Impact of Notch Filtering on Tracking Loops for GNSS Applications

POLITECNICO DI TORINO

Facoltà di Ingegneria dell’informazione Corso di Laurea in Ingegneria delle Telecomunicazioni

Tesi di Laurea Specialistica

Impact of Notch Filtering on

Tracking Loops for GNSS

Applications

Relatori:

prof. Mark Petovello - UoC

dr. Daniele Borio - UoC

prof.ssa Letizia Lo Presti - PoliTo

prof. Fabio Dovis - PoliTo

Candidato:

Giorgio Giordanengo

Gennaio 2009

List of Figures

1.1 Frequency plan for the different Global Navigation Satellite System

(GNSS) systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Clock misalignments in a GNSS system. . . . . . . . . . . . . . . . . 7

1.3 Example of a generic Direct Sequence Spread Spectrum (DSSS). . . . 8

1.4 Example of Auto Correlation Function (ACF) and Cross Correlation

Function (CCF) for a Gold code employed by the Global Positioning

System (GPS) C/A code. . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Ideal ACF of a GPS C/A code. . . . . . . . . . . . . . . . . . . . . . 13

1.6 General scheme of a GNSS receiver. . . . . . . . . . . . . . . . . . . . 13

1.7 Conceptual scheme of a tracking loop. . . . . . . . . . . . . . . . . . 15

1.8 Block scheme of a generic Delay Lock Loop (DLL). . . . . . . . . . . 16

1.9 Example of code correlation phase. . . . . . . . . . . . . . . . . . . . 17

1.10 Block scheme of a generic Phase Lock Loop (PLL) . . . . . . . . . . . 18

2.1 PSDs of a CWI and a GNSS signal. . . . . . . . . . . . . . . . . . . . 20

2.2 Notch Filter (NF) using the Overlapped FFT-based (OFFT) imple-

mentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Notch Filter (NF) using the Filter Bank interference suppression im-

plementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Adaptive Transversal Filter (ATF) scheme. . . . . . . . . . . . . . . . 24

2.5 Transfer function of the two-poles NF with different kα. . . . . . . . . 26

3.1 Simplified scheme of a GNSS receiver equipped with notch filter for

interference removal. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Filtering distortions on the CCF for different modulations. . . . . . . 31

3.3 Particular of the CCF depicted in Figure 3.2. . . . . . . . . . . . . . 31

3.4 Approximation of the CCF for different modulations and different

front-end bandwidths. . . . . . . . . . . . . . . . . . . . . . . . . . . 34

I

3.5 Approximation of the CCF for different modulations, different front-

end bandwidths and in the presence of NF with bandwidth of 100

kHz and fc of 0.5 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6 RMS-Bandwitdh (βrms) for different modulations and different front-

end bandwidths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.7 Asymmetry Coefficient (fa) for different modulations and different

front-end bandwidths. . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.8 Bias for different modulations and different front-end bandwidths. . . 40

4.1 Simulation and analytical techniques for the analysis of a complex

system and their relative computational complexity. . . . . . . . . . . 42

4.2 Block scheme of a digital DLL . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Equivalent representation of a digital DLL. The delay to be estimated

is represented along with its impact on the correlator outputs. . . . . 44

4.4 Tracking jitter for different E-L spacing as a function of the input

C/N0 under ideal conditions. . . . . . . . . . . . . . . . . . . . . . . . 48

4.5 Tracking jitter for different E-L spacing as a function of the input

C/N0 in the presence of front-end (Chebyshev, six taps, fco 24 MHz)

and notch (fc 0.5 MHz, bandwidth 100 kHz) filter. . . . . . . . . . . . 49

4.6 Jitter evaluated in different ways and for different chip spacing as a

function of the notch bandwidth and the central frequency. . . . . . . 51

4.7 Delay estimation for different filtering stages (front-end, fco 24 MHz,

notch filter, bandwidth 100 kHz, fc 0.5 MHz). . . . . . . . . . . . . . 52

4.8 Bias for different filtering stages and different chip spacing, (front-

end, fco 24 MHz, notch filter, bandwidth 100 kHz, fc 0.5 MHz). . . . 53

4.9 Delay estimated as a function of central frequency (fc) and spacing

(front-end, fco 24 MHz, notch filter, bandwidth 100 kHz). . . . . . . . 53

4.10 Bias introduced by the NF as a function of fc and bandwidth for

different chip spacing, (front-end, fco 24 MHz, notch filter, bandwidth

100 kHz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.11 Comparison between the Asymmetry Coefficient (fa) and the bias

introduced by the Notch Filter. . . . . . . . . . . . . . . . . . . . . . 55

5.1 Scheme used for the data collection. . . . . . . . . . . . . . . . . . . . 57

5.2 Antenna used to collect data. . . . . . . . . . . . . . . . . . . . . . . 57

5.3 Front view of the NI-PXI-5660 front-end. . . . . . . . . . . . . . . . . 58

5.4 Novatel OEM4 GNSS receiver. . . . . . . . . . . . . . . . . . . . . . . 58

II

5.5 Information provided by the Novatel receiver (tracking status 5.5a

and satellites available 5.5b). . . . . . . . . . . . . . . . . . . . . . . . 59

5.6 Power Spectral Density of the GPS-L1 signal. . . . . . . . . . . . . . 60

5.7 C/N0 for two PRNs in the presence of notch filtering for different fcs. 61

5.8 Position estimated by the GSNRxTM in the absence and presence of

the Notch Filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.9 Error position evaluated for different central frequencies of the NF. . 63

5.10 Doppler error evaluated as the difference between the cases with and

without notch filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

III

List of Tables

1.1 Major error sources for GNSS signals [1]. . . . . . . . . . . . . . . . . 7

1.2 Common DLL discriminators [2]. . . . . . . . . . . . . . . . . . . . . 18

2.1 Sources of interference for GPS civil signal [3]. . . . . . . . . . . . . . 21

2.2 Pre-correlation and post-correlation mitigation techniques [3, 4]. . . . 22

2.3 Adaptive Notch Filter algorithms. . . . . . . . . . . . . . . . . . . . . 27

3.1 Delay values for which the approximation has an error less than 10%

in the presence of front-end filtering (Chebyshev, six taps, fco 12 and

24 MHz). Frequency Integral (FI), Numerical Derivative (ND). . . . . 35

3.2 Delay values for which the approximation has an error less than 10%,

in the presence of front-end filtering (Chebyshev, six taps, fco 12 and

24 MHz) and notch filtering (IIR Notch Filter, fc 0.5 MHz, bandwidth

100 kHz). Frequency Integral (FI), Numerical Derivative (ND). . . . 37

3.3 Values of the RMS-Bandwitdh (βrms). . . . . . . . . . . . . . . . . . 39

5.1 Reduction of C/N0 introduced by the NF for different central fre-

quencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.2 Error position introduced by the NF for different central frequencies. 64

IV

Acronyms

ACF Auto Correlation Function

ADC Analog to Digital Converter

ATF Adaptive Transversal Filter

AWGN Additive White Gaussian Noise

BOC Binary Offset Carrier

BPSK Binary Phase Shift Keying

βrms RMS-Bandwitdh

C/A Coarse Acquisition

CAF Cross Ambiguity Function

CCF Cross Correlation Function

CCIT Calgary Center for Innovative Technologies

CDMA Code Division Multiple Access

C/N0 Carrier-to-Noise density ratio

CW Continuous Wave

CWI Continuous Wave Interference

DLL Delay Lock Loop

DSSS Direct Sequence Spread Spectrum

EGNOS European Geostationary Overlay Service

1

EXOR Exclusive-Or

fa Asymmetry Coefficient

FB Filter Bank

fc central frequency

fco cut-off frequency

FFT Fast Fourier Transform

FIR Finite Impulse Response

FLL Frequency Lock Loop

GBAS Ground Based Augmentation Systems

GNSS Global Navigation Satellite System

GSNRxTM GNSS Software Navigation Receiver

GLONASS Global’naya Navigatsionnaya Sputnikovaya Sistema

GPS Global Positioning System

I&D Integrate & Dump

IIR Infinite Impulse Response

IRNSS Indian Regional Nagational System

kα constrain factor

LNA Low Noise Amplifier

LPFN Linear Phase FIR Notch Filter

MC Monte Carlo

MSAS Multi-Functional Satellite Augmentation System

NCO Numerically Controlled Oscillator

NF Notch Filter

2

OFFT Overlapped FFT-based

PLAN Position Location and Navigation

PLL Phase Lock Loop

PRN Pseudo Random Noise

PSD Power Spectral Density

PVT Position Velocity Time

P(Y) Encrypted Precision

QZSS Quasi-Zenith Satellite System

RF Radio Frequency

RFI Radio Frequency Interference

RTK Real Time Kinematic

S-A Semi-Analytic

SBAS Satellite Based Augmentation Systems

SNR Signal to Noise Ratio

SS Search Space

UHF Ultra High Frequency

UoC University of Calgary

WAAS Wide Area Augmentation Systems

3

Contents

Acronyms 1

Introduction 1

Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1 Signals and Systems 4

1.1 GNSS overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 DSSS signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Auto Correlation Function (ACF) and Cross Correlation Function

(CCF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 GNSS receiver structure . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Digital Tracking Loops . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Notch Filter (NF) for Interference Removal 19

2.1 Continuous Wave Interference (CWI) . . . . . . . . . . . . . . . . . . 19

2.2 Notch Filter (NF) Characterization . . . . . . . . . . . . . . . . . . . 22

2.3 Adaptive Notch Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Theoretical Characterization of the Notch Filter Impact 28

3.1 ACF and CCF characterizations: Asymmetry Coefficient . . . . . . . 32

4 Simulation Analysis 42

4.1 Semi-Analytic model for digital DLL . . . . . . . . . . . . . . . . . . 43

4.2 Comparison of the results with the Theoretical Model . . . . . . . . . 48

4.2.1 Tracking Jitter Analysis . . . . . . . . . . . . . . . . . . . . . 49

4.2.2 Delay Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4

5 Real Data Analysis 56

5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Comparison of the results in the presence and absence of Notch Filter:

impact of the notch central frequency . . . . . . . . . . . . . . . . . . 60

6 Conclusions 66

Bibliography 68

5

Introduction

The Global Navigation Satellite System (GNSS) users continue to demand location-

based services everywhere at any time. In this respect, GNSS services should be

available also in hostile environments such as in the presence of Radio Frequency

(RF) interference. One of the most common interference sources is represented by

a Continuous Wave (CW), that is, all those RF signals that can be represented as

pure sinusoids. Since CWI has a narrow spectrum concentrated around a specific

frequency, it can be effectively removed by using a NF. A NF is a linear device

able to remove only a small portion of spectrum of the signal at its input. This

portion is concentrated around a specific frequency whereas all the other components

of the spectrum are left almost unaltered. For this reason the NF is an effective

solution for removing CWI. The drawback of notch filtering is represented by the

fact that also a portion of the useful GNSS signal is removed. This can introduce

distortions, especially in the Auto Correlation Function (ACF), because the classical

form (a triangle), is distorted and changed a little bit. This fact can degrade the

accuracy of a GNSS receiver. In the literature, different classes of NFs have been

considered and analyzed [5, 6, 7, 8], but the analysis has been essentially limited to

the acquisition stage [9, 10]. However, acquisition represents only the first stage of

a GNSS receiver and further investigations are required for fully characterizing the

impact of NF. For these reasons, the main topic of this thesis is the evaluation of the

Notch Filter impact on the receiver processing chain. The impact on the tracking

stage is analyzed in detail and some insight is provided on the corresponding bias

and distortions introduced in the position domain.

1

Objectives and Motivations

As explained above this work investigates the performance of a GNSS receiver when

a NF is inserted in the receiver chain.

More specifically, the following points will be investigated:

• definition of suitable metrics for the analysis of notch filtering;

• theoretical analysis of the distortions introduced by the NF on the correlation

function of the received signal. A GNSS receiver is able to estimate the prop-

agation time of the received signal by correlating it with a locally generated

replica. NF alters the correlation function obtained, thus biasing the estima-

tion process. The analysis of the correlation function can provide some insight

on the distortions caused by NF;

• performance analysis of a digital DLL, in the presence of NF;

• use of real data and a customized version of the University of Calgary (UoC)

software receiver GNSS Software Navigation Receiver (GSNRxTM) [11] for

assessing the performance of a GNSS receiver in the presence of CWI and

notch filtering.

Thesis Outline

The thesis is organized as follows:

• Chapter 1 provides an overview of Global Navigation Satellite System and

GNSS receivers. A brief introduction to digital tracking loops is provided

and different metrics for their characterization introduced. The correlation

properties of GNSS signals and the importance of ACF in the receiver chain

is also discussed.

• Chapter 2 introduces the main subject of the thesis, i.e., the Notch Filter (NF).

At first, a general description of CWI is provided, highlighting its impact on

a GNSS receivers. Different types of NFs (different implementations) are also

discussed.

• Chapter 3 deals with the theoretical analysis of the NF impact: power losses,

delay and frequency are studied as a function of the different filter parameters.

The asymmetry coefficient is introduced for the evaluation of the asymmetry

2

introduced by the NF on the CCF. The concept of rms bandwidth [2] is also

introduced and used for further evaluating the NF impact. This parameter

corresponds to the Gabor bandwidth [12] of the signal after notch filtering and

plays a significant role in the evaluation of the code tracking jitter [13, 14].

• In Chapter 4 simulation results are provided; in particular a semi-analytic

model for the study of code delay tracking loop, is presented.

• Chapter 5 describes the results obtained by using the PLAN group software

receiver (GSNRxTM) [11], for the analysis of live GPS data in the presence

of NF. The analysis is made using BPSK signals (GPS C/A), but it is to be

hoped that the results are equally applicable to other signals (e.g., BOC).

• Finally, Chapter 6 presents some conclusions and possible future directions are

outlined.

3

Chapter 1

Signals and Systems

1.1 GNSS overview

The more general definition of Global Navigation Satellite System (GNSS) is given

by [15]: “A worldwide position and time determination system that includes one or

more satellite constellations, aircraft receivers and system integrity monitoring, aug-

mented as necessary to support the required navigation performance for the intended

operation”. As stated before, GNSS systems provide services for navigation and the

following general classification can be made [2, 16]:

• Truly global GNSS:

– Global Positioning System (GPS) (US);

– Global’naya Navigatsionnaya Sputnikovaya Sistema (GLONASS) (Rus-

sia);

– Galileo (EU);

– Compass (China).

• Other systems composed by:

– Ground Based Augmentation Systems (GBAS);

– Regional Satellite Based Argumentation Systems including Wide Area

Augmentation Systems (WAAS) (US), European Geostationary Overlay

Service (EGNOS) (EU), Multi-Functional Satellite Augmentation System

(MSAS) (Japan) and GAGAN (India);

4

1 – Signals and Systems

– Regional satellite navigation systems such as Quasi-Zenith Satellite System

(QZSS) (Japan) and Indian Regional Nagational System (IRNSS) (India);

– Regional GBAS such as CORS networks;

– Local GBAS typified by a single GPS reference station operating for Real

Time Kinematic (RTK) corrections.

For more details about the new or renewed systems refer to [2, 16].

In Figure 1.1 the frequency allocation for the different GNSSs is reported. It is

noted that all the frequency bands used by GNSS are in the Ultra High Frequency

(UHF) range (300 MHz to 3 GHz) and that those frequencies have been chosen in

order to minimize interference with other RF systems.

Figure 1.1: Frequency plan for the different GNSS systems.

The main objective of a GNSS system is to provide position information to the

user. The position provided by a GNSS is based on the principle of trilateration.

The user measures their distance from the satellite and, by knowing the satellite

coordinates, determines the sphere of his possible positions. By repeating this oper-

ation for several satellites the user determines his position as the intersection of the

different spheres. In an ideal system, characterized by synchronized transmitters

and receiver, only three satellites are sufficient for determining the position. This is

because the unknown quantities are three (latitude, longitude and height). However,

in a real system, a fourth variable, the clock misalignment between satellites and

receiver, has to be accounted for. In this way, at least four measurements, from four

different satellites, are required. The distance between satellite and GNSS receiver

5


can be evaluated as:

R = c τ (1.1)

where:

• R is the distance [m];

• c is the speed of light [m/s];

• τ is the transit time needed required by the signal transmitted by a satellite

to reach the GNSS receiver [s].

Since the receiver clock is not synchronous with the satellite clock, the receiver

can measure only a biased version of the transit time, τ . However, since all the

satellites of a GNSS are synchronous amongst themselves (after correction of their

clock errors), the bias affecting the transmit time is constants to all measurements

and can be determined by the receiver. In this way, a GNSS receiver is able to

measure what is usually referred to as pseudorange, and it is given by:

ρ = c [τ + (δtRx − δtS)] (1.2)

where δtRx and δtS are the clock bias of the receiver and the satellite with respect

a the reference time. δtS can be quite large, but the error over δtS is usually small

because a model can be used to predict this value. For this reason δtS can be

neglected. Moreover, clock corrections for δtS are continuously broadcast by the

satellite. Finally, Eq. 1.2 can be simplified as follows:

ρ = c [τ + δtRx]

= c τ + c δtRx

= R + ǫR. (1.3)

For this reason, at least four satellites are required for determining the user position

and the clock bias, ǫR. In Figure 1.2, the time relationship described above and the

different clock misalignments are better illustrated.

There are also other types of errors that can affect the GNSS signals; these errors

could not be important for the tracking stage or are not the focus of this thesis, but

are summarized for completeness in Table 1.1 [1]:

6


Figure 1.2: Clock misalignments in a GNSS system.

Table 1.1: Major error sources for GNSS signals [1].

Major error sourcesSatellite Orbit & clock

PropagationIonosphere

Troposphere

Receiver

Code MultipathCode Noise

Carrier MultipathCarrier Noise

1.2 DSSS signals

GNSSs generally use a Direct Sequence Spread Spectrum (DSSS) modulation for the

transmission of the navigation signals. DSSS is a particular modulation where the

data message is multiplied by a Pseudo Random Noise (PRN) sequence (generally a

binary sequence). The duration of each element of the PRN is called a chip, whereas

the ratio between the duration of a data symbol of the navigation message and the

chip interval is called spreading factor. This type of transmission permits the signal

to occupy a bandwidth much larger than the one required by the data sequence; the

increase in bandwidth is equal to the spreading factor.

7


An example of DSSS modulation is shown in Figure 1.3. The first signal is the

data message that is multiplied by the the PRN sequence reported in the second

part of the figure. The last signal in Figure 1.3 is the result of the product of the

navigation message and PRN. This signal has the same rate and bandwidth of

the PRN sequence but it carries only the information provided by the navigation

message.

Figure 1.3: Example of a generic DSSS.

The advantages of employing DSSS modulations can be summarized as follows

[17]:

• since the signal is spread over a large frequency band, the signal Power Spectral

Density (PSD) becomes extremely low, reducing interference problems with

other communications systems;

• spreading and de-spreading makes the signal robust against Radio Frequency

Interference (RFI);

• since the bandwidth is much larger than the coherent bandwidth of the chan-

nel, the system is more robust to fading;

• finally there are some security aspects: without knowing the PRN, it is not easy

to recover the data sequence and besides, the signal may remain undetected

because the PSD is very low. This aspect is not applicable in GNSS because

the PRN sequences are known.

For more details about DSSS see [17].

8


1.3 Auto Correlation Function (ACF) and Cross

Correlation Function (CCF)

Several operations performed by a GNSS receiver exploit the correlation properties

of PRN sequence. The correlation function measures how similar two different se-

quences or waveforms are. The Auto Correlation Function (ACF) and the Cross

Correlation Function (CCF) are two specific types of correlation and are defined as

follows [16]:

1. the ACF measures the similarity between a sequence and a shifted version of

itself:

Rx,x(τ) =1

N

N∑

n=0

x[n]x∗[n − τ ]; (1.4)

2. the CCF measures the similarity between two sequences for different relative

delays:

Rx,y(τ) =1

N

N∑

n=0

x[n]y∗[n − τ ]. (1.5)

In (1.4) and (1.5), (·)∗ denotes complex conjugate.

One of the main properties of PRN sequences is that their autocorrelation func-

tion is close to a Kronecker delta, i.e., it assumes a significant value only for a delay

equal to zero. Similarly, the cross-correlation of two sequences from the same family

is almost zero. These properties make PRN sequences suitable for measuring the

transit time, τ : the receiver generates a local replica of the transmitted PRN and

correlates it with the incoming signal. The transit time is estimate from the delay

that maximizes the correlation function.

The most commonly used PRN sequences are Gold codes [18], that present good

correlation properties. These codes were proposed in 1967/1968 by [18], and are

constructed by the Exclusive-Or (EXOR) of two maximum length sequences (m-

sequences) [19] of the same length. A family of Gold codes is obtained by combining

one of two sequences with all possible shifts of the other. The most important

characteristics of these codes are the excellent correlation properties [20]:

• the ACF for any Gold code sequence can assume only four values:

Rx,x(τ) ∈{

1,−1

L,−β(N)

L,β(N) − 2

L

}

(1.6)

9


• the CCF between two different sequences can assume only three values:

Rx,y(τ) ∈{

−1

L,−β(N)

L,β(N) − 2

L

}

(1.7)

where

β(N) = 1 + 2

⌊N+2

2

⌋

.

L is the length of the Gold code and N is the size of the shift register used for the

generation of the m-sequences [19]. The CCF of an orthogonal code should be equal

to zero; these codes, instead, assume a value of CCF different from zero (even of

small). For this reason Gold codes are called quasi-orthogonal codes.

In the case of GPS, the Gold codes used for Coarse Acquisition (C/A) signal [2]

have the following properties:

• L equal to 1023 chips;

• shift registers of length N = 10.

The sequence for each satellite is chosen between the 1023 available and the values

of the CCF are:

Rx,y[n] =

{

−65

1023,−1

1023,

63

1023

}

.

These values are quite small, but different from zero (the ideal auto-correlation

function is zero outside the main peak) and the separation between the main and

the side peaks assume the values: ≈ −24, − 60, − 24 dB.

In Figures 1.4a and 1.4b an example of ACF and CCF of Gold code employed

by the GPS C/A signal is reported.

1.4 GNSS receiver structure

The signal at the input of a GNSS receiver can be modeled as [16]:

r(t) =K∑

i=1

√

2CiDi(t − τi)xi(t − τi)cos(2π(fC + fD,i)t + θi) + n(t) (1.8)

where:

• Ci is the received signal power from the i -th satellite;

10


(a)

(b)

Figure 1.4: Example of ACF and CCF for a Gold code employed by the GPS C/Acode.

• xi(t) is the code-sequence of the i -th satellite;

• Di(t) is the navigation data bit stream of the i -th satellite;

• fC is the carrier frequency;

• τi is the delay introduced by the transmission channel for the i -th satellite;

• fD,i is the Doppler frequency of the i -th satellite;

• θi is the phase offset of the i -th satellite;

11


• n(t) is the noise incoming into the receiver.

Due to the quasi-orthogonality of the PRN sequences a GNSS receiver is able to

process individually the signal transmitted by the different satellites. In this way

Eq. (1.8) can be simplified to consider a single satellite at a time, without loss of

generality, as follows:

y(t) =√

2CD(t − τ)x(t − τ)cos(2π(fC + fD)t + θ) + n(t) (1.9)

The signal x(t) is obtained from this sequence [16]:

x(t) =+∞∑

n=−∞

xnmodNSb(t − nTc)

=+∞∑

n=−∞

xnmodNδ(t − nTc)︸︷︷︸

gn

∗Sb(t) (1.10)

where:

• Tc is the chip period;

• gn is the periodic repetition of the PRN sequence;

• Sb is the sub-carrier.

The sub-carrier Sb determines the spectral characteristics of the signal and also the

form of the Auto Correlation Function (ACF). For example the GPS C/A code uses

Binary Phase Shift Keying (BPSK) to modulate the transmitted carrier such that

its sub-carrier can be expressed as [16]:

Sb(t) = ΠTc(t) (1.11)

where ΠTc(t) is the elemental chip waveform [2]:

ΠTc(t) =

{

1/√

Tc, −Tc/2 ≤ τ ≤ Tc/2

0, elsewhere.(1.12)

With this definition, the ACF assumes the classical form of a triangle (see Figure

1.5)

One of the main tasks of a GNSS receiver is to estimate the delay τ , the Doppler

frequency fD and the phase θ of the incoming signal. These operations are performed

by several stages. The general structure of a receiver is shown in Figure 1.6. The

main functional blocks of a GNSS receiver can be summarized as follows:

12


Figure 1.5: Ideal ACF of a GPS C/A code.

Figure 1.6: General scheme of a GNSS receiver.

• front-end: it is the first stage of the receiver and it is aimed at down-

converting, filtering and digitizing the analog signal collected by the antenna;

• acquisition: it gives approximate estimates of the Doppler frequency (fd) and

the code delay (τ);

• tracking: it produces an accurate estimation of the Doppler frequency (fd),

the code delay (τ) and the phase θ starting from the values coming from the

13


acquisition stage;

• Position Velocity Time (PVT) computation: all the delay and Doppler

estimates from the different satellites are then combined in order to solve for

the user position, velocity and time. This is the task of the PVT block and

different approaches (LMS!, Kalman filtering) can be adopted [2, 16, 21].

The signal at the output of front-end is, in general, a digital sequence down-

converted at an intermediate frequency fif . In this way, signal (1.9) becomes:

yif (nTs) =√

2CD(nTs − τ)x(nTs − τ) cos(2π(fd + fif )nTs + δθ) + n(nTs), (1.13)

where:

• Ts = 1/fs is the sampling period (fs in the sampling frequency of the receiver);

• x(t) = hfe(t) ∗ x(t), is the incoming filtered signal and it is obtained by the

convolution between the incoming signal x(t) and the impulse response of the

front-end filter hfe(t).

• n(t) = hfe(t) ∗ n(t), is the filtered noise, and it is obtained in the same way as

the filtered signal, with the difference being that the incoming noise (not the

code sequence) is being filtered.

The digital sequence described in Eq. (1.13) represents the signal model that will be

used in the rest of this thesis. Signal (1.13) enters the acquisition block that provides

first estimates, τ and fd, of the delay and Doppler frequency. These estimates are

used to initialize the tracking loops that are better detailed in next section.

Since the main focus of this thesis is the analysis of tracking loops, the acquisition

block and the navigation solution are no further detailed and the interested reader

is referred to the main textbooks in the literature [2, 16, 21].

1.5 Digital Tracking Loops

In this section a brief overview of digital tracking loops is provided. The acquisition

stage provides a rough estimate of the the delay τ and the Doppler frequency fD;

the objective of the tracking stage is to refine these estimates and to provide phase

information. This is usually achieved by using two tracking loops:

14


• Delay Lock Loop (DLL) that refines the code phase estimate and tracks its

changes by generating a local replica of the PRN and keeping it aligned with

the signal received from the satellite;

• Phase Lock Loop (PLL) refines the Doppler frequency estimate and provides

carrier phase information. In case of high dynamics the PLL is not able to

track the phase, so a Frequency Lock Loop (FLL) is required to lower the

frequency error.

A generic tracking loop is represented in Figure 1.7; the incoming signal k[n] rep-

resents the variable under consideration for the different loops and it can be the

delay τ , the Doppler frequency fD or the phase θ to be estimated by the loop. The

signal k[n] is the estimation of the same variable produced by the loop in order to

minimize the error estimation ǫ[n]. It is also possible to identify several functional

main blocks:

Figure 1.7: Conceptual scheme of a tracking loop.

• the Integrate & Dump (I&D) block reduces the noise impact by low-pass fil-

tering (integrate) the input signal and ,after a preset amount of time, give an

output of the integration step (dump);

• the loop discriminator implements a function able to extract the residual error

on the estimated code phase, Doppler frequency or carrier phase from the

output of the integrate and dump filter;

• the loop filter has the task to further reduce the noise coming from the dis-

criminator output;

15


• the NCO generates a local replica of carrier/code signal based on the informa-

tion provided by loop filter output.

This thesis is focused on the impact of notch filtering on tracking loops and in

particular on the loss and biases introduced on the delay domain, so below the DLL

and the PLL will be described, putting much more attention on the delay loop and

just making an overview of the PLL. Figure 1.8 depicts the general structure of a

Delay Lock Loop (DLL). The incoming signal y[n] is multiplied by the local carrier

generates by the PLL, to assume that only the code phase remains unknown. The

signal is separately multiplied by three copies of the local replica (cE, cP and cL),

each with a different delay (the multiplication is equivalent to the subtraction in

the model shown in Figure 1.7). The output of each branch, passed to the I&D,

produces as a result the evaluation of the correlation between the incoming signal

and the local replica evaluated by the loop. This result is used by the discriminator

to estimate the difference (error) between the incoming and local signals.

Figure 1.8: Block scheme of a generic DLL.

The expressions of the Early (E) Prompt (P ) and Late (L) signals are the fol-

lowing [22]:

E =√

2C/N0TcRx,y(τk − d/2)ejφk + ηE,k (1.14)

P =√

2C/N0TcRx,y(τk)ejφk + ηL,k (1.15)

L =√

2C/N0TcRx,y(τk + d/2)ejφk + ηL,k (1.16)

16


where:

• CN0

is the Carrier-to-Noise density ratio;

• Tc is the integration time;

• Rx,y(τk) is the cross-correlation between the incoming signal and the local

replica;

• τk is the code phase at the k -th instant of time;

• d is the spacing between the local replicas in chips;

• φk is the residual phase error at the instant k;

• ηk are the complex noise samples.

Figure 1.9 shows an example of the values assumed by the early, prompt and late

signals. As said, the DLL has to generate a local replica of the PRN code perfectly

synchronized with the incoming signal. Usually this is due comparing the output

of the early and late correlators in order to equalize the power/energy of the two

channels. The quality of this estimation depends on how the correlation function is

distorted. These distortions are mainly due to filtering, multipath and interference

effects. In Table 1.2 are summarized the most common discriminators using in a

Figure 1.9: Example of code correlation phase.

DLL [2]:

The Phase Lock Loop (PLL) works in the same way of the DLL, but ,in this case,

it has the task to refine the Doppler frequency fd and the carrier phase θ estimates.

17


Table 1.2: Common DLL discriminators [2].

Description Algorithm

Coherent 1/√

2(E − L)Non coherent early minus late envelope normalized 1/2(E − L)/(E + L)Non coherent early minus late power 1/2(E2 − L2)Quasi coherent dot product 1/2(E − L)P

The generic scheme of a Phase Lock Loop (PLL) is represented in Figure 1.10.

The Numerically Controlled Oscillator (NCO), controlled by the loop, generates a

sinusoidal signal with a frequency (fd) and a phase (θ), in order to estimate the

Doppler frequency (fd) and the phase (θ) of the incoming signal. The result still

contains the code and to eliminate it, the signal is multiplied by the prompt replica

coming from the DLL. In this way, the signal that passes into the I&D is a pure

sinusoid which contains a residual error of frequency and phase.

Figure 1.10: Block scheme of a generic PLL

As the DLL also the PLL can use different discriminators to refine the carrier

and the phase, but, basically, they are divided in two main classes (PLL and Costas-

PLL). The main different between the two is that the Costas-PLL is insensitive to

the data, so can be used also in the presence of data modulation. As said above

this thesis is focused on the performance on notch filtering over the CCF, then the

PLL is no more detailed. More details about this loop or the discriminators can be

found loops in [2, 16, 21].

18

Chapter 2

Notch Filter (NF) for Interference

Removal

2.1 Continuous Wave Interference (CWI)

The impacts of an interference (unintentional or jamming) on a GNSS system are

manifold. First of all it is necessary to distinguish between wide-band and narrow-

band interference. The terms wide and narrow are referred to the bandwidth of the

GNSS signal, because, for example, an interference can be considered wide for the

C/A code, but, at the same time, narrow for the Encrypted Precision (P(Y)) code,

due to the different Power Spectral Densities PSDs. Another distinction is referred

to the magnitude of the interference, because, if the power of the interference can

be compared with the noise, then its impact can be neglected. Instead, if the

interference is strong, the impact depends on its duration and on its PSD. All

these Radio Frequency Interference (RFI) affect the GNSS receiver and produce as

a result a degradation in terms of acquisition and tracking accuracy, which means

less precision when the position is determined or, in the worst case, the loss of the

useful signal. As seen in Chapter 1, GNSS signals employ DSSS modulation, that

can improve slightly the robustness of the signal against interference.

One of the most common sources of interference is the so named Continuous

Wave Interference (CWI) and they can be modeled as a sinusoidal wave in time

[16]:

j(t) =√

2Pj cos(2πfj + θj) (2.1)

where:

• Pj is the power of the interference;

19

2 – Notch Filter (NF) for Interference Removal

• fj is the frequency where the CWI is centered;

• θj is the phase of the interference.

Eq. (2.1) means that the PSD of a CWI is composed, ideally, of two lines. This is a

generalization that is confirmed showing Figure 2.1: the interference has some finite

(albeit small) bandwidth. So, in the field CWI can be included all the narrow-band

signals that can be assumed as a pure sinusoid respect to the GNSS signal. In [21]

it is said that a narrow-band interference can be considered such if the bandwidth

of this signal is much less then 1 MHz. In the context of this thesis interference with

a bandwidth less than 200 kHz is considered a reasonable approximation to CWI

for the GPS C/A signal. Figure 2.1 is an example of CWI superimposed to a GNSS

PSD (in this case the GPS-L1 signal).

Figure 2.1: PSDs of a CWI and a GNSS signal.

In [3] a lot of potential sources of interference are analyzed for the GPS-L1 band,

and Table 2.1 reports them: first column indicates the harmonic of the interfering

source that is located in the GPS L1 band, while the second column is the band

associated to the harmonic; finally the last column indicates for which services the

band is employed. In the same paper, the authors analyze deeply all these sources

of interference and they also evaluate the jamming distance between the GPS civil

signal and the interference.

As said above, the presence of interference in the GNSS band is not wanted, be-

cause it causes a degradation of the performance (i.e., the estimations of acquisition

20


Table 2.1: Sources of interference for GPS civil signal [3].

Order Band [MHz] UsageL1 1571.42 - 1579.42 GPS-C/A2th 785.710 - 788.710 UHF-TV3th 523.807 - 526.473 UHF-TV4th 392.855 - 394.855 Mobile station5th 314.284 - 315.884 Mobile station6th 261.903 - 263.237 Mobile station7th 224.488 - 225.631 Broadcasting8th 196.427 - 197.428 VHF-TV9th 174.602 - 175.491 VHF-TV10th 157.142 - 157.942 VHF-maritime11th 142.856 - 143.584 VHF-military12th 130.952 - 131.618 VHF-COM13th 120.878 - 121.494 VHF-COM14th 112.244 - 112.816 VOR/ILS15th 104.761 - 105.295 FM16th 98.214 - 98.714 FM

and tracking are less accurate) and, for this reason, many anti-jamming solutions

have been analyzed in the literature; in particular, in [3, 4], the main mitigation

techniques are cited and the advantages and disadvantages are compared. In Table

2.2 the main mitigation techniques are summarized; they are grouped in two parts:

pre-correlation and post-correlation mitigation techniques.

All these techniques have advantages and disadvantages and a compromise is

needed to decide which technique is better to use. Some implementations are more

efficient, but too expensive or they need a very complex realization, while the sim-

plest and low cost techniques do not perform as well against various types of inter-

ference or their response is slower if the interference changes in time.

In this thesis, the analysis is made using a Notch Filter (NF) as interference mit-

igation device, because has been demonstrated [10, 23] that it is a good compromise

between interference removal and preservation of the useful signal. The next section

analyzes this type of filter and different implementation techniques.

21


Table 2.2: Pre-correlation and post-correlation mitigation techniques [3, 4].

Pre-correlationFixed Frequency FilteringAdaptive Frequency FilteringAmplitude Domain Processing (ADP)ADP in frequency domainContinuous Look Through Filter (COLT)Adaptive transversal Filter (ATP)Adaptive Spatial Nulling AntennaPost-correlationExpended Adaptive Code LoopVector Tracking LoopIntegrated Inertial AidingAdaptive Tracking Loop Bandwidth

2.2 Notch Filter (NF) Characterization

A Notch Filter (NF) is a filter that passes all frequencies except those in a stop

or rejection band centered on a central frequency. From its definition, a NF is

characterized by two parameters [4]:

1. the bandwidth (B−3dB);

2. the central frequency (fc).

An ideal NF should have a bandwidth equal to zero and the magnitude of the

pass-band should be equal to one [24] and its transfer function is then:

HN(z) =

{

0, f = fc

1, elsewhere(2.2)

It can be demonstrated [24] that the narrower is the rejection bandwidth, the longer

is the transient time (in the ideal case the bandwidth is equal to zero and it means

that the transient is infinite). This means, as always, that is necessary to reach a

compromise, because a larger bandwidth implies also a larger degradation of the

useful signal, but the transient is shorter and vice-versa.

There are different implementations for a Notch Filter, but basically they can

be grouped in three main classes:

1. FFT-based [5];

22


2. Finite Impulse Response (FIR) implementations [5, 6];

3. Infinite Impulse Response (IIR) implementations [4, 23, 25, 26, 27];

In this thesis a NF belonging to the third class (IIR) will be used, so the first two

classes will be briefly described and more particulars can be found in the references.

In [5] two different implementations based on the FFT are analyzed:

1. Overlapped FFT-based (OFFT), which uses two weighted FFTs operating in

parallel, as it is possible to see in Figure 2.2.

Figure 2.2: Notch Filter (NF) using the Overlapped FFT-based (OFFT) implemen-tation.

2. Filter Bank (FB) interference suppression, that is an extension of the OFFT

filter. To do this the FFT-length is extended using filtering stages before and

after the conversions from time to frequency domain and vice-versa (see Figure

2.3).

Figure 2.3: Notch Filter (NF) using the Filter Bank interference suppression imple-mentation.

23


The functioning principle is the same for both the FFT-based techniques. The

excision of the interference is done setting to zero the frequency bins that pass

a certain threshold. The bins that remain under the threshold are unchanged.

The threshold is set by the suppression algorithm and it is usually proportional

to the mean noise in the absence of interference [5]. The difference between the two

algorithms is due to the fact that the second type (FB interference suppression) has

a lower implementation complexity given the same performance level, but it has a

low response time with respect to the OFFT technique.

The second class of filters (FIR implementation) has the advantage to not present

stability problems, but they are complicated to realize, because, for a very narrow-

band excision, they need a large number of taps, that means a large number of

additions and multiplications. Also for this type of filters there are different imple-

mentations:

1. Adaptive Transversal Filter (ATF) [5] is depicted in Figure 2.4.

Figure 2.4: Adaptive Transversal Filter (ATF) scheme.

The taps of this filter are evaluated by means of an algorithm [5], and it

can be demonstrated that, even if this filter has a very low response time

(due to the number of the taps), it has a very good performance when the

interference is not present (it introduces less loss than, for example, the FFT-

based algorithm) [5].

2. Linear Phase FIR Notch Filter (LPFN) [6], as the name says, is able to provide

linear phase on the pass-band in order to leave unchanged the components

24


outside the rejection bandwidth. In [6] three different approaches of the LPFN

are analyzed:

(a) The windowed Fourier series;

(b) The frequency sampling ;

(c) The optimal LPFN

In [6], the performance of these filters are compared and was found that the

best one is the optimal LPFN, but the simplest in terms of realization is the

windowed Fourier series.

The last class of NFs is based on the Infinite Impulse Response (IIR) implemen-

tation and also here there are different approaches to realize them, but, basically,

they can be divided as follows:

1. IIR lattice Notch Filter [25];

2. bilinear-second order IIR Notch Filter [26];

3. unconstrained poles and zeros IIR Notch Filter [4, 23];

4. constrained zeros IIR Notch Filter [27].

The general form of an IIR Notch Filters is given by [24]:

HN(z) = A1 − 2 cos ωz−1 + z−2

(1 + tan(B/2)) − 2 cos ωz−1 + (1 + tan(B/2))z−2(2.3)

where:

• A is the amplitude factor, that generally should be equal to one, but sometimes

it is different and this causes signal distortions;

• ω = 2πf is the notch frequency;

• B is the −3 dB rejection bandwidth.

All the types of IIR NF listed above can be expressed in the form given in Eq. (2.3).

Usually the IIR Notch Filter is preferred to other implementations because it

has a lower order than the equal FIR version and it uses fewer multipliers in its

realization.

25


In this thesis a two-poles IIR Notch Filter with constrained zeros is used (Case

4 ). Its generic transfer function can be written as [10]:

HN(z) =1 − 2ℜ{z0}z−1 + |z0|2z−2

1 − 2kαℜ{z0}z−1 + k2α|z0|2z−2

(2.4)

where:

• z0 = ej2πfj determines the position of the zero on the unit circle;

• kα is the pole-constrain factor, and it is the parameter that regulates the

bandwidth of the NF (0 < kα < 1).

The filter contains two complex conjugate zeros on the unit circle. Similarly, the

filter contains two complex conjugate poles with a magnitude in the range [0, 1).

The poles are able to partially offset the impact of the zeros on the useful portion

of the signal PSD.

In Figure 2.5, the transfer function of the two-poles NF with different values of

kα is depicted.

Figure 2.5: Transfer function of the two-poles NF with different kα.

This type of filter is necessary because, as explained in Section 2.1, a real CWI

presents a PSD composed, ideally, by two lines centered in +fj and −fj (see Figure

2.1).

26


2.3 Adaptive Notch Filter

As explained in the previous section, the Notch Filter (NF) is a device able to excise

a small portion of spectrum while the rest is largely unaffected. In this thesis a

NF with fixed parameters has been used; this means that the central frequency

and the bandwidth are decided a priori. This assumption is valid only under static

conditions (interference known in terms of frequency and bandwidth). As said this is

an assumption and it is restrictive, so, sometimes, it is useful to employ an adaptive

NF and the reasons below can clarify the motivations:

• the CWI can change in time and the NF needs to follow it, if it changes position

relative to the useful signal;

• the CWI can be present, but its power is comparable to the noise level, so it

is not useful to employ the NF;

• sometimes the number of CWIs is greater than one, so an adaptive algorithm

with multiple Notch Filters is able to detect and activate the right number of

filters to excise all the interference present.

For these reasons many algorithms have been proposed in the literature [25, 26, 27]

and, in Table 2.3, the main techniques are summarized.

Table 2.3: Adaptive Notch Filter algorithms.

Sign AlgorithmPlain Gradient AlgorithmNormalized Gradient AlgorithmRecursive Prediction AlgorithmLattice AlgorithmP-Power AlgorithmMemoryless non-linear Gradient Algorithm

27

Chapter 3

Theoretical Characterization of

the Notch Filter Impact

In Chapter 2 the Notch Filter (NF) has been introduced as a device able to remove

Continuous Wave Interference (CWI). The objective of this chapter is to characterize

the impact of the NF on the useful signal and especially on the CCF. To do this, a

simplified receiver scheme with notch filtering is introduced (see Figure 3.1).

Figure 3.1: Simplified scheme of a GNSS receiver equipped with notch filter forinterference removal.

28

3 – Theoretical Characterization of the Notch Filter Impact

In this analysis it is assumed that the parameters of the incoming signal are

known with the exception of the code delay; this means that the receiver is correctly

tracking the carrier phase and the Doppler frequency is properly removed. In Figure

3.1 several signals can be identified:

• y(t) is the incoming signal as expressed in Eq. (1.8);

• y(t) ∗ hFE(t) is the received signal after front-end filtering . In this case the

front-end is modeled using a Chebyshev filter with a certain cut-off frequency

(fco));

• yIF (nTs) is the digitalized and down-converted sequence as expressed in Eq.

(1.9);

• yIF (nTs) ∗ hN(nTs) is the intermediate-frequency signal filtered by a digital

Notch Filter (NF) with a certain central frequency (fc) and a certain band-

width;

• c(nTs) is the local code replica generated by the receiver.

The optimal strategy for recovering a RF signal in noise and transmitted over

a band-limited channel is represented by the matched filter [28]. In this respect,

acquisition and tracking are a sort of matched filter [10, 29], where the receiver tries

to produce a local replica of recovered signal. When the GNSS signal is corrupted by

a CWI, a NF is needed by the receiver to detect the useful signal. This filter removes

the interference, but introduces distortions and, for this reason, the correlation with

locally generated signal is no longer a matched filter. The distortions introduced

by the NF are usually preferable with respect to impact of the CWI and can be

summarized as follows:

1. Filtering loss or correlation loss [10], that measures the C/N0 degradation

introduced by filtering stages[30, 31]. This loss can be expressed as follows[10,

29]:

Lf =

∣∣∣

∫ fs/2

−fs/2GS(f)Hf (f)df

∣∣∣

2

∫ fs/2

−fs/2GS(f)

∣∣∣Hf (f)

∣∣∣

2

df(3.1)

where:

• fs is the sampling frequency;

• GS(f) is the Power Spectral Density (PSD) of the GNSS signal;

29


• Hf (f) is the transfer function of the input filters. For the case in exam

two filtering stages are considered: the front-end filtering and the notch

filtering (see Figure 3.1), so the composite transfer function assumes the

following form:

Hf (f) = HFE(f)HN(f) (3.2)

where HFE(f) is the transfer function of the front-end filter and HN(f)

is the transfer function of the NF.

These losses have been thoroughly examined in the literature [10, 29] and

won’t be further discussed in this thesis.

2. Correlation distortion: the notch filter distorts the correlation between the

input signal and the local code replica. The CCF is no longer a symmetric

function and some biases can introduced in the delay estimate.

Since this thesis is mainly focused on the analysis of the effect of notch filter

on tracking loops, this distortion on the Cross Correlation Function (CCF)

and its implications on the tracking loops is deeply analyzed. The distortion

introduced by the front-end filter is usually limited and consists of a smoothing

on the GNSS signal. Consequently, the CCF is also smoothed. The Notch

Filter (NF) introduces a distortion that depends on the central frequency and

the bandwidth of the notch. In Figure 3.2 two cases of CCF distorted by NF

are depicted. In particular, Figures 3.2a and 3.2b are the PSDs of two different

GNSS modulations, the BPSK(1) used by GPS-L1 and the BOC(1,1) used by

Galileo-E1 signal. The transfer function of the notch filter is plotted over

the signal PSDs, showing the frequency components excised by the filter. In

Figures 3.2c and 3.2d the corresponding CCFs overlapping to the ideal cross-

correlation are shown.

The represented CCFs can be considered as worst cases, since the filter notch

is in the main lobe of the respective PSDs. In this way, a significant portion

of the GNSS signal is excised causing a high distortion on the CCF. If the

Notch Filter falls in a null of the PSD, the CCF distortions are marginal and

the impact of the front-end filter becomes dominant. In Figure 3.3, the peaks

of the CCFs in Figure 3.2 are zoomed; around the mean peak the distortion

is limited and the most noticeable effect is the smoothing of the correlation

function.

The distortions discussed above produce an error on the tracking loop esti-

mations. For example, the DLL estimates the code delay τ by equalizing the

30


(a) (b)

(c) (d)

Figure 3.2: Filtering distortions on the CCF for different modulations.

(a) (b)

Figure 3.3: Particular of the CCF depicted in Figure 3.2.

power of the early and late correlators (Chapter 1) and the quality of the delay

estimation is deeply related to the quality of the CCF: if the peak is distorted

or no longer symmetric, the estimate is no more accurate and some bias can

be introduced. Also the Early-Late spacing plays a fundamental role, since

the distortion is more significant at the correlation base (see Figure 3.2), a

31


larger spacing introduces a larger estimation error. This aspect will be further

analyzed in Chapter 4, where a semi-analytic model is used for evaluating the

DLL tracking jitter in the presence of notch filtering.

.

The next section aims at characterizing the Notch Filter impact exploiting the

correlation properties discussed in Chapter 1 and relating it to the NF parameters.

3.1 ACF and CCF characterizations: Asymmetry

Coefficient

Starting from the definitions of Auto Correlation Function (ACF) and Cross Corre-

lation Function (CCF) seen in Chapter 1, it is possible to re-write the CCF using

the definition of Fourier transform:

R(τ) =

∫ fs/2

−fs/2

Φ(f)ej2πfτdf. (3.3)

Φ(f) is the Fourier Transform of the CCF and, in the absence of filtering, it corre-

sponds to the normalized Power Spectral Density (PSD) of the recovered GNSS:

Φ(f) =Φ(f)

∫ fs/2

−fs/2Φ(f)df

. (3.4)

In Eq. (3.4), Φ(f) is the unnormalized signal PSD. The normalization (3.4), follows

from the condition R(0) = 1. The CCF in Eq. (3.3) can be approximated by its

Taylor expansion:

R(τ) = 1 + a1τ + a2τ2 + O(τ 3) (3.5)

and, depending on which type of function (symmetrical or asymmetrical) is consid-

ered, the above expansion assumes different forms:

1. Symmetrical function: the odd coefficients are zero and the expansion (3.5)

becomes:

R(τ) = 1 + a2τ2 + O(τ 4). (3.6)

The coefficient a2 can be evaluated as the second derivative of the CCF in Eq.

(3.3):

a2 =1

2

d2R(τ)

dτ 2

∣∣∣∣∣τ=0

(3.7)

32


leading to:

d2R(τ)

dτ 2

∣∣∣∣∣τ=0

=d2

dτ 2

∫ fs/2

−fs/2

Φ(f)ej2πfτdf

= −(2π)2

∫ fs/2

−fs/2

f 2Φ(f)ej2πfτdf. (3.8)

Since the CCF function is expanded around τ = 0, the value of the exponential

term is equal to 1, and Eq. (3.8) becomes:

d2R(τ)

dτ 2

∣∣∣∣∣τ=0

= −(2π)2

∫ fs/2

−fs/2

f 2Φ(f)df

︸︷︷︸

(βrms)2

. (3.9)

The term βrms is called RMS-Bandwitdh and is defined as follows[2]:

βrms =

√∫ β/2

−β/2

f 2Φ(f)df. (3.10)

In this way, the coefficient a2 of Eq. (3.7) can be written as:

a2 = −2(π)2β2rms (3.11)

and by substituting this result in Eq. (3.6), the final expression for the ap-

proximated CCF is:

R(τ) ≈ 1 − 2(π)2β2rmsτ

2. (3.12)

The term βrms indicates how much the CCF is peaked; this means that high

values of βrms indicate more accurate code tracking [2]. There is also another

interpretation of this coefficient: from a statistical point of view βrms corre-

sponds to the standard deviation when the normalized PSD is considered as a

probability distribution. Thus, βrms is an indicator of signal frequency spread.

The approximation in Eq. (3.12) has been implemented in Matlabr in order

to obtain the same results obtained by Betz in [32]. These results are depicted

in Figure 3.4.

Figures 3.4a and 3.4c show the approximation of the BPSK(1) CCF when

two different front-end bandwidths (24 and 12 MHz) of the front-end filter.

The filter employed is a Chebyshev filter with six taps and in-band ripple of

0.4 dB. In Figures 3.4b and 3.4d, the approximation for the BOC(1,1) CCF

33


(a) (b)

(c) (d)

Figure 3.4: Approximation of the CCF for different modulations and different front-end bandwidths.

is depicted. Again two different front-end bandwidths are employed. In the

absence of front-end filter the BPSK and the BOC modulations have infinite

RMS-Bandwitdh over infinite bandwidth [32].

To confirm the validity of this approximation, the expansion has been imple-

mented using another approach based on the numerical derivative. Since all

the signals in a receiver are processed in the digital domain, the coefficient a2

can be approximated with the second numerical derivative of the CCF [33]:

a2 =1

2

d2R(τ)

dτ 2≈ R(τ + Ts) − 2R(τ) + R(τ − Ts)

(2Ts)2(3.13)

where Ts is the sampling period. In this way the approximation using the

numerical derivative can be written as:

R(τ) ≈ 1 +R(τ + Ts) − 2R(τ) + R(τ − Ts)

(2Ts)2. (3.14)

The results using this approximation are also depicted in Figure 3.4. It is noted

34


that those curves are superimposed to the approximations obtained using the

frequency domain definition of the RMS-Bandwitdh. In Table 3.1, the delay

values for which the approximations have an error less then 10% are reported.

Table 3.1: Delay values for which the approximation has an error less than 10%in the presence of front-end filtering (Chebyshev, six taps, fco 12 and 24 MHz).Frequency Integral (FI), Numerical Derivative (ND).

Error ≤ 10%Modulation fco 12 MHz fco 24 MHz

FI ND FI ND[ns] [ns] [ns] [ns]

BPSK(1) 140.1 140.1 87.1 87.1BOC(1,1) 105.3 104.8 62.9 62.7

As already pointed out, this approximation is valid only if the function is

symmetric and this is approximatively true when the NF is not active. Until

now, no attempts were made to characterize the impact of the Notch Filter

on the Cross Correlation Function and this part is one of the contributions of

this thesis.

2. Asymmetrical function: if the odd terms are no longer zero the expression

of Rτ is the same as Eq. (3.5):

R(τ) = 1 + a1τ + a2τ2 + O(τ 3) (3.15)

where the coefficient a2 is given by Eq. (3.11). Only the coefficient a1 is to

be determined. More specifically, it can be evaluated as the first derivative of

the CCF:

a1 =dR(τ)

dτ

∣∣∣∣∣τ=0

(3.16)

leading to:

dR(τ)

dτ

∣∣∣∣∣τ=0

=d

dτ

∫ fs/2

−fs/2

Φ(f)ej2πfτdf

= 2πj

∫ fs/2

−fs/2

f Φ(f)ej2πfτdf. (3.17)

35


This function is evaluated in τ = 0 and the value of e(•) is equal to 1. In this

way, Eq. (3.17) becomes:

dR(τ)

dτ

∣∣∣∣∣τ=0

= 2π j

∫ β/2

−β/2

f Φ(f)df

︸︷︷︸

fa

(3.18)

where the term fa is called frequency average or asymmetry coefficient. This

coefficient was not introduced before in the GNSS community and can be used

to provide a qualitative idea about the bias introduced by the Notch Filter.

In the same way as the βrms, the fa can be interpreted from a statistical point

of view. In fact it plays a similar role of the mean of random variable and

indicates how much the mean of the PSD is far from zero. Ideally, fa should

be equal to zero since an ideal CCF should be symmetrical. Now it is possible

rewrite Eq. (3.5) as:

R(τ) ≈ 1 + 2πfaτ − 2(π)2β2rmsτ

2. (3.19)

In the same way as Case 1, the approximation expressed by Eq. (3.19) has

been implemented in Matlabr and the results are depicted in Figure 3.5. The

front-end filter employed is the same Chebyshev filter of order six and with a

0.4 dB ripple. Different cut-off frequencies were considered and a NF was also

inserted. In this case, a notch with bandwidth of 100 kHz and fc of 0.5 MHz

was used. Also in this case the approximation has been evaluated using the

numerical derivative. Since the second derivative has the same expression of

Eq. (3.13), only the first derivative in the digital domain has to be evaluated

[33]:

a1 =dR(τ)

dτ≈ R(τ + Ts) − R(τ − Ts)

2Ts

. (3.20)

In this way, the approximation using a numerical derivative can be written as:

R(τ) ≈ 1 +R(τ + Ts) − R(τ − Ts)

2Ts

+

+1

2

R(τ + Ts) − 2R(τ) + R(τ − Ts)

2T 2s

.

(3.21)

The results using the numerical derivative method are also depicted in Figure

3.5 and also in this case it is possible to note that the curves are superimposed

to the approximations obtained using the frequency domain integral. To In

Table 3.2 the delay values for which the approximations have an error less then

10% are reported.

36


(a) (b)

(c) (d)

Figure 3.5: Approximation of the CCF for different modulations, different front-endbandwidths and in the presence of NF with bandwidth of 100 kHz and fc of 0.5MHz.

Table 3.2: Delay values for which the approximation has an error less than 10%,in the presence of front-end filtering (Chebyshev, six taps, fco 12 and 24 MHz)and notch filtering (IIR Notch Filter, fc 0.5 MHz, bandwidth 100 kHz). FrequencyIntegral (FI), Numerical Derivative (ND).

Error ≤ 10%Modulation fco 12 MHz fc 0.5 MHz fco 24 MHz fc 0.5MHz

FI ND FI ND[ns] [ns] [ns] [ns]

BPSK(1) 130.6 133.4 81.7 83.0BOC(1,1) 119.1 117.5 70.2 69.7

The two coefficients (βrms and fa) introduces above can be evaluated as a function

of the two parameters that characterize the Notch Filter (the central frequency and

the bandwidth) in order to better characterize how the NF impacts over a GNSS

signal in terms of distortion. The surfaces depicted in Figure 3.6 show the βrms for

37


different modulations and front-end bandwidths, while in Table 3.3 the ideal values

of βrms, i.e., in the absence of notch filter, are reported. Observing these figures,

it can be noted that outside of the main lobe, for both the modulations, the βrms

tends to its ideal value (represented by the dashed line). In particular, the surfaces

is oscillating around the ideal value and, when the fc of the notch corresponds to

the zero of the PSD or the fc is very far from the main lobe, the βrms assumes its

ideal value. The oscillations are more significant for larger notch bandwidth. This

is due to the fact that the filter eliminates more frequency components of the useful

GNSS signal.

(a) (b)

(c) (d)

Figure 3.6: RMS-Bandwitdh (βrms) for different modulations and different front-endbandwidths.

In Figure 3.7 the surfaces of the Asymmetry Coefficient are depicted as a function

of the notch bandwidth and the central frequency, for different front-end bandwidths

and different modulations. In the same way as the βrms, outside of the main lobe

of the respective PSDs the coefficient oscillates around its ideal value (zero). When

the NF is centered on a null of the signal spectrum, the fa is close to zero showing

that the NF has a limited impact on the useful signal. In the proximity of the main

38


Table 3.3: Values of the RMS-Bandwitdh (βrms).

Modulation fco 12 MHz fco 24 MHzβrms βrms

[MHz] [MHz]BPSK(1) 0.8 1.13BOC(1,1) 1.42 1.98

lobe of the signal PSD, fa presents an oscillating behavior and further analysis are

required to determine if this phenomena is due to numerical problems.

(a) (b)

(c) (d)

Figure 3.7: Asymmetry Coefficient (fa) for different modulations and different front-end bandwidths.

The model employed to analyze the fa and the βrms (Figure 3.1) can be also used

to analyze the delay introduced by notch filtering. Figure 3.8 shows the surfaces of

the bias introduced as a function of the central frequency and the bandwidth. The

surfaces have been evaluated measuring the delay of the peak of the CCF without

NF and with the insertion of the filter and calculating the difference between the

two values. It is possible to note that the value of the bias tends to zero when

39


the NF is far from the main lobe of the respective modulations and assumes also a

negligible value when the fc of the notch corresponds to a null of the signal PSDs.

The bias also depends on the bandwidths of the filters present in the system: the

larger is the bandwidth of the NF, the bigger is the delay introduced, while the

opposite happens for the front-end filter: the larger is its bandwidth, the smaller is

the bias introduced. This topic will be further analyzed in the next chapter using a

semi-analytic model for the DLL.

(a) (b)

(c) (d)

Figure 3.8: Bias for different modulations and different front-end bandwidths.

Now some conclusion can be made to summarize the theoretical results obtained

in this chapter:

1. to evaluate the impact of the NF in a GNSS receiver a new coefficient has been

introduced: the Asymmetry Coefficient (fa).

2. It Has been demonstrate that this coefficient is a function of the two param-

eters that characterize a Notch Filter that are the central frequency (fc) and

the bandwidth.

40


3. also the RMS-Bandwitdh (βrms) changes its values depending from the Notch

Filter parameters;

4. the two coefficients have quite the same trend and it is possible to assert that

outside from the main lobe of the GNSS PSDs the impact of the NF is not too

strong and this can be seen observing the values assumes by the coefficients,

that are very close to the ideal value (that in the case of the fa is zero).

5. Finally also a brief evaluation of the bias is reported to complete the analysis

of the NF impact over the Cross Correlation Function. Also in this case the

results are consistent with the others, so far from the main lobe of the GNSS

spectrum, the NF does not introduce bias and this can be also interpreted as

the fact that the distortions of the CCF are small.

41

Chapter 4

Simulation Analysis

The design of complex systems containing linear and/or non linear blocks can be

carried out using different approaches [34]:

• mathematical analysis;

• experimental analysis;

• simulation analysis.

If the objective is to evaluate a parameter of the system using simulation analysis, it

is possible to use two approaches: simulate the full system using a Monte Carlo (MC)

method [35], or adopt a Semi-Analytic (S-A) technique. S-A techniques try to exploit

the theoretical knowledge of the model for reducing the computational complexity

required by Monte Carlo (MC). In this way, a trade-off between the theoretical

model and full MC simulation (see Figure 4.1) is find.

Figure 4.1: Simulation and analytical techniques for the analysis of a complex systemand their relative computational complexity.

42

4 – Simulation Analysis

MC simulations can, in general, be applied to almost every system, provided that

enough computational power is available. The main drawback of MC is the long

simulation time that they can require. On the other side, theoretical analysis is often

difficult when non-linear blocks are present and its applicability is quite limited. S-A

techniques are a mix of these two methods and combine their advantages, leading to

short simulation time. To do this, the following principle can be adopted: the linear,

time invariant part of the system is analyzed theoretically, while the remaining

part, composed by non-linear blocks, is studied using MC techniques. A S-A model

has been suggested by [36, 37] for the analysis of digital Delay Lock Loop (DLL).

This model has been adopted and modified for evaluating the impact of the Notch

Filter (NF) on the DLL.

4.1 Semi-Analytic model for digital DLL

An equivalent model for a digital DLL [36] is depicted in Figure 4.2. In the following

it is assumed that the signal is completely locked in frequency, although a residual

phase error can be present.

Figure 4.2: Block scheme of a digital DLL

43


The incoming signal y[n] is the same used in Chapter 3 to evaluate the Cross

Correlation Function (CCF) in the presence of notch filtering, and it can be written

as:

y[n] = yIF [n] ∗ hFE[n] ∗ hN [n] (4.1)

where:

• yIF [n] is the intermediate-frequency signal;

• hFE[n] and hN [n] are the digital impulse responses of the front-end and notch

filter, respectively.

To simplify the notation, the following expression is used:

hf [n] = hFE[n] ∗ hN [n]. (4.2)

It can be noted that two different rates are present in the DLL: the signal rate

and the correlator rate (that is N times lower than the signal rate). The objective

of S-A techniques is to reduce the simulation rate and this can be applied to the

model in exam, substituting all the linear part (all the blocks inside the dashed

line in Figure 4.2) with an equivalent model that works at a lower rate and that is

modeled using analytical results. This model is depicted in Figure 4.3.

Figure 4.3: Equivalent representation of a digital DLL. The delay to be estimatedis represented along with its impact on the correlator outputs.

44


The discriminator used is a non-coherent early minus late (see Chapter 1) and

it is non-linear, so it will be evaluated using a MC simulation. A constant delay is

assumed at the system input. The loop tries to minimize the delay error generating,

at each instant, a new delay estimate. The noise component is analyzed separately

from the signal thanks to the linearity of the blocks and it is added after the corre-

lation block. The three components of the noise in Figure 4.3 are the early, prompt

and late components, as expressed in Eq. (1.14).

Knowing that the signal and the noise components can be analyzed separately,

the following considerations can be made [36]:

• the signal component is a complex base-band signal, which contains the delay

information coming from the acquisition stage and can be written as:

R(τ) =1

N

N−1∑

n=0

y[n]c[n − τ ]

=1

N(y[τ ] ∗ c[−τ ])

=1

N(c[τ ] ∗ c[−τ ]) ∗ hf [n]

=1

NR(τ) ∗ hf [n] (4.3)

where:

– R(τ) is the signal component as expressed in Eq. (1.14);

– y[n] is the signal of Eq. (4.1);

– c[n − τ ] is the local replica generated by the DLL;

Eq. (4.3) shows that the expression of the signal is a Cross Correlation

Function (CCF) convoluted with the impulse response of the filters.

• The noise component is a complex base-band signal with independent real

and imaginary parts, but the noise samples can be time correlated: the noise

processes are white in the time domain, but correlated along the delay direction

(correlation between the early, prompt and late). This correlation can be

45


written as a matrix [36]:

Ccelp =

1 Rc(ds/2) Rc(ds)

Rc(−ds/2) 1 Rc(ds/2)

Rc(−ds) Rc(−ds/2) 1

(4.4)

=

1 Rc(ds/2) Rc(ds)

Rc(ds/2) 1 Rc(ds/2)

Rc(ds) Rc(ds/2) 1

(4.5)

where Rc(•) is the correlation function of the code, that takes into account

the correlation of the noise processes in the delay domain:

Rc(τ) = hf [τ ] ∗ R(τ) ∗ hf [−τ ] (4.6)

Eq. (4.6) shows that every filter impacts twice on the correlation process of

the noise.

The model explained above has been simulated using Matlabr and the follow-

ing parameters have been adopted:

• BPSK-1 modulation;

• constant code delay;

• initial delay error (0.3Tc) from the acquisition block;

• 10 ms integration time;

• 1 Hz loop equivalent bandwidth;

• front-end modeled using a Chebyshev filter with six taps and a fco of 24 MHz;

• Notch Filter with different central frequency (generally between 0 and 4 MHz)

and bandwidth between 20 and 200 kHz.

To demonstrate the validity of this S-A model, the case under ideal conditions

is analyzed at first. This type of approach has been used in [38] to asses the perfor-

mance of a new discriminator and the results have been compared with full Monte

Carlo simulations. The same approach is used here for simulating the model under

ideal conditions and it consists in the generation of the signal without front-end and

notch filtering. The performance of this model are evaluated in terms of tracking

46


jitter. The tracking jitter is a measure of the variance of the residual error on the

delay estimated by the loop and it is a function of the input Carrier-to-Noise density

ratio. The tracking jitter in case of non-coherent early minus late discriminator and

under ideal conditions, can be written as [22]:

jtheo = c Tc

√

Beqds

2 CN0

√

1 +2

(2 − ds)TicCN0

(4.7)

where:

• c is the speed of light [m/s];

• Tc is the chip duration [s];

• Beq is the equivalent bandwidth of the DLL [Hz];

• ds is the spacing between early, prompt and late [chip];

• Tic is the coherent integration time [s];

• CN0

is the Carrier-to-Noise density ratio [dB/Hz].

The second√• is called squaring loss [22] and it is due to the fact that a non-

coherent discriminator is employed; this loss can be significant at low Signal to

Noise Ratios (SNRs). The simulation results are depicted in Figure 4.4; the curves

show the jitter calculated for different E-L spacings (ds = 0.1, 0.5, 1 chip), as a

function of the C/N0.

For each case tracking three jitters are evaluated:

1. the theoretical jitter as expressed in Eq. (4.7);

2. the true jitter, evaluated as the standard deviation of the residual delay error

multiplied by the speed of light c:

jtrue = c σ (4.8)

where σ is the standard deviation of ǫ[k]:

ǫ[k] = τ [k] − ˆτ [k] (4.9)

is the residual delay error, given by the difference between the true and the

estimated delay, τ [k] and ˆτ [k].;

47


Figure 4.4: Tracking jitter for different E-L spacing as a function of the input C/N0

under ideal conditions.

3. The measured jitter that is evaluated by propagating the standard deviation

of the discriminator output[39].

The tracking jitter is well defined only when the tracking loop is working under

lock conditions. When the DLL loses lock, the jitter diverges and a straight line is

observed in Figure 4.4.

4.2 Comparison of the results with the Theoreti-

cal Model

The extension of the general model accounting for the front-end and notch filtering

can be obtained by including the filter effects on the signal and noise correlations

as explained in Chapter 3. This part will be discuss in the next sections. The Semi-

Analytic model for the digital Delay Lock Loop will be employed to analyze the

tracking jitter and the bias introduced by the Notch Filter. The results obtained

will be compared with the theoretical analysis provided in Section 4.1 for the jitter

and in Chapter 3 for the bias.

48


4.2.1 Tracking Jitter Analysis

The formula of the theoretical jitter expressed in Eq. (4.7) is valid only under

assumptions of white noise and without filtering stages. In [30, 31], the code tracking

error variance has been evaluated in the presence of non-white noise and limited

receiver pre-correlation bandwidth. Also in this case some assumption has been

made; in fact it is considered that the signal is unaffected by multipath and other

channel effects that can introduce distortions on the CCF and the analysis is made

without considering dynamics. This formula is quite difficult to evaluate in closed

form, and only an approximate expression will be employed in the following. A first

approximation can be obtained by substituting in Eq. (4.7) the effect of the filtering

loss studied in Chapter 3:

jtheo = c Tc

√

Beqds

2 CN0

L

√

1 +2

(2 − ds)TicCN0

L. (4.10)

The input filters reduce the C/N0 by a factor L that is calculated using Eq. (3.1).

Besides also the CCF is changed, because, as seen in Chapter 3, the CCF is distorted

by the presence of filtering stages. That being so, the S-A model has been simulated

using Matlabr to evaluate the effect in term of jitter in the presence of filtering

stages. The results are depicted in Figure 4.5.

Figure 4.5: Tracking jitter for different E-L spacing as a function of the input C/N0

in the presence of front-end (Chebyshev, six taps, fco 24 MHz) and notch (fc 0.5MHz, bandwidth 100 kHz) filter.

49


It can be noted that, for high values of C/N0, there is a good matching between

the theoretical and the simulated jitters, while this is not quite true for low value

of C/N0 because the formula used is only an approximation. Moreover the figure

shows that the DLL becomes unstable for values of C/N0 higher than in the ideal

case.

Now the tracking jitter is evaluated as a function of the notch bandwidth and

the central frequency to better analyze the impact of this device. In Figure 4.6

different surfaces are depicted and each one represents a different tracking jitter

evaluated using different methodologies and for different parameters. Figures 4.6a

and 4.6b show the measured and the estimated jitter with a E-L spacing 0.1 chips,

while Figures 4.6c and 4.6d shows the same jitters for a spacing of 0.5 chips. Finally

Figures 4.6e and 4.6f refer to the case of 1 chip spacing. All these surfaces are

obtained using the same parameters and employing a strong signal (40 dB-Hz). The

surfaces for the same chip spacing present the same trend and tend to the same jitter

value, represented by the dashed line. This value is the theoretical jitter evaluated

when the NF is not inserted. This further validate of the effectiveness of the S-A

model employed.

It can be noted that the trend of all the surfaces is quite the same; the jitter is

bigger for large notch bandwidths and when the filter fc is on the main lobe of the

useful signal PSD. When the Notch Filter moves outside of the main lobe, the jitter

tends quickly at the theoretical value.

4.2.2 Delay Analysis

The model described in Section 4.1 can be also used to evaluate the bias in the delay

estimation introduced by the notch filters. This bias is defined as follows:

Bias = µ

= E[ǫ]

= E[τ − τ

](4.11)

In Figure 4.7 a full simulation run, for different filtering stages and with a fixed chip

spacing is depicted: the magenta line represents the code phase to be estimated

by the DLL, while the other lines are the estimations provided by the DLL when

different filters are employed: the blue line is the estimation without any filtering,

the green line is the DLL estimate with only the front-end filter. It is possible to

observe that the presence of front-end filtering is negligible and only a small bias is

introduced. The last two lines are evaluated when the notch filter is present: the

50


(a) (b)

(c) (d)

(e) (f)

Figure 4.6: Jitter evaluated in different ways and for different chip spacing as afunction of the notch bandwidth and the central frequency.

cyan line has been estimated without front-end filter, while the red contains both

the filters. It is important to note that the NF introduces a fixed bias given a the

chip spacing.

Further analysis is conducted by evaluating the bias as a function of the chip

spacing. For each simulation (so for each spacing), a value of bias is calculated and

the result is depicted in Figure 4.8. The curves are obtained for the same filtering

51


Figure 4.7: Delay estimation for different filtering stages (front-end, fco 24 MHz,notch filter, bandwidth 100 kHz, fc 0.5 MHz).

stages considered in Figure 4.7: the solid line represents the bias introduced without

filters and it is, as expected, equal to zero; the dotted line is the bias introduced by

the insertion of the front-end filter and it is important to note that has a constant

value varying the spacing; the dashed and the dash-dot curves show the effect of

notch filtering, with the only difference that the dash-dot curve is obtained with the

combination of the front-end and notch filtering, while the dashed is due to the NF

alone. It can be seen that the larger is the spacing, the higher is the bias introduced

by the Notch Filter. This effect can be explained recalling the CCF shown in Figure

3.2: the distortions introduced by the notch are more significant at the base of the

CCF and leading to significant error when large chip spacing are employed.

Figures 4.7 and 4.8 are evaluated for a fixed central frequency of the NF. In

Figure 4.9, the mean of delay estimated by the DLL is reported as a function of the

central frequency of the notch and the E-L spacing. In this case, the bandwidth of

the filter was B−3dB= 100 kHz. The estimate of the delay is accurate when the filter

is outside of the main lobe and the surface tends to the true value represented by

the dashed line).

Figure 4.9 was obtained for a fixed bandwidth of the NF. The impact of this

parameters is studied in Figure 4.10. Three surfaces, for different E-L spacing are

shown. The surfaces have similar trend: the only notable difference is due to the fact

that changing the spacing, the bias changes (see Figure 4.7), but, when the notch is

far from the signal main lobe, the bias introduced by the filter tends to zero.

It is interesting to compare the results obtained in Chapter 3 and the surfaces

52


Figure 4.8: Bias for different filtering stages and different chip spacing, (front-end,fco 24 MHz, notch filter, bandwidth 100 kHz, fc 0.5 MHz).

Figure 4.9: Delay estimated as a function of central frequency (fc) and spacing(front-end, fco 24 MHz, notch filter, bandwidth 100 kHz).

reported below. In Figure 4.11a the trend of the Asymmetry Coefficient is depicted,

whereas in Figure 4.11b the bias introduced by the NF is shown. It is possible to

notice that there is a correspondence between the fa and the bias estimated using

the S-A model. It is noted that the two surfaces have similar behaviors, indicat-

ing a possible relationship between the two parameters. Although the Asymmetry

Coefficient can be used for qualitatively determine the bias introduced by the notch

filter, further analysis is required for effectively relating those two parameters.

53


(a) (b)

(c)

Figure 4.10: Bias introduced by the NF as a function of fc and bandwidth fordifferent chip spacing, (front-end, fco 24 MHz, notch filter, bandwidth 100 kHz).

From the analysis reported above, the following conclusions can be drawn:

1. the tracking jitter is strongly influenced by the E-L spacing, the notch band-

width and the central frequency of the NF:

• the larger is the E-L spacing, the bigger is the jitter;

• the nearest is the fc to the main lobe of the GNSS signal, the larger is

the error introduced on the estimation of the jitter;

• the larger is the notch bandwidth, the bigger is the jitter, since a large

part of the useful GNSS signal is excised by the NF.

2. the delay bias is also influenced by the spacing between the replicas, the fc

and the bandwidth of the NF:

• larger spacing imply higher bias;

• the farther is the fc from the main lobe of the signal PSD, the lower is

the bias introduced by the filter;

54


(a)

(b)

Figure 4.11: Comparison between the Asymmetry Coefficient (fa) and the biasintroduced by the Notch Filter.

• wide notch bandwidth causes significant bias.

55

Chapter 5

Real Data Analysis

In the previous chapters, the impact of the Notch Filter (NF) on tracking loops has

been analyzed using both a theoretical and a simulation approaches. In this chapter

the results already obtained are further validated using real data (in particular live

data from the GPS C/A signal).

In the first section of the chapter, the experimental setup used for the data collec-

tion is described, whereas the second section shows the results obtained by process-

ing the collected data with the University of Calgary Software Receiver (GSNRxTM)

[11]. This software receiver allows one to evaluate the effects of a NF on the tracking

loops and to determine the biases introduced in the position solution.

5.1 Experimental setup

In Figure 5.1 the experimental setup used for the data collection is depicted. The

data have been collected on the 30th of September 2008 using a fixed active antenna

positioned on the roof of the Calgary Center for Innovative Technologies (CCIT)

(see Figure 5.2). In this way, a strong signal, with a high C/N0, has been collected

under open sky conditions. This aspect is important since it allows one to better

evaluate the effects of the Notch Filter (NF) only, neglecting the effect of other

disturbance (i.e., multipath).

The antenna has been connected to a National Instrument NI-PXI-5660 signal

analyzer that is a customized device composed by three front-ends, each one formed

by a down-converter (NI-5600) and a digitizer (NI-5620) (see Figure 5.3). Each

front-end is able to collect signals with a 20 MHz bandwidth and using selectable

intermediate and sampling frequencies [40].

56

5 – Real Data Analysis

Figure 5.1: Scheme used for the data collection.

Figure 5.2: Antenna used to collect data.

The NI front-end allows recording the GPS raw samples and to store them for

further processing. A Novatel-OEM4 receiver (see Figure 5.4) has been used for

determining which satellites where in view during the data collection and speeding

up the signal acquisition with the GSNRxTM software receiver. The Novatel-OEM4

receiver is able to track up to 24 channels for both GPS L1 and L2 signals and

its software (GPSolutionr), paired with a PC, can be used to have an idea of the

constellation available (the location and the PRN number of each tracked satellite is

displayed) along with the signal characteristics ( carrier Doppler, C/N0, lock-time,

reject-code, . . . ) [41]. Figure 5.5 depicts the information provided by the receiver

during the data collection.

As already mentioned, the NI PXI-5660 front-end allows to one to down-convert

and digitize the incoming signal at a fixed intermediate frequency. The collected

data have the following characteristics:

57


Figure 5.3: Front view of the NI-PXI-5660 front-end.

Figure 5.4: Novatel OEM4 GNSS receiver.

• 5 minutes of live GPS-L1 data;

• complex sampling;

• sampling frequency: 5 MHz;

• intermediate frequency: 0.42 MHz;

• 8 quantization bits.

In Figure 5.6 a representation of the spectrum of the useful signal is depicted . The

PSD is centered at 0.42 MHz. The choice of this intermediate frequency was dictated

by the NI-PXI-5660 characteristics: the signal collected can be converted only by a

frequency, integer multiple of 1 MHz.

58


(a)

(b)

Figure 5.5: Information provided by the Novatel receiver (tracking status 5.5a andsatellites available 5.5b).

The data collected have been analyzed using the GNSS Software Navigation

Receiver (GSNRxTM) developed by the Position Location and Navigation (PLAN)

Group of the University of Calgary [11]. The test on real-data has been performed

using a NF implemented in the GSNRxTM for different fixed central frequency and

in the absence of interference. This choice has allowed the analysis of the notch

filter alone. Moreover, it is possible to assume that, if a CWI were present, it would

be completely excised by the notch filter. In particular the following configuration

has been used:

• fc of the Notch Filter fixed at the first three null of the GPS-L1 spectrum (≈1.42, 2.42 and 3.42 MHz), and in the middle of the first three lobes (≈ 0.42,

1.92 and 2.92 MHz). The filter has been also tuned at 0.92 MHz, to evaluate

the effect of the notch in the main lobe, but not in the PSD peak.

59


Figure 5.6: Power Spectral Density of the GPS-L1 signal.

• a fixed value of constrain factor (kα) has been used: kα = 0.95.

5.2 Comparison of the results in the presence and

absence of Notch Filter: impact of the notch

central frequency

The results obtained using real data and in the presence of notch filtering are com-

pared with the ideal case of NF absence. The following metrics have been used for

comparing the different cases:

• estimated C/N0;

• position error;

• carrier Doppler.

At first, the effect of the filter in terms of Carrier-to-Noise density ratio (C/N0)

is reported. Figure 5.7 depicts two examples of C/N0 estimated for two different

satellites (PRN-04 and PRN-17). For each satellite the C/N0 has been estimated in

both the presence and absence of NF. Two different notch fc have been considered:

in the middle of the signal PSD main lobe (0.42 MHz) and in the third null of the

PSD (3.42 MHz)).

60


It is noted that, when the notch filter is present, the C/N0 is attenuated and

different attenuations are obtained for different values of central frequency; in par-

ticular, if the fc is far from the main lobe, the C/N0 is slightly less attenuated. This

fact is to be expected, since almost all the power of the signal is concentrated in

the main lobe and if the notch is tuned outside it, less useful signal components is

excised. At the same time, for fc far from the main lobe, a C/N0 almost equal to

the ideal one was expected. This result was not obtained and it could be attributed

to the way the receiver estimates the C/N0; in fact the algorithm employed is the

Narrow vs Wide band C/N0 estimator [21], that is based on the assumption of input

white noise. This assumption is no longer valid when the notch filter is active.

(a)

(b)

Figure 5.7: C/N0 for two PRNs in the presence of notch filtering for different fcs.

61


Since the reduction of the C/N0 introduced by the NF seems constant for a fixed

fc, this reduction can be evaluated as the mean of the difference between the C/N0

for a given fcs and the C/N0 in the absence of the filter. Table 5.1 reports the

reduction of the C/N0 for all the satellites tracked and for all the central frequencies

tested. Observing these values, it is possible to assert that the reduction of the

Table 5.1: Reduction of C/N0 introduced by the NF for different central frequencies.

fc [MHz] PRN02 04 09 12 17

C/N0 difference [dB]0.42 -3.1 -3.1 -3.2 -3.1 -2.80.92 -3.0 -3.0 -3.1 -2.9 -2.71.42 -2.9 -2.9 -3.0 -2.8 -2.71.92 -2.8 -2.8 -2.8 -2.8 -2.72.42 -2.7 -2.7 -2.8 -2.7 -2.62.92 -2.7 -2.6 -2.7 -2.7 -2.63.42 -2.6 -2.5 -2.7 -2.6 -2.5

C/N0 decreases when the fc of the NF is far from the main lobe.

The second way used to evaluate how a Notch Filter impacts a GNSS receiver is

the estimated position. The satellites tracked are used to solve the user position (see

Chapter 1). Again the filter has been tuned at different central frequencies. Figure

5.8 shows the position solution computed for the different time epochs. Latitude,

longitude and altitude in the presence of NF (tuned at 0.42 MHz and 3.42 MHz) are

compared with the case of absence of notch filtering. Also in this case it is possible

to notice that a certain bias is present and it is higher when the filter is in the

spectrum main lobe.

To better estimate this bias, the position error has been evaluated as the differ-

ence between the curves in the absence and presence of the filter. This means that

the curves obtained in the absence of notch filtering have been taken as a reference

(“correct position”). The results relative to the figure depicted above are shown in

Figure 5.9, where the error is expressed in meters.

The error introduced by the NF has a constant trend for a fixed fc, in the same

way as the C/N0, it is possible to evaluate a mean position error for the three

directions (North, East and Up relative). Table 5.2 reports these errors, for all the

central frequencies tested. Observing the relative values of North, East and Up, it

is clear that the error is higher for a fc closer to the main lobe, whereas the error is

62


Figure 5.8: Position estimated by the GSNRxTM in the absence and presence of theNotch Filter.

Figure 5.9: Error position evaluated for different central frequencies of the NF.

reduced when the fc is farther from the main lobe. The impact of the notch filter is

reduced when the fc is closed to the nulls of the signal PSD.

The last comparison is mades in term of carrier Doppler, in order to prove that

the Notch Filter does not essentially influence the carrier estimation. To do this,

the difference between the carrier Doppler without NF and the carrier Doppler with

NF for different values of central frequencies is evaluated. Figure 5.10 shows the

63


Table 5.2: Error position introduced by the NF for different central frequencies.

fc [MHz] δN δE δU[m] [m] [m]

0.42 6.1 0.3 -15.80.92 4.7 0.2 -10.51.42 4.2 -0.4 -8.61.92 3.7 -0.9 -7.02.42 3.1 -1.2 -5.22.92 2.4 -1.2 -3.13.42 1.5 -0.8 -0.7

carrier Doppler difference for two PRNs, each evaluated for two values of fc.

It is possible to observe that the Doppler error is always a process with zero

mean (represented by the continuous line) and showing that the NF has marginal

effect on the carrier estimation.

Finally, some conclusions, summarizing the results obtained from the real data

analysis, can be drawn:

• the Notch Filter degrades the useful GNSS signal, and this can be observed

from the drop of the Carrier-to-Noise density ratio;

• the notch filter introduces biases in the position domain and this error de-

creases when the NF is far from the main lobe of the PSD;

• the notch has marginal effect, on the carrier Doppler estimation.

64


(a)

(b)

Figure 5.10: Doppler error evaluated as the difference between the cases with andwithout notch filter.

65

Chapter 6

Conclusions

The focus of this thesis was the evaluation of impact of notch filtering on tracking

loops for GNSS applications. In the following, the obtained results are summarized

and some conclusions are drawn.

At first, a theoretical analysis of the distortions introduced by the Notch Filter

(NF) on the correlation function has been provided. The distortions on the Cross

Correlation Function (CCF) have been evaluated as a function of the NF parameters:

the bandwidth and the central frequency (fc). The Asymmetry Coefficient (fa) has

been introduced in order to give a qualitative estimate of the bias introduced by

notch filtering. To the best of the author’s knowledge, this coefficient was not used

before for characterizing the notch filter distortion and represents one of the results

obtained in this work.

The second part of this thesis consisted in analyzing the NF impact on a Delay

Lock Loop (DLL) using a semi-analytic technique. The results obtained have been

compared with the case of the notch filtering absence in terms of tracking jitter

and delay bias. In particular, a correspondence between bias introduced and the

Asymmetry Coefficient has been highlighted. Further analysis, better quantifying

the relationship between those two parameters, is required and left for future works.

Finally, real data have been used to validate the theoretical and simulation re-

sults. The impact of notch filtering on a GNSS receiver has been evaluated in terms

of Carrier-to-Noise density ratio, position error and carrier Doppler. The results

obtained are consistent with the theoretical and the simulation analysis. In particu-

lar, it has been found that the Notch Filter has only marginal effects on the carrier

Doppler. All the analysis has been made using the NF alone, in the absence of

interference. This means that the NF parameters were fixed before processing the

signal and it has been assumed that, if a Continuous Wave Interference (CWI) were

66

6 – Conclusions

present, it would be completely excised by the filter. The analysis in the presence

of interference and/or using an adaptive Notch Filter is left as a future work. From

this analysis, it emerges that, when a notch filter is present, receiver performance

can be seriously degraded, depending on the filter bandwidth and central frequency.

Biases in the delay estimation and in the final position can be introduced by the

notch filter and it use should be evaluated with respect to the degradations caused

by the CWI jamming the GNSS signal.

67

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Impact of Notch Filtering on Tracking Loops for GNSS Applications

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