS L1 C/A signal, 1024 serial operations

of the 2-point FFT block and two operations of the 1024-point FFT are performed

as explained in the Mixed-radix example described in the previous section. For the

Galileo E1 B/C signals, 1024 serial operations of the 8-point FFT block and eight

operations of 1024-point FFT are required. Note that the complex multiplication

7.7. CASE STUDIES AND DISCUSSION 179

8-point FFT

1024

1

12

Multipliers

Complex Coefficients

2-point FFT

1024

1

1024-point FFT

8

Figure 7.7. Example of the signal-time-sharing FFT architecture forCombination-I

and the coe�cients should accommodate the largest FFT under consideration, 8184-

point in this case.

Thanks to the Mixed-radix algorithm, the complex multiplication process and

coe�cients for the 2048-point FFT (the smaller FFT) are subsets of the 8184-point

FFT (the larger FFT), and hence no additional memory or multipliers are required

for the 2048-point FFT. Another advantage of the Mixed-radix algorithm is that

only a quarter of the number of coe�cients need to be stored, and hence the mem-

ory required is 4196 (2048 real and 2048 imaginary) locations in the current example.

Therefore in this signal-time-sharing approach the memory required is that of the

largest FFT when implemented in the standard power-of-two approach plus a mem-

ory of depth equal to a quarter of the length of largest FFT under consideration.

In addition there is no increase in the routing resources except for reading the 2048

depth memory into the multipliers. In the independent FFT block approach of the

Mixed-radix method, the hardware and routing resources and the memory of the

individual FFTs have to be added, and hence the requirements increase compared to

the signal-time-sharing approach. However, these numbers remain less when com-

pared to the standard power-of-two approach. Note that time-sharing is not possible

with the standard approach as each FFT block has to be independently instantiated.


L1 C/A + E1 B/C −SA L1 C/A + L2C L1 C/A + E1 B/C + L5/E5a/b0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

Signal Combination

Num

ber

of L

ogic

Ele

men

ts

StandardProposed − SimultaneousProposed − Time Shared

Figure 7.8. Comparison of number of LEs for di�erent signal com-binations

L1 C/A + E1 B/C −SA L1 C/A + L2C L1 C/A + E1 B/C + L5/E5a/b0

20

40

60

80

100

120

140

160

180

200

Signal Combination

Num

ber

of M

ultip

lier

Blo

cks

StandardProposed − SimultaneousProposed − Time Shared

Figure 7.9. Comparison of number of multipliers for di�erent signalcombinations

7.7.1. Resource utilisation results for di�erent combinations. Figs. 7.8

and 7.9 give the performance comparison of the standard and proposed approaches

for the selected signal combinations respectively. For Combination-I, the saving is

about 22% in the number of LEs, and for Combination-III the saving is around 35%.

7.7.2. Proposed FFT test result with the data collected from the real

signal: A case study. The proposed FFT method has been tested with data col-

lected from real signals for the Combination-I to acquire GPS L1 C/A and GIOVE-B

E1C signals in the same platform. The GeNeRx1 receiver from Septentrio was used

7.7. CASE STUDIES AND DISCUSSION 181

0 500 1000 1500 2000 25000

100

200

300

400

500

Cor

rela

tion

Val

ue

0 500 1000 1500 2000 2500−2

−1

0

1

2

Sample Number

Cor

rela

tion

Err

or (

%)

ProposedStandard

% Error

Figure 7.10. Acquisition results for the GPS L1 C/A signal; PRN17; 2048-point FFT; 1ms integration

0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

0

1000

2000

3000

Cor

rela

tion

Val

ue

0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

−2

−1

0

1

2

Sample NumberCor

rela

tion

Err

or (

%)

ProposedStandard

% Error

Figure 7.11. Acquisition results for the GIOVE-A E1 C signal;16364-point FFT realised using standard approach and the proposedMixed-radix (2*8*1024) approach; 8ms integration

to collect the IF signal samples. The IF samples were re-sampled to two samples per

chip (so as to obtain 0.5 chip spacing) and the samples were then fed to the Altera

FPGA for processing. The design with the FFT blocks used Altera DSPBuilder tool

in the Matlab Simulink and then programmed to the FPGA. The integration dura-

tion of one millisecond for the GPS L1 C/A and 8 ms for the GIOVE-B E1C signals

were selected. Figs. 7.10 and 7.11 show the correlation value (top half) from the

standard and the proposed FFT methods respectively. Note that the proposed ap-

proach (2*1024 point and 2* 8*1024 point FFTs) closely matched the standard (i.e.

2048-point and 16384-point FFTs) approach. The errors in the correlation values


Table 7.17. Some new GNSS signals and their parameters of interest

Signal name Centre

frequency

(typical

receiver

bandwidth) in

MHz

Modulation type Code length *

(memory

code? Y/N)

Chipping

rate (MHz)

GPS L1 C/A 1575.42 (2) BPSK 1023 (N) 1.023

GPS L2C 1227.6 (2) BPSK CM-20460 (N),

CL-7672501.023

GPS L5 1176.45 (20) BPSK 20460 (N) 10.23

GPS L1C,

Galileo E1,

Compass B1

1575.42 (4) MBOC / CBOC 1023 (N), 4096

(Y), **

1.023

Galileo E5,

Compass B2

1191.795 (50) AltBOC 10230 (N), ** 10.23

* Primary code only, ** Yet to be available for the Compass signal

are also plotted in the bottom half of Figs. 7.10 and 7.11. Note that the proposed

approach has less than two percent error compared to the standard approach. This

di�erence is due to the rounding used during the complex multiplications between

the stages in the Mixed�radix method. The loss due to this error is less than 0.1 dB

in the correlation value and hence negligible for all practical purposes.

7.8. E�cient Design of Core Correlator Blocks for Tracking

Table 7.17 revisits the centre frequency, bandwidth and code lengths of some of

the new open service signals. The important points to note here are: a) increased

signal bandwidths which demand higher sampling frequencies, which in turn in-

creases the minimum operating frequency of the baseband hardware; b) increased

spreading code lengths and chipping rates, which demand higher shift register clock

frequencies; c) use of multi-level sub-carriers, as in the case of AltBOC type of mod-

ulation, which increases the number of bits in the local reference signal; and d) use

of memory codes which demand additional memory to hold the spreading code for

all the satellites.

7.8.1. Generic baseband architecture for the tracking process in a

GNSS receiver. The GNSS baseband hardware in its usual de�nition is comprised

of all the signal processing circuit, bounded on the input side by the sampled and

digitised IF signal, and on the output side by the received signal measurements (car-

rier phase, code phase, navigation data bits, signal strength, etc.). Fig. 7.12 shows

the functional diagram of generic GNSS baseband hardware for a single component

of a signal. The functionality of each block is described in detail in the literature

7.8. EFFICIENT DESIGN OF CORE CORRELATOR BLOCKS FOR TRACKING 183

Carrier

Mixer

Local

Reference

Mixers

Carrier

Generator

Carrier

NCO

Code NCO

Code

Generator

Sub-carrier

Generator

Subcarrier

NCO

Subcarrier

Modulator

Shift

Register

... R1

...

1

2R

Nacc -bit

Accumulators

...

1

2R

Decision and

Feedback

ControlMeasurements

Convolution

Decoder

Navigation

data

Timing

Control (to

all

sequential

blocks)

CLK

Nif

Ncar

N1

Nref

N2

Nref

Nacc

Nnco1

Nnco2

Nnco3

Core hardware (correlator)

IF samples

Figure 7.12. A functional diagram of the baseband hardware (thicklines carry N• bits, dashed boxes are optional )

(e.g. Kaplan and Hegarty, 2006) and will not be discussed here. R is the number

of local reference signal �arms� (typically three).

7.8.2. Bit-width requirements of the correlator components. The pa-

rameter of interest for the complexity analysis of the core correlator is the number

of bits required to represent the intermediate signals, the bit-width of the accu-

mulator and other registers and the minimum frequency of operation required for

a particular signal (or any component of a signal thereof). The notations for the

number of bits at di�erent stages are shown in Fig. 7.12, as N• along with the

thick lines. In the following paragraphs a brief description on each of the underlying

modules is given and the number of bits required for the accumulator is derived.

ADC/IF (Nif): the signal loss due to the quantisation beyond 2-bits is in-

signi�cant as long as the sampling thresholds are sensibly set (Hegarty, 2009). How-

ever, 3-bits and more have been used to alleviate the problems with the AGC in

the presence of RF interference (Kaplan and Hegarty, 2006). Commercial mass-

market receivers normally use 2-bit uniform sign-magnitude quantisation with 4

levels {±1,±3}(Zarlink, 1999, 2001).Local Carrier Generator (Ncar): The loss due to the local carrier quanti-

sation is studied in Namgoong et al. (2000). Typically, 3-bit uniform NCO phase

quantisation and 2-bit amplitude quantisation with 4 levels {±1,±2} is used. More

bits in the phase and amplitude quantisation increases the Spurious-Free-Dynamic-

Range (SFDR) and reduces the quantisation noise. However this has a signi�cant

impact on the size of succeeding stages.


Carrier Mixer (N1): Since the resulting values will only have 8 levels {±1,±2,

±3,±6}, a 3-bit encoding is su�cient.

Subcarrier Generator & Subcarrier Modulator (Nref): Depends on the

number of levels in the subcarrier used for the modulation. BOC signals use a 2-

level {±1} subcarrier whereas AltBOC uses 4-levels (dominant component of the

sub-carrier) which can be approximated to {±1,±2}. In addition, the local spread-

ing code modi�es only the sign of the sub-carrier at the output of the subcarrier

modulation. Hence, upon encoding, Nref will be either 1 or 2.

Local Reference Mixer (N2): This can be easily determined from the number

of levels of the two inputs. However, the succeeding stage (the accumulator) is an

arithmetic operation and requires binary representation. This leads to an additional

bit at the output. For example, with the 8-level N1 {±1,±2,±3,±6} and the 4-levelNref{±1,±2}, the resultant set will have only 12 levels {±1,±2,±3,±4,±6,±12},but due to the later requirement of signed binary representation the output has to

be 5-bit wide. Let the sample-maximum (magnitude) of the output at this stage be

denoted by A2.

Accumulator (Nacc): The interval between two consecutive accumulator resets

is determined by the coherent integration duration. Let N′acc denote the number of

bits required to represent the worst-case value at the output of the accumulator.

Then

N′

acc =

⌈log−1

2

(A2

⌊fsfco

McL

⌋)+ C + 1

⌉(7.3)

where fs ∈ R+ is the sampling frequency in Hz, fco ∈ R+ is the chipping rate

(with any associated Doppler frequency) in Hz, L ∈ N is the primary code length,

M c ∈ Q is the number (or fraction) of primary code periods in the coherent integra-

tion and C is the complex modulation indicator, C ∈ {0 = Normal, 1 = Complex}.(7.3) clearly satis�es the Design-For-Test (DFT) guidelines, but it is an overkill as

all the samples may not end up with a value of A2. In reality the sample-maximum

is controlled by the input signal strength and the local carrier frequency. Hence the

required accumulator width Nacc < N′acc.

An R-arm correlator will have 2R(C+ 1) accumulators (due to the in-phase and

quadrature carrier components) and hence accumulator width plays a very important

role in correlator complexity. Some correlators use re-sampling prior to the local

reference mixer stage (e.g. Namgoong et al., 2000), to reduce the number of samples

input to the accumulator. However those special techniques are outside the scope

of the discussion here.

7.8.3. E�cient realisation of the correlator core. In the correlator core

of Fig. 7.12, all the blocks do not require sequential logic. The carrier mixer, sub-


16x4 LUT

IF Signal

Local Carrier

Local Code

Sample Correlation2

2

4

Accumulator

Add/Sub

Dat

a

Correlation value

16

Figure 7.13. Realisation of the core correlator block for the GPS L1C/A signal

carrier modulation and the local reference mixer are implemented as combinational

logic. For the single-bit reference signals, the circuit can be further simpli�ed by

feeding the local code to select the add or subtract operation of the accumulator.

Fig. 7.13 shows an e�cient realisation of this combinational logic using the Look-

Up-Table (LUT) method for the GPS L1 C/A signal.

7.8.4. Impact of signal structure on core correlator architecture.

Longer Codes (or Longer Code Period). Longer codes are usually obtained by a

shift register with more bits. For the shift register generated codes, more bits should

be allocated to the accumulator (i.e. the Nacc requirement increases), in addition to

increasing shift register bits.

Sub-carrier Modulation. With the sub-carrier modulation an additional NCO,

sub-carrier generator and subcarrier modulator may be required depending on the

tracking architecture. In addition the reference signal could be multi-valued, and

hence cannot be used directly as an input to the accumulator.

Modulation Type. The BOC family of signals have a narrow autocorrelation main

peak. As a result of this the spacing between the R delayed and advanced versions of

the reference signals should be reduced (Shivaramaiah and Dempster, 2009a). This

constrains the minimum clock frequency requirement of the sub-carrier NCO, and

hence constrains the overall operating frequency of the correlator.

Memory Codes. Memory codes eliminate the need for code generator shift regis-

ters. However the codes for all the satellites must be stored in a FIFO/ memory. In

addition, multiple GNSS correlator channels cannot use the same memory to acquire

or track the signal unless the memory block has multiple ports.

Multiple Signal Components. When a signal has more than one component (say

data carrying and data-less components), it is wise to compute the correlation values

independently for each signal component, thus allowing the subsequent processing

blocks to use the e�cient tracking techniques as described in Chapter5.

Receiver Bandwidth. Receiver bandwidth has a direct impact on the sampling

frequency and hence the operating frequency of the circuit. Bandwidth reduction


32x5 LUTsIrI

Carrier Mixer Output

Sample Correlation

3

3

32x5 LUTsQrQ

32x5 LUTsIrQ

32x5 LUTsQrI

sI

sQ

Reference Signal

22

rI

rQ

6

6+

+

Figure 7.14. Local reference mixer for the complex modulation signals

while processing the signal may result in blunt auto-correlation peaks, which in turn

result in noisier range measurements.

Complex Modulation. In the case of AltBOC signals the lines generated within

the core correlator portion in Fig. 7.12 carry complex signals. The local refer-

ence mixer LUT must cater for the complex correlation. An architecture for the

AltBOC(15,10) used in Galileo E5 and Compass B3 is shown in Fig. 7.14.

7.8.5. Core resource and power requirements for the new signals. In

order to gauge the resource requirements in terms of the number of registers and

combinational logic, the core correlators for the GPS and Galileo open service signals

have been implemented on the Altera Cyclone-III family device EP3C120F780C8.

The FPGA resource utilisation parameters are listed in Table 7.18.

The ratio of the power consumption estimate with respect to the GPS L1 C/A

is shown in Fig.7.15. The power consumption was estimated using the PowerPlay

Analyzer tool with the real IF signal samples provided as an input1 to the baseband

module. Major contributors in the resource utilisation for the new signals compared

to the GPS L1 C/A are listed below.

L2C - CM. Instead of the 10-bit code generator shift register in the L1 C/A

signal, the L2C - CM code generation requires a 27-bit shift register. This in turn

increases the code generator read /write and control register widths. The operating

frequency remains the same, and hence the power consumption is slightly more.

L2C (CM and CL). The only addition to the L2C - CM component is another

27-bit shift register. Since the CM and CL codes are time-multiplexed, the number

1The PowerPlay tool estimates the toggle rate of the internal nets and the output pins based onthe input signal and the associated clock-frequency


Table 7.18. Resource utilisation and power consumption estimatesof the core correlator for di�erent signals (single core)

Signal /

Component

Correlator

Operating

Frequency

(M Hz)

Resource Utilisation Power

estimate

(mW)

Registers Combina-

tional

Memory

(bits)

GPS L1 C/A 4 446 151 - 1.06

Galileo E1b or

E1c

8 436 149 4092 1.84

Galileo E1

(E1b and E1c)

8 631 176 8184 2.24

GPS L2C CM

only

4 478 210 - 1.13

GPS L2C (CM

and CL)

4 737 245 - 1.61

GPS L5 (Pilot

and Data)

40 701 204 - 11.93

Galileo E5a or

E5b

40 694 203 - 11.80

Galileo E5 100 1010 253 - 39.28

L2−CM L2 E1b E1 L5 E5a E50

5

10

15

20

25

30

35

40

Signal (Signal Component)

Pow

er C

onsu

mpt

ion

(rat

io w

.r.t.

GP

S L

1 C

/A)

1.07 1.52 1.74 2.11

11.25 11.13

37.06

Figure 7.15. Ratio of the power estimate for new signals with re-spect to GPS L1 C/A


of accumulators remain the same. Hence the increase in the power consumption

with respect to the L2C- CM signal is negligible.

E1b or E1c. The 10-bit shift register is absent and instead the local spreading

code is stored in memory. Hence the number of registers used is less by 10 compared

to the GPS L1 C/A. However, because of the 8 MHz sampling frequency requirement,

the power consumption increases.

E1b and E1c. Here two sets of memory codes are used each occupying 4092 bits.

In addition the local reference mixer and the accumulator need to operate on the

data of both the E1b and E1c signals. For this reason the power consumption is

close to twice that of the GPS L1 C/A.

L5 (Pilot and Data). The code generator shift register requires only 13 bits,

but the major contributors are the two signal components and the operating fre-

quency (which is 40 MHz). Hence the power consumption requirement is drastically

increased.

E5a or E5b (Pilot and Data). The code generator shift register requires only

14-bits and all the other circuit parameters remain the same as that of L5 signal.

Hence the estimated power consumption closely matches that of L5.

E5 Wideband. The code generators require only 14 bits. However, due to the

four signal components and the complex modulation the local reference mixer is

computationally intensive (more LUTs). In addition, a quadruple number of accu-

mulators are required. As mentioned earlier, independent correlation for all the four

signals is performed to allow design freedom for the subsequent stages in combining

these four components. As a result of a very high operating frequency the power

consumption shoots up to almost 37 times that of the GPS L1 C/A signal. The

power consumption for the E5 signal can be reduced a little bit further by focusing

more on how the complex mixers are realised as discussed in Chapter 8.

7.8.6. Complexity comparison results for di�erent baseband con�gu-

rations. Fig. 7.16 shows the power consumption of di�erent signals vs. the number

of channels. A �channel� comprises the core correlator, timing control, address data

multiplexer/demultiplexer (for a memory mapped interface to the subsequent stage),

and some housekeeping operations. Although the resource consumption is not de-

scribed in detail here, it should be mentioned that the two major memory spreading

code sets in the case of the Galileo signal occupy around 410K bits (E1, 4092 bits,

2 signal components, 50 satellites) of memory and 10K bits (E5 secondary code,

100 bits, 2 components, 50 satellites) which are totally new additions to the GNSS

receiver baseband hardware.

Fig. 7.17 shows the power consumption for di�erent combinations of signals

where each signal has been assumed to be using 12 channels. It is interesting to

7.9. SUMMARY 189

2 4 6 8 10 12 140

50

100

150

200

250

300

Number of channels

Pow

er c

onsu

mpt

ion

estim

ate

(mW

)

L1L2E1L5E5aE5

Figure 7.16. Power consumption of the entire baseband circuit

note that a GNSS receiver designed to process all the civilian signals of GPS and

Galileo would require slightly less than short one watt for the baseband hardware

(using the Altera Cyclone-III family device EP3C120F780C8), which is 38 times

that of GPS L1 C/A baseband hardware.

7.9. Summary

This chapter discussed the requirements for the FFT transform lengths used in

GNSS receivers to process multiple signals. The proposed method of factorising

large FFT transform lengths into smaller-point FFT blocks eliminates the need for

having separate FFTs for di�erent signals. It was demonstrated that the proposed

approach of combining the small-point blocks to build the required large FFTs pro-

vides bene�ts both in terms of reduced computational complexity and increased

resource sharing. It was also shown that for the GPS L1 C/A and Galileo E1B/C

signal combinations the reduction in complexity is about 22%. The percentage re-

duction is also high for other signal combinations.

From these results it can be concluded that the proposed approach has two

advantages:

• Code acquisition with longer codes can be achieved in practice via the FFT-based method (through the use of small-point FFT blocks) without having

to implement a FFT of large transform length.


0 200 400 600 800 1000Power consumption estimate (mW)

Sig

nal C

ombi

natio

n

L1

L1+L2

L1+L2+E1+L5 +E5

E1+E5a

L1+E1

L5+E5a

L1+L2+L5

L1+L2+E1+L5 +E5a

L1+E1+L5+E5

E1+E5

Figure 7.17. Power consumption for di�erent multi-signal con�gurations

• Multi-band GNSS receivers can make use of small-point FFTs from a com-

mon set of building blocks, hence reducing the design complexity and in-

creasing the re-usability.

It was shown that apart from reducing the resource requirements due to the employ-

ment of the small-point FFT blocks, the Mixed-radix method of combining small-

point FFT blocks to build the large required FFT block also reduces the number of

additions and subtractions compared to the direct FFT computation method.

The proposed method does not depend on the acquisition architecture and hence

can be used in conjunction with the other two design category options (acquisition

architecture modi�cation and assistance information) to further reduce the compu-

tational burden. Apart from the reduction in the computational complexity, the

proposed method is also useful for resource sharing in a multi-band receiver.

It was shown that the use of the proposed FFT architecture reduces the com-

putational load (or number of processor cycles) and increases the re-usability of the

acquisition search engine to process di�erent signals. The proposed method is a

potential candidate for acquisition engines in future multi-band GNSS receivers.

This chapter also analysed the core correlator complexities of modernised GNSS

receiver baseband hardware. A core correlator architecture description has been

given and the number of bits for the accumulator has been derived. Power con-

sumption estimates were provided for the new signals at the core correlator level

and at the channel level.

7.9. SUMMARY 191

It was shown that a multi-frequency all-civil signal GPS and Galileo receiver

baseband hardware would consume approximately 38 times the power of a GPS L1

C/A baseband hardware. The dominant contributor to this increased complexity

and power consumption is the Galileo E5 AltBOC signal. This calls for a thoughtful

and e�cient design for future GNSS signals rather than the use of a sophisticated

signal only targetting system performance with less emphasis on receiver complexity,

and is the topic of next chapter.

CHAPTER 8

Time-Multiplexed O�set-Carrier QPSK for GNSS

8.1. Introduction

In this chapter a new method of time-multiplexing QPSK signals modulated by

complex sub-carriers is proposed. This chapter contains the work from a journal

paper currently under review (Shivaramaiah et al., 2010) in the IEEE Transactions

on Aerospace and Electronics Systems, and an extension of that manuscript's work

published in ION ITM 2011 (Shivaramaiah et al., 2011).

First, a detailed description of the new modulation scheme called Time-Multiplexed

O�set-Carrier (TMOC) QPSK is given, in which two QPSK signals are combined

to form the transmitted signal. The design of the proposed TMOC-QPSK aims

to address the complexity and power consumption issues that a receiver faces with

the constant-envelope AltBOC currently used for the Galileo E5 signal. The signal

structure is described, focusing on the signal generation methodology, power spec-

tral density and correlation function in comparison with the AltBOC modulation. A

method to realise the correlator for the proposed TMOC-QPSK signal is described

and its complexity is compared to AltBOC.

Next, a Time-Multiplexed Multi-Carrier (TMMC) modulation scheme is pro-

posed where the multiplexing scheme in TMOC is extended to accommodate more

than two QPSK signals. The potential bene�ts of using TMMC for a wideband

GNSS signal are discussed.

This chapter is organised as follows. Sec. 8.2 brie�y discusses the AltBOC

modulation complexities used for the Galileo E5 signal. Sec. 8.3 describes time-

multiplexing in the context of other GNSS signals as well as the general QPSK

modulation variants. Sec. 8.4 describes the proposed TMOC-QPSK technique as

well as its signal properties. Sec. 8.5 describes the correlator architecture for the

proposed modulation scheme, followed by an analysis of the architecture details, re-

source utilisation and power consumption in sec. 8.6. Sec. 8.7 describes an extension

of the TMOC-QPSK modulation, for use in wideband GNSS signals. Conclusions

are provided in sec. 8.8.

193

194 8. TIME-MULTIPLEXED OFFSET-CARRIER QPSK FOR GNSS

8.2. Complexities with the AltBOC Modulation

8.2.1. The AltBOC transmitted signal. A generic expression for the trans-

mitted signal is (OSSISICD (2010) Sec. 2.3):

SX(t) =√

2P < [sX(t) · exp (jωXt)] (8.1)

where X represents a particular signal (or a component of the signal), P is the

transmitted signal power, ωX represents the angular carrier frequency, sX(t) =

sX−I(t) + jsX−Q(t) is the complex baseband signal and < is the real value func-

tion operator. In the case of Galileo E5 AltBOC, fE5 = 1191.795 MHz and the

baseband signal is (OSSISICD, 2010)

sE5(t) =1

2√

2(eE5a−I(t) + jeE5a−Q(t)) [scE5−S(t)− jscE5−S(t− Tsc,E5/4)] +

1

2√

2(eE5b−I(t) + jeE5b−Q(t)) [scE5−S(t) + jscE5−S(t− Tsc,E5/4)] +

1

2√

2(eE5a−I(t) + jeE5a−Q(t)) [scE5−P (t)− jscE5−P (t− Tsc,E5/4)] +

1

2√

2(eE5b−I(t) + jeE5b−Q(t)) [scE5−P (t) + jscE5−P (t− Tsc,E5/4)] (8.2)

Note a set of four individual binary signals EE5 = {eE5a−I , eE5a−Q, eE5b−I , eE5b−I}are combined in the above equation. The �rst two lines of (8.2) contain the E5a

and E5b signals and the last two lines help in generating the constant-envelope and

contain 14.64% of the transmitted energy. eX is the product-signal and is given by

eX =∏

(EE5\X). The sub-carrier frequency fsc,E5 = 1/Tsc,E5 = 15.345 MHz. The

individual binary signals are comprised of the primary code, the secondary code and

the data; scE5−S and scE5−P are the special sub-carrier waveforms whose details are

given in Sec 2.3.1 of (OSSISICD, 2010). The constellation diagram of the baseband

signal in (8.2) is equivalent to that of an 8-PSK modulation. Each phase point is

determined by the pattern involving all the binary signals as well as the phase of the

sub-carrier. For the rest of this chapter, the su�x E5 is omitted unless explicitly

required.

It is interesting to note from (8.2) that in AltBOC modulation the real and imag-

inary components of the baseband signal do not directly re�ect the corresponding in-

phase and quadrature phase components of the E5a and E5b signal. In other words,

sI(t) 6= F(ea−I(t), eb−I(t)), but sI(t) = F(ea−I(t), eb−I(t),ea−Q(t), eb−Q(t), scS(t)),

and similarly sQ(t) = H(ea−I(t), eb−I(t), ea−Q(t), eb−Q(t), scS(t)) where F and H are

functions dictated by the AltBOC modulation.

8.2.2. Complexities in processing the full Galileo E5 AltBOC signal.

Since the transmitted signal energy is distributed equally among the four signals, it

8.2. COMPLEXITIES WITH THE ALTBOC MODULATION 195

Reference

Baseband Signal

Generator

Complex

Carrier

( )IFr t

Code

Loop

Filter

Code

NCO

,mz( )y t

( )y t

( )x t

0( ) [ ( ) ( ) ( )]y t y t y t y t

1

1( 1)

kT

k T

dt

1T

1T

,mz * ˆs t

* ˆs t

*

0ˆs t

Carrier

Loop

Filter

Carrier

discrimin

ator

Carrier

NCO

0 ( )y t 0,lz

2T

Code

discrimin

ator

Filter and

Resampler

(optional)

1

1( 1)

kT

k T

dt

2

2( 1)

kT

k T

dt

, 0, ,[ ]m m l mz z z z

Figure 8.1. A generalised tracking architecture for AltBOC signals

would be bene�cial if the received signal energy input to the acquisition or tracking

processes can be maximised by combining the individual signal components. There

are several ways to combine the signal components, each with its own advantages and

disadvantages (for details see Chapter 5). Fig. 8.1 shows an architecture where the

reference baseband signal dictates the signal component(s) tracked. The input to the

acquisition/tracking stage is assumed to be a wideband signal (> 50 MHz) centred

around 1191.795 MHz. The following is a summary of the key items that contribute

to the complexity and power consumption requirements of AltBOC receivers:

• To generate the AltBOC sub-carrier waveform scS(t) or the code modulated

sub-carrier waveform s(t), at least 4-bit amplitude quantisation is required

for a su�ciently accurate representation. This is because the values in

the sub-carrier waveform have four levels {±1.2071, ±0.5}. In addition,

due to unequal spacing between the amplitudes, anything less than 4-bit

quantisation will result in correlation loss. As mentioned in the sec. 2.11

section, using a 1-bit (two level) sub-carrier, such as those used in BOC

modulation, as a reference sub-carrier will incur about 1.38 dB correlation

loss (Soellner and Erhard, 2003).

• Multiple bits used for the sub-carrier in turn generate additional require-

ments (dealt with in detail in Sec. 8.6). First, mixing the code (or tiered

code while tracking) with the sub-carrier to generate the local reference sig-

nal requires a Look-Up-Table (LUT) instead of simple XOR logic. Second,

the reference signal shift register must carry more than 1 bit, and hence code

mixers demand either large LUTs or multipliers with adders/subtractors.

This is true even if the AltBOC code-plus-sub-carrier replica is gener-

ated like an 8-PSK constellation (OSSISICD, 2010) with output values of

{0,±0.7071,±1} which requires 4 bits. Third, the bit-width requirement

increases in the case of the wideband (or full) signal frequency domain code

acquisition technique.


e1(tk) e2(tk)

sc1(t)

s(t)

e1(tk+1) e2(tk+1)

s1(tk) s2(tk)

s(t)

s1(tk+1) s2(tk+1) sc1(tk) sc2(tk)

e(t)

s(t)

sc1(tk+1) sc2(tk+1)

t t

t

(a) (b) (c)

tk = t - kTM TM is the multiplex interval

Figure 8.2. Time-multiplexing methods to construct the basebandsignal (a) spreading codes with optional data are time-multiplexed;(b) sub-carriers are time-multiplexed; and (c) spreading codes withsub-carriers are time-multiplexed

• Even if the computational resources (logic gates in hardware or processor

load in software) can be provided in a receiver these complexities result in

a signi�cant increase in the power consumption per correlation compared

to that of a BOC modulation (as described in Chapter 7).

Since most of the parameters that trigger the complexities are related to making

the phase-multiplexing a constant-envelope signal, the next section revisits time-

multiplexing techniques to explore possibilities of creating a signal that would permit

a reduction in receiver complexity.

8.3. Time-Multiplexed Modulations

This section discusses existing time-multiplexed modulations. A �time-multiplexed

modulation� within the scope of this chapter refers to the modulation in which the

transmitted signal is constructed by time-multiplexing some or all of the compo-

nents that constitute the signal. Further, the time-multiplex operation could have

taken place at any one or more stages of the signal construction.

8.3.1. Existing time-multiplexing methods. A transmitted ranging signal

S(t) in general comprises of at least one independent stream of primary spreading

code c(t) and a carrier exp(jωt). Additional streams of primary spreading codes,

secondary or overlay codes, sub-carriers sc(t) and data d(t) may be present in the

signal. Usually, the primary code and secondary (or overlay) codes are combined

to form a tiered code and hence in this chapter, the notation c(t) is retained to

represent the tiered code as well (and distinguished appropriately when required).

As mentioned in the previous section, c(t) & d(t) constitute e(t) and the baseband

signal s(t) is constructed using e(t) and sc(t).

8.3.1.1. Time-multiplexing during the baseband signal generation. Fig. 8.2 shows

three possible time-multiplexing methods for constructing a GNSS baseband signal,

TM representing the multiplex interval. In Fig. 8.2(a), two spreading codes with

optional data are time-multiplexed and combined with the optional (the dashed box)

8.3. TIME-MULTIPLEXED MODULATIONS 197

sub-carrier. GPS L2C modulation is an example of this method with TM = Tc, the

chip period (there is no overlay code - one signal carries data, and no sub-carrier

is used). In Fig. 8.2(b), a single spreading code bit stream is combined with the

time-multiplexed sub-carrier. GPS L1C modulation is an example of this method

with TM = Tc, but the multiplex operation occurs at irregular intervals (but with

a repeated pattern of irregularity). Fig. 8.2(c) generalises (a) and (b), s1(t) and

s2(t) appropriately representing the code and sub-carrier combinations. In addition,

8.2(c) also accommodates two independent codes combined with two independent

sub-carrier streams. This last option is discussed later in detail.

8.3.1.2. Time-multiplexing during the carrier modulation. In general, the base-

band signal in GNSS is phase modulated onto the carrier using the Phase Shift

Keying (PSK) technique. A signal with a single component uses BPSK, and those

with two signal components (or two signals modulated on to the same carrier) use

QPSK. For example, the C/A code and the P(Y) codes on the GPS L1 carrier are

modulated onto the quadrature phases of the carrier, and the GPS L5 signal also

uses quadrature phases for its pilot and data signal components.

To deal with more than two signals either the number of phase points can be in-

creased or some other technique must be used. One such technique for multiplexing

three signals moves the signal energy of the third signal away from the centre such

that it does not overlap the dominant lobe of the �rst two signals. This spectral

separation requirement is one of the main reasons for adopting BOC modulation in

GNSS. The M code on the GPS L1 frequency is an example of this third signal.

For the M code the BOC (10,5) modulation was preferred mainly to enable inde-

pendent control of the power level of the L1M signal without a�ecting the C/A and

P(Y) signals. A complicated situation is encountered in situations when more than

three signals have to be combined without compromising the spectral separation

performance (Dafesh and Cahn, 2009).

Using additional frequency spectrum may not always be possible (due to fre-

quency availability or bandwidth and signal design constraints), in which case in-

creasing the number of phase points on the same carrier becomes inevitable. The

Galileo E5 signal faced this situation initially (Lestarquit et al., 2008) while trying

to combine four signal components. One of the design requirements was to exploit

the available bandwidth (~50 MHz) and at the same time ensure QPSK-like demod-

ulation of both E5a (which would allow the receivers to process E5a with GPS L5

(ISGPS705, 2010)) and E5b signals. Hence using more phase points by employing

the AltBOC modulation was the preferred choice.

In communication systems, a variant of the QPSK known as π4-QPSK has been

used (Proakis, 2000; Rappaport, 2002) to combine two QPSK signals. This is an


π/4S1=(0,0)

S2=(0,1)

S2=(1,1) S2=(1,0)

S1=(0,1)

S1=(1,1)

S1=(1,0)

may be alternate symbols of a single bit streamS1 S2and

two independent bit streamsS1 S2and

OR t

S1k

S2=(0,0)

S2k S1k+1 S2k+1 S1k+2

S2k S2k+1 S1k+2

S1k S1k+1 S1k+2

cos[ωt+Φ(S1k)] cos[ωt+Φ(S2k)] cos[ωt+Φ(S1k+1)] cos[ωt+Φ(S2k+1)]

Ts

Ts Modulated Carrier

t

Figure 8.3. Illustration of phase points in π4-QPSK modulation

alternative option to 8-PSK to obtain eight phase points. In π4-QPSK, the two sym-

bol (two-bit) streams that use QPSK have a carrier phase o�set of π4radians and a

time o�set of half a bit between them. As a result the constellation diagram resem-

bles that of the 8-PSK. Fig. 8.3 shows the phase points in a π4-QPSK modulation.

Observe that the two symbols can come from the same bit stream or di�erent bit

streams. It can be inferred from the �gure that the π4-QPSK modulation can be

considered as a time-multiplexed modulation with the time-multiplexing operation

performed during the carrier modulation. An advantage of the π4-QPSK modulation

over the 8-PSK modulation is that the successive phase change between any two

symbol transitions is limited to 135◦ instead of 180◦. As a consequence, zero cross-

ings are avoided, and hence the higher frequency components in the signal spectrum

are suppressed. This chapter makes use of a concept similar to π4-QPSK modulation,

but using sub-carriers instead of carriers.

8.3.2. Time-multiplexing example - GPS L2C. One of the constraints for

the GPS L2 civilian signal design was to place both the data and pilot components

of the signal onto the same phase of the carrier (Cheung et al., 2001; Fontana

et al., 2001). This resulted in combining the L2-CM (data carrying) and the L2-

CL (pilot) baseband signal components via a time-multiplexing method followed

by carrier modulation to form the GPS L2C signal (ISGPS200E, 2010). Other

constraints such as the spectral separation between GPS L2 M and the GPS L2C

signals also in�uenced the signal design, but the time-multiplexing methodology and

its consequences are more important in the context of this chapter.

The L2-CM and L2-CL signals are multiplexed chip-by-chip, each having a chip-

ping rate of 511.5 kHz (Fig. 8.4). Acquisition and tracking techniques have been

described in Fontana et al. (2001). It is shown in Fontana et al. (2001) that the

reduction in the chipping rate does not a�ect the code tracking performance of the

signal.

8.3. TIME-MULTIPLEXED MODULATIONS 199

CMk CLk CMk+1 CLk+1

CMk

CLk

CMk+1

CLk+1CLk-1

CMk+2

511.5 kHz CM Code eL2CM (t)

511.5 kHz CL Code eL2CL (t)

1.023 MHz L2C Code sL2C (t)

Figure 8.4. Code-multiplexing in GPS L2C signal

The baseband L2C signal can be represented in the time domain as

eL2C(t) =+∞∑

k=−∞

[cL2CM,|k|LL2CM

dL2CM,[k]NL2CMpTc,L2C/2 (t− kTc,L2C)

+cL2CL,|k|LL2CLpTc,L2C/2 (t− kTc,L2C + Tc,L2C/2)

](8.3)

=sL2C(t)

where cX,k is the kth chip of the spreading code; LX is the spreading code length;

dX,k is the kth symbol of the navigation data; NX is the number of code chips per

data symbol; Tc,X is the code chipping rate of the resultant signal (= twice the code

chipping rate of the two signal components); pT (t) is the rectangular function which

is unity from 0 < t < T and zero elsewhere; X represents the signal components L2-

CM and L2-CL in this case. Observe that the �nal baseband signal sL2C(t) = eL2C(t)

since there is no sub-carrier modulation onto the binary signal eL2C(t).

8.3.3. Time-multiplexing example - GPS L1C. The GPS L1C signal was

proposed for use in Block III satellites (Avila-Rodriguez et al., 2008) under an agree-

ment between the GPS Program O�ce and the Galileo Program O�ce. The com-

mon criterion for the Galileo E1 open service signal and the GPS L1C signal was to

transmit the signal with similar spectral characteristics using the Multiplexed Binary

O�set Carrier (MBOC) modulation technique. The mutually agreed MBOC modu-

lation dictates a power spectrum formed by combining a BOC(1,1) and a BOC(6,1)

signal with power ratio of 10/11 and 1/11 respectively (Avila-Rodriguez et al., 2008;

ISGPS800, 2010) to form the �nal signal. In the case of the Galileo E1 signal this

power requirement is handled by multiplexing the BOC(1,1) and BOC(6,1) signals

with appropriate power levels and is called a Composite BOC (CBOC) signal. More

discussions on CBOC modulation can be found in OSSISICD (2010).

Of interest in the context of this chapter is the GPS L1C signal. The pilot com-

ponent of this signal, L1C-P, adopts Time-multiplexed BOC (TMBOC) to achieve

the power spectrum properties of the MBOC modulation1(ISGPS800, 2010). Since

1Detailed theoretical background for the TMBOC and the related time multiplexing BOC modu-lations are available in Hegarty et al. (2004); Spilker (2010).


L1CPm L1CPm+2

COk COk+1 COk+2

L1CP TMBOC Sequence

COk+3 COk+4 COk+5L1CO Overlay Code (1800 bits @100 bps)

10 ms

33 Chips

L1CP Code (10230 chips @1.023 Mbps)

Time Multiplexed Subcarrier BOC (1,1) Subcarrier

BOC(6,1) Subcarrier

0 4 6 29

XOR operation

L1CPm+1

Figure 8.5. The proposed L1C pilot code generation scheme usingthe TMBOC technique

the L1C-D channel uses BOC(1,1) modulation with 25% of the total power, the L1C-

P, which uses 75% of the power employs BOC(6,1,4/33) for the time-multiplexing

scheme to achieve the desired power ratio between BOC(1,1) and BOC(6,1). Fig.8.5

illustrates the code generation scheme for the L1C-P signal. In this particular time-

multiplexing scheme the L1C-Overlay code is modulated with the time-multiplexed

sub-carrier. The time-multiplexed sub-carrier comprises 4 cycles of the BOC(6,1)

sub-carrier and 29 cycles of the BOC(1,1) sub-carrier for every 33 chips.

8.4. Time-Multiplexed O�set-Carrier QPSK : The Signal Structure

This section introduces the proposed TMOC-QPSK modulation. The signal

structure is explained by referring the task of combining four signal components. The

Galileo E5 signal components E5a-I, E5a-Q, E5b-I and E5b-Q, denoted henceforth

in this chapter as aI,aQ,bI and bQ respectively, are used as examples for the four

signal combination task.

8.4.1. Signal generation. Since two signals can be combined with QPSK

modulation, the task is now to formulate these two signals from the four that are to

be combined. A straightforward way is to combine two of these signals via a time-

multiplexing method. Under the constraint that the I and Q signal components

should occupy the in-phase and quadrature phase of the carrier and that the phase

relation between I and Q signal components should be identical for a and b signals,

there are two possibilities:

• I = TI(aI, bI) and Q = TQ(aQ, bQ) referred to as �TMOC-QPSK-ab� : in

this case, aI and aQ components are transmitted on in-phase and quadra-

ture phase respectively in the �rst (even) slot , bI and bQ components are

transmitted on in-phase and quadrature phase respectively in the second

(odd) slot; and

mailto:@1.023

8.4. TIME-MULTIPLEXED OFFSET-CARRIER QPSK : THE SIGNAL STRUCTURE 201

eP1(tk) eP2(tk) eP1(tk+1) eP2(tk+1)

Subcarrier

TC

ePx(t)

scL(tk) scH(tk) scL(tk+1) scH(tk+1)

eP1(tk) eP2(tk) eP1(tk+1)ePx(t) eP2(tk+1)

scL(tk)Subcarrier

TC

s(t)= sI(t) +j sQ(t)

scH(tk) scL(tk+1) scH(tk+1)

(a) TMOC-QPSK-ab

(b) TMOC-QPSK-IQ

(c) Subcarrier Waveform

eD1(tk) eD2(tk) eD1(tk+1)eDy(t) eD2(tk+1)

scL(tk)Subcarrier

TC

scH(tk) scL(tk+1) scH(tk+1)

eD2(tk) eD1(tk) eD2(tk+1) eD1(tk+1)

Subcarrier

TC

eDy(t)

scH(tk) scL(tk) scH(tk+1) scH(tk+1)

scL(t) = sc(t) – j sc(t-Tsc/4) = scI(t) – j scQ(t)

scH(t) = sc(t) + j sc(t-Tsc/4) = scI(t) + j scQ(t)

x,y {1,2}

x,y {1,2}


sc(t)1

-1

0Tsc

sco(t)1

-1

0TSC

(d) π/4 phase shifted Subcarrier

Tsc /2 Tsc /2Tsc /8

Figure 8.6. Time-multiplexing methods and the corresponding sub-carrier waveform: (a) TMOC-QPSK-ab multiplexing method; (b)TMOC-QPSK-IQ; (c) one cycle of sub-carrier waveform

• a = Ta(aI, aQ) and b = Tb(bI, bQ) referred to as �TMOC-QPSK-IQ�: in

this case, aI and bI components are transmitted on in-phase and quadra-

ture phase respectively in the �rst (even) slot, bQ and aQ components are

transmitted on in-phase and quadrature phase respectively in the second

(odd) slot;

where Ta, Tb, TI , TQ are the functions representing the time-multiplexing operations

with the requirement of an ordered pair of signal components as the parameters. In

order to avoid low frequency inter-modulation components, chip-by-chip multiplex-

ing has been used in the case of the L2C signal, and the same approach is adopted

for the proposed time-multiplexing method. In addition it is assumed that all four

signal components possess similar properties (code length, chipping rate and indi-

vidual signal transmit power) as in the case of the Galileo E5 signal components.

This allows the use of an identical time-multiplexing method for both the groupings

i.e. Ta= T b = TI = TQ = T . (Note: It not a necessary condition that all four signal

components possess similar properties but further discussion is beyond of the scope

of this thesis). Fig. 8.6 shows the two multiplexing schemes in detail, tk is used as a

time parameter to represent one chip duration of the corresponding signal. It is clear

that the TMOC-QPSK-ab method would result in an alternating spectrum corre-


sponding to the a and b signals. In other words, a and b signals, when considered

individually, will have discontinuities in the time domain. However the duration of

this discontinuity depends on the chipping rate. When the chipping rate is high,

for example 10.23 MHz as in the Galileo E5a or E5b signal, the RF front-end can

still see the signal as nearly-continuous because the typical Automatic Gain Control

(AGC) time constant is much higher than 97.75ns (i.e. 110.236

) (Zarlink, 1999). In the

TMOC-QPSK-IQ method, the pilot signals are time-multiplexed with each other

(the other group has data signals).

8.4.1.1. TMOC-π4-QPSK : a variant of the TMOC-QPSK. It is possible to make

use of eight phases of the sub-carrier with the TMOC-QPSK method. In TMOC-π4-QPSK, the time-multiplexing scheme is exactly the same as the corresponding

TMOC-QPSK method but the sub-carrier phase assignments for the signal compo-

nents in odd slots are shifted by π4. This is equivalent to advancing the sub-carrier

phase by a factor of π4, i.e. sco(t) = sc(t). exp(j π

4) = sc(t + Tsc

8) (see Fig. 8.6). The

advantage of TMOC-π4-QPSK is that successive sub-carrier phase changes between

the even and odd chips are limited to 135 degrees instead of 180 degrees for the

TMOC-QPSK method.

8.4.2. Signal representation, phase assignments and the transmitted

signal. The generalised complex baseband TMOC-QPSK signal for a combination

of four signal components (assuming two data and two pilot signal components) can

be written as

s(t) =1

2√

2eP1(t)

[sc(t)− jsc(t− Tsc

4)

]+

1

2√

2· jeD1(t)

[sc(t)− jsc(t− Tsc

4)

]+

1

2√

2eP2(t)

[sc(t) + jsc(t− Tsc

4)

]+

1

2√

2· jeD2(t)

[sc(t) + jsc(t− Tsc

4)

](8.4)

where

ePx(t) =+∞∑

k=−∞

cPx,|k|LCpTM/2 (t− kTM) , x ∈ {1, 2}

eDy(t) =+∞∑

k=−∞

cDy,|k|LCdy,|k|NC pTM/2 (t− kTc + TM/2) , y ∈ {1, 2} (8.5)

where e• represents the spreading code with optional data, su�xes P and D denote

the pilot and data respectively, sc(t) denotes the sub-carrier with period Tsc, TM is

the multiplex interval and Tc is the chip period. Observe from (8.4) and (8.5) that

the TMOC-QPSK-ab is achieved when x = y and TMOC-QPSK-IQ is achieved

when x 6= y. In the case of TMOC-π4-QPSK, s(t) should be replaced with sco(t).

The transmitted signal equation and the constellation diagram are shown in Fig. 8.7.


Time Slot

Even Chip

Odd Chip

Transmitted Signal S(t)

sI(t) cos(jωct+Φ) - sQ(t) sin(jωct+Φ)


SubcarrierUsed

(a) TMOC-QPSK

sc(t)

sc(t)

Time Slot

Even Chip

Odd Chip

Transmitted Signal S(t)



SubcarrierUsed

(b) TMOC-π/4-QPSK

sc(t)

sc(t+Tsc/8)

Figure 8.7. TMOC-QPSK and TMOC-π4-QPSK transmitted signal

Table 8.1 lists the phase assignments that results from a TMOC-QPSK modulation.

It can be observed that only four phases are used due to the time-multiplexing

method.

8.4.2.1. Nomenclature. In line with the AltBOC modulation, the TMOC-QPSK

modulation described in this chapter is denoted as TMOC-QPSK(m,n), e.g. TMOC-

QPSK(15,10) meaning fsc = 15.345 MHz and fco = 10.230 MHz.

8.4.3. Power spectral density. The power spectral density (PSD) of a constant-

envelope AltBOC signal is given in Rebeyrol et al. (2005), and repeated here for com-

pleteness. For 2fscfco

odd, the normalised PSD of the constant-envelope AltBOC(m,n)

is given by

GAltBOC(f) =4

π2f 2Tc

cos2(πfTc)

cos2(πf Tsc2

)(8.6)[

cos2

(πf

Tsc2

)− cos

(πf

Tsc2

)− 2 cos

(πf

Tsc2

)cos

(πf

Tsc4

)+ 2

]The TMOC-QPSK essentially contains a sine-phased sub-carrier and a cosine-phased

sub-carrier. This is because (see Fig. 8.6) if the in-phase sub-carrier component

scI(t) corresponds to the cosine sub-carrier, scQ(t) = sc(t − Tsc4

) corresponds to

the sine sub-carrier. In other words, this is equivalent to the spectrum of a non-

constant-envelope AltBOC (AltBOC-NCE). The PSD of a TMOC-QPSK signal can

be obtained by combining the PSDs of Sine-BOC and Cosine-BOC (Betz, Winter

2001-2002; Rebeyrol et al., 2005). Thus for 2fscfco

odd the PSD of TMOC-QPSK(m,n)

is given by (Appendix J):

GTMOC−QPSK(f) =4

π2f 2Tc

cos2(πfTc)

cos2(πf Tsc2

)

[sin2

(πf

Tsc2

)+

{cos

(πf

Tsc2

)− 1

}2]

(8.7)

For the TMOC-π4-QPSK signal, the sub-carrier in the alternate slots are phase

shifted by π4w.r.t their phases in the preceding slots. The equation for the PSD of a

204 8. TIME-MULTIPLEXED OFFSET-CARRIER QPSK FOR GNSSTable8.1.Tran

smitted

sub-carrier

signal

phases

intheprop

osedmodulation

Meth

od

TMOC-Q

PSK-IQ

/TMOC-π4 -Q

PSK-IQ

TMOC-Q

PSK-ab

/TMOC-π4 -Q

PSK-ab

aI

11

-1-1

--

--

11

-1-1

--

--

bQ1

-11

-1-

--

--

--

-1

-11

-1bI

--

--

11

-1-1

--

--

11

-1-1

aQ

--

--

1-1

1-1

1-1

1-1

--

--

Even

chip

sc=1

π47π4

3π4

5π4

--

--

π47π4

3π4

5π4

--

--

sc=-1

5π4

3π4

7π4

π4-

--

-5π4

3π4

7π4

π4-

--

-Oddchip

sc=1

--

--

π47π4

3π4

5π4

--

--

π47π4

3π4

5π4

sc=-1

--

--

5π4

3π4

7π4

π4-

--

-5π4

3π4

7π4

π4

Oddchip

sc=1

--

--

π22π

π3π2

--

--

π22π

π3π2

π4 -QPSK

sc=-1

--

--

3π2

π2π

π2-

--

-3π2

π2π

π2

Even

chip

constella-tion

E5aI

E5bI

E5aI

E5aQ

Oddchip

constella-tion

E5bQ

E5aQ

E5bI

E5bQ

Oddchip

constellation

π4 -QPSK

E5bQ

E5aQ

E5bI

E5bQ


−100 −50 0 50 100−110

−105

−100

−95

−90

−85

−80

−75

−70

−65

−60

Frequency (MHz)

Pow

er/fr

eque

ncy

(dB

/Hz)

AltBOCAltBOC−NCETMOC−QPSKTMOC−pi/4−QPSK

Figure 8.8. PSD of the AltBOC, AltBOC-NCE, TMOC-QPSK andTMOC-π

4-QPSK; analytical

TMOC-π4-QPSK(m,n) signal when 2fsc

fcois odd (derived in Appendix J) is given by:

GTMOC−QPSK(f) =2

π2f 2Tc

cos2(πfTc)

cos2(πf Tsc2

)

[sin2

(πf

Tsc2

)+

{cos

(πf

Tsc2

)− 1

}2

+ cos2

(πf

Tsc2

)− 2 cos

(πf

Tsc2

)cos

(πf

Tsc4

)+ 1

](8.8)

Fig. 8.8 plots (8.6),(8.7) and(8.8) for the case with fco = 10.23 MHz and fsc = 15.345

MHz (i.e. 2fscfco

= 3). In Fig. 8.9 the PSD is plotted for the three types of signals

simulated using the GIOVE-B primary and secondary codes. It is interesting to note

that the magnitude spectrum of the AltBOC(15,10) and the TMOC-π4-QPSK(15,10)

match very closely; this is because both the modulations are making use of eight

phases. However, as explained in the previous section, in TMOC-π4-QPSK all the

spectrum energy is due to the four component signals only (i.e. does not include

any product signals).

8.4.4. Transmit power level. It is evident from the previous sections that it

is possible to keep the total transmit power level for the TMOC-QPSK modulation

the same as the AltBOC modulation, and hence at the receiver utilise the complete

transmit power - as opposed to AltBOC which loses 15% due to the product signal

component. Table 8.2 compares the transmit power levels of AltBOC and TMOC-

QPSK signals assuming that the wideband AltBOC signal transmitted power is

100%.


−100 −50 0 50 100−100

−95

−90

−85

−80

−75

−70

−65

−60

Frequency (MHz)

Pow

er/fr

eque

ncy

(dB

/Hz)

Welch Power Spectral Density Estimate


Figure 8.9. PSD of the AltBOC, AltBOC-NCE, TMOC-QPSK andTMOC-π

4-QPSK; simulation

Table 8.2. Comparison of relative transmit signal power levels

Signal

Component /

Modulation

AltBOC TMOC-QPSK-ab TMOC-QPSK-IQ

Wideband

Signal

100% 100% 100%

aI 21.34% 25% (even chip

50%, odd chip

0%)

25% (even chip 50%,

odd chip 0%)

aQ 21.34% 25% (even chip 50%,

odd chip 0%)

25% (even chip 0%,

odd chip 50%)

bI 21.34% 25% (even chip 0%,

odd chip 50%)

25% (even chip 0%,

odd chip 50%)

bQ 21.34% 25% (even chip 0%,

odd chip 50%)

25% (even chip 50%,

odd chip 0%)

a (aI and aQ) 42.68% 50% (even chip 100%,

odd chip 0%)

50% (even chip 50%,

odd chip 50%)

b (bI and bQ) 42.68% 50% (even chip 0%,

odd chip 100%)

50% (even chip 50%,

odd chip 50%)

both a and b 85.36% 100% 100%

8.5. CORRELATOR ARCHITECTURE FOR THE TMOC-QPSK SIGNAL 207

Carrier Generator

( )IFr t

( )IF dj te

Carrier Mixer

Code Mixer

Ref. signal Generator

ˆ( )s t

ˆ( )ks t 1

ˆ( )ks t

( )kx t1( )kx t ( )x t

ˆ( )s t

( ) ( )k ks t n t( )x t contains 1 1( ) ( )k ks t n t

Code mixer input

Reference signal

( )x t

2Tc

Even/Odd Chip Select (EOCS)

( )y t I Accumulator Q Accumulator

Corr. Values

Figure 8.10. Correlator architecture to process the widebandTMOC-QPSK signal

8.5. Correlator Architecture for the TMOC-QPSK Signal

This section describes the correlator architectures for processing the TMOC-

QPSK signal and compares the correlator output with that of the AltBOC modu-

lation, both for wideband processing and sideband processing.

8.5.1. Correlator architecture : wideband processing. Fig. 8.10 shows

the architecture to process the wideband TMOC-QPSK signal, with all lines carrying

complex signals. The signal at IF plus associated Doppler is mixed with the local

carrier to obtain the baseband signal x(t), which comprises two sidebands separated

at equal spacings from the centre (zero frequency plus any residual Doppler) and

carries both a and b signal components. The reference signal generates s(t− τ) and

comprises the code and the sub-carrier. The Even/Odd Chip Select (EOCS) is a

gating signal of duration Tc used to select the reference signal a or b. The complex

code mixer multiplies the signal to produce y(t) which is fed to the accumulators.

The detailed implementation of the reference signal generator and code mixer is

described in the next section.

8.5.2. Correlator output and the correlation waveform : wideband

processing. The received IF signal for a particular satellite employing a complex

sub-carrier modulation (AltBOC or TMOC-QPSK) can be written as:

rIF (t) =√

2Pr [sI(t− τ) cos (ωIF t+ ωdt+ θ)− sQ(t− τ) sin (ωIF t+ ωdt+ θ)]+nW (t)

(8.9)

where Pr = A2

2is the power, τ is the delay (s), θ is the phase (rad) and ωd is

the Doppler frequency (rad/s) of the received signal and nW is the additive white

Gaussian noise with two-sided power spectral density N0

2W/Hz. The carrier signal


√2 exp

[−j(ωIF t+ ωdt+ θ

)]generated in the receiver with the estimated Doppler

ωd and the estimated phase θ is mixed with rIF (t) to obtain:

x(t) =A√2{[sI(t− τ) cos (∆ωdt+ ∆θ)− sQ(t− τ) sin (∆ωdt+ ∆θ)]

j [sQ(t− τ) cos (∆ωdt+ ∆θ) + sQ(t− τ) sin (∆ωdt+ ∆θ)]}+ nI(t) + jnQ(t)

(8.10)

where ∆ωd = ωd − ωd and ∆θ = θ − θ . The local reference signal generated is

s(t− τ) = sI(t− τ)− jsQ(t− τ). The signal s(t) di�ers from s(t) due to the absence

of secondary codes and the data. However, assuming that the correlation opera-

tion is considered within one secondary code period (or data bit period whichever

is minimum) allows one to write s(t− τ) = s∗(t− τ) where ∗ denotes the conjugateoperation (see Chapter 5). The output of the reference signal mixer y(t) is accumu-

lated (typically over one code period duration) to obtain the correlation result. The

output of the accumulator is (see Appendix K for the derivation):

zk+ =√

2ATcohR (∆τ)Rsc (∆τ) sinc

(∆ωd

Tcoh2

)ej∆θ +R′ + n′ (t) (8.11)

where R (∆τ) denotes the autocorrelation of the codes (all four codes are assumed

to have equal autocorrelation values), Rsc (∆τ), the sub-carrier correlation denotes

the combined correlation of the sub-carrier component pairs, R′ contains the cross

correlation noise contributed by all the code pairs, n′(t) contains the noise terms

obtained via the mixing operation of nI(t) + jnQ(t) with s∗(t− τ), Tcoh = LTc is the

coherent integration duration and L is the primary code length.

For the TMOC-QPSK and the TMOC-π4-QPSK modulations the equation for

the output of the correlator remains the same as in (8.11) except that the sub-

carrier correlation is di�erent. The number of autocorrelation and the number of

cross correlation terms remain the same. The sub-carrier correlation for the AltBOC

modulation Rsc,AltBOC (∆τ) is provided in (K.5) in Appendix K. It is the task now to

derive the sub-carrier correlation for the two modulations. The complete derivations

of all the terms have been carried out individually for all the three modulations, but

only the results are discussed here.

8.5.2.1. Incoming and reference sub-carrier correlations in TMOC-QPSK. In

this case, the sub-carrier is a complex square wave, i.e. sc(t) can be written as

sc(t) = sgn (cos (ωsct)) + j sgn (sin (ωsct)). Moreover, the correlation between the

sub-carrier phases are equal because integration of the product of both the cosine and

sine square wave functions yields a triangular function. Following the same proce-

dure as for the AltBOC modulation described in Appendix K, Rsc,TMOC−QPSK (∆τ)


is found to be:

Rsc,TMOC−QPSK (∆τ) = 8RscI ,scI (∆τ) (8.12)

8.5.2.2. Incoming and reference sub-carrier correlations in TMOC-π4-QPSK. In

TMOC-π4-QPSK, there are essentially two sets of sub-carriers. The �rst set used

in even numbered slots is the same as the sub-carriers used in TMOC-QPSK. The

second set used in the odd numbered slots is o�set by π4rads with respect to. the

�rst set and is given by sco(t) = sgn(cos(ωsct+ π

4

))+ j sgn

(sin(ωsct+ π

4

)). The

sub-carrier correlation contains the correlation due to eight pairs RscI ,scI , RscQ,scQ ,

RscoI ,scoI , RscoQ,scoQ , RscI ,scoI , RscQ,scoQ , RscoI ,scI and RscoQ,scQ . Since the correlation

depends on the relative phase shift, the �rst four pairs are the same. In the last four

pairs, there are two types, one with the phase advance and the other with the phase

delay, and hence result in corresponding phase shifted triangular waves. Finally,

Rsc,TMOC−π4−QPSK (∆τ) is found to be:

Rsc,TMOC−π4−QPSK (∆τ) = 4RscI ,scI + 2RscI ,scoI + 2RscoI ,scI (8.13)

The sub-carrier correlations for all three types of modulations are plotted in Fig.

8.11. Rsc,AltBOC (∆τ) is the correlation between two quantised cosine waves and re-

sembles a smoothed triangular wave. Rsc,TMOC−QPSK (∆τ) is the correlation of pairs

of square waves and is a triangular wave. Rsc,TMOC−π4−QPSK (∆τ) not only resembles

a smoother triangular wave, but also closely matches the sub-carrier correlations of

the AltBOC modulation. The Alternate Linear O�set Carrier (AltLOC) uses Sine

and Cosine sub-carriers and hence the cosine waveform (which is the correlation be-

tween incoming and the reference sub-carrier) is shown for reference. Observe that

the cosine function serves as a good approximation for all three sub-carrier types.

8.5.3. Correlator output waveform : wideband processing. Fig. 8.12

shows the normalised auto-correlation waveforms of the modulations under consid-

eration. Observe that the auto-correlation waveform of TMOC-QPSK matches that

of the AltBOC-NCE and the auto-correlation waveform of TMOC-π4-QPSK matches

that of the AltBOC. At the boundaries (close to +1 or -1 chip shifts) the AltBOC

and TMOC-π4-QPSK exhibit peaks of very low magnitude which are absent in the

TMOC-QPSK. This is justi�ed by noting that those small peaks are also absent in

the autocorrelation waveform of AltBOC-NCE. The di�erences in autocorrelation

values among the modulations are greatly reduced when the �ltering is considered,

as shown in Fig. 8.13.

8.5.4. Correlator architecture : independent sideband processing. Be-

cause the spectrum of the TMOC-QPSK signal closely resembles the AltBOC signal,

the individual signals can be obtained following the same procedure as in the Al-

tBOC case (the following discussion applies to TMOC-π4-QPSK as well). That is,


−1.5 −1 −0.5 0 0.5 1 1.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time delay in number of subcarrier cycles

Nor

mal

ized

Cor

rela

tion

Val

ue

AltBOCTMOC−QPSKTMOC−pi/4−QPSKAltLOC

Figure 8.11. Incoming and reference sub-carrier correlations for Al-tBOC, TMOC-QPSK, TMOC-π

4-QPSK and AltLOC modulations

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Chip Shift

Nor

mal

ized

Aut

ocor

rela

tion

Val

ue


Figure 8.12. Normalised auto-correlation waveforms ofAltBOC,TMOC-QPSK,TMOC-π

4-QPSK and AltBOC-NCE; in�-

nite bandwidth; simulation


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Chip Shift

Nor

mal

ized

Aut

ocor

rela

tion

Val

ue


Figure 8.13. Normalised auto-correlation waveforms ofAltBOC,TMOC-QPSK,TMOC-π

4-QPSK and AltBOC-NCE; 50

MHz bandwidth; simulation

the signal components a and b can be obtained by translating the individual bands

to an appropriate IF (or zero-IF) and then treating them as QPSK signals. The

e�ect of time-multiplexing is that the �signal-plus-noise� and �noise-only� alternate

with every chip slot in the case of TMOC-QPSK-ab, as seen from the centre of

any one sideband. However this e�ect is absent in the case of TMOC-QPSK-IQ as

the spectrum is either always occupied by the I or the Q signal component. Fig.

8.14 shows a correlator architecture for obtaining the individual signals a and b

independent of each other; all lines carry complex signals. In Fig. 8.14(a), the re-

ceived IF version of the TMOC-QPSK-ab signal is mixed with the locally generated

carrier at a frequency of ωIF + ωd ± ωsc, where the sign used for the sub-carrier

frequency in the equation determines whether signal a is being processed or signal

b. The resultant signal x(t) contains the baseband signal (with residual Doppler)

and noise components without any sub-carrier. The signal-plus-noise component

e′a(t) = ea(t) + n(t) {or e

′

b(t) = eb(t) + n(t)} and the noise-only term n(t) alternate

between chips. The gating signal �Code Enable� (CEN) will ensure the exclusion of

the noise-only component during the correlation process.

The correlator in Fig. 8.14(b) is suitable for the TMOC-QPSK-IQ modulation

and follows the same process as in the case of TMOC-QPSK-ab for the carrier re-

moval. The di�erence would be in the code mixing stage where individual I and

Q code enable signals (ICEN and QCEN respectively) are used to enable the cor-

responding code mixing and the subsequent accumulation path. Observe from Fig.


Carrier Generator

( )IFr t

( )IF d scj te

Carrier Mixer

Code Mixer

Code Generator

( )c t

( ) ( ) ( )( ) ( ) ( )

I Qc t c t jc te t e t n t

Code Enable (CEN)

I Accumulator Q Accumulator

Carrier Generator

( )IFr t

( )IF d scj te

Carrier Mixer

Code Mixer

Code Generator

( )c t ICENQCEN

ICEN = I Code Enable QCEN = Q Code Enable(a) for TMOC-QPSK-ab (b) for TMOC-QPSK-IQ

( )kc t 1( )kc t

1

0

1

0

( )kx t1( )kx t ( )x t

( )c t

( )xI ke t( )x t contains ( )xQ ke t1( )xI ke t

1( )xQ ke t

Code mixer input

Reference code

ICEN

QCEN

( )kc t 1( )kc t

1

0

( )kx t1( )kx t ( )x t

( )c t

( )x ke t( )x t contains ( )kn t1( )x ke t

1( )kn t

Code mixer input

Reference code

CEN

‘x’ represents component ‘a’ if -ωsc is used or component ‘b’ if +ωsc is used

( )x t ( )x t

2Tc

2Tc

Corr. Values( )y t ( )y t I Accumulator

Q Accumulator

Corr. Values

Figure 8.14. Correlator architecture to process the individual signalcomponents (a and b) in TMOC-QPSK modulation

8.14 that for both types of modulations the code generation clock cycle requirement

is 1/2Tc. As an example, a 10.23 MHz code chipping rate requires a code generation

(or memory read in the case of stored codes) module clocked at 511.5 kHz. Fig.

8.15 shows the output of the correlator obtained through simulation (without any

added noise), and note that the waveform matches that of a BPSK modulation.

8.6. Resource Utilisation and Power Consumption

8.6.1. Architecture details: reference signal generator. Since the TMOC-π4-QPSK modulation is a variant of the TMOC-QPSK modulation, the following dis-

cussion uses the generic term �TMOC-QPSK modulation� to refer to both TMOC-

QPSK and TMOC-π4-QPSK modulations. The two core components that di�er

between the correlator for the AltBOC modulation and the TMOC-QPSK mod-

ulation are the reference signal generator and the code mixer. The complexity

reduction in these two core components for TMOC-QPSK(15,10) compared to that

of AltBOC(15,10) is explored below.

8.6.1.1. AltBOC modulation. The tasks of the reference signal generator are to:

• generate (or read from memory) the aI, aQ, bI, bQ codes,

• generate the sub-carrier, and• combine the codes and the sub-carriers according to the modulation type

to obtain the reference �phase points� s(t− τ)

There are two categories of techniques for the combination of the code and the

sub-carrier (the third task). First, the direct (arithmetic) computation method in

8.6. RESOURCE UTILISATION AND POWER CONSUMPTION 213

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.2

0

0.2

0.4

0.6

0.8

1

Chip Shift

Nor

mal

ized

Cor

rela

tion

Val

ue

’a’ Component’b’ Component

Figure 8.15. Output of the correlator for independent sideband pro-cessing; simulation

which the complex code and the complex sub-carrier are multiplied to obtain the

complex output s(t − τ). This method is suitable for software implementations,

e.g. Matlab/C on a desktop or embedded processor. Since �oating point arithmetic

is expensive in terms of computation, scaled integer arithmetic often replaces such

computations. Fig. 8.16 shows the direct computation method of generating the

reference signal in the case of AltBOC(15,10). The �rst two troughs of the scaled

integer approximation error function result in representing the values taken by the

sub-carrier as {±2,±5} and {±5,±12} with errors of 3.5% and 0.6% respectively

(tables of di�erent representations and errors are provided in Appendix L). Hence

the complex multiplication operation results in eight sign change operations on 4 (or

5)-bit numbers and four addition/subtraction operations on 6 (or 7)-bit numbers.

Even if the �rst approximation is chosen, 6 bits are required to accommodate the

result. This is an enormous amount of computation compared, to say, a complex

BOC(m,n) modulation case where the result is 1 bit and two XOR gates (one each

for I and Q) are required to combine the code and the sub-carrier. (Note: to simplify

the discussion this model omits the product-signal components and incurs a 14.64%

loss in received signal energy; including the product-signal components would further

increase the computation requirements.)

The second method is the Look-Up-Table (LUT) method, better suited for hard-

ware implementations, e.g. in an FPGA or an ASIC. (Note: the LUT method can

also be used in software implementations especially for an embedded processor, but

often it is not the choice for higher level programming languages.) Fig. 8.17 shows


-1

Codes

+1

-0.5Subcarriers

+1.2071

0.5

-1.2071

Operation ( )( ) ( )( )aI aQ I Q bI bQ I Qc jc sc jsc c jc sc jsc

Integer Arithmetic

(Approximation #1)

Integer Arithmetic

(Approximation #2)

±1 (2 bits)

{±0.5, ±1.2071}

~= {±2, ±5}

=> 4 bits

3.5% error

±1 (2 bits)

{±0.5, ±1.2071}

~= {±5, ±12}

=> 5 bits

0.6% error

Max. I & Q result

= ±20 => 6 bits

Max. I & Q result

= ±48 => 7 bitsEight sign change operations on 4(or 5)-bit numbers,

Four add/sub operations on 6(or 7)-bit numbers

Figure 8.16. Direct computation method of AltBOC reference sig-nal generation

AltBOC

Modulat

ion

Sub-carrier

NCO

ˆ ˆ( ) ( )I Qs t js t aI

aQ

bI

bQ

cc

cc

3 MS bits

(subcarrier phase)

{0, ±0.7071, ±1}

Arithmetic Representation

(Approximation #1)

{0, ±0.7071, ±1} ~=

{0, ±2, ±3} => 3 bits

5.7% error

Arithmetic Representation

(Approximation #2)

{0, ±0.7071, ±1} ~=

{0, ±5, ±7} => 4 bits

1% error

128x3

128x3

LUT

128x4

128x4

LUT

Mapped Representation of

Approximation #2

{0, ±5, ±7}

=> 5 levels

=> 3 bits

128x3

128x3

LUT

Two 128x3 LUTs Two 128x4 LUTs Two 128x3 LUTs

Figure 8.17. LUT method of AltBOC reference signal generation

the LUT method of AltBOC signal generation. OSSISICD (2010) suggests this

method because it is comparatively simple to implement. The three most signif-

icant (MS) bits of the sub-carrier NCO (which represent the eight phases of the

sub-carrier) and the four codes act as address inputs to the LUT. The output (in-

cluding the product components) can be represented using approximated integer

arithmetic. Representing {0,±0.7071,±1} as {0,±5,±7} incurs only 1% error but

requires 4 bits (Appendix L). In most situations, the code mixer which is the next

stage is also implemented as a LUT and hence s(t− τ) which is an input to the code

mixer, does not require true value representation (see sec. (7.8)). The third option

for mapping the values can now be presented. Since there are only 5 levels, 3 bits

are su�cient to represent s(t− τ). As a result, the LUT method requires two (one

for I and one for Q) 128x3 LUTs. This requirement is also very expensive compared

to a complex BOC(m,n) case.

8.6.1.2. TMOC-QPSK modulation. Fig. 8.18 illustrates the direct computation

method of generating the reference signal. The sub-carriers in the TMOC-QPSK

modulation are two-valued and hence logical representation is possible. s(t − τ)

requires only 1 bit for the representation, and, interestingly the computation load is

just four 3-bit XOR gates (two for each I and Q). Additional two 2 bit XOR gates

are required to incorporate the EOCS signal. The LUT method of implementation

shown in Fig. 8.19 would be an overkill for the TMOC-QPSK modulation, as

it requires a 128x1 LUT. The sub-carrier phase identi�cation in TMOC-π4-QPSK


-1

Codes

+1

Subcarriers

Operation ˆ +

ˆ + I aI I bI I

Q aQ Q bQ Q

s c EOCS sc c EOCS sc

s c EOCS sc c EOCS sc

Logical

Representation

1,0 => 1-bit

I and Q result

1-bit

-1

+1

1,0 => 1-bit

0

EOCS

1

1,0 => 1-bit

Four 3-bit XOR operations, Two 2-bit XOR operations

Figure 8.18. Direct computation method of TMOC-QPSK referencesignal generation

TMOC-

QPSK

Modulat

ion

Sub-carrier

NCO

ˆ ˆ( ) ( )I Qs t js t aI

aQ

bI

bQ

cc

cc

2 MS bits

(subcarrier phase)

{±1}

Logical Representation

(Also a mapped representation)

±1 => 0,1 =>

1-bit

128x1

128x1

LUT

Even/Odd Chip

Select (EOCS)

Two 128x1 LUTs

Figure 8.19. LUT method of TMOC-QPSK reference signal generation

requires 3 MS-bits, however the amplitude is still represented using 1 bit and hence

the size of the LUT does not change.

8.6.1.3. Implications for architectures for correlation in the frequency domain.

Reducing the number of bits in the local reference signal has a huge impact on the

architectures that employ the frequency domain approach for correlation. Usually,

computing the FFT of the reference signal is required as part of the frequency domain

correlation approach. It is well known (Oppenheim and Schafer, 1998) that the

number of guard bits necessary to compensate for the maximum possible bit growth

for an N-point FFT is log2N + 1. Given that the code length of AltBOC signals is

10230, a 10230-point FFT requires 14 guard bits. A 2-bit input (representing ±1,

which would be the case for TMOC-QPSK signal) will require 16-bit words for the

FFT. On the other hand, for the AltBOC case, 20-bit words would be required and

this results in a larger footprint.

8.6.2. Architecture details: Code mixer. For a 2-bit IF signal representing

the levels {±1,±3} and a 2-bit local carrier representing {±1,±2}, the carrier mixer

output will be {±1,±2,±3,±6} requiring 4 bits for an arithmetic representation and


Code Mixer

ˆ( )s t

( )x t{±1,±2,±3,±6}

(4 bits)

Max. ±20 (6 bits)

Six Multiplications, Ten Additions/Subtractions

Max. ±240 (9 bits)

Code Mixer

ˆ( )s t

( )x t

{±1,±2,±3,±6}(8 levels, 3 bits)

{0, ±5, ±7} (5 levels, 3 bits)

Four 64x8 LUTs, Two Addition/Subtractions

Max. ±84 (8 bits)

(a) Direct (Arithmetic) Computation Method (b) LUT Method

I Accumulator Q Accumulator

Correlation Values

Correlation Values( )y t ( )y t I Accumulator

Q Accumulator

Figure 8.20. Direct computation and LUT method of code mixerimplementation in AltBOC modulation

Code Mixer

ˆ( )s t

( )x t{±1,±2,±3,±6}

(4 bits)

1-bit

Four sign change operations, Two Additions/Subtractions

Max. ±12 (5 bits)

Code Mixer

ˆ( )s t

( )x t

{±1,±2,±3,±6}(8 levels, 3 bits)

1-bit

Four 16x3 LUTs, Two Additions/Subtractions

Max. ±12 (5 bits)

(a) Direct (Arithmetic) Computation Method (b) LUT Method

I AccumulatorQ Accumulator

Correlation Values

Correlation Values( )y t ( )y t I Accumulator

Q Accumulator

Figure 8.21. Code mixing operation in TMOC-QPSK modulation

3 bits (eight levels) for a mapped representation (see Chapter 7). The direct com-

putation method and the LUT method of code mixing for the AltBOC modulation

are shown in Fig. 8.20. If the reference signal generation uses an arithmetic rep-

resentation then the code mixer must use the direct computation method, and in

that case the code mixer requires six multiplications and ten addition/subtraction

operations (two complex multiplications, one each for I and Q). In the case of the

LUT method, four 64x8 tables are required. The output of the code mixer, y(t)

grows to 8 bits, thus resulting in wider accumulators to obtain the correlation val-

ues (note: the accumulators have to accumulate a large number of samples; typically

the duration is one code period). In the case of TMOC-QPSK it is interesting to

note that the computation is greatly reduced (Fig. 8.21). Given that for the wide-

band AltBOC(15,10), the number of samples per integration period is very high due

to a high sampling frequency, the lower bit-width requirement for y(t) results in a

signi�cant advantage over the AltBOC modulation.

8.6.2.1. Correlator with multiple taps. In the architecture discussions in this sec-

tion, the role of the shift register has not been explicitly mentioned. However, when

the correlator is used for tracking the signal, the delay locked loops require more

than one tap, typically three; the �early�, the �prompt� and the �late�. For this

reason, s(t− τ) will be a 3-element vector (for the early, prompt and late reference

signals) and the bit-width of the shift registers is a�ected by the number of bits used

to represent the reference signal. As a result, the requirement of the number of code

mixers and the number of accumulators also increases.


Table 8.3. Correlator complexity comparison summary for AltBOCand TMOC-QPSK modulations

AltBOC TMOC-QPSK

Ref.signalgen-erator

DirectCom-puta-tion

8 sign changeops, 4

Add/Sub(excludingthe product-

signalgeneration)

Four 3-bitXOR, two2-bit XOR

LUTMethod

Two 128x3LUTs

Two 128x1LUTs

Code mixer DirectCom-puta-tion

6 Mult, 10Add/Sub

4 signchange ops,2 Add/Sub

LUTMethod

Four 64x8LUTs, 2Add/Sub

Four 16x3LUTs, 2Add/Sub

Table 8.4. Logic resource and estimated power consumption for Al-tBOC and TMOC-QPSK correlators

Number of Logic Elements Power Con-sumptionEstimate

AltBOC 1186 31.98 mWTMOC-QPSK 802 24.59 mW% Reduction 32.37 23.10

8.6.3. Complexity comparison summary. Table 8.3 summarises the com-

parison and highlights (in green) the best options to realise the reference signal gen-

erator and the code mixer in both the AltBOC modulation and the TMOC-QPSK

modulation.

8.6.4. Hardware resource utilisation and power consumption estimate.

In order to gain more insight into the complexity reductions the correlators for the

AltBOC and TMOC-QPSK modulations were implemented on a Altera Cyclone-

III FPGA. With the help of Altera's Power Play Power Analyzer tool the power

consumption for both types of modulations were estimated and are listed in Table

8.4. Best options (minimum resource, those highlighted in Table 8.3) were chosen

to implement the reference signal generator and the code mixer.


8.7. On E�cient Wideband GNSS Signal Design

This section aims at extending the TMOC-QPSK signal to e�ciently utilise a

given bandwidth. This modulation, referred to as Time-Multiplexed Multi-Carrier

(TMMC), divides the available bandwidth intoN sub-bands such that each sub-band

resembles a Quadrature Phase Shift Keying (QPSK) modulation and at the same

time allows the receiver to exploit the bene�ts of a wideband signal. The generation

of the new signal and its properties are described. The bene�ts of using TMMC

modulation for wideband signals in order to overcome the errors due to propagation

channel impairments, continuous wave interference mitigation and receiver design

complexity is discussed.

8.7.1. Wideband GNSS signals.

8.7.1.1. How wide is a wideband signal? Since this chapter deals with wideband

signals, it is necessary to discuss what bandwidth can be referred to as �wideband�

in the context of GNSS or more generally, in the context of navigation and ranging

signals.

The accuracy of the position, velocity and time solution in GNSS depends on

many factors, both internal and external to the system. The internal error sources

are from the space, ground and control segments of GNSS and the rest arise form

external factors, such as the atmosphere and other ground-based systems that in-

terfere with GNSS. Let these errors be grouped into two categories, those that are

attributed to the �signal structure� and those that are not. In general, wideband

signals o�er better performance than narrowband signals. As the signal bandwidth

increases, there would be a �breakeven point� around which the errors due to the

signal structure are comparable to the combined contribution of all other errors.

Beyond the breakeven bandwidth the errors due to the signal structure diminish

further (though it may asymptotically reach a constant, eventually), but the other

errors remain the almost the same. It can be argued that the wideband signals are

the signals whose bandwidth is greater than this breakeven bandwidth.

As an example, consider the GPS L1 C/A code signal, which has a certain

multipath-free code ranging accuracy, certain multipath performance. etc., which

can be attributed to the signal structure. Consider a hypothetical signal �L1 X�

in the same system, that has ten times the signal bandwidth (only the dominant

frequency lobes) compared to the GPS L1 C/A but at the same carrier frequency.

Further, assume that the L1 X signal o�ers better ranging accuracy and multipath

mitigation capabilities that result in lower code ranging errors compared to the L1

C/A. However, the L1 X signal experiences almost the same errors as does the L1

C/A due to the ground segment, the ionosphere, etc., that are not related to the

signal structure. Assuming a breakeven bandwidth of 20MHz, the GPS L1 C/A

8.7. ON EFFICIENT WIDEBAND GNSS SIGNAL DESIGN 219

can be considered a narrowband signal and the L1 X can be considered a wideband

signal. The discussions in this chapter continue this assumption of 20MHz as the

breakeven bandwidth.

8.7.1.2. Drawbacks of current wideband GNSS signal modulations. A given fre-

quency band can be used to transmit one of the several possible variants of the

existing signals. For example, assuming an integer multiple of 1.023MHz for the

code chipping rate, a bandwidth of 20.46MHz can accommodate any of the follow-

ing modulations: BPSK(10), BOC(5,5), BOC(6,4), BOC(8,2), BOC(9,1) or any of

the AltBOC counterparts of the corresponding BOC (details will be provided later in

this section, see Fig. (8.22)). However, each of these options has its own advantages

and disadvantages. On the one hand BPSK(10) with a 10.23MHz spreading code

occupies the entire band, o�ers a simple correlation function but is more prone to

continuous wave (CW) interference. On the other hand AltBOC(9,1) has dominant

lobes present at the boundaries of the spectrum which can be independently de-

modulated. This allows the receiver to mitigate CW interference to some extent (by

excluding the interference-a�ected lobe, resulting in only half of the signal energy

being used) but su�ers from the problem of several strong correlation side-peaks.

Considering AltBOC modulation as an example, it was discussed in Chapter

6 that the total group delay error due to signal propagation (including that due

to multipath) as seen from the centre of the band (E5 signal in this case) can

be compensated for by estimating the di�erential phase delays of two equidistant

points from the centre (E5a and E5b in this case). In other words, independently

demodulating and estimating the phase delays of QPSK signals like those on the E5a

and E5b bands can help in compensating for the group delay error of the AltBOC

E5 signal. The two-point estimate is a trivial solution and cannot address the second

order errors. Moreover having several such virtually independent sub-bands allows

a receiver to isolate the interference-a�ected sub-bands.

Hence the modulation options mentioned in the previous paragraph do not o�er

any signi�cant advantage in dealing with frequency selective channel impairments.

The Time-Multiplexed Multi-Carrier modulation introduced in this chapter can be

used to generate a wideband signal that has the potential to address these drawbacks.

8.7.1.3. Desired signal modulation type and properties. The question now is : is

it possible to have a signal within the 20.46MHz bandwidth that has the following

features?

• The signal should have several sub-bands, that can be independently de-

modulated as QPSK(1) signals and also as a wideband signal,

• a combination of the ten QPSK(1) signals or the single whole signal should

produce a narrow correlation triangle with suppressed side-peaks,


• the baseband signal should o�er a constant-envelope when the individual

sub-bands are combined onto a single central carrier at the transmitter,

• the sub-bands should not interfere with each other, and

• processing the signal, especially the baseband signal processing, should be

simple and relatively inexpensive.

The AltBOC modulation currently used for the Galileo E5 signal exhibits features

similar to the above for a two sub-band case. Hence it would be wise to use AltBOC

as a starting point. However, extending the AltBOC modulation scheme to include

more than two sub-bands (i.e. using more than one complex sub-carrier) a�ects the

constant-envelope property. This is because all the components should be combined

onto one single central carrier so as to use a single transmit ampli�er, and the

combination of several components result in varying amplitudes in the complex

plane. Even if there is a possibility to obtain a constant-envelope, the sub-carrier

waveform design becomes extremely di�cult. In other words, the AltBOC type of

modulation is not scalable.

8.7.2. Some wideband signal candidates with AltBOC modulation.

Within a given bandwidth the AltBOC modulation can be used to accommodate

some of the signals that satisfy at least a sub-set of the desired signal features. In-

vestigating these signals will provide an insight into the gap between the desired

signal and AltBOC.

The AltBOC(m,n) modulation results in two dominant frequency lobes with

null-null bandwidth 2 · fco Hz and 2mn− 2 suppressed frequency lobes with null-null

bandwidth fco Hz. Thus a bandwidth of B Hz can be used to accommodate any

AltBOC signal that satis�es the criterion m = B2·1.023e6

− n, with the constraint of n

and m to be integers and B to be a multiple of fco.

Fig. 8.22 shows the possible AltBOC signals within a 20.46MHz bandwidth

(�rst two dominant lobes) choosing n to be an integer. The BPSK(10) modulation

is also shown for reference. The correlation functions for the two extreme cases

AltBOC(5,5) and AltBOC(9,1) are shown in Fig. 8.23.

As mentioned in sec. (8.7.1), although AltBOC(9,1) provides a narrow corre-

lation peak, there are several side-peaks. This is due to the fact that the useful

information in AltBOC(9,1) is contained only in the two 2.046MHz lobes at the

extreme ends of the band and hence that modulation does not e�ciently utilise

the entire spectrum. On the other hand, AltBOC(5,5) is closer in its features to

BPSK(10) due to the presence of wide frequency lobes which are more prone to

interference, though the spectrum usage is more e�cient compared to AltBOC(9,1).

The next section discusses a multi-carrier modulation that combines the bene�ts of


−10 −5 0 5 10−85

−80

−75

−70

−65

−60

−55

Frequency (MHz)

Pow

er/F

requ

ency

(dB

/Hz)

BPSK(10)AltBOC(5,5)AltBOC(6,4)AltBOC(7.5,2.5)AltBOC(8,2)AltBOC(9,1)

Figure 8.22. Possible AltBOC signals within a 20.46MHz band

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Chip Shift

Nor

mal

ized

Aut

ocor

rela

tion

Val

ue

BPSK(10)AltBOC(5,5)AltBOC(9,1)

Figure 8.23. Correlation functions for BPSK(10), AltBOC(5,5) andAltBOC(9,1)


−10 −5 0 5 10−85

−80

−75

−70

−65

−60

−55

Frequency (MHz)

Pow

er/F

requ

ency

(dB

/Hz)

AltBOC(1,1)AltBOC(3,1)AltBOC(5,1)AltBOC(7,1)AltBOC(9,1)

Figure 8.24. A 20.46MHz band used with AltBOC variants; odd m

better spectral usage and at the same time preserves a narrow correlation function

with suppressed side-peaks.

8.7.3. Multi-carrier modulation. The multi-carrier actually refers to the use

of more than one sub-carrier, and the �sub-� is left out - this is also the case for modu-

lation schemes used in other wireless communication systems (Pun et al., 2007). The

use of multiple sub-carriers seeks to �ll the gaps in the frequency spectrum within the

given bandwidth. The time-multiplexing technique aims to generate a signal that

uses multiple sub-carriers, which is otherwise not possible with the AltBOC mod-

ulation. This section explains the proposed multi-carrier modulation and the next

section describes the generation of the proposed signal using a time-multiplexing

method.

Instead of the AltBOC modulation �lling in the whole of the given bandwidth

with one signal, it is possible to use the given bandwidth in smaller non-overlapping

portions, each portion used by one signal. This is the basis of the requirement

that the desired signal should allow independent demodulation of smaller portions

of itself (how it can be �independently� demodulated will be discussed later in this

chapter). To do this one can start from the AltBOC(1,1) signal which �lls the

4.092MHz (-2.046MHz to +2.046MHz) at the centre of the given 20.46MHz band.

Next, the AltBOC(3,1) signal places its dominant lobes beyond 4.092MHz, and now

the coverage is (including the �rst signal) -4.092MHz to +4.092MHz. This can be

continued until the whole of the 20.46MHz is occupied by dominant lobes of di�erent

AltBOC variants.

Fig. 8.24 shows the spectrum obtained with all the variants with �odd� m from

AltBOC(1,1) to AltBOC(9,1) such that the dominant lobes do not overlap. This

non-overlapping is a constraint at this stage because the individual signals should


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.5

0

0.5

1

Chip Shift

Nor

mal

ized

Aut

ocor

rela

tion

Val

ue

AltBOC(1,1)+AltBOC(3,1)+AltBOC(5,1)+AltBOC(7,1)+AltBOC(9,1)BPSK(10)

Figure 8.25. Combined correlation waveform of AltBOC variants; odd m

−10 −5 0 5 10−85

−80

−75

−70

−65

−60

−55

Frequency (MHz)

Pow

er/F

requ

ency

(dB

/Hz)

AltBOC(2,1)AltBOC(4,1)AltBOC(6,1)AltBOC(8,1)

Figure 8.26. A 20.46MHz band used with AltBOC variants; even m

not interfere with each other (later in the chapter it will be shown that the time-

multiplexing eliminates this constraint). Now it is of interest to look into the corre-

lation waveform of the signal which combines all these individual signals. Fig. 8.25

shows the correlation waveform along with that of BPSK(10) for reference. Observe

that the correlation main peak is narrow, but there are two side-peaks at a distance

of 0.5 chips.

Instead of using the odd values for m the other option is to use �even� values for

m which gives AltBOC(2,1) to AltBOC(8,1) signals. The power spectral densities

are shown in Fig. 8.26 and the correlation waveform obtained by combining the

signals is shown in Fig. 8.27.

Observe again that the main correlation peak is narrow but there are two side-

peaks at 0.5 chip o�sets. The combination of odd m variants or the even m variants


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Chip Shift

Nor

mal

ized

Aut

ocor

rela

tion

Val

ue

AltBOC(2,1)+AltBOC(4,1)+AltBOC(6,1)+AltBOC(8,1)BPSK(10)

Figure 8.27. Combined correlation waveform of AltBOC variants; even m

−10 −5 0 5 10−85

−80

−75

−70

−65

−60

−55

Frequency (MHz)

Pow

er/F

requ

ency

(dB

/Hz)

AltBOC(1,1)AltBOC(2,1)AltBOC(3,1)AltBOC(4,1)AltBOC(5,1)AltBOC(6,1)AltBOC(7,1)AltBOC(8,1)AltBOC(9,1)

Figure 8.28. A 20.46MHz band used with AltBOC variants; bothodd and even m

produce side-peaks because there is still the gap between the dominant lobes of

the signals. However, there is a di�erence, the side-peaks in the odd type have a

negative correlation value, and they are positive in the case of the even type. The

very next option would be to combine both the types, thus having all the variants

from AltBOC(1,1) to AltBOC(9,1) and allowing the overlap of dominant lobes. Fig.

8.28 shows the power spectral density and Fig. 8.29 shows the correlation waveform

of combining all the odd and even variants.

Observe now that the side-peaks have been highly suppressed and also that


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Chip Shift

Nor

mal

ized

Aut

ocor

rela

tion

Val

ue

BPSK(10)AltBOC(1,1)+AltBOC(2,1)+AltBOC(3,1)+AltBOC(4,1)+AltBOC(5,1)+AltBOC(6,1)+AltBOC(7,1)+AltBOC(8,1)+AltBOC(9,1)

Figure 8.29. Combined correlation waveform of all AltBOC vari-ants; both odd and even m

the correlation main peak is narrower than that of the BPSK(10). The centre

2.046MHz (assuming a similar overlap like other lobes in the spectrum) of the band

is still un�lled, but this would require a BPSK(1) modulation unlike other sub-

bands which are the byproducts of AltBOC modulation. Interestingly, BPSK(1)

modulation can be treated as AltBOC(0.5,1) and hence becomes a speci�c case of

the AltBOC modulation. The AltBOC(0.5,1) signal is included as the tenth signal

and the correlation waveform is only slightly altered (not plotted in the �gures in

this chapter) with the addition of this signal. Hence, the total number of sub-bands

becomes 19 and the total number of signals is now 10.

This section used the term �combined� without going into details of how the

individual signals are combined. The next section discusses a time-multiplexing

method to realise the combination of individual signals.

At this point it is worth mentioning that the ideal spectrum for the desired signal

would have a rectangular shape in the power spectral density, which means almost

no gap between the sub-bands. Realisation of such a signal would require a very high

number of sub-bands (in other words sub-carriers) like those used in other wireless

communication systems and the feasibility of generation of such a signal has to be

studied in detail, which is beyond the current scope of this thesis.

8.7.4. Time-multiplexed multi-carrier modulation. One of the require-

ments for the desired signal is to ensure that there is little or no interference between

sub-bands. However the previous section discussed the bene�ts of overlapping the

sub-bands in suppressing the side-peaks. Avoiding interference by using di�erent

spreading codes for di�erent side-bands (say one code for one sub-carrier) would

have two issues at the outset. First, the cross-correlation between the codes will not


AltBOC (5,1)

exp{ jq(k)ωc(t-kTc) }

q=[-4 4], k = [0 8], q,k Ƶ

Multiplexedsignal

Figure 8.30. AltBOC(5,1) covering a 20.46MHz band in the �scantype� time-multiplexing

be eliminated completely just outside the dominant lobe, and may lead to correla-

tion sub-peaks (Issler et al., 2003). Second, the requirement for numerous spreading

codes that should be employed increases as the number of side-bands increases. For

these reasons, multiplexing the individual signals in time would be is a good option

to explore further.

A method of multiplexing the individual signals for a single complex sub-carrier

(two side-bands), with the example of replacing the AltBOC modulation, has been

proposed earlier in this chapter.

There are many ways to realise the multi-carrier signal whose spectrum is shown

in Fig. 8.28. This section describes three straightforward schemes.

8.7.4.1. AltBOC signals are time-multiplexed. The sub-bands can be time-multip

-lexed, chip-by-chip, one sub-band pair at a time, if each pair uses the constant-

envelope AltBOC modulation.

Case 1. Spread type: The sequence of signals transmitted in this case is Al-

tBOC(1,1), AltBOC(2,1), AltBOC(3,1), AltBOC(4,1), AltBOC(5,1), Al-

tBOC(6,1), AltBOC(7,1), AltBOC(8,1), AltBOC(9,1), and then this se-

quence repeats for the next chip. This is called �spread type� because in

the frequency domain, the sequence of side-bands transmitted follows the

two �nger points of a �spread� operation on a touchscreen device.

Case 2. Scan type: In this case, only AltBOC(5,1) modulation is used for all

the sub-band pairs. In the �rst slot, AltBOC(5,1) is multiplied by a -4MHz

complex tone to shift the baseband signal such that the left dominant lobe

occupies the space of leftmost desired signal lobe. In the second slot the

translation is -3MHz, and so on up to +4MHz in the ninth slot. This process

is depicted in Fig. 8.30.

However, doing so requires four spreading codes and a multi-level sub-carrier wave-

form. Since one of the requirements is to reduce receiver complexity it would be

preferable to use a simple two-level (square wave) sub-carrier waveform and to use


eP(t)

sc1LSubcarrier

TC


sc1H sc2L sc2H

eD(t)

TC

Subcarrier

+ =

eP(tk+1)

scNH sc1L sc1H

eD(tk+1)

2NTC

scNL

sciL= sci(t)-jsci(t-Tsci/4) sciH= sci(t)+jsci(t-Tsci/4)

eP(tk)

eD(tk)

sc1L sc1H sc2L sc2H scNH sc1L sc1HscNL

Figure 8.31. An illustration of time-multiplexing with multiple sub-carriers: �spread type�

only two spreading codes, one for the I channel and the other for the Q channel. This

leads to a scheme of extending TMOC-QPSK to multiplex the sub-bands, instead

of AltBOC.

8.7.4.2. TMOC-QPSK signals are time-multiplexed. TMOC-QPSK can be used

instead of AltBOC in the two cases similar to those ones mentioned in the previous

sub-section.

Case 1. Spread type: Fig. 8.31 shows the time-multiplexing considering N

complex sub-carriers (2N sub-bands). The sequence illustrated in Fig. 8.31

results in transmitting TMOC-QPSK(1,1), TMOC-QPSK(2,1), TMOC-QPSK

(3,1), ..., TMOC-QPSK(9,1), and then the sequence repeats.

Case 2. Scan type: A variant of the above method is also possible where the

two bands in the TMOC-QPSK pair are not transmitted one after the other.

In this case the sequence of multiplexing is sequential when viewed from the

frequency domain perspective. Fig. 8.32 shows the �scan type� of time-

multiplexing considering N complex sub-carriers (2N sub-bands or 2N + 1

sub-bands if BPSK(1) signal is also included). The sequence can be written

as QPSK−9(1), QPSK−8(1),...,QPSK−1(1), QPSK1(1),QPSK2(1),...,QPSK9(1),

where the subscript denotes the multiplier (in multiples of fcerror is reduced)

used to o�set of the QPSK signal from the centre of the band. Observe that

the sequence is repeated after all the sub-carriers are used.

8.7.4.3. Nomenclature. TMMC can be identi�ed with the bandwidth (B) and

the chipping frequency (fco) of the individual components, both represented as in-

teger multipliers of 1.023MHz. For example, the 20.46MHz bandwidth, 1.023MHz

chipping frequency case considered in the previous sections, which is implemented


eP(t)

sc9LSubcarrier

TC


sc8L sc(N-1)H

eD(t)

TC

Subcarrier

+ =

eP(tk+1)

scNH sc9L

eD(tk+1)

2NTC

sciL= sci(t)-jsci(t-Tsci/4) sciH= sci(t)+jsci(t-Tsci/4)

eP(tk)

eD(tk)

sc1L sc8L

sc9L sc8L sc(N-1)H scNH sc9L sc8L

scNHsc1H sc2H

sc1L sc1H sc2H

Figure 8.32. An illustration of time-multiplexing with multiple sub-carriers: �scan type�

Tc

c(t)

Re [sc(t)]

Figure 8.33. Code and sub-carrier for a partial sequence ofTMMC(10,1); only the real component of the complex sub-carrieris shown

with 10 sub-carriers (including the AltBOC(0.5,1) signal) signal with 19 sub-bands

of 2.046MHz width (with 1.023MHz overlap between sub-bands) is represented as

TMMC(10,1).

8.7.4.4. Receiver baseband signal processing. This section discusses a method to

acquire and track a TMMC signal, speci�cally TMMC(10,1). During the discussion,

BPSK(10) with (I and Q channels) and AltBOC(5,5) are referred to when necessary

to emphasise the di�erences.

Acquiring the complete signal: The acquisition of a TMMC(10,1) signal

requires the generation of one spreading code and nine sub-carrier waveforms as

shown in Fig. 8.33.

Acquiring individual sub-bands: Individual sub-bands can be acquired by

translating the desired band to baseband (frequency translation) and then corre-

lating with the local spreading code. However, any one particular sub-band is not

continuous in time, for example, QPSK−5 will only be present in every 5th slot and

repeats every 18 slots (19 slots if the BPSK(1) is also included). This is not a prob-

lem for three reasons: (i) the same spreading code is used for all the sub-bands; (ii)


0 10 20 30 40 50 60−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Multipath delay (m)

Cod

e m

ultip

ath

erro

r (m

)


E5a−20MHz; δ=0.15TMMC(10,1)−20MHz; δ=0.15

Figure 8.34. Comparison of code multipath error-envelope forTMMC(10,1) and BPSK(10) signals

the local code repeats in each slot; and (iii) the time-multiplexing is carried out chip-

by-chip. As a result the only issue is that the initial code phase ambiguity due to

time-multiplexing is up to one chip duration and has to be resolved. A non-coherent

combination of all the individually acquired sub-bands will produce a BPSK(1)-like

correlation waveform.

8.7.5. Potential performance bene�ts of using TMMC for wideband

signals.

8.7.5.1. Code multipath error envelope comparison. Fig. 8.34 shows the code

multipath error-envelope for BPSK(10) and TMMC(10,1) signals. Observe that

even though the error for low multipath delays is reduced, there are oscillations

in the error-envelope for multipath delays beyond one chip duration. This is also

evident from the correlation waveform in Fig. 8.29. This is one drawback of the

TMMC modulation that needs to be addressed in future work.

8.7.5.2. Continuous wave (CW) interference. When the signal is a�ected by

narrow-band CW interference only the sub-band where the CW interference is

present, sideband would have the signal strength attenuated, as illustrated in Fig.

8.35. In the worst case if the frequency of the CW interference happens to lie midway

between two sub-bands, then two sub-bands will be a�ected. Since any one sub-

band has 1/19th of the total weight in a TMMC(10,1), the total signal degradation

due to one CW interfering tone will be approximately 5.2% or about 0.47dB in the

best case and about 0.95dB in the worst case. When there is knowledge about the

sub-band being a�ected, the local code/sub-carrier portion for that sub-band could

be initialised to zeros, thus avoiding any noise creeping into the processing. When

there is no knowledge about the a�ected sub-band, it is also possible to compare the


CW Interference

Figure 8.35. Illustration of CW interference a�ecting only one sub-band

signal strengths of all the sub-bands by tracking the individual sub-bands. This may

allow the receiver to isolate that particular sub-band. However tracking individual

sub-bands may lead to additional processing requirements.

8.7.5.3. DME/TACAN interference. The DME/TACAN interference is known

to be present in the L5/E5 band and occurs at multiples of 1MHz. It is interesting

to note that if the centre of the band is aligned to a frequency mid-way between the

1MHz boundary (i.e. the centre frequency is not a multiple of 1MHz) then all the

multiples of 1MHz frequencies will coincide with the centre of the gap that exists

between the sub-bands in the TMMC(10,1). However, design practice in GNSSs

will allow only integer multiples of 10.23MHz to be chosen as carrier frequencies.

For example the L5 carrier frequency is 1176.45MHz, which is 115 x 10.23MHz. So

in this example there is only a slight o�set between the 0.5MHz and 0.45MHz and

the interfering frequency will be �just inside� the 3dB bandwidth of any sub-band,

which is better than the interfering signal a�ecting the peak of the frequency lobe.

This situation is illustrated in Fig. 8.36.

8.7.5.4. Frequency selective propagation delay distortion. GNSS signals su�er

from several types of propagation delay distortions. Some of the distortions that

seriously a�ect the ranging performance are frequency dependent, ionosphere and

multipath being the two major sources.

The ionosphere causes phase advances and time delays for signals that travel

through it. The ionosphere delay is inversely proportional to the square of the carrier

frequency of the signal. When there are multiple carriers within a wideband signal,

the parameter of interest is the ionosphere dispersion. Signals with a bandwidth

of 20.46MHz at about 1.2GHz carrier frequency experience a dispersion of about

0.34 metres for a 10m ionosphere delay up to 3.4 metres for a 100m ionosphere

delay. For a 51.15MHz signal these dispersion values become about 0.85 m and

about 8.5 m respectively. The code measurement noise standard deviation for a

20.46MHz BPSK(10) signal is about 0.28 meters (Sleewaegen et al., 2004). For


1176.45 MHz

1175 MHz 1178 MHz1170 MHz

Interfering frequencies

Figure 8.36. DME/TACAN interference at the gaps between sub-bands

the TMMC(10,1) signal, the code measurement noise standard deviation is about

0.2 meters. Hence it would be di�cult to isolate the ionospheric errors when the

ionosphere delay is small. However, large ionospheric errors show up as di�erences

in the code measurements of the sub-bands. In addition, higher bandwidths will

also help identify ionosphere dispersion, and hence ionospheric delay.

The phase of the re�ected signal that is superimposed onto the direct signal in

multipath scenarios will depend on the frequency and the multipath delay. Di�erent

sub-bands in the TMMC(10,1) modulation experience multipath at di�erent phases.

Assuming for a moment that the other frequency selective errors such as ionosphere

are absent, the output of the tracking loops provides a multipath-a�ected phase pro-

�le throughout the band. Note that even without the presence of re�ected signals

there exists a phase pro�le across the sub-bands. Now, a comparison between the

multipath-free and multipath-a�ected phase pro�les should provide more informa-

tion on the re�ected signal, corresponding to the combined error due to the re�ected

signal.

When both the ionosphere and multipath errors are present, the receiver sees the

combined e�ect in terms of total phase delay that would be di�erent for di�erent sub-

bands. This is illustrated in Fig. 8.37. Since TMMC(10,1) can also be demodulated

as a single signal the receiver can also estimate the group delay at the centre of the

band. The group delay at the centre of the band can also be estimated using the

phase delay pro�le. Using the two pieces of information the receiver can compensate

for the group delay error at the centre of the band as explained in Chapter 6.


Pha

se d

elay

Figure 8.37. Illustration of forming a phase delay pro�le with theaim of aiding the compensation of group delay at the centre of theband.

8.8. Summary

A new constant-envelope modulation technique TMOC-QPSK, and its vari-

ant TMOC-π4-QPSK, which use a time-multiplexing method to combine complex

sub-carrier modulated spreading codes were presented in this chapter. The power

spectral density and correlation function of TMOC-QPSK and TMOC-π4-QPSK are

shown to match that of the non-constant-envelope AltBOC and constant-envelope

AltBOC respectively. The TMOC-π4-QPSK modulation technique uses the time-

multiplexing scheme with two QPSK signals separated in time and sub-carrier phase

to achieve the AltBOC-like correlation function. Within the TMOC-QPSK and

the TMOC-π4-QPSK modulation techniques, two methods of arranging the signal

components for time-multiplexing, TMOC-QPSK-ab and TMOC-QPSK-IQ are dis-

cussed. It is shown that the correlators for the TMOC-QPSK methods are simpler to

realise and that the proposed modulation techniques utilise ~32% less logic resources

and consume ~23% less power compared to the constant-envelope AltBOC modu-

lation. The proposed modulation technique is elegant yet simple and an alternative

to the AltBOC modulation.

Starting with the desired properties of a wideband signal this chapter also ex-

plored the possible options for a wideband signal and their drawbacks. Then a new

modulation scheme called Time-multiplexed Multi-carrier (TMMC) was proposed

to address some of the drawbacks of existing wideband signals. Four ways of gen-

erating a TMMC signal were described with a discussion on their advantages and

disadvantages. The receiver processing (especially the places where it di�ers from

8.8. SUMMARY 233

Table 8.5. Comparison summary of AltBOC, TMOC-QPSK andTMMC modulations

AltBOC TMOC-QPSK

TMMC

Constant Envelope? Yes Yes YesMultiplexing Technique Phase Time TimeCarrier modulation

(Complex)Single Carrier Single Carrier Multi-carrier

Subcarrier levels Multi-level Two-level(square)

Two-level(square)

Baseband Complexity High Low MediumRobustness to frequency

selective propagation delaydistortions and CW

interference

Low Low High

Suitable for narrow tomoderatebandwidth

narrow tomoderatebandwidth

moderate towide

bandwidthSpectrum resembles -NA- AltBOC MC-CDMA /

OFDM

the processing of other GNSS signals) of a TMMC signal was discussed. Finally,

the potential use of TMMC in mitigating narrow band CW interference and the

frequency selective propagation delay distortion was discussed. It was shown that

a single CW interfering tone degrades the TMMC signal by less than 1 dB, making

the rest of the band still usable for navigation. Future work will involve an anal-

ysis of the similarities of TMMC with other modulation schemes such as OFDM,

MC-CDMA and OFDMA (Pun et al., 2007; Mateu et al., 2010)used in wireless com-

munications. Table 8.5 provides a comparitive summary of AltBOC, TMOC-QPSK

and TMMC modulations.

CHAPTER 9

Conclusions and Recommendations

9.1. A Review of the Objectives

The Galileo E5 signal not only has the most sophisticated signal structure among

all the current GNSS signals but also o�ers unprecedented ranging performance.

The performance advantages of the AltBOC(15,10) modulation used for Galileo E5

have already attracted the attention of the Chinese Compass navigation system

designers, who have indicated that they will use the same modulation scheme at the

same carrier frequency of 1191.795 MHz in the L5 band for their B2 signal planned

for Phase III. Moreover, with GPS L5, Galileo E5, Compass B2, GLONASS L3OC

and the planned GLONASS L5OC signals, the L5/E5/L3 band will be crowded with

high performance signals.

All of the developments mentioned above, except the L5OC, will happen in the

next 4-5 years, and a GNSS receiver designed to receive wideband signals in the

L5/E5 band will have access to these signals from a signi�cant number of satellites.

Since these signals are �open service� /civilian signals, there is likely to be a drive

from the user community to track these high performance signals. Semiconductor

technology advancements will accelerate demand and propel low form-factor and

low-power requirements in such receivers. Due to its performance bene�ts, Galileo

E5 AltBOC is likely to get more attention in the L5/E5 band.

The primary objectives of the thesis as stated in chapter 1 were to:

� explore e�cient signal acquisition and signal tracking algorithms for the Galileo

E5 AltBOC(15,10) signal,

� exploit the unique features of AltBOC such as frequency diversity, independent

demodulation of the sidebands in acquisition, tracking and multipath mitigation,

� study the complexity of Galileo E5 AltBOC(15,10) baseband hardware,

� investigate acquisition engine architectures to realise multi-frequency receivers

including Galileo E5, and

� explore the possibility of devising a new signal in place of the existing Alt-

BOC(15,10) that reduces the GNSS receiver complexity.

The outcomes of this thesis are summarised in the following sections.

235

236 9. CONCLUSIONS AND RECOMMENDATIONS

9.2. Acquisition, Tracking and Multipath Mitigation

9.2.1. Galileo E5 signal acquisition. Di�erent acquisition methods described

in the literature that exploit the features of the AltBOC modulation were categorised

into two main groups: the �search strategy� based methods and the �correlation

scheme� based methods. Since the acquisition performance depends directly on the

strength of the received signal, the foremost criterion for selection between the can-

didate acquisition methods was the degree of utilisation of the complete received

signal power. Though the Double Sideband (DSB) acquisition method o�ers almost

the same amount of received signal energy as the Direct AltBOC acquisition method,

a detailed inspection of the acquisition methods revealed that the Direct AltBOC

acquisition method needed further investigation for two reasons. First, the Direct

AltBOC acquisition approach avoids the use of frequency translators and �lters that

are required for the DSB method; second, the Direct AltBOC correlator can be im-

plemented using Look-Up-Tables (LUTs) instead of correlators for individual signal

components.

However, the e�ect of code search step size during code acquisition in direct

acquisition of the Galileo E5 AltBOC(15,10) signal was shown to have a serious

e�ect on the acquisition performance, degrading the probability of detection by

between 2.2 dB (average case) and 6.3 dB (worst case) for 0.5 chip steps. Then it

was shown that combining the correlation values separated in time by one quarter of

a sub-carrier period can reduce the e�ect of code search step size on the acquisition

performance. In the AltBOC(15,10) case this method not only reduces the loss from

2.2 dB to 0.4 dB in the average case, but also improves the mean acquisition time

by about 41% compared to the Direct AltBOC approach.

It was also shown that, unlike the navigation data, the knowledge of secondary

codes can be exploited to increase the coherent integration duration during the ac-

quisition. A novel way of applying the principles used in decoding the convolutional

codes was introduced, and a method to de�ne and determine the �characteristic

length� of memory codes (analogous to the register length of pseudorandom codes

generated using shift registers) was developed. It was shown that the chip shift of

the 100-chip length Galileo E5 secondary codes (memory codes) can be identi�ed by

looking at less than 20 consecutive chip values. As a consequence, the acquisition

time of the Galileo E5 secondary codes is able to sped up by more than 80%.

9.2.2. Galileo E5 signal tracking. As in the case of acquisition, the pros and

cons of di�erent tracking architectures for the Galileo E5 signal were discussed, each

architecture employing a di�erent local reference signal. A generalised tracking

architecture was introduced and an equation for the code tracking error of this

generalised tracking architecture was derived. In order to overcome the issue with

9.2. ACQUISITION, TRACKING AND MULTIPATH MITIGATION 237

two data carrying signals (while the other two are pilot signals) three hybrid tracking

architectures were studied. Two of these, the data wipe-o� and the coherent pilot

signal components are, in principle, extensions of similar architectures used for the

GPS L5. However, formulating the inputs to the AltBOC LUT in the data wipe-o�

architecture was a novel contribution. The data wipe-o� architecture uses 12 code

mixers and accumulators, per correlation output, but provides the best code and

carrier tracking error performance. The coherent pilot architecture uses only 5 code

mixers and accumulators per correlator but has 38% higher core tracking error and

43% higher carrier tracking error at moderate to low signal strengths (<35 dB-Hz)

compared to the data wipe-o� architecture.

Another contribution was devising a third architecture referred to as the �pre-

correlation combination� architecture. It was shown that the pre-correlation com-

bination architecture o�ers a good trade-o� between resource utilisation and per-

formance. The pre-correlation combination architecture requires the same number

of code mixers and accumulators as the coherent pilot tracking architecture (i.e. 5)

but has only 15% higher noise in code and about 18% in carrier tracking; in other

words, a reduction in performance loss of more than 50% compared to the coherent

pilot architecture. Apart from the lower resource utilisation the biggest advantage

of the pre-correlation combination method compared to the data wipe-o� method is

that the integration duration can be extended beyond the data-bit boundary (4ms in

Galileo E5), which helps recover performance losses. In principle the correlation out-

puts (of one data bit duration) can also be accumulated across data bit boundaries

in the data wipe-o� architecture, however the �max� operation has to be performed

only on the correlation values within one data bit duration, which would have an

adverse e�ect on the performance - especially at low signal strengths (depending on

the integration duration).

It was shown that the tracking range of the delay-locked-loop (DLL) is limited

by the shape of the correlation waveform and AltBOC(15,10), which has a narrow

correlation main peak, will su�er from this problem. A novel way of combining

the discriminator outputs that produce an AltBOC(15,10) correlation function (e.g.

the wideband 8-PSK AltBOC tracking architecture) and the BPSK(10) correlation

function (e.g. sideband tracking architecture) was proposed. This hybrid DLL

increases the code tracking range of the E5 signal from 0.33 chips (±0.167 chips

from the main peak) up to 1 chip (±0.5 chips) without being a�ected by the side-

peaks. Moreover it was shown that within ±0.167 chips from the peak, the code

tracking error performance of the hybrid DLL remains the same as that of the 8-PSK

AltBOC tracking.


9.2.3. Galileo E5 code phase multipath mitigation. An innovative method

to mitigate the code phase multipath in Galileo E5 AltBOC receivers was presented.

This method, called the Sideband Carrier Phase Combination (SCPC) method, com-

bines the carrier phases of the E5a and E5b tracking loops with the code phase mea-

surement of the wideband E5 tracking loop to mitigate the code multipath errors in

the wideband E5 code phase measurement. It was shown that the maximum error

due to a single re�ected signal half the amplitude of the direct signal can be reduced

to 0.5 metres with the SCPC method, compared to about 2 metres (max) error of

the standard 8-PSK AltBOC tracking architecture.

The underlying concept that makes the SCPC method mitigate the multipath

error was further investigated. It was shown that the code phase error at the centre

of the E5 band is related to the group delay and the carrier phase at the centre of

the E5a and E5b bands is related to the corresponding phase delays. The SCPC

method was shown to be based on the principle that the di�erence in the phase

delay between two equidistant points from the centre of the band represents the

group delay at the centre frequency.

The study on the group delay compensation suggests that frequency diversity

in GNSS plays an important role in mitigating the frequency selective propagation

delay distortions. Closer the frequency bands, better will be the ability of the

receiver to distinguish relative carrier phase errors that result due to multipath.

However it is well known that the carrier phase error due to ionosphere can be

estimated by having at least two frequency bands that are far away (but not too far

away) from each other in frequency, L1/E1 - L5/E5 is a good example. Therefore it

can be concluded that it would be bene�cial for a GNSS to have both �inter-signal�

and �intra-signal� frequency diversities. Though the same modulation scheme can

be used for the inter-signal frequency diversity property, the multipath mitigation

capability depends on this intra-signal modulation scheme. AltBOC modulation,

which o�ers a basic frequency diversity is a good candidate for the intra-signal

modulation.

9.3. Baseband Hardware Complexity

During the signal acquisition stage in a multi-frequency GNSS receiver the need

for FFTs with varying transform lengths was identi�ed. It would be di�cult for any

current or near future digital hardware (in an embedded system) to accommodate

a requirement for varying transform lengths in either a dedicated or programmable

way, due to the high resource (area and power consumption) demands. In order to

reduce the resource utilisation of the FFT blocks, a novel application of Mixed-radix

FFT algorithms to GNSS signals was described. The key steps are:

9.3. BASEBAND HARDWARE COMPLEXITY 239

• to �factorise� the various transform lengths required by various signals and

make a union of the set of all the factors,

• e�ciently implement the small-point FFTs corresponding to these factors,

and

• combine the small-point FFT blocks using the Mixed radix FFT algorithm

to realise the required large FFT.

The set of small factors that can best satisfy the FFT requirements of GPS L1 C/A,

GPS L2C, GPS L5, GPS L1C, Galileo E1B/C and Galileo E5a/E5b/E5 were found

to be {2, 3, 4, 5, 8, 1024}. Two variants of the proposed FFT method were described:

the �simultaneous� and the �time shared� approaches. Three case studies of multi-

frequency receivers were conducted, Case-I: GPS L1 C/A and Galileo E1, Case-II:

GPS L1 C/A and Galileo E1, and Case-III: GPS L1 C/A, Galileo E1, GPS L5 and

Galileo E5a/E5b. The combined (adders and multipliers) reduction in the resource

utilisation for the simultaneous approach in the proposed FFT method compared to

the dedicated FFT method ranges from 33% to 41% for the combinations studied.

For the time shared approach the resource savings are much larger, ranging from

53% to 64%.

In addition, optimised core correlation logic (complex correlation circuit com-

prised of carrier generator, carrier mixers, code generator, code mixers and accumu-

lators) for the Galileo E5 that is generally useful for the signal tracking was described

and an equation for the number of accumulator bits required for the correlation was

derived. The core correlators for the GPS and Galileo civil signals (except GPS

L1C) were implemented on an FPGA to estimate the resource utilisation and power

consumption during the tracking stage. It was shown that the core correlator of an

E5 AltBOC(15,10) signal requires almost double the logic resource and has about

37 times the power consumption of a GPS L1 C/A core correlator.

As a consequence it was shown that a 12 channel GPS L1C/A + Galileo E1

+ GPS L5 + Galileo E5 baseband hardware (baseband hardware = core corre-

lator + timing control + address data multiplexer/demultiplexer + housekeeping

operations) would require approximately 19 times the power of a 12 channel GPS

L1 C/A baseband hardware, and a 12 channel all civil signal (except GPS L1C)

GPS+Galileo baseband hardware would require about 38 times the power. As a

rough guide, the core GPS L1 C/A correlator implemented on an Altera Cyclone-

III family FPGA consumed around 1 mW (core voltage 1.2V), and a 12 channel

baseband hardware consumed around 25 mW, which implies a value of close to half

a Watt for the L1/E1-L5/E5 case and close to 1 Watt for the GPS+Galileo all-civil

signal baseband hardware.


9.4. TMOC-QPSK and TMMC Modulation Schemes

The AltBOC core correlator blocks were studied in detail and the contributors to

the receiver complexity were identi�ed. It was found that the root of the complexity

arises from modi�cations made to the non-constant-envelope AltBOC modulation

to convert it to a constant-envelope AltBOC modulation. Due to the limitation of

the phase-multiplexing scheme employed in the AltBOC modulation, the designers

of the AltBOC modulation had to change the two-level square wave sub-carrier to a

four-level special sub-carrier waveform - which became the main contributor to the

complexity. It was envisioned that other multiplexing schemes could help combine

four signal components and produce a constant-envelope signal with square sub-

carrier waveform.

A new constant-envelope modulation scheme based on a chip-by-chip time- mul-

tiplexing method and square wave sub-carrier was developed, and was referred to as

Time-multiplexed o�set-carrier quadrature-phase-shift-keying (TMOC-QPSK). The

power spectral density and the correlation function of TMOC-QPSK was found to

exactly match that of a non-constant-envelope AltBOC. A variant of the TMOC-

QPSK called TMOC-π4-QPSK was described where the phase of the sub-carrier for

odd slots was shifted by π4with respect to the phase of the sub-carrier used for

even slots. The power spectral density of TMOC-π4-QPSK was derived and found

to exactly match that of a constant-envelope AltBOC. The core correlator for the

TMOC-π4-QPSK(15,10) was implemented on an FPGA and found to reduce the

logic resource requirement by at least 32% and power consumption by 23% com-

pared to the constant-envelope AltBOC(15,10). Being simple and elegant, the pro-

posed TMOC-π4-QPSK modulation has the potential to replace the existing AltBOC

modulation.

The time-multiplexing scheme used for four signal components in TMOC-QPSK

was extended to multiple signal components to design a multi-carrier modulation

suitable for future wideband GNSS signals, and this modulation was named Time-

multiplexed multi-carrier (TMMC). TMMC divides the available bandwidth into

several sub-bands, where each sub-band can be independently demodulated as a

QPSK signal and the whole band can be demodulated as a wideband signal. The

correlation waveform of the TMMC wideband signal was found to possess a narrow

main peak with highly suppressed side-peaks. The potential bene�ts of TMMC

modulation were discussed. The TMMC is superior to AltBOC in a given bandwidth

because: 1) more than four signals can be combined in the TMMC modulation (thus

eliminating the limitation of AltBOC modulation), 2) TMMC is constant envelope

by design, 3) TMMC modulation allows the re-use of same spreading codes for

all the sub-bands since the underlying technique is time-multiplexing rather than

9.5. RECOMMENDATIONS FOR FUTURE WORK 241

phase-multiplexing, 4) TMMC is more robust to interference, multipath and other

frequency selective sources of errors and 5) TMMC is simpler to realise, (i.e. more

�receiver friendly� than AltBOC). Hence TMMC modulation is a better candidate

than AltBOC for the intra-signal modulation discussed in sec. 9.2.3.

9.5. Recommendations for Future Work

9.5.1. Multi-frequency, Multi-GNSS receiver baseband signal process-

ing algorithms. The Galileo E5 AltBOC signal can be considered a super-set of

all the other signals. The research work carried out in this thesis pertaining to the

acquisition and tracking has focussed on the following:

1) Maximising the received signal energy at the output of the correlator (such

as the pre-correlation combination architecture).

2) Combining di�erent signal components to tap the bene�ts o�ered by each

component (such as combining the higher robustness of the wider BPSK-like corre-

lation waveform and higher performance o�ered by the narrow main peak, as in the

hybrid DLL method).

3) Developing simple, but generalised and con�gurable architectures to accom-

modate more than one signal or signal component (such as the hybrid tracking

architecture and the Mixed-radix FFT approach).

The objectives mentioned above can be thought of as the initial design criteria for

the baseband signal processing modules in a multi-frequency, multi-GNSS receiver,

and the methods described in this thesis can be extended.

In addition, speci�c sections that can be improved are:

• A theoretical expression for the threshold can be derived that helps to end

the iterations in the branch elimination method proposed (sec. 4.7) for

extended integration with secondary codes

• A feasibility study of the application of the pre-correlation combination

method (sec. 5.5.3 ) to other GNSS signals could be carried out and its per-

formance evaluated. This is signi�cant because the proposed pre-correlation

combination architecture for the Galileo E5 AltBOC signal o�ers the ad-

vantage of extending the coherent integration duration beyond the data-bit

period

9.5.2. Recon�gurable (hardware / software-de�ned hardware / soft-

ware) multi-GNSS receiver baseband processor. Even if a dedicated Applica-

tion Speci�c Integrated Circuit (ASIC) replaces the FPGA baseband hardware, as a

rule of thumb, and the author's own experience with multiple generations of GPS L1

C/A correlator ASIC design, there will be a best case reduction of the FPGA power

consumption by a factor of 5. In other words, a baseband ASIC will consume about


100 mW for the L1-L5 and about 200-mW for the all civil GPS+Galileo baseband.

This power consumption is very high given that it is only for the baseband hardware

and not for the entire receiver. Finally, if other global and regional satellite naviga-

tion systems (such as GLONASS, Compass, QZSS, IRNSS) are included, then, the

�200 times� estimate of Dempster (2007) mentioned in sec. 2.10.1 would not be far

away. Hence it can be concluded that development of a commercial general purpose

multi-GNSS receiver is still a challenging task.

To address this issue a recon�gurable / time sharing correlator module could

be developed extending the FFT level recon�gurability work carried out for the

acquisition module described in Chapter 7. The author envisages that the following

types of recon�gurability options can be explored:

1) Code generation level recon�gurability - (such work has already commenced

and has been reported in Mumford et.al. (2011).

2) Core correlator or channel level recon�gurability (recon�guring the channel

comprising of carrier generator, code generator, carrier mixer, code mixer and accu-

mulator).

3) Baseband hardware level or signal component level recon�gurability - that is,

a baseband hardware from digitised intermediate frequency bits up to the correlation

values / status/ control passed through a memory mapped interface, is completely

recon�gured for other signals as the need arises.

9.5.3. Performance evaluation of wideband GNSS signals with TMMC

modulation. Chapter 6 suggests that the frequency diversity present in AltBOC

can be used to measure the carrier phases at two points in the frequency spectrum

in order to compensate for the frequency selective code delay distortions midway

between the two points. Using the TMMC modulation, which allows independent

demodulation of all the sub-bands spanning the entire bandwidth in a wideband sig-

nal, more accurate estimation of the frequency selective channel impairments can be

obtained. This needs further investigation. In addition, the potential bene�ts of the

TMMC modulation as explained in sec. 8.7.5 such as continuous wave interference

and DME interference, could be investigated further.

There is an active search for a wideband GNSS signal in the S-band as described

in Mateu et al. (2010). The feasibility of using the TMMC modulation can be anal-

ysed and the advantages with respect to other modulations could also be evaluated.

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APPENDIX A

Fundamentals of AltLOC and AltBOC-NCE modulation

This appendix describes the Alternate Linear O�set Carrier (AltLOC) and the

non-constant envelope Alternate Binary O�set Carrier (AltBOC-NCE) with a slightly

di�erent treatment to that available in Lestarquit et al. (2008).

The baseband signal - AltLOC modulation. The baseband signal is a com-

plex signal which comprises four codes modulated onto the orthogonal components of

a complex subcarrier. In order to understand the basic concept of AltBOC modula-

tion, it is useful �rst to understand the AltLOC equivalent, because the sub-carriers

in AltLOC modulation are pure tones and hence simpler to deal with than AltBOC.

The baseband signal in AltLOC can be expressed as (CAL = CosAltLOC)

sCAL(t) = sb(t) ejωsct + sa(t) e

−jωsct

= (sbI(t) + jsbQ(t)) ejωsct + (saI(t) + jsaQ(t)) e−jωsct

= sbI(t) ejωsct + saI(t) e

−jωsct + sbQ(t) ej(ωsct+π2

) (A.1)

+saQ(t) e−j(ωsct−π2

)

The modulated signal can thus be represented as

SCAL(t) =√

2PT<[(sbI(t) ejωsct + saI(t) e

−jωsct (A.2)

+ sbQ(t) ej(ωsct+π2

) + saQ(t) e−j(ωsct−π2

)) ejωct]

=√

2PT<[sbI(t) ej(ωc+ωsc)t + saI(t) e

j(ωc−ωsc)t

+ sbQ(t) ej((ωc+ωsc)t+π2

) + saQ(t) ej((ωc−ωsc)t+π2

)]

= sbI(t) cos((ωc + ωsc)t) + saI(t) cos((ωc − ωsc)t)− sbQ(t) sin((ωc + ωsc)t)

− saQ(t) sin((ωc − ωsc)t) (A.3)

= sbI(t) cos(ωbt)− sbQ(t) sin(ωbt) + saI(t) cos(ωat)− saQ(t) sin(ωat)

(A.4)

where ωb = (ωc + ωsc) and ωa = (ωc − ωsc) are the E5b and E5a centre frequencies

(see Table (1.1)) respectively.

From the above equation it can be inferred that the transmitted signal spectrum

is real and has two lobes centred at ωb and ωa. A pictorial representation of this

equation is shown in Fig. A.1. Also note (through (A.3)) how the spreading code

(with data) phase modulates the sum and di�erence carriers.

255

256 A. FUNDAMENTALS OF ALTLOC AND ALTBOC-NCE MODULATION

ω-ωc

-ωc-ω

s

-ωc+ω s

E5a

-I

E5b

-I

E5a-QE5b-Q

ω c

ω c-ωs

ω c+ω s

E5a

-I

E5b

-I

E5a-QE5b-Q

0

Figure A.1. spectrum of the cosine-AltLOC

Note that because the baseband complex signal is formulated by addition of

two exponential functions, this modulation is in fact Cosine-AltLOC (CAL). If the

baseband complex signal is formulated by the di�erence of two exponential functions,

then the modulation will be SineAltLOC. The equation for the Sine-AltLOC (SAL)

is then

sSAL(t) = −j ·(sb(t) e

jωsct − sa(t) e−jωsct)

= e−jπ2 ·(sb(t) e

jωsct − sa(t) e−jωsct)

(A.5)

= (sbI(t) + jsbQ(t)) ej(ωsct−π2

) − (saI(t) + jsaQ(t)) e−j(ωsct+π2

)

= sbI(t) ej(ωsct−π2 ) − saI(t) e−j(ωsct+

π2

) (A.6)

+sbQ(t) ejωsct − saQ(t) e−jωsct

The corresponding transmitted signal becomes

SSAL(t) =√

2PT<[(sbI(t) ej(ωsct−π2 ) − saI(t) e−j(ωsct+

π2

) (A.7)

+ sbQ(t) ejωsct − saQ(t) e−jωsct) ejωct]

=√

2PT<[sbI(t) ej((ωc+ωsc)t−π2 ) − saI(t) ej((ωc−ωsc)t−

π2

)

+ sbQ(t) ej(ωc+ωsc)t − saQ(t) ej(ωc−ωsc)t]

= sbI(t) sin((ωc + ωsc)t)− saI(t) sin((ωc − ωsc)t) + sbQ(t) cos((ωc + ωsc)t)

− saQ(t) cos((ωc − ωsc)t) (A.8)

= sbI(t) sin(ωbt) + sbQ(t) cos(ωbt)− saI(t) sin(ωat)− saQ(t) cos(ωat)

(A.9)

The spectrum representation of SAL is shown in Fig. A.2. The spectrum is

similar to that of the CAL except that the baseband components occupy di�erent

phases of the carrier.

The baseband signal - AltBOC-NCE modulation. The subcarrier in the

case of AltBOC-NCE modulation is a complex square wave, unlike the pure tone

used for AltLOC. Using the square wave instead of a pure tone greatly reduces

the burden on the subcarrier generation hardware (or software). Mathematically,

A. FUNDAMENTALS OF ALTLOC AND ALTBOC-NCE MODULATION 257

ω

E5b

-Q

0 ω c

ω c-ωs

ω c+ω s

E5a

-Q

E5a-I

E5b-I-ω

c

-ωc-ω

s

-ωc+ω s

E5a

-Q

E5a-I

E5b-I

Figure A.2. spectrum of the sine-AltLOC

the square wave is obtained by passing the corresponding exponential through the

signum function. Thus the baseband signal in the case of AltBOC-NCE can be

represented as follows (CABN = CosAltBOC-NCE):

sCABN(t) =1

2

[sbI(t) sgn(ejωsct) + saI(t) sgn(e−jωsct) + sbQ(t) sgn(ej(ωsct+

π2

))

+saQ(t) sgn(e−j(ωsct−π2

))]

(A.10)

Following steps similar to those for AltLOC case, one can write the expression

for the transmitted signal as

SCABN(t) =√

2PT

[<{

1

2[sbI(t) (sgn(cos(ωsct) + j sin(ωsct))) (A.11)

+sbQ(t) (sgn(cos(ωsct)− j sin(ωsct)))

+saI(t) (sgn(− sin(ωsct) + j cos(ωsct)))

+saQ(t) (sgn(sin(ωsct) + j cos(ωbt)))] ejωct}]

=√

2PT

[<{

1

2[(sbI(t) + sbQ(t)) sgn(cos(ωsct))

+j(sbI(t)− sbQ(t)) sgn(sin(ωsct))

+(saQ(t)− saI(t)) sgn(sin(ωsct))

+j(saQ(t) + saI(t)) sgn(cos(ωsct))] ejωct}]

(A.12)

In the above equation, the subcarrier is a square wave of angular frequency

ωsc. Sine terms can be replaced by time-delayed versions of the cosine terms. Let

258 A. FUNDAMENTALS OF ALTLOC AND ALTBOC-NCE MODULATION

Tsc = 1fsc

be the period of the square wave. Then (A.12) can be written as

SCABN(t) =√

2PT

[<{

1

2[(sbI(t) + sbQ(t)) sgn(cos(ωsct))

+ j(sbI(t)− sbQ(t)) sgn(cos(ωsct− Tsc4

))

+ (saQ(t)− saI(t)) sgn(cos(ωsct− Tsc4

))

+j(saQ(t) + saI(t)) sgn(cos(ωsct))] ejωct}]

(A.13)

=√

2PT

[<{

1

2

[(sbI(t) + sbQ(t)) sc(t) + j(sbI(t)− sbQ(t)) sc(t− Tsc

4)

+ (saQ(t)− saI(t)) sc(t− Tsc4

)) + j(saQ(t) + saI(t)) sc(t)]ejωct

}](A.14)

where sc(t) = sgn(cos(ωsct)) is the subcarrier component. As an example, a single

cycle of the subcarrier waveform for AltBOC-NCE(15,10) which has fsc=15.345 MHz

and the code frequency fco=10.23 MHz is shown in Fig. A.3.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Am

plitu

de

tTsc

sc(t)

sc(t − Tsc

4 )

Figure A.3. subcarrier in AltBOC-NCE

Note that the complex baseband signal may take the values of the sum and

di�erence of {−1, 0, 1}. In the complex domain this gives the normalised values

{−1, 1, 0, j,−j, 12

+ j 12, 1

2− j 1

2,−1

2+ j 1

2,−1

2− j 1

2}. The constellation diagram of the

baseband signal is shown in Fig. A.4. The magnitude of these values is not constant.

This modulation is known as �non-constant envelope AltBOC modulation�.

When this baseband signal is used to modulate the carrier at the transmitter,

the ampli�er has to cater to varying amplitudes of the signal. Also there is a zero

in the baseband signal which results in no energy transmitted. Such an ampli�er

design will be ine�cient and also non-linearities creep in during the modulation.

For this reason, AltBOC-NCE modulation is not used in practice.

The Power Spectral Density (PSD). The PSD of an non-constant envelope

AltBOC signal is derived in Rebeyrol et al. (2005) with 2fscfco

representing the number

A. FUNDAMENTALS OF ALTLOC AND ALTBOC-NCE MODULATION 259

1

j

-1

-j

0

Figure A.4. AltBOC-NCE modulation : constellation diagram

of sub-carrier periods in one chip period,

GAltBOC−NCE(f) =8

Tcπ2f 2

sin2(πfTc)

cos2(πf Tsc

2

) (1− cos(πf

Tsc2

)),

2fscfco

even (A.15)

GAltBOC−NCE(f) =8

Tcπ2f 2

cos2(πfTc)

cos2(πf Tsc

2

) (1− cos(πf

Tsc2

)), n odd (A.16)

Equations (A.16) and (2.15) (which describes the PSD for the Galileo E5 AltBOC

signal) are plotted in Fig. A.5 for the parameters fco = 10.23MHz and fsc =

15.345MHz.

−6 −4 −2 0 2 4 6

x 107

−110

−105

−100

−95

−90

−85

−80

−75

−70

−65

−60

frequency (Hz)

Am

plitu

de (

dBW

)

AltBOC − Non Const. Env.AltBOC − Const. Env.

Figure A.5. PSD of the constant envelope AltBOC(15,10)

APPENDIX B

Signi�cance of the Product Signal in AltBOC(15,10)

It is observed that in order to make the envelope constant, two special sub-carrier

waveforms were chosen. Of these two waveforms, the product sub-carrier waveform

scp carries the product codes. The power spectral density for an extended frequency

range is shown in Fig. B.1. It can be seen that the e�ect of the product sub-

−1 −0.5 0 0.5 1

x 108

−110

−105

−100

−95

−90

−85

−80

−75

−70

−65

−60

Frequency (Hz)

Am

plitu

de (

dBW

)

AltBOC − Non Const. Env.AltBOC − Const. Env.

Figure B.1. PSD of the AltBOC-NCE(15,10) in a wider frequency range

carrier is to re-arrange the signal into slightly di�erent frequency ranges (Lestarquit

et al., 2008). The two main lobes remain almost the same as in AltBOC-NCE

(Fig.2.5). This is due to the fact that the product sub-carrier has its centre frequency

component at ±45 MHz (six zero crossings instead of two of the single subcarrier).

The product signal is formulated as (using the same notations as the Galileo

ICD):

E5prod =1

2√

2·(

eaI(t− τ)±j · eaQ(t− τ)

)· scp(t− τ)+

1

2√

2·(

ebI(t− τ)±j · ebQ(t− τ)

)· sc∗p(t− τ) (B.1)

The product signal has a very interesting feature to play in the total signal, apart

from that of helping in producing a constant envelope modulation. The auto-

correlation function (ACF) of the product signal for the in�nite bandwidth and

70 MHz receiver bandwidth is shown in Fig. B.2. Because the correlation function

261

262 B. SIGNIFICANCE OF THE PRODUCT SIGNAL IN ALTBOC(15,10)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.1

−0.05

0

0.05

0.1

0.15

Time delay (chips)

Nor

mal

ized

cor

rela

tion

valu

e

E5prod

: Inf BW

E5prod

: 70 MHz BW

Figure B.2. ACF of the product signal

has a very sharp peak, it in�uences the sharpness of the ACF of AltBOC (15,10).

Figs. (B.3) and (B.4) show the correlation function of the AltBOC(15,10) signal

with and without considering the product component, along with the zoom ver-

sion around the peak. One can observe that neglecting the product signal yields

slightly inferior performances (due to a less sharp main correlation peak) for very

high bandwidths. However the same is not true for lower bandwidths.

Because the product sub-carrier frequency is thrice that of the sum-sub-carrier

(6 zero crossings as against 2 of the sum sub-carrier), the product signal energy will

be concentrated around +/- 45 MHz from the centre. Hence a 70 MHz �ltering

(i.e. +/- 35 MHz) will �lter out the product signal. Due to this reason, the ACF

of AltBOC (15,10) in 70 MHz bandwidth with and without considering the product

signal will be very close to each other, as observed in Fig. (B.4).

B. SIGNIFICANCE OF THE PRODUCT SIGNAL IN ALTBOC(15,10) 263

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time delay (chips)

Nor

mal

ized

cor

rela

tion

valu

e

With E5prod

Without E5prod

−0.1 0 0.10.5

0.6

0.7

0.8

0.9

1

Figure B.3. ACF of the AltBOC(15,10) signal with and without theproduct signal with in�nite bandwidth; zoom version around the peak(inset)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time delay (chips)

Nor

mal

ized

cor

rela

tion

valu

e

With E5prod

Without E5prod

−0.1 0 0.10.5

0.6

0.7

0.8

0.9

Figure B.4. ACF of the AltBOC(15,10) signal with and withoutthe product signal with 70 MHz bandwidth; zoom version around thepeak (inset)

APPENDIX C

Factorisation of the FFT Transform Lengths

This section brie�y describes the prime factor and Mixed-radix approaches. De-

tailed descriptions of the prime factor and Mixed-radix algorithms can be found in

(Smith 1995) and citations therein.

Prime-factor FFT algorithm. The basic idea is to factor N (for an N -Point

FFT) into two or more relatively prime numbers, implement the small-point building

blocks, and combine them to obtain the �nal result. Hence prime factor algorithms

are characterised by small-point building blocks. Thus if N can be factored into

N = P · Q then the transform can be implemented as shown in Fig. C.1. The

small-points can be further factored along the same lines. The algorithm involves

Q P -point FFT computations, P Q-point FFT computations and data re-ordering

in between. If is N factored into n relatively prime factors Pi then the number of

real additions and real multiplications is given by

AP =n∑i=1

N

Pi· Ai (C.1a)

MP =n∑i=1

N

Pi·M i (C.1b)

where Ai is the number of real additions in the Pi-point building block and Mi is

the number of real multiplications in the Pi-point building block.

Mixed-radix FFT algorithm. The basic idea is similar to the prime factor

approach except that there is no constraint on the factors. The penalty paid is that

complex-multiplications should be used while combining the results of small-point

blocks instead of just re-ordering. Thus if N can be factored into N = P · Q then

the transform can be implemented as shown in Fig. C.2. The small-points can

be further factored along the same lines. The algorithm involves Q P -point FFT

computations, and P Q-point FFT computations, data re-ordering and complex

Data Re-ordering P-Point FFT

Data Re-ordering Q-Point FFT

Figure C.1. Prime factor FFT approach

265

266 C. FACTORISATION OF THE FFT TRANSFORM LENGTHS

Data Re-ordering P-Point

FFT

Data Re-ordering Q-Point

FFT

Complex Multipliers

Figure C.2. Mixed radix FFT approach

multiplications in between. For the two-factor approach the number of real additions

and real multiplications is

AM = P · AQ +Q · AP + 3 · (P − 1) · (Q− 1)

MM = P ·MQ +Q ·MP + 3 · (P − 1) · (Q− 1)

where A(orM) and P (or Q) indicates the number of real additions (or multiplica-

tions) in the P (or Q)-point building blocks respectively. In the above equations it

is assumed that the complex multiplications are achieved by three multiplications

and three additions.

APPENDIX D

Frequency Selective Propagation Delay Distortions

Phase delay, group delay and the frequency selectivity. It is well known

that the channel through which the GNSS signal traverses from the satellite to the

receiver acts like a �lter. Let Θ(ω) be the radian phase shift experienced by each

sinusoid component of the signal when the signal passes through a �lter (ω denoting

the angular frequency). The phase delay is de�ned by

tp(ω) , −Θ(ω)

ω(D.1)

i.e. the phase delay expresses the phase response as time delay. The group delay is

nothing but the time delay of the amplitude envelope of a sinusoid at frequency ω

and is de�ned by

tg(ω) , − d

dωΘ(ω) (D.2)

If the phase response is linear over the observation bandwidth, the group delay at a

particular frequency ω gives the slope of the phase response.

The channel for the GNSS signal in practice is not a Linear Time Invariant

(LTI) �lter. In addition, for wideband signals like Galileo E5 AltBOC(15,10), the

frequency selectivity within the signal bandwidth plays a major role in determining

the overall channel response to the signal. The two main properties of the channel

that a�ect the channel response are the ionosphere and multipath propagation.

Phase delay, group delay and Ionospheric errors. The ionosphere is a

dispersive medium such that whenever a signal passes through the ionosphere, it

speeds-up the signal (compared to that in the free space). This results in a phase

advance of the signal and the amount of that phase advance is dictated by the

electron density of the ionosphere (at that particular pierce point and time). For

the sake of dealing with causal systems, usually the ionospheric e�ect is speci�ed in

terms of the propagation delay which is given by Won and Lee (2005)

tpiono(ω) = −bc· TECω2

(D.3)

where c is the speed of the signal in free space, TEC represents the total electron

content and the constant b = 40.3× 4π2.

The e�ect of ionospheric dispersion in the E5 AltBOC(15,10) is studied in Slee-

waegen et al. (2004). If I0 is the delay experienced by the signal at the centre of

267

268 D. FREQUENCY SELECTIVE PROPAGATION DELAY DISTORTIONS

the band (1191.795 MHz), then the phase delay and the group delay at any other

frequency is given by

tpiono(ω) = I0

(ωE5

ω

)2

(D.4)

tgiono(ω) = −I0

(ωE5

ω

)2

(D.5)

It is very clear from (D.4) and (D.5) that the phase delay and the group delay

ionosphere errors are proportional to the square of the ratio of frequencies referenced

to the centre frequency.

APPENDIX E

Envelope and Squared Envelope Detectors

This section provides a brief background to the two basic types of detectors, the

Envelope detector and the Squared-envelope detector. Fig. E.1 shows the functional

diagram of a typical receiver depicting detector outputs for both the types and the

nomenclature is self explanatory. The �nal summation is �non-coherent�. These de-

tectors are sometimes referred to as �Linear Law� and �Square-Law� in the literature

(Bird, 1995; Marcum, 1960). Di�erent terminology is used here since the envelope

operation is not a linear process. Results of earlier analysis of these detectors (Bird,

1995; Marcum, 1960; McDonough and Whalen, 1995) applied to the unambiguous

detector architecture, are discussed below.

Let the input to the detector be represented as

rs(t) = Ac(t− τ)d(t− τ) cos ((ωc + ωd) t+ θ) + nW (t) (E.1)

where A= signal amplitude, c(t)= PRN code modulation, d(t)= data modulation,

ωc = 2πfc= carrier frequency (E5), ωd=Doppler frequency, τ is the unknown delay, θ

is the unknown carrier phase and nW (t) is the AWGN process. The received signal is

multiplied by a locally generated complex carrier, correlated with a locally generated

replica of the spreading code, integrated coherently over a speci�ed duration and

the result is sampled. Typical integration duration is Tcoh= one code period and the

sampling interval is ∆Tc where Tc is the chip duration ( 1/10.23e6 in case of E5) and

∆ ∈ (0, 1] is the code search step size (which is a design parameter). The in-phase

and quadrature phase correlation results are then squared and added to obtain the

�Squared Envelope� output or the magnitude is computed to obtain the �Envelope�

output. These results can then be accumulated over M samples, �non-coherently�

Code

Generator

Complex

Carrier

2

)(ny

2

Envelope

Sq.

Envelope

t Tc

ze

zs

t Tc

)(trs

Figure E.1. Typical quadrature detector

269

270 E. ENVELOPE AND SQUARED ENVELOPE DETECTORS

to obtain the �nal decision statistic ze or zs:

ze =M∑k=1

ek, ek =√I2k +Q2

k (E.2a)

zs =M∑k=1

sk, sk = I2k +Q2

k (E.2b)

Squared envelope detector. The two correlation outputs can be considered

as independent and identically distributed (i.i.d.) Gaussian processes. With this

assumption, the squared envelope detector statistic will have a Chi-squared distri-

bution in the noise-only case with 2M degrees of freedom. In the signal plus noise

case, this distribution will be a non-central Chi-square with 2M degrees of freedom

and non-centrality parameter N = 2M · SNR · R2(τ − τ), τ being the estimated

code delay. The probability of detection (Fischer et al., 2004; Simon, 2002) is:

Pd = QM

(a

σ,

√η

σ

)(E.3)

where a =√N , σ is the noise standard deviation, η is the decision threshold and

QM is the generalised Marcum's Q function of order M .

Envelope detector. Under the same i.i.d. Gaussian process assumption, the

input to the non-coherent integration, ek will have a Ricean distribution (Simon,

2002) where signal is present (non-zero mean) along with the noise. In the noise

only situation (zero-mean), ek will have a Rayleigh distribution. To the authors'

knowledge, there is no closed form expression for the probability density function

of a sum of any M Ricean (or Rayleigh, except for M=2 (Altman and Sichak,

1956)) distributed processes. Only closed form approximations to the in�nite series

expressions are available (Hu and Beaulieu, 2005b,a). Hence the computation of the

probability of detection is not straightforward.

This problem has been analysed and the e�ect of these two detectors on the

probability of detection compared (Bird, 1995; Marcum, 1960). When M=1, both

detectors perform alike. The envelope detector performs better for small values

of M< 70 with a peak di�erence of 0.11 dB at M=10. For M> 70, the squared

envelope detector performs better asymptotically reaching 0.19 dB as M →∞.

APPENDIX F

Code Phase Jitter for the Generalised Tracking Architecture

The steps followed to arrive at the code phase error variance for the code tracking

loop, that uses a non-coherent early-late power discriminator are (similar to the

analysis given in Holmes (2007);pp. 483-492):

(1) Multiply rIF (t) given in (5.1) with the local carrier x(t) = exp(−j(ω0t+ θ

))to get y(t).

(2) Obtain y1(t) = y(t) · s1(t) and y2(t) = y(t) · s2(t).

(3) Obtain y1 and y2 from y1(t) and y2(t) respectively, after the integrate and

dump operation: y1m =∫ mT1

(m−1)T1y1(t) dt and y2m =

∫ mT1(m−1)T1

y2(t) dt.

(4) Assuming that the integration over T1 is su�cient to form the autocorrela-

tion function, form the discriminator output for the non-coherent early-late

discriminator as: E(t) = |y1|2−|y2|2. Note that the subscript m is omitted

since the outputs correspond to only one integration duration.

(5) The error in code delay estimate is ε, which is actually a time varying pa-

rameter. The discriminator output is a function of the ε(t). De�ning the

code tracking loop transfer function, forming a stochastic di�erential equa-

tion, assuming that the error contribution is only from the loop noise (i.e.

assuming that the input timing is constant over the integration duration)

and that the loop is much less than the integration duration, the linearised

tracking loop error variance due to thermal noise can be approximated to

σ2ε u 2BLSN (0)

K2P 2T 41

where K is the slope of the S-curve, BL is the one sided

close-loop noise bandwidth and SN(0) is the noise power spectral density

at zero frequency.

(6) Evaluate SN(0) using the noise-related terms of E(t) to obtain σ2ε .

The result of each step is given below.

y(t) = rIF (t) ·√

2(

cos(ω0t+ θ

)− j sin

(ω0t+ θ

))=ysignal(t) + ynoise(t) (F.1)

271

272 F. CODE PHASE JITTER FOR THE GENERALISED TRACKING

ysignal(t) =√P ·

s′c(t−τ)︷︸︸︷

sc (t− τ) · cos (θe)− ss (t− τ) · sin (θe)

j

s′s(t−τ)︷︸︸︷

(−sc (t− τ) · sin (θe) + ss (t− τ) · cos (θe))

(F.2)

where it is assumed that during tracking ∆ω0 = ω0 − ω0 = 0, θe = θ − θ and

s′(t) = s

′c(t) + js

′s(t).

ynoise(t) =

n′c(t)︷︸︸︷

nc (t) · cos (θe)− ns (t) · sin (θe) + j

n′s(t)︷︸︸︷

(−nc (t) · sin (θe) + ns (t) · cos (θe))

(F.3)

with n′(t) = n

′c(t)+n

′s(t). Using the subscript z = 1, 2 to indicate both the reference

signals,

yz =√P

∫ T1

0

(s′(t− τ) + n

′(t))sz(t− τ) dt

=√PT1Rz + nz (F.4)

where Rz indicates the correlation between input signal and the reference signal,

and nz indicates the correlation between the noise and the reference signal.

E(t) = T 21P[|R1|2 − |R2|2

]+[|n1|2 − |n2|2

]+ 2√PT1 [R1n1c −R2n1c] (F.5)

where the �rst term is related to �signal/signal� , second term is �noise/noise� and

the third term is related to �signal/noise�. Now

SN(0) u T1

[σ2noise/noise + σ2

signal/noise

](F.6)

and the noise variances can be computed from (F.5) as

σ2noise/noise = 2N2

0T21

[1− |Rr|2

](F.7)

σ2signal/noise = 2PT 3

1 [R1R∗2 (1−R∗r) +R∗1R2 (1−Rr)] (F.8)

Substituting (F.7) and (F.8) in (F.6), according to step 6 one obtains (5.7):

σ2ε =

4N0BL

K2P

[(|R1|2 + |R2|2 −R1 ·R∗2 ·R∗r −R∗1 ·R2 ·Rr

)+N0

(1− |Rr|2

)PT1

](F.9)

APPENDIX G

EMLP Discriminator Function for AltBOC Signals in

Multipath

Consider the direct signal with one re�ected signal component at the input of

the base band processing where the IF signal is:

rIF (t) =√

2P ·(sc (t− t0) · cos (ω0t+ θ0)−ss (t− t0) · sin (ω0t+ θ0)

)+

√2aiP ·

(sc (t− t0 − τi) · cos (ω0t+ θ0 + φi)−ss (t− t0 − τi) · sin (ω0t+ θ0 + φi)

)(G.1)

The output of the complex carrier mixer can be written as:

y(t) = rIF (t) · x(t)

=rIF (t) ·√

2(

cos(ω0t+ θ0

)− j sin

(ω0t+ θ0

))(G.2)

Assuming perfect carrier frequency synchronisation (for the DLL analysis purpose)

with ω0 = ω0, neglecting the second harmonics and after some trigonometric sim-

pli�cation one obtains:

y(t) =√P ·(sc (t− t0) · cos (θe)− ss (t− t0) · sin (θe)−jsc (t− t0) · sin (θe) + jss (t− t0) · cos (θe)

)+

√α1P ·

(sc (t− t0 − τ1) · cos (θe + φ1)− ss (t− t0 − τ1) · sin (θe + φ1)−jsc (t− t0 − τ1) · sin (θe + φ1) + jss (t− t0 − τ1) · cos (θe + φ1)

)(G.3)

where θe = θ0 − θ0 is the error in the carrier phase estimate. Multiplication with

the reference signals produces the outputs y1(t) and y2(t). For the case of early and

273

274 G. EMLP DISCRIMINATOR FUNCTION FOR ALTBOC SIGNALS IN MULTIPATH

late arms, the reference signals are given by (5.4a) and (5.4b).

y1(t) = y(t) · s∗ (t− τ + δTc)

=y(t) · (sc (t− τ + δTc)− jss (t− τ + δTc))

=√P ·

sc (t− t0) · sc (t− τ + δTc) · cos (θe) +

jsc (t− t0) · sc (t− τ + δTc) · sin (θe) +

jss (t− t0) · ss (t− τ + δTc) · sin (θe)−ss (t− t0) · ss (t− τ + δTc) · cos (θe)

+

√P ·

sc (t− t0 − τ1) · sc (t− τ + δTc) · cos (θe + φ1) +

jsc (t− t0 − τ1) · sc (t− τ + δTc) · sin (θe + φ1) +

jss (t− t0 − τ1) · ss (t− τ + δTc) · sin (θe + φ1)−ss (t− t0 − τ1) · ss (t− τ + δTc) · cos (θe + φ1)

(G.4)

Note that in (G.4) the sc(·)xss(·) terms have been neglected because they are the

result of combination of di�erent PRN codes modulated onto the orthogonal carriers

and hence have negligible correlation value. Strictly speaking, these terms should

have been omitted in the next step, but is done so here to shorten the equation.

Let ε = t0 − τ be the error in the code delay estimate. After the integration

followed by the sample and hold operation, assuming that carrier phase error θe is

zero, one gets (after some algebraic manipulation):

y1m =√P

Rc (ε+ δTc) +Rs (ε+ δTc) +√α1 [Rc (ε+ δTc + τ1) +Rs (ε+ δTc + τ1)] cos (φ1)

+j√α1 [Rc (ε+ δTc + τ1) +Rs (ε+ δTc + τ1)] sin (φ1)

(G.5)

where Rc/s(ε) is the normalised correlation function. (G.5) is normalised with the

integration time T1. Following similar steps for the second output:

y2m =√P

Rc (ε− δTc) +Rs (ε− δTc) +√α1 [Rc (ε− δTc + τ1) +Rs (ε− δTc + τ1)] cos (φ1)

+j√α1 [Rc (ε− δTc + τ1) +Rs (ε− δTc + τ1)] sin (φ1)

(G.6)

The EMLP DLL forms the error:

Demlp (ε) = |y1m|2 − |y2m|2 (G.7)

G. EMLP DISCRIMINATOR FUNCTION FOR ALTBOC SIGNALS IN MULTIPATH 275

Now the square of the �rst output is (after some algebraic steps and grouping the

terms together to obtain terms):

|y1m|2 = P

|R (ε+ δTc)|2 + α1 |R (ε+ δTc + τ1)|2 +

2√α1 cos (φ1)





(G.8)

The last term inside the brackets, comprising the correlation functions, can be

thought of as the correlation indicator of the complex correlations and is de�ned

as:

R′((ε+ δTc) , τ1) =





(G.9)

|||ly the square of the second output gives:

|y2m|2 = P

(|R (ε− δTc)|2 + α1 |R (ε− δTc + τ1)|2 +

2√α1 cos (φ1)R

′((ε− δTc) , τ1)

)(G.10)

Substituting (G.8) and(G.10) into (G.7), (6.6) is obtained.

Even though the above derivation uses the early and late correlator concept,

recall that the reference signals can take any form and the equation still holds good.

In this generic case:

Demlp (ε) = T 21P

|R1 (ε)|2 − |R2 (ε)|2 + α1

(|R1 (ε+ τ1)|2−|R2 (ε+ τ1)|2

)+2√α1 cos(φ1)

(R′1 (ε, τ1)−R′2 (ε, τ1)

) (G.11)

where R1(·) and R2(·) are the complex correlation functions between the input and

�rst reference signal and the input and the second reference signal respectively.

Similarly, R′1 and R

′2 can be de�ned.

With (G.5) and (G.6), the discriminator error for the coherent early minus late

discriminator can be derived and it follows that

Dceml (ε) = P

(|R1 (ε)| − |R2 (ε)|+

√α1 cos(φ1) (|R1 (ε+ τ1)| − |R2 (ε+ τ1)|)

)(G.12)

APPENDIX H

Carrier Phase Multipath Error for AltBOC Signals

Under the same assumption of single re�ected signal component one can start

with (G.2). Let ωe = ω0 − ω0 be the error in frequency estimation. Without loss

of generality it can be assumed that the phase of the direct signal θ0 is zero. Then

(neglecting the second harmonic terms):

y(t) =√P ·

sc (t− t0) · cos(ωet− θ

)− ss (t− t0) · sin

(ωet− θ

)−

jsc (t− t0) · sin(ωet− θ

)+ jss (t− t0) · cos

(ωet− θ

) +

√α1P ·

sc (t− t0 − τ1) · cos(ωet− θ + φ1

)− ss (t− t0 − τ1) · sin

(ωet− θ + φ1

)+

jsc (t− t0 − τ1) · sin(ωet− θ + φ1

)+ jss (t− t0 − τ1) · cos

(ωet− θ + φ1

) (H.1)

Multiplication with the generic reference signals (5.4c), followed by integration

and sample and hold produces the output:

y0m =1

T2

nT2∫(n−1)T2

y(t)· s∗(t− τ)dt

=√P ·

(Rc(ε) + jRs(ε)) ·(

cos (ωet) cos(θ) + sin (ωet) sin(θ))

+

(Rc(ε) + jRs(ε)) ·(

sin (ωet) cos(θ)− cos (ωet) sin(θ))

+√α1P ·

(Rc(ε+ τ1) + jRs(ε+ τ1)) ·(

cos (ωet) cos(θ + φ1) + sin (ωet) sin(θ + φ1))

+ (Rc(ε+ τ1) + jRs(ε+ τ1)) ·(sin (ωet) cos(θ + φ1)− cos (ωet) sin(θ + φ1)

)

(H.2)

The discriminator of the carrier tracking loop (PLL) drives the sin(ωet) terms to

zero. In addition, the loop estimates the composite phase of the signal. Hence

θ = θc. Equating sin(ωet) terms to zero:

T2

√P

( (R(ε) +

√α1R(ε+ τ1) cos(φ1)

)sin(θc)

−√α1R(ε+ τ1) sin(φ1) cos(θc)

)+

jT2

√P

( (R(ε) +

√α1R(ε+ τ1) cos(φ1)

)cos(θc)

+√α1R(ε+ τ1) sin(φ1) sin(θc)

)= 0 (H.3)

277

278 H. CARRIER PHASE MULTIPATH ERROR FOR ALTBOC SIGNALS

Since the output has real and imaginary terms, both of them should be zero to satisfy

the equality. In addition, it can be veri�ed that both result in the same expression

for the composite phase. Equating the real term to zero and rearrangement will

give:

tan(θc) =

√α1 |R (ε+ τ1)| sin (φ1)

|R (ε)|+√α1 |R (ε+ τ1)| cos (φ1)(H.4)

For any non-zero phase shift of the direct signal, φc = θc − θ0 gives (6.8).

APPENDIX I

Group Delay Error Caused by Multipath

The multipath phase error observed at the output of the correlator when the

direct signal is a�ected by a single re�ected signal can be obtained by re-writing

(6.8) as

tpmulti =1

ωarctan

[AR(ε+ δ) sin θ


](I.1)

where ε is the code phase error caused by the delay locked loop, and δ is the time

di�erence between the direct and the re�ected signal, θ is the phase di�erence be-

tween the direct and the re�ected signal and A is the amplitude ratio of the re�ected

signal to the direct signal.

Using the de�nition of group delay, the error due to multipath is

tgmulti = − d

dω[tpmulti ]

tgmulti = − d

dω

[arctan

(AR(ε+ δ) sin θ


)]The phase di�erence θ can be written as θ = βω where β = δTc

C, C being the speed

of light. Using the di�erentiation rules for the nested functions,

tgmulti = − 1(1 +

(AR(ε+δ) sin(βω)

R(ε)+AR(ε+δ) cos(βω)

)2) .

d

dω

(AR(ε+ δ) sin(βω)

R(ε) + AR(ε+ δ) cos(βω)

)

tgmulti =−AβR(ε)R(ε+ δ) cos(βω)− A2βR2(ε+ δ)

R2(ε) + A2R2(ε+ δ) + 2R(ε)AR(ε+ δ) cos(βω)

Substituting β = −(tg2 − tg1) and reverting back to the phase di�erence representa-

tion θ,

tgmulti =AR(ε+ δ) (tg2 − tg1) [AR(ε+ δ) +R(ε) cos θ]

R2(ε) + A2R2(ε+ δ) + 2R(ε)AR(ε+ δ) cos θ

279

APPENDIX J

Power Spectral Density of TMOC-QPSK

Power Spectral Density of TMOC-QPSK, for 2fscfco

odd. Denoting sc(t)

as scc(t) and sc(t − Tsc4

) as scs(t), the autocorrelation expression for the baseband

signal in (8.4) can be written as

Rs(τ) = ReP1,scc(τ) +ReP1,scs(τ) +ReD1,scc(τ) +ReD1,scs(τ)

+ReP2,scc(τ) +ReP2,scs(τ) +ReD2,scc(τ) +ReD2,scs(τ) (J.1)

Under the assumption that the autocorrelations of di�erent codes are equal (sim-

ilar assumption as in Rebeyrol et al. (2005)),

GTMOC−QPSK(f) =4

Tc|SCc(f)|2 +

4

Tc|SCs(f)|2 (J.2)

where SCc(f) and SCs(f) are the Fourier transforms of scc(t) and scs(t) respectively

over [0, Tc). The Fourier transform of scc(t) is

SCc(f) =

(−jπf

)e−jπfTc

sin2(πf Tsc

4

)cos (πfTc)

cos(πf Tsc

2

) (J.3)

from which one can obtain

|SCc(f)|2 =1

π2f 2

cos2 (πfTc)

cos2(πf Tsc

2

) {cos

(πf

Tsc2

)− 1

}2

(J.4)

From (A−12) of Betz (Winter 2001-2002) for the sine phased subcarrier component,

|SCs(f)|2 =1

π2f 2

cos2 (πfTc)

cos2(πf Tsc

2

) sin2

(πf

Tsc2

)(J.5)

Substituting (J.4) and (J.5) in (J.2) (8.7) is obtained.

Power Spectral Density of TMOC-π4-QPSK, for 2fsc

fcoodd. Denoting sco(t)

as scco(t) and sco(t− TSc4

) as scso(t), the autocorrelation expression for the baseband

signal in (8.4) can be rewritten as (taking TMOC-π4-QPSK-IQ as the example)

Rs(τ) = ReP1,scc(τ) +ReP1,scs(τ) +ReD1,scco(τ) +ReD1,scso(τ)

+ReP2,scc(τ) +ReP2,scs(τ) +ReD2,scco(τ) +ReD2,scso(τ) (J.6)

281

282 J. POWER SPECTRAL DENSITY OF TMOC-QPSK

Hence, the power spectral density can be expressed as

GTMOC−π4−QPSK(f) =

2

TC|SCc(f)|2 +

2

TC|SCs(f)|2 +

2

TC|SCco(f)|2 +

2

TC|SCso(f)|2

(J.7)

Now with 2p = 2fscfco

,

sc(t) =

2p−1∑k=0

(−1)k µTsc2

(t− kTsc

2

)(J.8)

where, for the cosine phased π4phase-shifted subcarrier scco(t),

µTsc2

(t) =

1[0, Tsc

8

)−1

[Tsc8, Tsc

2

) (J.9)

The Fourier transform SCco(f) can be derived as

SCco(f) =

(1

πf

)e−jπf

Tsc8 e−j(

2TcTsc−1)πf Tsc

2cos (πfTc)

cos(πf Tsc

2

) ·[sin

(πf

Tsc8

)− e−jπf Tsc2 sin

(3πf

Tsc8

)](J.10)

For the sine phased π4phase-shifted subcarrier scs(t),

µTsc2

(t) =

1[0, 3Tsc

8

)−1

[3Tsc

8, Tsc

2

) (J.11)

and it can be shown that SCso(f) evaluates to the same expression as (J.10). Hence

|SCco(f)|2 = |SCso(f)|2 =1

π2f 2

cos2 (πfTc)

cos2(πf Tsc

2

) ·[cos2

(πf

Tsc2

)− 2 cos

(πf

Tsc2

)cos

(πf

Tsc4

)+ 1

](J.12)

Substituting (J.12), (J.4) and (J.5) in (J.7) (8.8) is obtained.

APPENDIX K

Output of the Correlator and Reference Signal Correlations

With (8.10) as one input and s(t− τ) = sI(t− τ)− jsQ(t− τ) as the other input,

the output of the reference signal mixer can be written as

y(t) =A√2{[sI(t− τ)sI(t− τ) cos (∆ωdt+ ∆θ) + sQ(t− τ)sQ(t− τ) cos (∆ωdt+ ∆θ)]

+j [sQ(t− τ)sQ(t− τ) sin (∆ωdt+ ∆θ) + sI(t− τ)sI(t− τ) sin (∆ωdt+ ∆θ)]

+ [sI(t− τ)sQ(t− τ) sin (∆ωdt+ ∆θ)− sQ(t− τ)sI(t− τ) sin (∆ωdt+ ∆θ)]

− j [sI(t− τ)sQ(t− τ) cos (∆ωdt+ ∆θ)− sQ(t− τ)sI(t− τ) cos (∆ωdt+ ∆θ)]}+ny(t) (K.1)

The output of the accumulator is

zk =

kTcoh∫(k−1)Tcoh

y(t)dt (K.2)

The individual correlations between the baseband component of the input signal

and the corresponding component of the reference signal have to be evaluated. Since

the baseband component and the reference signal component depend on the type of

modulation, one needs to evaluate all the four correlations present in (K.1) for the

three types of modulations viz. AltBOC, TMOC-QPSK and TMOC-π4-QPSK. One

example is shown here in detail and the results are provided for others. Consider the

�rst of the eight terms of (K.1) populated in (K.2). For AltBOC, with the subcarrier

signal representing the corresponding waveform (and neglecting the inter-modulation

product terms for simplicity), the baseband component of the input signal (real part

here) is

sI,AltBOC(t− τ) = eaI(t− τ)scI(t− τ) + eaQ(t− τ)scQ(t− τ)

+ebI(t− τ)scI(t− τ)− ebQ(t− τ)scQ(t− τ)

and the reference signal (real part) is

sI,AltBOC(t− τ) = eaI(t− τ)scI(t− τ) + eaQ(t− τ)scQ(t− τ)

−ebI(t− τ)scI(t− τ) + ebQ(t− τ)scQ(t− τ)

283

284K. OUTPUT OF THE CORRELATOR AND REFERENCE SIGNAL CORRELATIONS

Hence the �rst term of zk can be evaluated as

kTcoh∫(k−1)Tcoh

sI,AltBOC(t− τ)sI,AltBOC(t− τ) cos (∆ωdt+ ∆θ) dt =

Tcoh2

[RaI,aI (∆τ)RscI ,scI (∆τ) + 3Rc +RaQ,aQ (∆τ)RscQ,scQ (∆τ) + 3Rc

RbI,bI (∆τ)RscI ,scI (∆τ) + 3Rc +RbQ,bQ (∆τ)RscQ,scQ (∆τ) + 3Rc]·

sinc

(∆ωd

Tcoh2

)cos (∆θ)

=Tcoh

24R (∆τ)Rsc,AltBOC (∆τ) sinc

(∆ωd

Tcoh2

)cos (∆θ) + 12Rc (∆τ) (K.3)

where Rsc,AltBOC (∆τ) represents the combined correlations of all the subcarrier

pairs and Rc (∆τ) represents the cross correlation value of a single primary code

pair. It can be shown that the correlation between the subcarrier pairs are equal

i.e. RscI ,scI (∆τ) = RscQ,scQ (∆τ).

Similarly, the correlation due to sQ,AltBOC(t − τ)sQ,AltBOC(t − τ) can be shown

to be equivalent to (K.3). The correlation due to sI,AltBOC(t − τ)sQ,AltBOC(t − τ)

and sQ,AltBOC(t− τ)sI,AltBOC(t− τ) result in only cross correlation noise 12Rc. The

correlation between noise and the reference signal can be grouped together as n′(t) =∫ kTcoh(k−1)Tcoh

ny(t)s(t− τ)dt . Finally the output of the correlation can be written as

zk =√

2ATcohR (∆τ)Rsc,AltBOC (∆τ) sinc

(∆ωd

Tcoh2

)ej∆θ +R′ + n′ (t) (K.4)

where

Rsc,AltBOC (∆τ) = 8RscI ,scI (∆τ) (K.5)

and

R′ =A√2

24Rc (∆τ) sinc

(∆ωd

Tcoh2

)ej∆θ (K.6)

is the combined contribution due to the cross correlation terms.

APPENDIX L

Approximation Tables

The criteria for the best possible representation are (a) the error for all possible

representations for a given bit-width should be small and (b) the representation

values themselves should be small. Moreover if the errors in any two possible repre-

sentations are small and comparable to each other, then criterion (b) is given higher

priority. This is because, it is better to use the smaller values so that the growth

of the �nal correlation value after code mixing is kept to a minimum. For example

given 4 bits, {±1.2071,±0.5} can be represented either as {±5,±2} which results

in an error of 3.55% or as {±7,±3} which results in an error of 3.35%. Since the

di�erence between both the errors are negligible, {±5,±2} is selected according to

criterion (b). In the last column, the percentage error is calculated as (actual ratio -

integer representation ratio)/actual ratio.

Table L.1. Scaled integer approximations for {±1.2071,±0.5}

Bit-width Best possible representation % Error

(2's complement) ±1.2071 ±0.52 ±1 ±1 58.58

3 ±2 ±1 17.16

4 ±5 ±2 3.55

5 ±12 ±5 0.59

6 ±29 ±12 0.10

7 ±41 ±17 0.10

Table L.2. Scaled integer approximations for {±1,±0.7071}

Bit-width Best possible representation % Error

(2's complement) ±1 ±0.70712 ±1 ±1 29.29

3 ±3 ±2 6.07

4 ±7 ±5 1.00

5 ±14 ±10 1.00

6 ±17 ±12 0.17

7 ±58 ±41 0.03

285

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