On Filter Bank Based MIMO Frequency Multiplexing and ...

On Filter Bank Based MIMO Frequency Multiplexing and Demultiplexing

Master thesis performed in Electronics Systems division by

Amir Eghbali

Report number:LiTH-ISY-EX--06/3911--SE September 2006

iii

Title On Filter bank Based MIMO Frequency Multiplexing and

Demultiplexing

Master thesis in Electronics Systems at Linköping Institute of Technology

by

Amir Eghbali

LiTH-ISY-EX--06/3911--SE

Supervisor: Prof. Håkan Johansson

Examiner: Prof. Håkan Johansson Linköping: 26 September 2006

v

Presentation Date 2006-09-26 Publishing Date (Electronic version) 2006-10-02

Division of Electronics Systems Department of Electrical Engineering

Abstract

The next generation satellite communication networks will provide multimedia services supporting high bit rate, mobility, ATM, and TCP/IP. In these cases, the satellite technology will act as the internetwork infrastructure of future global systems and assuming a global wireless system, no distinctions will exist between terrestrial and satellite communications systems, as well as between fixed and 3G mobile networks. In order for satellites to be successful, they must handle bursty traffic from users and provide services compatible with existing ISDN infrastructure, narrowcasting/multicasting services not offered by terrestrial ISDN, TCP/IP-compatible services for data applications, and point-to-point or point-to-multipoint on-demand compressed video services. This calls for onboard processing payloads capable of frequency multiplexing and demultiplexing and interference suppression.

This thesis introduces a new class of oversampled complex modulated filter banks capable of providing frequency multiplexing and demultiplexing. Under certain system constraints, the system can handle all possible shifts of different user signals and provide variable bandwidths to users. Furthermore, the aliasing signals are attenuated by the stopband attenuation of the channel filter thus ensuring the approximation of the perfect reconstruction property as close as desired. Study of the system efficient implementation and its mathematical representation shows that the proposed system has superiority over the existing approaches for Bentpipe payloads from the flexibility, complexity, and perfect reconstruction points of view. The system is analyzed in both SISO and MIMO cases. For the MIMO case, two different scenarios for frequency multiplexing and demultiplexing are discussed.

To verify the results of the mathematical analysis, simulation results for SISO, two scenarios of MIMO, and effects of the finite word length on the system performance are illustrated. Simulation results show that the system can perform frequency multiplexing and demultiplexing and the stopband attenuation of the prototype filter controls the aliasing signals since the filter coefficients resolution plays the major role on the system performance. Hence, the system can approximate perfect reconstruction property by proper choice of resolution. Number of pages: 95

ISBN (Licentiate thesis) ISRN: LiTH-ISY-EX--06/3911—SE Title of series (Licentiate thesis) Series number/ISSN (Licentiate thesis)

Language English Number of Pages 95

Type of Publication Licentiate thesis ● Degree thesis Thesis C-level Thesis D-level Report Other (specify below)

Publication Title On Filter Bank Based MIMO Frequency Multiplexing and Demultiplexing Author Amir Eghbali

URL, Electronic Version http://www.ep.liu.se

Keywords Frequency Band Reallocation, Filter Bank, Multirate Signal Processing, MIMO, Satellite Communications

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Abstract

The next generation satellite communication networks will provide multimedia services supporting high bit rate, mobility, ATM, and TCP/IP. In these cases, the satellite technology will act as the inter-network infrastructure of future global systems and assuming a global wireless system, no distinctions will exist between terrestrial and satellite communications systems, as well as between fixed and 3G mobile networks. In order for satellites to be successful, they must handle bursty traffic from users and provide services compatible with existing ISDN infrastructure, narrowcasting/multicasting services not offered by terrestrial ISDN, TCP/IP-compatible services for data applications, and point-to-point or point-to-multipoint on-demand compressed video services. This calls for onboard processing payloads capable of frequency multiplexing and demultiplexing and interference suppression.

This thesis introduces a new class of oversampled complex modulated filter banks capable of providing frequency multiplexing and demultiplexing. Under certain system constraints, the system can handle all possible shifts of different user signals and provide variable bandwidths to users. Furthermore, the aliasing signals are attenuated by the stopband attenuation of the channel filter thus ensuring the approximation of the perfect reconstruction property as close as desired. Study of the system efficient implementation and its mathematical representation shows that the proposed system has superiority over the existing approaches for bentpipe payloads from the flexibility, complexity, and perfect reconstruction points of view. The system is analyzed in both Single Input single Output (SISO) and Multiple Input Multiple Output (MIMO) cases. For the MIMO case, two different scenarios for frequency multiplexing and demultiplexing are discussed.

To verify the results of the mathematical analysis, simulation results for SISO, two scenarios of MIMO, and effects of the finite word length on the system performance are illustrated. Simulation results show that the system can perform frequency multiplexing and demultiplexing and the stopband attenuation of the prototype filter controls the aliasing signals since the filter coefficients resolution plays the major role on the system performance. Hence, the system can approximate perfect reconstruction property by proper choice of resolution.

ix

Acknowledgments

First, I would like to thank my supervisor Prof. Håkan Johansson for the invaluable guidance and incredible patience in answering my questions. I could ask any questions at any time.

Special thanks go to my family for all the support they provided. I will never forget their kindness.

I would also like to thank all the apples and bananas that kept me alive during the time I was working on my thesis!

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Foreword

The next generation information society will include telecommunications, computing, video, TV, videoconferencing, and consumer electronics in every building and requires wideband services to provide multi-application networks at rates around 2 Mbps accessible to everybody everywhere [1]. The terrestrial networks, even with the large bandwidth available due to optical fiber technology, cannot meet these requirements. However, satellites play an important role since if a satellite is in orbit, the subscriber only has to install a satellite terminal and subscribe to the service. To solve the problem of the next generation networks, network technicians suggest asynchronous transfer mode (ATM) comprised of a multiplexer with a high-rate output having every possible lower rate at the input side. On the other hand, telecommunications managers try to provide temporary solutions such as asynchronous digital subscriber line (ADSL) and high-rate DSL (HDSL) [1].

One of the disadvantages of geostationary communications satellites, is the large delay for one up- and downlink, which is disturbing for voice. However, the terrestrial copper, optical fiber, and the cellular radio networks carry most voice traffic. Thus, the satellites can be a suitable choice for interactive data services and delivery of a large amount of data on request. In addition, low earth orbit (LEO) systems such as GLOBALSTAR and ICO are competing with the terrestrial networks for voice applications. Therefore, for wideband multimedia applications, geostationary satellites with several high-gain spot-beam antennas, OnBoard Processing (OBP), and switching seem to be a logical step in migration from pure TV broadcast to interactive multimedia services. The functionality that the OBP system offers is suited to provide the services required by the information society. The elements making up the OBP system are [1]:

• The User Station (UTS): The UTS consists of an outdoor unit and an indoor unit with a capability of being equipped with Integrated Services Digital Network (ISDN), Electronic Network Systems (ENS), and packet switch (TCP/IP) interface.

• The Master Control Center (MCC): The MCC translates the subscriber terminal’s protocols and algorithms into commands. It also controls the communication flow inside the broadband satellite communications network. This block is also responsible

xii

for the compatibility of the new networks with the existing protocols and algorithms.

• The switching payload: The payload consists of DSP functions such as digital beamforming, frequency multiplexing and demultiplexing, interference suppression, signal level control and, in a regenerative system, modems [2].

This thesis focuses on digital signal processing of satellite payloads which has two major categories as [2]:

1. Onboard regeneration and baseband processing: Examples for this type are data buffering and multiple access reformatting, data rate conversion, coding, and encryption. These systems decouple noise and interference on the uplink and downlink and are able to optimize access, modulation, and coding techniques for the uplink and downlink.

2. Onboard non-regenerative processing: Here, signals are sampled with appropriate precision and sampling rate. Subsequent processing is performed as arithmetic operations on the signal samples. In particular, such techniques allow the digital demultiplexing of narrowband channels and processing of individual channels to include level control and beamforming. Hence, we need a transparent payload architecture where signals are not regenerated onboard. The system level advantages of this system are power efficiency, frequency reuse, flexibility in response to changing traffic, reproducibility, and lack of sensitivity to temperature changes [2].

This thesis proposes a bentpipe payload architecture that handles all possible frequency shifts and all possible user data rates, has low complexity, achieves high level of parallelism, and is easy to analyze and design. The system uses a new class of oversampled complex modulated Filter Banks (FB), which brings superiority over previously proposed architectures. In particular, it outperforms the regular modulated FB based networks from the flexibility point of view and has better performance over the tree-structured FB based networks in terms of flexibility and complexity. Furthermore, the proposed system outperforms the overlap/save DFT/IDFT based networks if perfect reconstruction property is important. The key features of the proposed system are:

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• Use of oversampled filter banks: This choice makes the suppression of aliasing easier and allows the combination of smaller subbands into wider subbands without introducing large aliasing distortion. This property brings full flexibility to the system.

• More FB channels than granularity bands: This feature brings the ability to generate all possible frequency shifts and reduces the complexity of the system.

• Complex modulated filter banks: These filter banks result in very low complexity and simplicity in terms of analysis, design, and implementation.

The report is organized in three chapters. In the first chapter, basic building blocks of multirate systems i.e. interpolators, decimators, and polyphase decomposition are introduced. Since the proposed structure uses filter banks, building blocks of filter banks and their mathematical representations are derived. Based on the structure and parameters, the maximally decimated and oversampled filter banks are discussed. Next, the concept of paraunitariness followed by DFT and cosine modulated filter banks in maximally decimated systems is covered. The distinction between uniform and non-uniform filter banks is treated mathematically but the focus is on the uniform filter banks. The oversampled filter bank analysis starts with the definition of frame theory followed by the example on oversampled DFT modulated filter banks. Having discussed the time invariant systems, basics and properties of the time varying filter banks are investigated. The chapter ends with common issues in design of filter banks from a system point of view described as constraints in a hierarchical manner.

The second chapter discusses the basics of transmultiplexers, as duals of filter banks, and derives their mathematical representation. Next, perfect reconstruction, cancelling of multiuser and interblock interference, and channel equalization are discussed. As special cases of transmultiplexers, the multiple access schemes such as Code Division Multiple Access (CDMA), Time Division Multiple Access (TDMA), and Frequency Division Multiple Access (FDMA) are introduced. Having discussed the transmultiplexers, different architectures of payloads i.e., bentpipe, partial processing, full processing, and hybrid systems used in satellite applications and their features are studied. The chapter ends with a review of filter bank applications in payload systems.

The third chapter illustrates the proposed system for frequency multiplexing and demultiplexing. First, the problem of frequency

xiv

multiplexing and demultiplexing is formulated followed by the introduction to a new class of online variable oversampled complex modulated filter banks. Based on the problem formulation and the filter bank definition, the constraints of the architecture are derived. Next, the characteristics of the filter bank blocks namely analysis/synthesis banks and channel switch are defined. In order to decrease the implementation complexity, the polyphase decomposition is applied to derive the new system architecture. In reality, there are several users in the uplink which must be multiplexed to different downlink spot beams. This calls for a MIMO system capable of performing the multiplexing and demultiplexing and satisfying the defined Perfect Reconstruction (PR) properties. The extension of the proposed system to a MIMO case is covered in two scenarios. The last part of the chapter illustrates simulation results of the proposed architecture from the functionality and performance points of view. To do so, the system test setup and the error measurement algorithm which is Mean Square Error (MSE) are described. Next, examples on SISO and MIMO cases verifying the system functionality to multiplex and demultiplex signals are presented. To evaluate the system performance, the finite word length effects are introduced and examples for a 64-QAM signal with different resolutions are illustrated. The chapter ends with conclusion and topics for future research.

xv

Outline of Tasks

The tasks assigned in this thesis work were as follows:

1. Study of the multirate signal processing basics. 2. Literature review on different filter bank architectures. 3. Literature review on different satellite payload systems. 4. Implementation of a MIMO polyphase Frequency Band

Reallocation network in MATLAB including the finite word length effects.

5. Evaluation of the MIMO polyphase network from the BER point of view.

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Table of Contents ABSTRACT ................................................................................................................VII ACKNOWLEDGMENTS........................................................................................... IX FOREWORD ............................................................................................................... XI OUTLINE OF TASKS ...............................................................................................XV LIST OF ABBREVIATIONS .................................................................................. XXI

CHAPTER ONE: OVERVIEW OF MULTIRATE SYSTEMS AND FILTER BANKS ............................................................................................................................1 1. INTRODUCTION ..................................................................................3

1.1. BASIC BUILDING BLOCKS OF MULTIRATE SYSTEMS ..................................... 3 1.1.1. Polyphase Decomposition........................................................................ 5

1.2. DIGITAL FILTER BANKS ................................................................................ 9 1.2.1. Analysis Filter Bank................................................................................. 9 1.2.2. Downsamplers ....................................................................................... 10 1.2.3. Subband Processing............................................................................... 10 1.2.4. Upsamplers ............................................................................................ 11 1.2.5. Synthesis Filter Bank ............................................................................. 11

1.3. GENERAL FILTER BANK ARCHITECTURE..................................................... 11 1.4. MAXIMALLY DECIMATED FILTER BANKS ................................................... 13 1.5. PARAUNITARY FILTER BANKS..................................................................... 16

1.5.1. Properties of Paraunitary PR Filter banks............................................ 17 1.6. DFT MODULATED FILTER BANKS............................................................... 18

1.6.1. Uniform and Non-uniform Filter Bank .................................................. 18 1.6.2. Uniform DFT Modulated Filter Banks .................................................. 19

1.7. COSINE MODULATED FILTER BANKS .......................................................... 24 1.8. OVERSAMPLED PR FILTER BANKS .............................................................. 29 1.9. TIME VARYING FILTER BANKS.................................................................... 34 1.10. DIFFERENCES BETWEEN TIME VARYING AND LTI FILTER BANKS............... 38 1.11. FILTER BANK DESIGN ISSUES...................................................................... 39

1.11.1. Filter Issues....................................................................................... 39 1.11.2. Filter Bank Issues.............................................................................. 39 1.11.3. Analysis/Synthesis Issues .................................................................. 39 1.11.4. Total System Issues ........................................................................... 40

CHAPTER TWO: OVERVIEW OF TRANSMULTIPLEXERS AND SATELLITE PAYLOAD SYSTEMS .........................................................................41 2. INTRODUCTION ................................................................................43

2.1. TRANSMULTIPLEXERS ................................................................................. 43 2.1.1. Mathematical Representation of Transmultiplexers .............................. 43 2.1.2. Perfect Reconstruction in Transmultiplexers......................................... 45 2.1.3. Canceling InterBlock Interference in Transmultiplexers....................... 46 2.1.4. Canceling Multi User Interference in Transmultiplexers ...................... 46 2.1.5. Time Frequency interpretation .............................................................. 48 2.1.6. CDMA System Based on Transmultiplexers .......................................... 49

xviii

2.1.7. TDMA System Based on Transmultiplexers........................................... 50 2.1.8. FDMA System Based on Transmultiplexers........................................... 51

2.2. SATELLITE PAYLOAD ARCHITECTURES ....................................................... 51 2.2.1. Bentpipe Payload................................................................................... 52 2.2.2. Full Processing Payload........................................................................ 52 2.2.3. Partial Processing Payload ................................................................... 53 2.2.4. Hybrid Payload...................................................................................... 54

2.3. FREQUENCY MULTIPLEXING/DEMULTIPLEXING USING FILTER BANKS....... 54 CHAPTER THREE: PROPOSED BENTPIPE SYSTEM AND SIMULATION

RESULTS......................................................................................................................55 3. INTRODUCTION ................................................................................57

3.1. PROBLEM FORMULATION ............................................................................ 58 3.2. CLASS OF ONLINE VARIABLE OVERSAMPLED COMPLEX MODULATED FILTER BANKS 59

3.2.1. System Constraints................................................................................. 59 3.2.2. Constraints on Sampling Rate Converters and Number of Channels .... 60 3.2.3. Analysis Filters ...................................................................................... 61 3.2.4. Synthesis Filters..................................................................................... 62 3.2.5. Application of Switch in the FFBR Network .......................................... 63 3.2.6. Efficient Implementation........................................................................ 64

3.3. MIMO FFBR NETWORK ............................................................................. 65 3.3.1. K-Input K-Output FFBR Networks ........................................................ 65 3.3.2. S-Input K-Output FFBR Networks......................................................... 66

3.4. SIMULATION RESULTS................................................................................. 66 3.4.1. System Parameters Selection ................................................................. 67 3.4.2. Transmitter/Receiver Filter Design ....................................................... 67 3.4.3. Implementation of the SISO System ....................................................... 69 3.4.4. Implementation of the MIMO System..................................................... 71

3.5. FINITE WORD LENGTH EFFECTS ON THE FFBR NETWORK.......................... 74 3.6. CONCLUDING REMARKS AND FUTURE TOPICS ............................................ 77

REFERENCES .............................................................................................................79 APPENDIXES...............................................................................................................83 APPENDIX A: MATLAB PROGRAM TO DESIGN THIRD AND SIXTH BAND

FILTERS.......................................................................................................................85 APPENDIX B: MATLAB PROGRAM TO GENERATE USER SIGNALS ..........87

APPENDIX C: MATLAB PROGRAM TO IMPLEMENT THE SYSTEM IN FIGURE 29....................................................................................................................89

APPENDIX D: MATLAB PROGRAM TO IMPLEMENT THE SYSTEM IN FIGURE 31....................................................................................................................91

APPENDIX E: MATLAB PROGRAM TO DESIGN PROTOTYPE FILTERS USING MINIMAX ALGORITHM.............................................................................95

xix

LIST OF FIGURES

Figure 1: Effect of Aliasing and Imaging in Upsamplers and Downsamplers........................................................................................... 5 Figure 2: Noble Identities in Multirate Systems. ...................................... 6 Figure 3: Efficient Polyphase Decimator and Interpolator Implementation. ........................................................................................ 7 Figure 4: Filer Realization Using Subband Decomposition. .................... 8 Figure 5: Typical Analysis and Synthesis Banks...................................... 9 Figure 6: Typical Frequency Responses of Analysis Filters. ................. 10 Figure 7: General Filter Bank Architecture. ........................................... 12 Figure 8: Realization of the Analysis and Synthesis Banks Based on Polyphase Matrices. ................................................................................ 13 Figure 9: Simplified Realization of Filter Banks Using Noble Identities.................................................................................................................. 14 Figure 10: Filter Characteristics for Uniform and Non-Uniform Filter Banks....................................................................................................... 18 Figure 11: Analysis Bank Polyphase Realization of DFT Modulated Filter Banks. ............................................................................................ 21 Figure 12: Analysis Bank Polyphase Realization of DFT Modulated Filter Banks. ............................................................................................ 22 Figure 13: Simplest Case of the DFT Modulated Filter Banks.............. 23 Figure 14: Analysis Filters for the Cosine Modulated Filter Banks. ...... 26 Figure 15: Polyphase Realization Analysis Bank for the Cosine Modulated Filter Banks........................................................................... 26 Figure 16: Architecture of Oversampled DFT Modulated Filter Bank. . 32 Figure 17: Polyphase Realization of the Oversampled DFT Modulated Filter Bank............................................................................................... 33 Figure 18: General Architecture of Time Varying Filter Banks. ............ 35 Figure 19: Different Stages of a Time Varying Filter Bank. .................. 35 Figure 20: General Architecture of a Transmultiplexer.......................... 44 Figure 21: Architecture of Transmultiplexer with Transmit and receive Filters. ..................................................................................................... 45 Figure 22: Modeling the Channel to Cancel InterBlock Interference..... 46 Figure 23: Time Frequency Tilde of a General Discrete Time Function.49 Figure 24: CDMA System Based on Transmultiplexer. ......................... 50 Figure 25: Simple TDMA System Based on Transmultiplexer.............. 50 Figure 26: Transmultiplexer Synthesis/Analysis Filter Characteristics for FDMA System. ....................................................................................... 51 Figure 27: Illustration of Guard and Granularity Bands in the FFBR System..................................................................................................... 58

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Figure 28: FFBR system with Fixed Analysis and Adjustable Synthesis Bank. ....................................................................................................... 59 Figure 29: FFBR system with Fixed Analysis/Synthesis Banks and Channel Switch. ...................................................................................... 63 Figure 30: Polyphase Implementation of the FFBR Network. ............... 64 Figure 31: K-Input K-Output MIMO FFBR with Fixed Analysis and Synthesis FBs.......................................................................................... 65 Figure 32: S-Input K-Output MIMO FFBR with Fixed Analysis and Synthesis FBs.......................................................................................... 66 Figure 33: Transmit and Receive Filter Characteristics to Evaluate the FFBR Network........................................................................................ 68 Figure 34: Test Setup for FFBR Network Evaluation. ........................... 69 Figure 35: Example Channel Switch for SISO Case. ............................. 70 Figure 36: Input, Output, and Analysis Filters for SISO Polyphase FFBR Network................................................................................................... 71 Figure 37: Example Channel Switch for Two-Input Two-Output MIMO FFBR Network........................................................................................ 71 Figure 38: Inputs and Outputs for MIMO FFBR Network with two Inputs and two Outputs. ..................................................................................... 72 Figure 39: Input and Outputs of the FFBR Network without Channel Switch. .................................................................................................... 73 Figure 40: Example One-Input/Two-Output Channel Switch for MIMO FFBR Network........................................................................................ 73 Figure 41: Input and Outputs of the FFBR Network with Channel Switch of Figure 40............................................................................................. 74 Figure 42: Quantization in the Polyphase FFBR Network. .................... 75 Figure 43: Multiplexed 64-QAM Data Constellation for Three Filter Coefficient Lengths................................................................................. 76 Figure 44: FFBR Network Noise Variance for Channels in Figure 38. . 77

xxi

List of Abbreviations Abbreviation Comments AFB Analysis Filter Bank ATM Asynchronous Transfer Mode BER Bit Error Rate CDMA Code Division Multiple Access DCT Discrete Cosine Transform DFT Discrete Fourier Transform DSL Digital Subscriber Line DSP Digital Signal Processing ENS Electronic Network Systems ESA European Space Agency FB Filter Bank FDM Frequency Division Multiplexing FDMA Frequency Division Multiple Access FFBR Flexible Frequency Band Reallocation FIR Finite Impulse Response GDFT Generalized Discrete Fourier Transform HDSL High bit rate Digital Subscriber Line IDFT Inverse Discrete Fourier Transform IIR Infinite Impulse Response ISDN Integrated Services Digital Network ISI Inter Symbol Interference ISP Internet Service Provider LEO Low Earth Orbit LP Low Pass LTI Linear Time Invariant LTV Linear Time Variant MCC Master Control Center MIMO Multiple Input Multiple Output MSE Mean Square Error MUI Multi User Interference OBP OnBoard Processing PFBR Perfect Frequency Band Reallocation PR Perfect Reconstruction PU ParaUnitary QAM Quadrature Amplitude Modulation SFB Synthesis Filter Bank SISO Single Input Single Output SNR Signal to Noise Ratio SS/TDMA Satellite-Switched Time Division Multiple Access TCP/IP Transmission Control Protocol/Internet Protocol TDMA Time Division Multiple Access TM TransMultiplexer TVFB Time Varying Filter Banks UTS User Station VPN Virtual Private Network

1

Chapter One: Overview of Multirate Systems and Filter Banks

3

1. Introduction

Multirate digital filters and filter banks find wide application in areas such as speech processing, communications, analog voice privacy systems, image compression, antenna systems, and digital audio industry. This applicability has excited immense amount of research leading to a substantial progress in multirate systems including decimation and interpolation filters, polyphase structures, and several types of analysis/synthesis filter banks with specific properties that suit some applications. To analyze different systems mathematically, it is useful to have some blocks that are common among the systems and furthermore, ease the analysis process. In the analysis of the multirate systems and filter banks, which is the subject of this chapter, the basic building blocks are the interpolators and decimators, which used along with the concept of the polyphase decomposition, reduce the implementation complexity.

In this chapter, we start with the definition of these building blocks, and then we proceed to define the basics of filter bank theory. In this context, different types of maximally decimated and oversampled filter banks are discussed. Furthermore, a brief introduction to time varying filter banks is provided.

1.1. Basic Building Blocks of Multirate Systems

In the area of the multirate signal processing, interpolators and decimators are the basic blocks that alter the sampling frequency at different parts of the system leading to name “Multirate”. An interpolator is a combination of an upsampler and a lowpass filter where the upsampler inserts 1−M zeros between consecutive samples of the original signal. Doing so, the output signal spectrum is a compressed version of the input signal spectrum. In the mathematical representation of an upsampler, we have [3]

4

)()()()( MjjM eXeYorzXzY ωω == , 1.1

where )(ny and )(nx are the output and input sequences, respectively. If

)( ωjeX is periodic with π2 , then )( ωjeY will be periodic with M

π2 [3].

On the other hand, a decimator is the combination of a lowpass filter and a downsampler where the downsampler retains only the Mth samples of the input signal. In the mathematical representation, assuming the notations on interpolator, we have

MjM

k

Mkj

jM

k

kM eWeXM

eYorWzXM

zYππω

ω21

0

)2(1

0

1

,)(1)()(1)(−−

=

−−

=

=== ∑∑ . 1.2

Hence, )( ωjeY is a sum of M uniformly shifted versions of an −M fold stretched version of )( ωjeX [3]. An important issue in the analysis of these blocks is imaging and aliasing.

Looking at Equation (1.2), one can conclude that if )(nx is band limited to MM

πωπ <<− (more generally Mπαωα 2+<< [3]), the

original signal can be recovered from )(ny by the use of a lowpass filter. Otherwise, the problem of aliasing can occur damaging the information. So, an interpolator can cause imaging due to compression of the input signal spectrum, which must be removed by a lowpass filter following the upsampler.

Similarly, a decimator can cause aliasing due to the stretching of the input signal spectrum. To deal with this problem, a lowpass filter must remove unnecessary signals before the downsampler. The imaging and aliasing effects and the characteristics of the lowpass filters for a system with a decimation and interpolation ratio of three are shown in

Figure 1.

5

Figure 1: Effect of Aliasing and Imaging in Upsamplers and Downsamplers.

It must be added that in reality, the brick wall filters can not be realized, so the filters should have transition bands. This can be solved by considering the fact that data signals are not also strictly band limited which allows for filters to have transition bands. To reduce the complexity of the interpolator and decimator implementation, the idea of polyphase decomposition is used and will be discussed in the next section.

1.1.1. Polyphase Decomposition

Polyphase decomposition realizes any lowpass filter as the sum of polyphase components [3]. Any finite or infinite length sequence { })(nh with a z-transform )(zH can be written as [4]

[ ]

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

===

−

−−−−

=

−∞

−∞=

− ∑∑

)(...

)()(

...1)()()(

1

1

0

)1(11

0

MM

M

M

MM

k

Mk

k

n

n

zH

zHzH

zzzHzznhzH. 1.3

The right hand side of the Equation (1.3) is called the polyphase decomposition. In general, there are two types of polyphase

Interpolation Filter

23π π2

23π−π2−

Aliasing in the absence of the filter

π2 π2− 2

π+2

π−

Decimation Filter )( ωjeX

)( ωjeX

π2π+ π−π2−

32π

3π

3π−

Images to be removed

6

decompositions. As the first type, any lowpass filter with cutoff frequency at

Mπ can be written as

∑−

=

−=1

0

)()(M

i

Mi

i zHzzH , 1.4

where )(zHi are the polyphase components. In the time domain, the impulse responses of the polyphase components can be derived as

10),()( −≤≤+= MiMnihnhi . It must be noted that the polyphase components can have different lengths. As an example, the 2-fold and 3-fold polyphase components of a 6th order filter with transfer function

)(zH can be derived as

.3,])5[]2[(])4[]1[(])6[]3[]0[(

2,])5[]3[]1[(])6[]4[]2[]0[(

]6[]5[]4[]3[]2[]1[]0[)(

)(

32

)(

31

)(

63

)(

421

)(

642

654321

32

31

30

21

20

foldhzhzhzhzhzhzh

foldhzhzhzhzhzhzh

hzhzhzhzhzhzhzH

zEzEzE

zEzE

−++++++=

−++++++=

++++++=

−−−−−−

−−−−−−

−−−−−−

44 344 2144 344 214444 34444 21

4444 34444 21444444 3444444 21

1.5

The second type of the polyphase decomposition can be derived as

∑−

=

−−−=1

0

)1( )()(M

i

Mi

iM zRzzH and is useful in the analysis of synthesis bank filters

[3]. The relationship between these types is )()( 1 zEzR iMi −−= [5]. The advantage of polyphase components can be better understood by the use of two noble identities shown in Figure 2 whose properties are proved in [5]. It must be added that these noble identities are different in the case of time varying systems and are defined in [6].

Figure 2: Noble Identities in Multirate Systems.

Having these tools, we can derive the efficient decimation and

interpolation filter implementations as shown in Figure 3.

≡

≡

[ ]nx

[ ]nx [ ]my

[ ]my

[ ]mv1

[ ]mv1 L

M )(zH

)( LzH[ ]my [ ]nx

[ ]my [ ]nx

[ ]nv2 L )(zH

)( MzH M [ ]nv2

7

Figure 3: Efficient Polyphase Decimator and Interpolator Implementation.

In these structures, the filters run at lower sampling rates compared

to the input signal sampling rate i.e. T1 . Since the samples across the adders are phased by T seconds and hence they do not interact in the adder, some commutator models as described in [7] can be used to avoid the adders for easier implementation. A generalization of the polyphase decomposition is called the structural subband decomposition given by [4]

[ ]

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−

−−−

)(...

)()(

...1)(

1

1

0

)1(1

MM

M

M

M

zV

zVzV

TzzzH, 1.6

where [ ]ijtT = is an MM × non-singular matrix. A non-singular square matrix is one that has a matrix inverse. In other words, a square matrix is nonsingular if and only if its determinant is nonzero. The relationship between the polyphase components and the generalized polyphase components is as

Polyphase Interpolator

.

.

.

L

L

)(0 zH

)(2 zHL−

)(1 zHL− 1−z

1−z [ ]my+

+

L [ ]nx

Polyphase Decimator

1−z

1−z

.

.

.

[ ]nx [ ]myM

)(1M

M zH −

)(1MzH

)(0MzH +

8

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

− )(...

)()(

)(...

)()(

1

1

0

1

1

1

0

MM

M

M

MM

M

M

zX

zXzX

T

zH

zHzH

. 1.7

As with the case for the polyphase decomposition, the structural

decomposition can be used to realize an FIR filter. Suppose )(zH is an FIR filter with an impulse response of length MPN ×= , where P and M are positive integers. One can apply the structural subband decomposition and write the filter as [4]

[ ]

.1,...,1,0,)(

,)()(

)(...

)()(

...1)(

1

01,1

1

0

1

1

0

)1(1

−==

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

∑

∑

−

=

−++

−

=

−

−−−

MkztzI

zVzI

zV

zVzV

TzzzH

M

j

jjkk

M

k

Mkk

MM

M

M

M

1.8

Finally, the filter can be realized as shown in Figure 4.

Figure 4: Filer Realization Using Subband Decomposition.

where 1,...,1,0),()()( −== MizVzIzF M

iii . It must be mentioned that by choosing simple invertible transform matrices T , the complexity can further be reduced. Polyphase decomposition reduces the complexity of the filter realization and hence finds extensive use in the analysis and implementation of filter banks, as discussed in the next sections.

)(1 zF. . .

+)(nx )(ny

)(0 zF

)(1 zFM −

9

1.2. Digital Filter Banks

The idea of filter banks is to split the input signal )(nx into subband signals )(nxk through the use of analysis filters )(zHk . The subband signals can then be processed which is usually called subband processing. The last stage is the reconstruction to approximate the output signal )(

^nxk

by the use of synthesis filters )(zFk to combine the subband signals [3]. The typical system diagram is shown in Figure 5.

Figure 5: Typical Analysis and Synthesis Banks.

In this section, we will introduce the main blocks of the filter banks

and their properties for specific types of filter banks namely maximally decimated, oversampled, and time varying filter banks which will be discussed in the later subsections. Generally, a filter bank has five main blocks namely analysis bank, downsampler, subband processing, upsampler, and synthesis bank. These blocks will be discussed in the next subsections.

1.2.1. Analysis Filter Bank

This block is a collection of M so called analysis or decimation filters with a common input signal. The typical frequency responses of these filters can be overlapping, marginally overlapping, and non-overlapping as shown in Figure 6 .

)(nx

)(1 zH

)(0 zH

)(1 zHM−

)(0 nx

)(1 nxM −

)(1 nx. . .

Analysis Bank

.

.

.

)(0 zF

)(1 zFM −

)(1 zF

+

+

)(^

nx

)(1 ny

)(1 nyM −

)(0 ny

Synthesis Bank

10

Figure 6: Typical Frequency Responses of Analysis Filters.

1.2.2. Downsamplers

In order to increase the subband processing efficiency, the sampling rate can be reduced. The choice of down sampling ratio leads to two types of systems as:

• Maximally decimated filter banks: In this case, the number of the subband channels is equal to the down sampling ratio leading to equal number of samples in the subband and full band signals. Although this seems to bring maximum efficiency, but it causes aliasing.

• Oversampled filter banks: Contrary to the maximally decimated case, one can choose the decimation ratio to be less than the number of subband channels. The draw back here is that the number of subband samples is larger than the number of full band samples. This has some advantages though and will be discussed later.

1.2.3. Subband Processing

In this block, the subband signals are processed according to the requirements. Examples of the processing can be coding, decoding, etc. In the design of filter banks, this part is usually ignored and the prefect reconstruction properties are defined for the filter bank only. Throughout this document, we will assume the frequency response of the subband processing block to be unity for all frequencies.

Overlapping

Marginally Overlapping

Non-Overlapping

Tω

Tω

Tω

11

1.2.4. Upsamplers

In order to have the data at the original sampling rate, upsampling which simply inserts a number of zeros in between every two samples is used.

1.2.5. Synthesis Filter Bank

As discussed in Section 1.1, upsampling causes imaging and must be removed by an interpolation filter. The synthesis bank is a collection of M so called synthesis or interpolation filters with a summed output which is simply a combination of the subband signals. In order to have perfect reconstruction, the frequency responses of the synthesis filters must be matched to frequency responses of the analysis filters. The waveform )(th is said to be matched to the waveform )(ts if [8]

∆−∆− =−=−∆= fjfj efjkSefjkSfjHorksth ππ πππτ 2*2 )2()2()2()()( , 1.9

where k and s are arbitrary constants.

In other words, ignoring the delay and amplitude factors, the transfer function of a matched filter is the complex conjugate of the spectrum of the filter to which it is matched. The use of a matched filter gives the maximum Signal to Noise Ratio (SNR). However, in most cases, the synthesis and filters are exactly the same as the analysis filters.

1.3. General Filter Bank Architecture

As a conclusion of the previous discussion, the filter bank architecture can be drawn as shown in Figure 7.

12

Figure 7: General Filter Bank Architecture.

In general, the decimation and interpolation ratios mR can be

different resulting in the aliased channel outputs as [9]

)()(1)()2(1

0

)2(mm

mmm R

kR

j

m

R

k

Rk

Rj

m

jm eHeX

ReY

πωπωω

−−

=

−

∑= . 1.10

The set { })(nym forms a critically sampled time-frequency

representation of the original signal. To construct the input signal and assuming there is no processing, the signals { })(nym must be upsampled and filtered through the synthesis filters )(zFm . The reconstructed signal can be written as

)()()(1

)()()(

1

0

1

0

)2(2(

1

0

^^

ωπωπω

ωωω

jm

M

m

R

k

Rkj

mR

kj

m

M

m

jm

Rjm

j

eFeHeXR

eFeYeX

mmm

m

∑ ∑

∑−

=

−

=

−−

−

=

=

=

. 1.11

The drawback of this system is that the information about the

aliased signals in one channel is available in the other channel signals. However, it is possible to design exactly reconstructing analysis and synthesis systems despite existence of aliasing in every individual channel [9]. A special case can be derived letting 10, −≤≤= MiMRi and is called a maximally decimated filter bank where the number of samples in the set of { })(nym and )(nx is equal. This type of filter bank will be discussed in the next section.

0Y

1Y

1−MY

.

.

.

)(zX

)(1 zHM −

)(1 zH

)(0 zH 0R

1R

1−MR Processing

Processing

Processing . . .

1

^

−MY

0

^Y

1

^Y

1−MR

0R

1R

)(1 zFM −

)(1 zF

)(0 zF

+ )(

^zX

13

1.4. Maximally Decimated Filter Banks

As stated before, a simplification by setting 10, −≤≤= MiMRi in the general filter bank system of Figure 7 leads to maximally decimated case. In this system, the number of samples for full band and subband signals is equal. To analyze this system, the input-output relationship can be written as [10]

)()}()({1)(})()({1)(1

1

1

0

1

0

lM

l

M

kk

lk

M

kkk zWXzFzWH

MzXzFzH

MzY ∑ ∑∑

−

=

−

=

−

=

+= . 1.12

The output signal has two parts as follows:

• The first term represents the amplitude and phase distortion and its distortion function is as })()({

1

0∑−

=

M

kkk zFzH . For PR, the distortion

function should be a pure delay. • The second term represents the aliased signal and its transfer

function is as ∑−

=

1

0)().(

M

kk

lk zFWzH which in the ideal case, must be

zero. The system can be analyzed by the use of the polyphase

representation. To do this, the architecture is redrawn according to the polyphase matrices as shown in Figure 8.

Figure 8: Realization of the Analysis and Synthesis Banks Based on Polyphase Matrices.

In this architecture, the matrices )( MzE and )( MzR represent the

polyphase components of the analysis and synthesis filters in the sense that, the ith row of )( MzE and the ith column of )( MzR have the polyphase components of the )(),( zFzH ii respectively. In the mathematical form, this can be shown as [10]

)1( +−Mny )(nx ...

.

.

. 1

)1( −− Mz)2( −− Mz

1 1−z

)1( −− Mz

M

M

M

)( MzE )( MzR

M

M

M

+

14

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−−−

−

−−−−−

−

−

)(...)(::

)(...)(.

1:

)(:

)(

:1

.)(...)(

::)(...)(

)(:

)(

1,10,1

1,00,0)1(

1

0

)1(1,10,1

1,00,0

1

0

NNN

NN

NN

NTNT

N

NNNN

NN

NN

N

N

zRzR

zRzRz

zF

zF

zzEzE

zEzE

zH

zH

. 1.13

Using the noble identities, this system can further be simplified to ease the extraction of perfect reconstruction conditions as shown in Figure 9.

Figure 9: Simplified Realization of Filter Banks Using Noble Identities.

In this system, the only part affecting the PR is the product

)()( zRzE since the rest can be proved to be a PR system. It can be shown that the system is a PR system if this product is a pseudo circulant matrix. A pseudo circulant matrix is a circulant matrix i.e., a matrix whose rows are cyclically shifted versions of a sequence, but the elements below the main diagonal are multiplied by 1−z . So, the matrix is of the form [10]

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−

−−

−−−

−−

)(...)()()()()(

)(...)()()(...)()(

021

11

011

21

2011

110

zpzpzzpzzpzpzzpz

zpzpzpzzpzpzp

MM

MM

M

. 1.14

In this case, the first row is comprised of the polyphase components

of distortion function )(...)()( 1)1(

11

0M

MMMM zpzzpzzp −−−− +++ which must be a

pure delay for a PR system. As a conclusion, the condition for PR can be derived as [10]

)(nx M

M

M M

M

M

+)(zE )(zR

1 1−z

)1( −− Mz 1

)1( −− Mz )2( −− Mz )1( +−Mny.

.

.

.

.

.

15

10 ,0

0)()( 1 −≤≤⎥

⎦

⎤⎢⎣

⎡=

−−−

−

NrIz

IzzEzR

r

rNδ

δ 1.15

where NI is the identity matrix with r being a constant. The PR condition can be stated in another way. If we have the set of power complementary analysis filters, by a proper choice of the synthesis filters [3], the subband signals can be combined in a way to produce the original input signal at the output. In general, a set of filters )(zHk is said to be complementary of order p if we have [11]

.1)(1

0=∑

−

=

pM

k

jk eH ω 1.16

Here p is a positive integer. In special cases, the magnitude and

power complementary filters are the set which satisfy the general equation for values of 2,1=p as

.1)(,1)(21

0

1

0== ∑∑

−

=

−

=

M

k

jk

M

k

jk eHeH ωω 1.17

It can be shown [11] that the higher order complementary filters

can generate ordinary magnitude and power complementary filters while maintaining superior cut-off characteristics.The procedure to design a maximally decimated filter bank has the following steps [10]:

• An appropriate method should be chosen to design all the analysis filters.

• Having designed the analysis filters, polyphase matrix )(zE can be determined.

• The polyphase matrix of the synthesis filters )(zR can be determined by inverting )(zE .

In general, we prefer the Finite Impulse Response (FIR) solutions which are guaranteed to be stable despite having larger delays compared to their Infinite Impulse Response (IIR) counterparts. However, the inverse of a FIR matrix generally leads to IIR solutions which necessitate stability checks. For a special case of FIR matrices, called unimodular matrices, FIR inverse solutions exist. A polynomial matrix is called unimodular, if [12] its determinant is a nonzero constant. It must be mentioned that for ParaUnitary (PU) matrices, there exist FIR inverses

16

also. Usage of paraunitary matrices, leads to paraunitary PR filter banks which will be discussed in the next section.

1.5. Paraunitary Filter Banks

As stated before, paraunitary filter banks constitute a special class of the maximally decimated filter banks where the polyphase matrices are paraunitary. The definition of paraunitariness needs the concept of paraconjugation to be defined. This property can be defined for two types of transfer matrices as follows [10].

1. In the case of a scalar transfer function )(zH , the paraconjugate is

defined as )()( 1~

−∗= zHzH . Thus, to obtain the paracojugate, one

has to replace z by 1−z and also replace each coefficient by its complex conjugate. On the unit circle, paraconjugation is equivalent to complex conjugation since we have

*1

*

~})({)()( ωω

ωjj

jezez

ez

zHzHzH==

−

=

== . 1.18

2. In the case of a matrix transfer function )(zH , paraconjugate is

defined as )()( 1*

~−= zHzH T . To obtain the paracojugate, one has to

transpose the matrix, replace z by 1−z , and replace each coefficient by its complex conjugate. On the unit circle, paraconjugation is equivalent to transpose conjugation since we have

T

ezez

T

ezjj

j

zHzHzH })({)()( 1*

~

ωωω

==

−

=

== . 1.19

Having these definitions, a matrix transfer function )(zH , is

defined to be paraunitary if IzHzH =)()(~

. In the case of a square matrix function, and using the concept of the inverse matrix, we have

{ } 1~

)()( −= zHzH . So the paraconjugate can be derived from the inverse matrix.

17

Another property of the paraunitary matrices is that if two matrices )(),( 21 zPzP are paraunitary, then the product )()( 21 zPzP will also be

paraunitary. This fact can be used to make a conclusion about the system in Figure 9 . If the polyphase matrices satisfy the relationship

NIzEzR =)()( , and assuming )(zE to be paraunitary i.e. IzEzE =)()(~

, the

PR system can be obtained choosing )()(~

zEzR = . In this case, If )(zE is FIR, then )(zR will also be FIR and there is no concern about the stability. In the next section, some properties of the paraunitary filter banks will be introduced.

1.5.1. Properties of Paraunitary PR Filter banks The choice of matrices being paraunitary brings some useful properties as follows:

1. If the polyphase matrix )(zE is paraunitary, then )( NzE is paraunitary also. Hence, assuming

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−

−

−)1(

1,10,1

1,00,0

1

0

:1

)(...)(::

)(...)(

)(:

)(

NNNN

NN

NN

N

N zzEzE

zEzE

zH

zH, 1.20

it can be shown that the vector transfer function )(zH composed of all the analysis filters is paraunitary.

2. If the vector transfer function )(zH is paraunitary and assuming that its components are power complementary as

. )(1

0

2consteH

N

k

jk =∑

−

=

ω 1.21

then, we have a lossless system with one input and N outputs.

3. If we have the set of the analysis filters )(zHk of order L , the synthesis filter coefficients can be computed by conjugating the analysis filter coefficients and reversing their order as

10 ],[][ * −≤≤−= NknLhnf kk . Using the Fourier series properties, it can be shown that the magnitude response of the synthesis filter

)(zFk is the same as the magnitude response of the analysis filter

18

)(zHk while its phase is negative. In this sense, they satisfy the definition of the matched filters [13].

4. As a straightforward result, if the analysis filters are power complementary, then the synthesis filters are power complementary also.

A uniform DFT filter bank is a system where a cascade of DFT and IDFT matrices replaces the polyphase matrices and will be discussed in the next section.

1.6. DFT Modulated Filter Banks

Before moving to the discussion of DFT modulated filter banks, we will define the concept of uniform and non-uniform filter banks.

1.6.1. Uniform and Non-uniform Filter Bank

Based on the characteristics of the data signals, one can choose to shift the analysis and synthesis filters uniformly or non-uniformly along the frequency axis. This leads to new classes of filter banks whose sample filter characteristics are shown in Figure 10.

Figure 10: Filter Characteristics for Uniform and Non-Uniform Filter Banks.

In the uniform case, the channel filters are derived from a real linear-phase LowPass (LP) prototype filter )(ng of length L by modulation as [14]

,1,...,0,)()()()

2)1()(5.0(*)

2)1()(5.0(

−=+=−

−+−

−+−Miengaenganh

LniM

j

i

LniM

j

ii

ππ

1.22

Uniform

Non-uniform

Tω

Tω

19

where subscript * denotes the complex conjugation. In this system, since the LP prototype has real coefficients, the channel filters are obtained by cosine modulation and will be discussed in Section 1.7. The complex multiplying factors ia define the modulation phase. The synthesis filters are similar to the analysis filters but with a different modulation phase usually chosen so that the resulting filters are the time-reverse of the analysis filters. In this case, the overall filter bank response will have linear phase. By appropriate design of the prototype filter, the overall frequency response can be made flat also.

On the other hand, for the case of non-uniform filter banks, the analysis filter for channel i is generated by modulation of a possibly complex lowpass prototype )(ngi of length iL , as [14]

.1,...,0,)()()()

2)1(

)(5.0(**

)2

)1()(5.0(

−=+=−

−+−

−+−

Miengaenganhi

ii

ii

i

Lnk

Mj

ii

Lnk

Mj

iii

ππ

1.23

Hence, the synthesis filters are given as

.1,...,0,)()()()

2)1(

)(5.0(**

)2

)1()(5.0(

−=+=−

−+−

−+−

Miengbengbnfi

ii

ii

i

Lnk

Mj

ii

Lnk

Mj

iii

ππ

1.24

Here, the term )5.0( +± ii

kMπ defines the ith channel center frequency, ik is

an integer, and iM is the decimation factor. The coefficients ii ba , are complex and define the modulation phase. The choice of different decimation factors iM gives the possibility of having narrow channels at low frequencies and wider channels at high frequencies or vice versa.

1.6.2. Uniform DFT Modulated Filter Banks

DFT filter banks can realize linear-phase analysis and synthesis filters using a proper complex modulation of a real-valued lowpass prototype filter. In an N-channel uniform filter bank, the prototype filter

)(zP is uniformly shifted on the unit circle. To analyze this system, we use the z transform properties [15] and define the set of analysis filters

)(),...,(),( 110 zHzHzH N− in the time and z domain as [16]

20

)()(,][][22Nkj

kN

nkj

k zePzHenpnhππ

−== . 1.25

Assuming a non-causal prototype filter and in order to obtain causal analysis and synthesis filters, the impulse responses are delayed by

21−N

samples. Therefore, the time-domain representation of the analysis filters will be [16]

1,...,0,1,...,1,0,]2

1[][)

21(2

−=−=−

−=−

−MkNneNnpnh

NnNkj

k

π

. 1.26

The synthesis filters are identical to the analysis filters. It can be

shown [16] that if the prototype filter has the zero phase property, then all the analysis and synthesis filters will be linear-phase. In the implementation phase, the polyphase decomposition can be used to reduce the implementation complexity. The polyphase components of the prototype filter can be written as

∑−

=

−=1

0

)()(N

l

Nl

l zEzzP . 1.27

So, the analysis filters can be written as

)()()()(1

0

1

0

/2/2/2 ∑∑−

=

−−−

=

−−− ===N

l

Nl

kllN

l

NkNjNl

NkljlNkjk zEWzezEezzePzH πππ , 1.28

and can be arranged in a matrix formulation as

),(

)(.:

)(.)(.

)(

...::::

...

...

...

)(

)(:

)()()(

11

22

11

0

)1()1(2)1(0

)1(2420

)1(210

0000

1

2

1

0

2

zX

zEz

zEzzEz

zEIDFT

WWWW

WWWWWWWW

WWWW

zX

zH

zHzHzH

NN

N

N

N

N

NNN

N

N

N⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−+−

−

−

−−−−−−

−−−−

−−−−

−

4444444 84444444 76

1.29

where NjeW /2π−= .

As a conclusion, the whole analysis bank can be implemented at the cost of one filter plus an IDFT as shown in Figure 11. At the design

21

phase, we only need to design the prototype filter since the other filters are shifted versions of the prototype filter.

Figure 11: Analysis Bank Polyphase Realization of DFT Modulated Filter Banks.

To analyze the system in another way, we can use the definition of

a transfer function. The relationship between signals )(nxi and )(nyk can be written as [3]

∑−

=

−=1

0)(1)(

N

i

ikik Wnx

Mny . 1.30

By defining the transfer function

)()()( zX

zYzH kk = , it can be verified that

)(zHk is a shifted version of a prototype response )(zP through Equation

1.26 namely )()( kk zWPzH = [3].

To analyze the synthesis side, we assume the synthesis filters to be )(),...,(),( 110 zFzFzF N− that satisfy the time reverse property as

)()( /2/2 NkjNkj

k zePezF ππ −= . 1.31

Again, using the polyphase representation, we have

)()(1

0∑−

=

−=N

l

Nl

l zRzzP . 1.32

In general, for all of the synthesis filters, we have

)()(1

0

)1(∑−

=

−−−=N

l

Nl

lNklk zRWzzF . 1.33

Thus, the reconstructed output signal )(ny in the matrix domain can be written as

)(nx 1−z

1−z

N

N

N

.

.

.

.

.

.

IDFT

)(0 ny

)(1 ny

)(1 nyN−

)(0 nx

)(1 nxN−

)(1 nx)(1 zE

)(0 zE

)(1 zEN−

22

[ ]

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−

−

)(:

)()()(

)(...)()()()(

1

2

1

0

1210

zY

zYzYzY

zFzFzFzFzY

N

N

. 1.34

which using Equation 1.34 can be rewritten as

[ ] .

)(:

)()()(

...::::

...

...

...

)()()(...)(

1

2

1

0

)1()1(2)1(0

)1(2420

)1(210

0000

011

22

11

2 ⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

−

−

−−−

+−

zY

zYzYzY

DFT

WWWW

WWWWWWWW

WWWW

zRzRzzRzzRz

NNNN

N

N

NNNNN

N

4444444 84444444 76

1.35 Finally, the architecture of the synthesis bank can be drawn as Figure 12.

Figure 12: Analysis Bank Polyphase Realization of DFT Modulated Filter Banks.

where )()( 1

1 zEzR lNl−

−−= and is the second type of the polyphase representation [3]. In this scheme, the first filter should be centered at

0=ω but using a Generalized Discrete Fourier Transform (GDFT) [7], this constraint can be removed. It must be mentioned that by appropriate choices of GDFT matrix, one can obtain a filter bank with

2M filters

having real coefficients for even M . In the GDFT case, the analysis filters )(nhk are derived from a

real-valued LP prototype FIR filter )(np that has even length L as [17]

Nnkenpnhnnkk

Mj

k ∈=++

,,)()())((2

00π

. 1.36

DFT

)(0 ny

)(1 nyN −

)(1 ny

.

.

.

)(1 zRN −

)(2 zRN −

)(0 zR

+

+ )(ny

1−z

1−zNN

N

23

Here, the offsets 0k and 0n are introduced leading to the name GDFT. Choosing a linear-phase prototype filter and setting 0n in a way to have a transform symmetric to 21−L , the modulated filters will have the linear-phase property also. If we choose 5.00 =k , the frequency range )2,0( π will be covered by 2M subbands for even M . In this case, the remaining subbands are complex conjugate versions and can be ignored in the processing reducing the complexity. So, we have a filter bank with 2M filters. The synthesis filters can be obtained by time reversion of the analysis filter as )1()( * +−= nLhnf kk . Thus, all filters can be derived from one single prototype. The procedure to design these filter banks can be summarized in the following steps [10]:

• The prototype filter must be designed according to the system requirements.

• Having the prototype filter, the polyphase components )(zEk can be achieved.

• Assuming that )(zEk can be inverted, the synthesis filters can be chosen as )()( 1

1 zEzR kNk−

−−= . It can be shown that the maximally decimated DFT filter banks at the same time satisfy perfect reconstruction, have FIR analysis and synthesis filters, and are paraunitary.

As a simple case, assume a prototype filter of the form

121 ...1)( +−−− ++++= NzzzzP , 1.37

which leads in polyphase components as delays reducing the filter bank structure to Figure 13.

Figure 13: Simplest Case of the DFT Modulated Filter Banks.

Keeping in mind the fact that the cascade of the IDFT and DFT matrices is equal to a unity matrix i.e. NIDFTIDFT =−1. , one can assume the overall

)(nx )1( +−Mny

M

M

M M

M

M

+ IDFT DFT)2( −− Mz

1

)1( −− Mz1

1−z

)1( −− Mz

.

.

.

.

.

.

24

system response to be still a delay but doing so, the filtering order must be changed. The analysis to derive equations for the analysis and synthesis filters is similar to the general case. It can be shown that the analysis filters can be modeled as

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

−

−

−−−

−

−

−1

2

1

)1()1(210

)1(2420

1210

0000

1

2

1

0

:

1

...::::

...

...

...

)(:

)()()(

2 N

DFT

NNN

N

N

N z

zz

WWWW

WWWWWWWWWWWW

zH

zHzHzH

444444 8444444 76

. 1.38

In other words, we can assume the analysis filters as

)()( )/2( Nkjjk ePeH πωω −= , 1.39

which are obviously the uniformly shifted versions of the prototype filter.

1.7. Cosine Modulated Filter Banks

In uniform DFT modulated filter banks, assuming )(zP to be the prototype lowpass filter with a cutoff frequency at N/π± , the analysis filters can be derived as )()(

2Nkj

k zePzHπ

−= . Cosine modulated filter banks

are defined by the use of Discrete Cosine Transform (DCT) which has four types as [18]

25

⎪⎪⎩

⎪⎪⎨

⎧

=

≠=

−=⎥⎦⎤

⎢⎣⎡ +

+

−=⎥⎦⎤

⎢⎣⎡ +

−=⎥⎦⎤

⎢⎣⎡ +

=⎥⎦⎤

⎢⎣⎡

.0,21

0,1

1,...,0,,))5.0()5.0(cos(2:

1,...,0,,))5.0(cos(2:

1,...,0,,))5.0(cos(2:

,...,0,,)cos(2:

Noriif

Noriifc

NkiiN

kN

IVTypeDCT

NkiN

kicN

IIITypeDCT

NkiN

ikcN

IITypeDCT

NkiNkicc

NITypeDCT

i

k

i

ki

π

π

π

π

1.40

It can be shown [18] that only types IVII , can be used for cosine

modulated filter banks and there is a relationship between the filter banks defined using these two types of modulation. To be specific, suppose we have a type IV cosine modulated filter bank with ),(),( zHzP k and )(zFk being its prototype, analysis, and synthesis filters, respectively as

}4

.)1()2

)(12(2

cos{)(.2)(

}4

.)1()2

)(12(2

cos{)(.2)(

ππ

ππ

kk

kk

LnkN

npnf

LnkN

npnh

−−−+=

−+−+=, 1.41

where N and L are respectively the number of the channels and order of the prototype filter. Having this, the type II cosine modulated filter bank can be derived as [18]

}4

.)1()2

1)(12(2

cos{)(2)(

}4

.)1()2

1)(12(2

cos{)(2)(

^^

^^

ππ

ππ

kk

kk

LnkN

npnh

LnkN

npnh

−−+

−+=

−++

−+=. 1.42

Hence, if the prototype real-coefficient lowpass filter is )(zP with a cutoff at N2/π± , the analysis filters are

)()()()5.0(*)5.0(N

kj

kN

kj

kk zePzePzHππ

αα++−

+= . 1.43

26

In the time domain, we have

})1(4

)2

)(5.0(cos{][2][ kk

LnkN

npnh −+−+=ππ

. 1.44

This is obviously a cosine modulation instead of exponential modulation. So, if the prototype filter is a lowpass filter, the analysis filters are bandpass filters with the characteristics of the prototype filter as shown in Figure 14 [10].

Figure 14: Analysis Filters for the Cosine Modulated Filter Banks.

Exploiting the advantages of the polyphase decomposition, and

assuming that )(zP has N2 -fold polyphase, we have

∑∑∑∞

−∞=

−−

=

−

=

− +===k

kl

N

l

Nl

lL

k

k lNkpzzEzEzkpzzP ]2[)(,)(][)(12

0

2

0

. 1.45

If )(zP has a length of Nm..2 , then the analysis filters realization can be shown in a matrix form as in Figure 15.

Figure 15: Polyphase Realization Analysis Bank for the Cosine Modulated Filter Banks.

Here, the matrix NNT 2× is [10]

)(kx )( 20

NzE −

)( 21

NzE −

)( 212

NN zE −−

)(0 zH)(1 zH

)(1 zH N−

NNT 2×

1−z

1−z

N2π

0H

π2

P

π2N2π

1H

π2

27

[ ]

.

...00:::0...00...0

00...1:::01...010...0

,

1...00:::0...100...01

)()(

})5.0(cos{

})5.01(cos{

})5.0(cos{

22

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=Λ

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

+−−Λ=

−

+

××

mN

m

m

JI

JIJICNT NNNN

π

π

π

1.46

It can be verified that ))5.0)(5.0(cos(2 −−= ji

NNCij

π are the elements of an

NN × type four DCT matrix as discussed before. In this case, the implementation cost of the analysis block is the prototype filter plus the DCT modulation.

If the prototype filter has 1+L real coefficients where mNL 21 =+ for some integer m and it is linear-phase, then it can be shown that the FIR cosine modulated analysis bank is paraunitary if and only if the polyphase components )(, )( zEzE Nkk + of the prototype filter are power complementary. This means that we have a lossless one input-two output system. Having this, the FIR synthesis bank can be obtained by paraconjugation to satisfy the PR condition.

In general, the length of the prototype can be arbitrary [18]. To illustrate this, assume )(),( zRzE be the polyphase component matrices of the analysis and synthesis filter banks and the linear-phase prototype filter has a length of 12 mmNL += with a transfer function

∑−

=

−=12

0

2 )()(N

l

Nl

l zGzzH , 1.47

with )( 2N

l zG being the polyphase components. It can be shown [18] that )(zGk satisfies

28

⎪⎪⎩

⎪⎪⎨

⎧

>

−≤=

−−+−

−−

.),(

1),()(

1121

11~

1

1

mkzGz

mkzGzzG

kmN

kmm

k . 1.48

Thus, the polyphase matrices )(),( zRzE can be written as

T

CzgzgzzR

zgz

zgCzE

~2

1

~2

0

~1

21

1

20

^

)()()(

)(

)()(

⎟⎠⎞

⎜⎝⎛ −−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−=

−

−, 1.49

where

.4

)1()2

(2

)12(cos2

))(...)()(()())(...)()(()(

,

^

1211

1100

⎟⎠⎞

⎜⎝⎛ −+−+=⎥⎦

⎤⎢⎣⎡

==

−+

−

ππ k

lk

MMM

M

LlN

kC

zGzGzGdiagzgzGzGzGdiagzg

1.50

To derive the PR conditions, we define )()()( zEzRzP

∆

= . It can be shown [18] that for PR, we must have

⎟⎟⎠

⎞⎜⎜⎝

⎛=

+−

−

00

)(2

)1(1

IzIz

zP v

v

, 1.51

where v is a positive integer and the dimensions of unity matrices 21, II add to N . Further simplification can be made leading to the necessary and sufficient conditions for PR as [18]

10,21)()()()(

~~−≤≤=+ ++ Nk

NzGzGzGzG kNkNkk . 1.52

As a conclusion, the property for PR in cosine modulated filter bank with arbitrary prototype filter length is similar to the case when the length is an even multiple of N .The design procedure is as follows [10]:

29

1. First, the lossless systems )(, )( zEzE Nkk + must be parameterized according to the structures.

2. The next step is to optimize parameterization to achieve the linear-phase prototype filter based on these polyphase components satisfying the given specifications.

As a comparison between cosine modulated and DFT modulated

filter banks, it must be mentioned that, in a maximally decimated cosine modulated filter bank, two polyphase components of the prototype filter replace only one polyphase component in the DFT modulated case. In such a system, in order to satisfy paraunitariness, or equivalently having FIR system to be PR, each such pair of polyphase filters should form a power complementary pair. In other words, they must be a lossless system. On the other hand, for a DFT modulated system to be paraunitary, each polyphase component must separately be lossless which means that each polyphase component should be an allpass transfer function. There is a problem here since allpass functions are usually IIR and concerns about stability arise. To further simplify the PR conditions, the oversampled filter banks are used which will be discussed in the next section.

1.8. Oversampled PR Filter Banks

In the previous sections, we have discussed maximal decimation, which causes aliasing and makes the PR property hard to achieve. A solution to this is the choice of oversampled FBs that easily suppress aliasing and allow the combination of smaller subbands into wider subbands without introducing large aliasing distortion. To analyze the oversampled FBs, the concept of frame expansion needs to be defined. Signal decomposition in )(2 Zl , is the expansion of the signal through a sliding window using a selected set of elementary waveforms as [12]

∑∑−

=

∞

−∞=

=1

0,, )()(

K

i jjiji ncnx ϕ , 1.53

where the vectors )(, njiϕ are the translated versions of K waveforms

30

KNjNnn iji ≤−= ),()(, ϕϕ . 1.54

Any signal in )(2 Zl can be represented using such an expansion if and only if the family Φ

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∈−=−==Φ ZjKijNnn ijiji ,1,...,1,0),()(: ,, ϕϕϕ , 1.55

forms a frame in )(2 Zl . A family of vectors Φ given in Equation 1.56 is said to be a frame if for any )(2 Zlx∈ ,

∞<>≤≤ ∑ ∑−

=

∞

−∞=

BAxBxxAK

i jji ,0,, 2

1

0

2

,2 ϕ . 1.56

The constants BA, are called the frame bounds and the frame is tight if

BA = . If the family Φ is a frame, there is another frame as

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∈−=−==Ψ ZjKijNnn ijiji ,1,...,1,0),()(: ,, ψψψ . 1.57

In this case, the coefficients of the new signal decomposition can be written as

∑∑−

=

∞

−∞=

=1

0,, )(,)(

K

i jjiji nxnx ϕψ . 1.58

In addition,

jix ,,ψ stands for the inner product. Furthermore, frames

ΨΦ, can be interchanged in a way that any signal in )(2 Zl can also be written as

∑∑−

=

∞

−∞=

=1

0,, )(,)(

K

i jjiji nxnx ψϕ . 1.59

Using the concepts of duality and tightness [19], the expansion formula can be written similar to orthogonal expansions as

31

∑∑−

=

∞

−∞=

=1

0,, )(,1)(

K

i jjiji nx

Anx ϕϕ . 1.60

The starting point to relate this expansion to filter banks is the fact that the inner product of a signal x with the vectors of the family Φ can be obtained as the outputs of an analysis bank )(),...,(),( 110 zHzHzH K−

followed by downsampling ratio of KN ≤ . As discussed before, the analysis filters are complex conjugates of the time-reversed versions of a prototype filter. In this case, the prototype filter becomes the elementary waveforms iϕ . So, the analysis filters are constructed as

1,...,1,0),()( * −=−= Kinnh ii ϕ . 1.61

It must be added that the case KN = leads to critically sampled case. Extending this idea to filter banks, if Φ is a frame, any signal expanded by the analysis bank can be reconstructed from the subband components. This reconstruction is performed by the use of a synthesis filter bank

)(),...,(),( 110 zGzGzG K− whose impulse responses are derived as )()( nng ii ψ= .

Hence, the synthesis bank can be expressed as

∑∑−

=

∞

−∞=

−=1

0, )()()(

K

i jjii kknynx ψ , 1.62

where )(nyi is the input of the ith channel of upsampler and synthesis filter. As with almost all the filter banks, the polyphase idea can be used to reduce the implementation complexity. The polyphase matrix of the analysis and synthesis banks can be written as

32

∑

∑

−

=

−

−−−

−

−

−

=

−−−

−

−

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

1

0

)1)(1(0)1(

)1(110

)1(000

1

0

)1)(1(0)1(

)1(110

)1(000

)()(,

)(...)(.........

)(...)()(...)(

)(

)()(,

)(...)(.........

)(...)()(...)(

)(

N

j

Nij

ji

NKK

N

N

N

j

Nij

ji

NKK

N

N

zGzzG

zGzG

zGzGzGzG

zG

zHzzH

zHzH

zHzHzHzH

zH

. 1.63

Some important theorems about the properties of the oversampled filter banks based on these matrices can be summarized as follows [12]:

1. A filter bank implements a frame expansion if and only if its polyphase analysis matrix is full rank on the unit circle. If A is an

NM × matrix, its rank is the largest number of columns of A forming a linearly independent set. This set of columns is not unique, but the number of elements of this set is unique. A matrix is Full Rank if ),min()( NMArank = .

2. A filter bank implements a tight frame expansion if and only if its polyphase analysis matrix is paraunitary.

3. For a frame associated with an FIR filter bank with the polyphase analysis matrix )(zH , its dual frame consists of finite length vectors if and only if )()(

~zHzH is unimodular.

A special case is oversampled DFT modulated filter banks which

are FIR, PR, DFT modulated, and paraunitary as we shall see. To analyze this system, we assume Figure 9 with the property NIzEzR =)()( . As a special case, one can choose different dimensions for )(),( zEzR leading to the system in Figure 16 [10].

Figure 16: Architecture of Oversampled DFT Modulated Filter Bank.

.

.

.

.

.

.

)1( +−Mny

)(nx 1

1−z

)1( −− Mz

M

M

M

MNzE ×)(

1

)1( −− Mz)2( −− Mz

M

M

M

NMzR ×)(+

33

In this system, the condition NIzEzR =)()( still guarantees the system to be PR. It can be shown that the PR condition becomes easier to achieve if

NM > . As an example, if NM 2= , we have [10]

[ ] .)()(

)()()()(2

121 ⎥

⎦

⎤⎢⎣

⎡==

zEzE

zRzRIzEzR N 1.64

It is important to note that this equation does not necessarily imply

)()( 111 zEzR −= , so inverses may be avoided and we can expect FIR PR

filter banks. In general, an analysis filter bank with M channels and the decimation ratio of N can be realized based on an M -point IDFT cascaded with an NM × polyphase matrix containing the N -fold polyphase components of the prototype filter )(zP . It can easily be verified that if NM = , then B in Figure 17 is a diagonal matrix.

Figure 17: Polyphase Realization of the Oversampled DFT Modulated Filter Bank.

So, the polyphase components of the prototype filter can be derived as [10]

∑∞

−∞=

− +=k

kl lkNhzzE ][)( '

0 , 1.65

with

),gcd('

NMMNN = and ),gcd( NM representing the Greatest Common

Divisor of numbers NM , . In this case, the polyphase matrix will be

1−z

1+−Nz

1

NMNzB ×)( IDFT

)(0 zH

)(1 zH

)(1 zHM −

)(nx

34

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−−

+−

−

−

)(...0

.0...

.)(00)(

)(..

.0..

....)(00...0)(

)(

124

81

1

1

0

MN

NN

MN

N

MN

M

M

N

zEz

zEzzEz

zE

zEzE

zB

. 1.66

It is shown that if )()( zBIDFTzE ×= is paraunitary, then with the choice

of DFTzBzR ×= )()(~

, the PR property can be satisfied. This means that the analysis bank will be paraunitary if )(zB is paraunitary. The paraunitariness of )(zB can be proved if and only if

1,...,1,0),(),( −=+ NkzEzE Nkk are power complementary. Hence, oversampled PR FBs can at the same time be DFT modulated, FIR, and paraunitary.

All the filter banks discussed up to now are Linear Time Invariant (LTI) systems. However, for some applications, we may require time varying systems to improve the efficiency. In the next section, we will discuss the time varying filter banks and their properties.

1.9. Time Varying Filter Banks

In Time Varying Filter Banks (TVFB), the analysis/synthesis filters, the number of bands, the decimation ratios, and the frequency coverage of the bands change in time. This is in contrary to the FBs discussed in the previous sections, where the system structure does not change with time. The typical structure of a TVFB is shown in Figure 18.

35

Figure 18: General Architecture of Time Varying Filter Banks.

The advantage of such a system is that, we can modify the analysis

section according to the input signal properties and hence improve the system performance. In this case, the PR property is the same as the regular FBs i.e. )()(

^∆−= nxnx where ∆ is an integer. The design

problem is to choose the system parameters in a way that the PR property holds for all times. TVFBs can be analyzed using the time-domain formulation [20] where the time varying impulse response of the entire filter bank is derived in terms of the analysis and synthesis filter coefficients. To do this, the filter bank is divided into three stages namely the analysis filters, the down/up samplers, and the synthesis filters as shown in Figure 19.

Figure 19: Different Stages of a Time Varying Filter Bank.

The analysis filters’ output is [ ]TnM nvnvnvnv )(),...,(),()( 1)(10 −= , where

)(nvi is the output of the ith analysis filter at time n . Furthermore, the down/up samplers output at time n is [ ]TnM nwnwnwnw )(),...,(),()( 1)(10 −= . Assuming the length )(nN input signal at time n to be

[ ]TN nNnxnxnxnxnx )1)((),...,2(),1(),()( +−−−= , we have

)(0 nw

)(1 nw

)(1)( nw nM −

)(nx )(nx∧

.

.

.

)(nQ

Synthesis Filters

)(1)( nv nM −

)(1 nv

)(0 nv

)(nP

Analysis Filters

)(nΛ

Down/Up Samplers

+

)(nx +. . .

),(0 znH

),(1)( znH nM −

),(1 znH )(nx

∧

Processing

Processing

Processing

)(0 nR

)(1 nR

)(1)( nR nM −

)(0 nR

)(1 nR

)(1)( nR nM −

),(0 znG

),(1)( znG nM −

),(1 znG. . .

36

)()()( nxnPnv N= , 1.67

where )(nP is an )()( nNnM × matrix whose mth row contains the coefficients of the mth analysis filter at time n . Similarly, we have

)()()( nvnnw Λ= , 1.68

where )(nΛ is a diagonal matrix of size )()( nMnM × with mth diagonal element at time n being one if the input and output of the mth down/up sampler are identical. The last stage is the contribution of the synthesis filters, modeled by a matrix as

[ ])(...)()()(

)1)(,(...)2,()1,()0,(.........

)1)(,()2,()2,()0,()1)(,(...)1,()1,()0,()1)(,(...)0,()1,()0,(

)(

1)(210

1)(1)(1)(1)(

2222

1111

0000

nqnqnqnq

nNngngngng

nNngngngngnNngngngngnNngngngng

nQ

nN

nMnMnMnM

−

−−−−

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−−−

= , 1.69

where [ ]TnMi ingingingingnq ),(),...,,(),,(),,()( 1.)(210 −= , and ),( jngi is the jth coefficient of the ith synthesis filter at time n . Having all these, the

FB output at time n can be written as

∑−

=

−=1)(

0

^)()()(

nN

i

Ti inwnqnx 1.70

To derive a matrix equation for the output, we first define [ ]TT

nNTTT nqnqnqnqns )(),...,(),(),()( 1)(210 −= . So, we have

37

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−+−+−Λ

−−−Λ−−−Λ

Λ

=

)1)(()1)(()1)((...

)2()2()2()1()1()1(

)()()(

)()(^

nNnxnNnPnNn

nxnPnnxnPn

nxnPn

nsnx

N

N

N

N

T . 1.71

Using the fact that the last 1)( −nN elements of vector )( inxN − are identical to the first 1)( −nN elements of vector )1( −− inxN , this equation can be decomposed as

[ ][ ]

[ ]

[ ]

,

)1)(2(...

)2()1(

)(

)1)(()1)((0...0........0...0)2()2(000...0)1()1(00...0)()(

)()(^

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

−−

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−+−Λ

−−Λ−−Λ

Λ

=

nNnx

nxnx

nx

nNnPnNn

nPnnPn

nPn

nsnx T

1.72 where 0 is the zero column vector of length )(nM . In this case, the input/output relationship of the TVFB becomes [20]

[ ]1)(2

)1(),...,2(),1(),()(),()()(^

−=+−−−==

nNIInxnxnxnxnxnxnznx T

IIT

. 1.73

The time varying impulse response vector of the FB is defined as

)()()( nsnAnz = . In this case, the [ ] [ ])()(1)(2 nMnNnN ×− matrix )(nA is defined as

38

[ ][ ]

[ ]⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−Λ+−

−Λ−Λ

=

)1)(()1)((0...00.......0.)1()1(0

00)()(

)(

nNnnNnP

nnPnnP

nA

TTT

T

T

TT

TTT

. 1.74

In order for the system to act as a delay of integer ∆ , it is necessary

and sufficient [20] that all elements except the th)1( +∆ in )(nz be zero at all times. Having a desired impulse response )(nb , PR property holds [20] if and only if )()()( nbnsnA = for all n .

1.10. Differences between Time Varying and LTI Filter Banks

Since the nature of LTI systems is different from that of the time

varying systems, there are some differences as outlined below [6]: 1. For an LTI system, if a FB is PR, then the FB with analysis and

synthesis filters interchanged is also PR. This does not apply in the LTV case.

2. For an LTI PU system, the analysis and synthesis banks are lossless whereas in the LTV case, the losslessness of analysis bank does not imply losslessness of the synthesis bank. A system with a transfer function )(zH is lossless if it preserves signal energy for all inputs. Mathematically, if the system input and output are )(),( nynx respectively, we have

22

)()( ∑∑∞

−∞=

∞

−∞=

=nn

nynx . 1.75

1. Replacing the delay 1−z in the implementation of an LTI PU

system with Lz− for some integer L does not change the PU property. However, this is usually not true for a LTV lossless system.

39

1.11. Filter Bank Design Issues

Regardless of the different types of filter banks, the general system of Figure 7 can be viewed as a hierarchical system [9] having specific requirements and issues at each level. In the next subsections, we will discuss these hierarchies and outline some of their important constraints on the overall system design.

1.11.1. Filter Issues

This is the lowest level where we face the filter design problems and tradeoffs. In the analysis and synthesis side, the stop band, pass band, and transition band characteristics are important. Furthermore, for the synthesis part, we must consider the reconstruction issues as well. There are some common issues such as the implementation complexity and numerical sensitivity also.

1.11.2. Filter Bank Issues

In this context, the quality of the frequency coverage of the analysis filter bank must be considered. In order to save in the realization complexity, it must be noted that one can implement the whole filter bank requiring less operations than the sum of the operations for the individual filters and must be checked thoroughly. As the last issue, the capability of the system to reconstruct the data in the presence of distortion, forces some constraints on the individual filter characteristics in a top down approach.

1.11.3. Analysis/Synthesis Issues

If we do not consider the processing, the system goal is to reconstruct the input at the output. In this case, the analysis/synthesis level distortions can be modeled as LTI distortions in the form of magnitude and phase as well as distortions caused by aliasing. Here, there is a problem whose goal is to minimize these distortions. This minimization problem imposes further constraints on the design.

40

1.11.4. Total System Issues

The desired goal in the system design is to maximize system performance. To increase the performance, one can do improvements in the analysis/synthesis filters to reject out of band and out of time processing distortion.

41

Chapter Two: Overview of Transmultiplexers and Satellite Payload Systems

43

2. Introduction

Digital filter banks find applications in subband coders for speech signals, frequency domain speech scramblers, image coding, and frequency multiplexing/demultiplexing. In this chapter, we will describe the mathematical theory of the transmultiplexers as duals of filter banks followed by issues such as channel equalization and interference cancellation. As a special case of transmultiplexers, TDMA, CDMA, and FDMA systems will be studied. Next, we will discuss and compare different payload architectures for satellite applications. The chapter ends with introduction to applications of filter banks in payload systems. This topic will be studied in detail in the third chapter.

2.1. Transmultiplexers

By definition, Transmultiplexer (TM) converts the time multiplexed components of a signal into a frequency multiplexed version and back [21]. A TM can also be used for applications such as channel equalization, channel identification, etc. In [22], it was shown that a FB and a TM are duals and the transposition of the analysis/synthesis banks gives the dual TM. Using the duality, at the transmitter side, M different source signals are multiplexed into one transmit signal by upsamplers and synthesis filters. On the receiver side, the received signal is decomposed into M source signals by analysis filters and downsamplers. As it can be predicted, non-ideal synthesis/analysis filters result in crosstalk between channels. Since analysis/synthesis filters are reversed, analysis bank removes crosstalk introduced by synthesis bank. However, the perfect reconstruction theory still applies as we shall see.

2.1.1. Mathematical Representation of Transmultiplexers

Suppose we have a series of symbol streams, either generated by different users or parts of a signal generated by one user, and we want to transmit these signals through a channel. As shown in Figure 20, we can pass the signals through a series of transmitter (pulse shaping) filters

)(zFk to produce the signals [21]

44

∑ −=

ikkk iPnfisnx )()()( . 2.1

The term pulse shaping comes from the fact that the filters take each sample of )(nsk and put a pulse )(nfk around it [21].

Figure 20: General Architecture of a Transmultiplexer.

Here we have M users transmitting through a channel described by

a linear time invariant filter )(zC followed by additive noise. The constraint of being time invariant may not be valid in the case of mobile communications, but as we will see, equalization of the channel is possible even in these cases. Finally, at the receiver side, the filters )(zHk separate the signals and only a downsampling by P is needed to get the original symbol streams. In this system, M signals are multiplexed into one channel which necessitates the constraint MP > giving the name of redundant transmultiplexer [21] as opposed to minimal transmultiplexers where MP = . Ignoring the effects of the channel, the input-output relationship can be written as [23]

Mj

i

M

lk

M

k

Nk

Ni eWNizWHzWFzS

MzS

π21

1

0

11

0

^,10),()()(1)(

−−−

=

−−

=

=−≤≤= ∑∑ . 2.2

The transfer function )()()( 11

0

1 −−

=

−∑= zWHzWFzT i

M

lk

Nki relates the

output signal )(^

Ni zS to the input signal )( N

k zS . In general, due to the existence of Multi User Interference (MUI), Inter-Symbol Interference (ISI) caused by the channel linear distortion, and the additive noise, there is always a difference between the transmitted and the received signals. In order to decrease the Bit Error Rate (BER) of the system, channel equalization is needed. For the case MP > , channel equalization can be done by the use of transmit and receive filters. The idea is shown in

)(1 nx

)(1 nxM −

)(0 nx

P

P

P

)(zC

)(0 ns

)(1 ns

)(1 nsM−

P

P

P

)(0 zF

)(1 zF

)(1 zFM −

+

Noise

+)(1 zH

)(0 zH

)(1 zHM −

)(^

1 ns

)(1

^nsM −

)(^

0 ns

45

Figure 21, where M different channels )(zCk and a common additive noise replace the channel in the previous model.

Figure 21: Architecture of Transmultiplexer with Transmit and receive Filters.

If the channel transfer functions are the same, the system is

equivalent to the system in Figure 20. Interestingly, in the case of multiuser communications over wireless channels, the new representation becomes useful. In the next section, we will discuss the perfect reconstruction property of the system from a mathematical point of view.

2.1.2. Perfect Reconstruction in Transmultiplexers

Assuming the system in Figure 21 and using a mathematical result, one can derive the perfect reconstruction constraint. To further simplify the analysis, we deploy the fact that, if an LTI filter )(ng is placed between an upsampler and a downsampler of ratio P , the overall system is equivalent to the decimated version of the filter impulse response which becomes )(nPg [21]. In this case, designing the transmit/receive filters in each branch, so that the decimated version of )()()( zFzCzH mmk becomes a pure delay, the MUI can be cancelled and the system is a PR system. From the duality property mentioned in Section 2.1, and according to the PR condition discussed in Section 1.4, if )(),( zGzH PP are the polyphase matrices of the analysis and synthesis filters, then the FB is PR if and only if IzHzG PP

T =)()( . On the other hand, a TM is PR if and only if IzGzH P

TP =)()( [24]. If the decimation ratio and the number of

channels are the same, the PR conditions are identical. It must be mentioned that, the PR properties are independent of filter lengths, causality of filters etc., and can be satisfied for both minimal and redundant transmultiplexers. However, for the minimal case, there may not always exist FIR or stable IIR solutions. So, allowing some redundancy will make the solutions feasible.

)(0 ns

)(1 ns

)(1 nsM− Additive Noise

+)(1 zH

)(0 zH

)(1 zHM −

P

P

P

)(^

1 ns

)(1

^nsM −

)(^

0 ns

+ P

P

P

)(0 zF

)(1 zF

)(1 zFM −

)(0 zC

)(1 zC

)(1 zCM −

Channels

Transmitter Filters

Receiver Filters

46

2.1.3. Canceling InterBlock Interference in Transmultiplexers

Using the polyphase realization of the transmit/receive filters, we

can derive the channel between the mth transmitter and kth receiver in matrix form. The new system diagram is shown in Figure 22.

Figure 22: Modeling the Channel to Cancel InterBlock Interference.

In this figure, )(),( ,, zEzR ikmi are the polyphase components of the transmitter and receiver filters, respectively. It is shown in [21] that the InterBlock Interference defined as the interference between input vectors

occurring at different times at the input of ∑=

−=L

n

nmm znczC

0

)()( can be

cancelled through two methods as follows: • Zero Padding: In this scheme, a block of L zeros is inserted at the

end of each block of LP − symbols. In other words, we have 0)(...)( ,1, === −− zRzR mPmLP .

• Zero Jamming: In this scheme, a block of L samples at the beginning of each block of P successive received symbols are set to zero. Mathematically, we have 0)(...)( 1,0, === − zEzE Lkk .

2.1.4. Canceling Multi User Interference in Transmultiplexers

In reality, the signal )(nsk

∧ is affected by both )(nsk and kmnsm ≠),( ,

the latter being called MUI. To cancel this, the mth channel matrix is derived to be [21]

Channel between the mth transmitter and kth receiver

)(nsm

)(,1 zR m

)(,1 zR mP−

)(,0 zR m

P

P

P

++

1−z

1−z

1−z

)(zcm

1−z

1−z

1−z z

z

z

)(1, zEk

)(1, zE pk −

)(0, zEk

P

P

P

+ +

)(^

nsk

Channel

47

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

)(...0.........

)(0)(

....

....

....0...)0()1(0...0)0(

Lc

LcLc

ccc

A

m

m

m

mm

m

m

. 2.3

This matrix is a banded Toeplitz matrix. An NN × matrix

[ ]1,...1,0,;, −== NjktT jkN is Toeplitz if jkjk tt −=, . In other words, the matrix has constant values along negative slope diagonals. The matrix is banded if there is a finite m for which mktk >= ,0 . A banded Toeplitz matrix has a finite number of diagonals with nonzero entries and zeros everywhere else [25]. Hence, for any nonzero number kρ we have [ ] [ ])1(1)1(1 ...1)(...1 −−−−−−− = LP

kkkmmP

kk CA ρρρρρ . 2.4

In this case, if we choose transmit/receive filters as

)1(,1

2,2

1,1,0

)1()1(221

...)(

)...1()(−−−

−−−−

−−−−−

++++=

++++=LP

mLPmmmm

PPkkkkk

zrzrzrrzF

zzzazH ρρρ, 2.5

The transfer function from )(nsm to )(nsk

∧ becomes

)()()( kmkmkkm FCazT ρρ= which in the ideal case, must equal )( mk −δ . If

the multipliers ka are chosen to be )(

1

kkC ρ, the constraint to satisfy the

MUI cancellation becomes

1,0),()( −≤≤−= MmkmkF km δρ . 2.6

Some conclusions can be made as follows:

• The values kρ can be arbitrary numbers but they must be distinct.

48

• The receive filter )(zFm must at least have an order of M leading to LMP +≥ .

• The MUI can be cancelled even if the channels are unknown since only their order is important.

The idea of transmultiplexers can be used in several well known multiple access schemes such as TDMA, FDMA, and CDMA. To describe these architectures, we will define the time-frequency interpretation and link the definition to the transmultiplexer theory. Doing so, we can understand the relationship between the transmultiplexer theory and these schemes.

2.1.5. Time Frequency interpretation

The “uncertainty principle” states that no function can at the same time be centered in both the time and frequency domains. To get around, two types of spread for a discrete time function are defined as [26]:

• Time Spread: The time spread of a function { })(0 nh is defined as

∑∫∑∑ ===−=− n

jw

nnn nhn

EndweHnhEnhnn

E2

0

_2

02

02

0

2_2 )(1,)(

21)(,)()(1 π

ππσ , 2.7

where _

, nE are the energy and time centers respectively. • Frequency Spread

The frequency domain spread of a function { })(0 nh is defined as

∫∫−−

=−=π

π

π

π ππσ dweHw

EwdweHww

Ejwjw

w

2

0

_2

02

_2 )(

21,)()(

21 , 2.8

with _w being the frequency center. The general frequency-time tilde of

the signal is shown in Figure 23. The shape and the location of the tilde can be modified according to the time and frequency centers.

49

Figure 23: Time Frequency Tilde of a General Discrete Time Function.

In the next sections, we will discuss CDMA, TDMA, and FDMA schemes as results of expanding the time frequency interpretation to design orthogonal structures.

2.1.6. CDMA System Based on Transmultiplexers

In Code Division Multiple Access (CDMA) systems, each user is assigned a pseudo-random code sequence i

Niii cccc ,...,,, 321 chosen from a

set of orthogonal codes called spreading sequences. A set of codes are said to be orthogonal, if for any jicc i

mj

m ≠,, we have [13]

ji

N

m

jm

imcc ,

1δ=∑

=

. 2.9

These codes are simultaneously spread in time and frequency which means they are both allpass like and spread in time domain [26]. These codes act as keys in transmission since they spread the undesired signal spectrum improving the system immunity towards noise and jamming. On the other hand, these codes if used properly despread the signal coded by the same spreading sequence. In the transmission, each user symbol

][kui is replaced with the sequence ][],...,[],[],[ 321 kuckuckuckuc iiN

iiiiii . This is equal to time domain multiplication of the user data with the spreading sequence. By setting the filter coefficients of a TM equal to orthogonal user codes, a CDMA transmitter and receiver can be achieved. In this sense, the transmitter code multiplication may be viewed as filtering operation, with FIR transmit filter as i

NNiii czczczC 1

21

1 ...)( +−− +++= .On the receiver side, the receiver code multiplication and summation can also be treated as a filtering operation, whose receive filter is

wσ2

nσ2

_n

_w

n

w

50

)...()1( 11

221 iNiNi

NNi czczcz

zC +−+−− +++= .The architecture of the system with

a simple example is depicted in Figure 24 [10].

Figure 24: CDMA System Based on Transmultiplexer.

2.1.7. TDMA System Based on Transmultiplexers

In TDMA, each user occupies the whole channel frequency and transmits in a dedicated time slot. In the extreme case, the allocation of the time slot can be done at sample level with a transfer function as

ωω jkjk eeF −=)( . In other words, we have spectrally spread synthesis filters

1,...,1,0,0,1)( −=≤≤= MieF jk πωω [26]. So, we replace the synthesis

and analysis filters by delay operators as shown in Figure 25 [10].

Figure 25: Simple TDMA System Based on Transmultiplexer.

In this case, the channel will have series of the signals [ ] [ ] [ ] [ ] [ ],...1,1,,...,, 2121 ++ kukukukuku N which can be retrieved after passing the

synthesis filters. In general, the time slot can be done at a frame level consisting of several samples.

.

.

.

.

.

.

[ ] [ ]1, 11 +kuku

[ ] [ ]1, 22 +kuku

[ ] [ ]1, +kuku NN

Channel

N

N

N )( 12 −zC

+

N

N

N

)(1 zC

)(2 zC

)(zC N

[ ] [ ]kuku 22 ,1+

11−z

1+−Nz

Nz−

1+−Nz

1−z

.

.

.

.

.

.

[ ] [ ]1, 11 +kuku

[ ] [ ]1, 22 +kuku

[ ] [ ]1, +kuku NN

Channel

N

N

N +

N

N

N

51

2.1.8. FDMA System Based on Transmultiplexers

In FDMA systems, each user is assigned a portion of the available channel frequency. To do this, we can choose the synthesis and analysis filters as frequency selective filters that add up to cover the full channel frequency band. The filter characteristics for the ideal case are shown in Figure 26 [26].

Figure 26: Transmultiplexer Synthesis/Analysis Filter Characteristics for FDMA System.

2.2. Satellite Payload Architectures

In order to provide transponded satellite connectivity among terminals with a wide range of data rates, wideband payloads will play an important role in the next generation communications satellites. The fundamental topics in this area are channelization and satellite routing. In order to combat the effects of low radiated power and receiver sensitivity at the mobiles, there is a need to increase the size and complexity of the satellite antennas. The satellites must also generate a large number of spot beams in order to cover the full field-of-view from the satellite [27]. It is necessary to have onboard switching or equivalently FDM multiplexing and demultiplexing to direct the received carriers to the desired spotbeams. In literature, four types of payload architectures have been proposed namely Bentpipe, Full Processing, Partial Processing, and Hybrid payloads [28]. The following sections will study and compare these architectures.

π0

Mπ

ω

)( ωji eF

……

52

2.2.1. Bentpipe Payload

This type of payload is the simplest architecture and converts the uplink carrier frequency to another carrier frequency for downlink signal without any processing. In this system, there is no information about the bit level data. Hence, the routing is not intelligent. In order to increase the efficiency and cover all possible beam-to-beam connections, a transponder-hopping technique [28] must be used. However, the drawback of transponder hopping solution is that, the number of transponders is the square of the number of beams, which makes it impractical for a large number of beams. Another solution is the onboard beam switching technique based on a so-called Satellite-Switched Time Division Multiple Access (SS/TDMA) [28]. However, the bentpipe system has some drawbacks irrespective of the routing algorithms used. In this system, the traffic from one beam to another beam may not always be at the full capacity of the transponder. This means that a portion of the capacity is always wasted, reducing the payload efficiency.

The main drawback of the bentpipe payload is the transfer of the uplink noise to downlink, since there is no error correction algorithm in the transponder. It is shown in [28] that the overall Signal to Noise Ratio (SNR) of the system is

1

0

1

0

1

0

−−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

downlink

b

uplink

b

overall

b

NE

NE

NE . 2.10

This obviously shows degradation in the end-to-end performance. However, adding some processing in the payload can improve the performance as will be discussed in the next sections.

2.2.2. Full Processing Payload

As opposed to the bentpipe technique, in a full processing payload, the uplink signal is demodulated and decoded, so the routing can be done at the packet level according to the destination information provided in the transmitted user data packets. To transmit the routed data in the downlink, there is a need for encoding and modulation, which can be done considering the propagation characteristics of the downlink. This method is usually called Satellite Based Asynchronous Transfer Mode

53

(Satellite-ATM). The advantage of this type of payload is that there is an intelligent routing at the packet level, which increases the efficiency. The routing subsystem for a full processing payload can be implemented using a combination of analog and digital components reducing the weight and power consumption. An important feature of the full processing system is the decoupling of the uplink and downlink noise since channel coding is applied twice. This results in an improvement on the overall end-to-end BER performance. On the other hand, a full processing system is complicated and there is a trade-off between reduction of weight and power consumption, complexity, and ability of using digital components.

2.2.3. Partial Processing Payload

As discussed before, the main features of bentpipe payload are simplicity and degradation of overall SNR. On the other hand, a full processing payload is complicated but it has better performance from the SNR point of view. According to the applications, a compromise on memory size, onboard decoder, speed, and power consumption can be made using a partial processing system. A partial processing system includes demodulator and modulator, but not decoder and encoder. Therefore, channel corruption in uplink and downlink are decoupled but there is no coding gain. Furthermore, a hard decision has to be applied in the uplink signal demodulation, which will further destroy the soft information of the received uplink signal since a detection error made during the processing of the uplink signal cannot be corrected. From the end-to-end BER performance, a partial processing system is generally superior to a bentpipe system, but inferior to a full processing system. However, the routing is not still in the bit level. To solve this, we can add an uncoded header that contains the routing information. However, to make the system less vulnerable to the uplink noise, we need a long header to compensate for the lack of coding gain. Another approach can be to include a simpler full processing subsystem to process headers thus allowing for coded headers.

54

2.2.4. Hybrid Payload

In a hybrid [28] structure, a combination of bentpipe and full processing architecture or bentpipe and partial processing is used. This system has some of the disadvantages and advantages of the individual architectures as discussed before. The next sections will discuss filter bank solutions for frequency multiplexing/demultiplexing.

2.3. Frequency Multiplexing/Demultiplexing Using Filter Banks

Filter banks provide solutions to frequency multiplexing and

demultiplexing problems in the satellite communications. The main solutions reported in the literature are as [29]:

• Channel-individual digital filtering with single- or multi-step decimation: In this algorithm, center frequency and bandwidth of each channel is independent of adjacent channels, which brings the highest flexibility. The drawback is that, this approach has the maximum computational load.

• Tree-structured filter bank: This architecture has cascaded directional filter cells, where the complex FDM input is first downsampled by two followed by a separation into two complex subsignals with half bandwidth. Therefore, each directional filter cell is a four channel oversampled complex modulated uniform filter bank. However, only two channels are used for subsequent processing. Before the whole filter bank, a Hilbert transform converts the real FDM signal to its associated analytic representation.

• Complex modulated uniform DFT filter bank: As discussed in Section 1.6, a single-step decimation and a polyphase filtering is used. This approach has the highest efficiency in arithmetic operations and storage. The previous versions have no flexibility to channel allocation and bandwidth, but it will be shown in the next chapter that a new solution utilizing a channel switch can bring full flexibility to the system making it a suitable choice for frequency multiplexing and demultiplexing.

55

Chapter Three: Proposed Bentpipe System and Simulation Results

57

3. Introduction

In order to provide a solution to the increasing demands on multimedia services supporting high bit rates and mobility, the European Space Agency (ESA) has proposed three major network structures for broadband satellite-based systems as [30]:

• Distributed bentpipe satellite internet access network: In this scenario, user terminals combine one or few user traffics producing unbalanced forward/reverse link traffic. The bentpipe architecture has simplicity and easy system evolution support.

• Meshed type of regenerative satellite network for professional users: This type of network will support different classes of professional users. A set of earth stations will support different classes and the system will be able to build Virtual Private Networks (VPN).

• Meshed type of regenerative satellite network for backbone connectivity: The main advantage of this network over terrestrial networks is the capability to interconnect several Internet Service Provider (ISP) access points. This calls for a regenerative onboard processor to flexibly direct spot beams creating an add-on to terrestrial networks.

In all, the aim is to have a globally interconnected digital society, with multimedia applications, information on demand, and low cost delivery of advanced data services which is the user’s expectation and the operator’s promise [31]. In these systems, satellites communicate with users through multiple spot beams, which necessitate efficient use of the limited available frequency spectrum. This calls for satellite onboard signal processing to support frequency band reusage among the beams and bring flexibility in bandwidth and transmission power allocated to each user. To support services at different data rates and bandwidths, a dynamic frequency reusage system is required. Consequently, there is a need for digital Flexible Frequency Band Reallocation (FFBR) networks (also referred to as frequency multiplexing and demultiplexing networks [29]). These networks should bring Perfect Frequency Band Reallocation (PFBR), flexibility, low complexity, parallelism, and implementation simplicity.

In this chapter, a new class of FFBR networks based on variable oversampled complex modulated filter banks (FBs) is studied. This system uses some of the properties of the alternatives discussed in

58

Section 2.3 and can outperform the existing structures from flexibility, low complexity, parallelism, PFBR property, and simplicity points of view. The proposed system can be deployed in any communications environment that requires transparent (bentpipe) reallocation of information. In the next sections, we will start with the formulation of the problem followed by the proposed SISO network. Next, we will study the system from the implementation point of view with an extension to the MIMO case where we will discuss different scenarios of the MIMO case. We will illustrate the simulation results of the system to evaluate the architecture for different input scenarios.

3.1. Problem Formulation

We assume the input signal is divided into Q fixed granularity bands as shown in Figure 27. Any user can occupy one or several (at most Q ) of these granularity bands. Consequently, the input signal contains a variable number of user subbands q where Qq ≤≤1 [32]. In the extreme cases, Qq = and 1=q , which means the user can occupy up to whole the available frequency band and the system can support highest possible data rates on demand.

Figure 27: Illustration of Guard and Granularity Bands in the FFBR System.

The value of q can change during operation corresponding to a

specific reallocation scheme at any time. Furthermore, frequency guard bands (or equivalently filter transition bands) are only present between different user subbands, and ensure the realizability. In brief, the SISO FFBR network has three major tasks:

• Separate the input signal into the desired user subbands: This is similar to an analysis bank as discussed in Section 1.2.1.

• Shift the user subbands in frequency to the desired positions: The use of a switch can accomplish this task as will be discussed later.

Qπα2

QQπαπ 22

+QQπαππ 222 +−

Qπα2

Q Granularity Bands Granularity Band Guard Band

59

• Combine the frequency-shifted user subbands into the output signal: As discussed in Section 1.2.5, a synthesis bank can perform this task.

As a conclusion, to complete the FFBR system, a filter bank needs to be chosen. In the next section, we will discuss a class of filter banks used for the proposed FFBR system.

3.2. Class of Online Variable Oversampled Complex Modulated Filter Banks

This section introduces the proposed class of variable oversampled

complex modulated FBs used in the proposed FFBR network. We will start with constraints of the system followed by the structure of the proposed filter bank. Finally, we will discuss the implementation issues of the system.

3.2.1. System Constraints

As discussed in Section 3.1, the input signal consists of variable q neighbouring users with Q being the fixed number of granularity bands. Furthermore, the input/output sampling rates are the same and the input/output subbands have unique positions. Consequently, the problem becomes reallocating the subbands in the input spectrum to the desired positions in the output spectrum and can be solved by using the filter bank shown in Figure 28 [32].

Figure 28: FFBR system with Fixed Analysis and Adjustable Synthesis Bank.

)(nx . . .

)(0 zH

)(1 zH

)(1 zH N−

M

M

M

.

.

.

M

M

M

)(0 zG

)(1 zG

)(1 zGN−

+ )(ny

)(0 ny

)(1 ny

)(1 nyq−

Fixed Analysis FB Adjustable Synthesis FB

Channel Combiner

60

In this system, the analysis filter bank splits the input signal into subbands. Furthermore, the combination of downsamplers, upsamplers, and synthesis filter bank with adjustable synthesis filters generates the required frequency shifts and recombination of FB subbands into the q shifted user subbands 1,...,1,0),( −= qinyi . To satisfy the system requirements, specific constraints on MN , must be posed, which will be discussed in the next section.

3.2.2. Constraints on Sampling Rate Converters and Number of Channels

As discussed in Section 3.2.1, the choices of M and N play an

important role for the system to satisfy its properties. For instance, if QNM == , the system becomes a maximally decimated FB and hence the

variable subband widths and zero aliasing cannot be achieved simultaneously. As discussed in Section 1.8, letting a slight oversampling by choosing NM < , makes the PR conditions milder. To generate all integer frequency shifts of the granularity frequency, decimation and interpolation by M can be used. Thus,

.int,1, BBBQM ≥= . 3.1

Since NM < , the number of uniform-band channels cannot equal the number of granularity bands [33]. Instead, N must be a multiple of Q as

.int,, ABAB

AMAQN >== . 3.2

It is shown in [33] that for a fixed N , the complexity is minimized by selecting M as large as possible without introducing aliasing. Hence, from Equations 3.1 and 3.2, B is selected as [33]

.int,11, KAKKAB −≤≤−= , 3.3

61

where

KQNM −= . 3.4

In addition, K is the smallest integer allowed without introducing aliasing. In practice, it is possible to make aliasing components arbitrarily small by the stopband attenuation of the analysis filters [33]. Precisely, if the filter bandwidth and transition band are

Nπ2 and ∆2 respectively, the

filters attenuate the aliasing and the minimum value for K is [33]

)(π+∆

∆≥

NN

QNK . 3.5

The next sections will study the building blocks of the system including analysis filters, synthesis filters, channel switch, and the channel combiner.

3.2.3. Analysis Filters

As discussed in the filter bank theory, the analysis and synthesis filters are obtained from the prototype filter

∑=

−=D

n

nznpzP0

)()( , 3.6

where D is the filter order with a constraint that it must be linear-phase and symmetric so that, )()( nDpnp −= . Hence, its frequency response can be written in the form of real zero-phase frequency response as [15]

)()( 2 TPeeP R

TjDTj ω

ωω

−= . 3.7

To obtain the set of analysis filters, the complex modulation can be used as

1,...,1,0),()( −== + NkzWPzH kNkk

αβ , 3.8

62

where

2)(2

,Dk

NkN

j

N WeWαπ

β+−

== . 3.9

In addition, α is a real valued constant to shift the filters to the desired center frequencies and the constants kβ compensate the phase rotations caused by replacing the D th-order linear-phase filter )(zH with

)( α+kNzWH . In this way, all analysis filters become linear-phase FIR filters

with the same delay as the prototype filter.

3.2.4. Synthesis Filters

To obtain the synthesis filters, in this specific case, we are interested both in PR and ability of shifting the signals to the desired location in the frequency spectrum. It is shown in [33] that the choice of the synthesis filters as

MNDm

Nkrrkrckrk

r

krWAskczHzG 2,),()( =+== µµ , 3.10

with

⎪⎪

⎩

⎪⎪

⎨

⎧

<+

≥

=

0,

0,

rr

rr

r

sBsM

sBs

m , 3.11

satisfies these constraints. Here, rs is the number of granularity band shifts for each signal. In general, the synthesis filters are obtained by replacing the D th-order linear-phase filters )(zHk with )( rm

Mk zWH . The constants krµ compensate the resulting phase rotations. For simplicity, the constants can be made 1=krµ if [33]

63

.int2

=MDmr . 3.12

As a result of this discussion, we are able to transform the adjustable synthesis filter bank to the combination of a set of fixed filters, an adjustable switch, and a series of multipliers. In the next section, the adjustable switch operation that shifts the signals will be discussed.

3.2.5. Application of Switch in the FFBR Network

To decrease the complexity of the system and considering the fact that variable filter banks are expensive to implement, with an appropriate choice of filters and parameters in the FFBR network [33], it is possible to implement the same function using variable channel switch and fixed FBs according to the scheme in Figure 29.

Figure 29: FFBR system with Fixed Analysis/Synthesis Banks and Channel Switch.

In this system, the outputs from the analysis banks are connected to the inputs of the synthesis bank. In this way, the complexity can be reduced since fixed filters are less complex to implement. Furthermore, the fixed analysis/synthesis FB, can be implemented using only one filter and an IDFT/DFT block [3]. In the next section, the efficient implementation of the system making use of the polyphase decomposition will be discussed.

)(0 ny

)(1 ny

)(1 nyq−

+ )(ny

Channel Switch

M

M

M

Fixed Analysis FB

)(0 zH

)(1 zH

)(1 zH N−

)(nx

M

M

M

Fixed Synthesis FB

)(0 zG

)(1 zG

)(1 zGN−

Channel Combiner

krµ

.

.

.

64

3.2.6. Efficient Implementation

The polyphase decomposition reduces the implementation complexity of filters and filter banks. The starting point in the analysis is the construction of the polyphase components of the prototype filter as

∑−

=

−=1

0)()(

N

i

Ni

i zPzzP , 3.13

where )(zPi are the polyphase components. Using this, the analysis filters

)(zHk can be written as [32]

[ ] iNi

kiN

N

i

NN

Nii

ikk WWWzPzzH αα ααβ −−

−

=

− == ∑ ,)()(1

0

. 3.14

As a result of the discussions in Section 1.6, it can easily be verified that the system can be implemented using an N -point IDFT and an N -point DFT as shown in Figure 30.

Figure 30: Polyphase Implementation of the FFBR Network.

Here k

Nkk Wβγ = compensate for the phase rotations and B

AL = is chosen

to be an integer. For the cases when L is not integer, a more general polyphase implementation of the polyphase components followed by downsampling has to be used [34]. The system in Figure 30 is a SISO network. In general, we may have several inputs and outputs leading to a MIMO network. In the next sections, extension of the SISO case to MIMO will be studied.

krµ

)(0N

NLWzP α

)(1N

NLWzP α

)(1N

NL

N WzP α−

Channel Switch

IDFT

0α

1α

1−Nα

0β

1β

1−Nβ

0γ

1γ

1−Nγ

1−Nα

2−Nα

0α

)(1N

NL

N WzP α−

)(2N

NL

N WzP α−

)(0N

NLWzP α

DFT

)(ny

+

+

M

M

M

1−z

1−z

.

.

.

.

.

.

)(nx1−z

1−z

M

M

M

Analysis Bank Synthesis Bank

65

3.3. MIMO FFBR Network

In general, it is desired to have a system with several inputs/outputs since the proposed system will be used in the satellite payloads for the next generation communications networks. To deal with these requirements, two scenarios and their corresponding system architectures are derived. For the first case, the number of inputs and outputs are equal, while for the latter case, the number of outputs will be larger.

3.3.1. K-Input K-Output FFBR Networks

Generalizing the SISO system considered so far to a MIMO system with equal number of inputs and outputs, the system in Figure 31 can be used [32].

Figure 31: K-Input K-Output MIMO FFBR with Fixed Analysis and Synthesis FBs.

In this system, the analysis FBs (AFBs) and synthesis FBs (SFBs)

are instances of the fixed FBs used so far but the channel switch can redirect the outputs from one input beam to another output beam. If the SISO FFBR network is designed to satisfy the required BER, the overall performance for each output subband in the MIMO network will be the same as in the SISO network. In general, the satellite payload may have different number of inputs and outputs which is the topic of the next section.

Channel Swtich

In 1

In 2

In K

Out 1

Out 2

Out K

AFB

AFB

AFB

SFB

SFB

SFB

MIMO FFBR

66

3.3.2. S-Input K-Output FFBR Networks

To handle different number of inputs and outputs, the system shown in Figure 32 can be used [32].

Figure 32: S-Input K-Output MIMO FFBR with Fixed Analysis and Synthesis FBs.

In this system, RSK = and the output beam’s bandwidth is R

times narrower than that of the input beam [32]. However, this case can be generalized to allow outputs with different data rates requiring different downsampling factors at the outputs. In the implementation, different instances of polyphase synthesis FBs must be used with some of the DFT inputs set to zero. This keeps the required signal branches only and the task of the channel combiners is to add the necessary SFB outputs to form the desired signals. The channel switch can direct the signals to baseband giving more flexibility in the FFBR network. The next section will illustrate the simulation results on system functionality and performance.

3.4. Simulation Results

To test the system functionality and quality, some issues were considered as follows:

• Selection of system parameters • Construction of a transmitter/receiver pair with low BER • Implementation of the SISO system • Implementation of the MIMO system

The system performance was measured in the Mean Square Error (MSE) sense. In other words, the variance of the error between the transmitted

In 1

In 2

In S

Channel Switch

R

R

R

Ch Co

Ch Co

Ch Co

SFB

SFB

SFB

AFB

AFB

AFB

Out 1 to 1K

Out 11 +K to2K

Out 1+rK to K

MIMO FFBR

Channel Combiners

67

and the received signals was calculated, which can be converted into BER. The next sections will discuss these issues and illustrate the simulation results.

3.4.1. System Parameters Selection

As discussed in Section 3.2.2, and in order to control the aliasing by the stopband attenuation of the filters, the system parameters must satisfy Equations 3.1 to 3.5. In the simulation of the system, the following parameters were considered.

No. Parameter Value 1 Number of Granularity Bands (Q ) 4 2 Number of FB Channels ( N ) 8 3 Downsampling Factor ( M ) 4 4 Transition Band Width (∆ ) Q

π125.0

5 Number of Subbands ( q ) 3 6 Prototype Filter Order ( D ) 134 7 Phase Rotation Factor ( L ) 2 8 Frequency Offset (α ) 0.5

It must be added that according to Equation 3.12, the choice of the prototype filter order to be a multiple of M2 will make krµ equal to unity since rm will in any case be an integer.

3.4.2. Transmitter/Receiver Filter Design

The purpose of the transmitter/receiver pair is to evaluate the quality of the FFBR system for different types of input data i.e. M-QAM or Gaussian signals. Transmit and receive filters design is a traditional communication problem. In brief, the transmit filter constrains the transmitted signal’s spectrum to a limited bandwidth while the receive filter rejects out of band noise thus maximizing SNR. Furthermore, the cascade of the transmit and receive filter must minimize ISI, which means that the convolution of the transmit and receive filters’ impulse responses must satisfy [35]

68

⎪⎪

⎩

⎪⎪

⎨

⎧

−+

=±−=

−=

=

12

1,...,2,1,21,0

21,1

)(

MLkkMLn

LnM

nh , 3.15

where ,, Ln and M are the filter impulse index, filter length, and oversampling factor respectively. The oversampling factor relates the sampling rate with the baud rate through

RFM s= . 3.16

Furthermore,

ML

21+ must be an integer. This type of filter is called a

Nyquist filter. Having designed the equiripple linear-phase Nyquist filter with a nonnegative frequency response, one can use the standard spectral factorization methods [36] to extract transmit and receive filters. However, the main focus of the thesis was to evaluate the FFBR system. To do so, the designed receive filter passband covered passband and the transition band of the M-band transmit filter with a sharp transition band and large attenuation in the stopband as shown in Figure 33. This leads to high order filters resulting in expensive receivers and a future research topic will be to use the spectral factorization methods to reduce the order.

Figure 33: Transmit and Receive Filter Characteristics to Evaluate the FFBR Network.

In this thesis and for simulation purpose, third and sixth band filters

whose characteristics satisfied the constraints of the system were designed. The MATLAB program to design these filters can be found in Appendix A. These filters are used to form the user signals. To do so, three sets of Gaussian or M-QAM signals are generated. The signals are then upsampled and filtered with the third or sixth band filters, respectively. Finally, different users are modulated to appropriate

Mπ Tω

)( ωjeHDashed Line: Receive Filter Solid Line: Transmit Filter

69

positions in the frequency spectrum and summed to form a beam of signals. The MATLAB program constructing two test beams for the general MIMO case can be found in Appendix B. Having constructed input beams, the transmit and receive filters with characteristics shown in Figure 33 were designed. The designed transmit/receive filters had a MSE in the order of 1110− , so it could detect larger errors caused by the FFBR system. The test setup to verify the performance was as Figure 34.

Figure 34: Test Setup for FFBR Network Evaluation.

3.4.3. Implementation of the SISO System

As discussed before, the FFBR has two types of implementation as shown in Figure 29 and Figure 30. One aim in this thesis was to verify the equivalence of these systems in the presence of the channel switch. To implement the system in Figure 29, the MATLAB program in appendix C is used. It must be mentioned that this program implements a general MIMO case. However, setting the number of inputs to one will result in a SISO case. As mentioned before, the polyphase decomposition can reduce the implementation complexity. To verify the system in Figure 30, the MATLAB program in Appendix D is used. This program implements the general MIMO case shown in Figure 31 and can easily be converted to a SISO system by setting the number of inputs to one. Regarding the polyphase implementation of the FFBR network, some issues must be mentioned as follows:

• Prototype Filter Polyphase Decomposition Since the designed prototype filter in this thesis was linear-phase and it had symmetry, the polyphase decomposition resulted in polyphase components with different lengths. Otherwise, the prototype filter would lack symmetry resulting in nonlinear-phase characteristics. Consequently, the vectors at the outputs of the filter blocks in Figure 30 were of different lengths. Since the DFT/IDFT operation was

Input Data

Transmit Filter

Receive Filter FFBR Network

MSE

Transmit/Receive Pair

70

implemented in a matrix multiplication form, the outputs of the filter blocks were zero padded to ease the multiplication process. However, in a sample-based implementation, zero padding can be avoided thus allowing vectors of different lengths.

• DFT/IDFT Implementation The DFT and IDFT operations can be modelled as the multiplication of a column vector whose elements are the values of each branch in Figure 30 at time n , with square matrices given by Equations (1.30) and (1.36). The result would also be a column vector whose elements are the values of branches in Figure 30 at time n .

An important issue in the design of filter banks is the procedure to design the prototype filter. Generally, two techniques namely minimax and least-squares are used to design the prototype filter for filter banks. In this thesis, the minimax algorithm was used and the MATLAB program to design the prototype filter can be found in Appendix E.

To test the SISO system, a data pattern composed of three user subbands was generated at the input of the FFBR Network. A channel switch directs outputs of the analysis filters to different synthesis filters according to the reallocation scheme. For the SISO case, redirection occurs between branches of one filter bank. An example switch scheme is shown in Figure 35.

Figure 35: Example Channel Switch for SISO Case.

The input, output, and the analysis/synthesis filters of the FFBR

network with this switch is shown in Figure 36.

Analysis Bank Outputs

Synthesis Bank Inputs

71

Figure 36: Input, Output, and Analysis Filters for SISO Polyphase FFBR Network.

As it can be seen, three user signals have been shifted to different locations in the spectrum.

3.4.4. Implementation of the MIMO System

Extension of the SISO case to MIMO is done by increasing the instances of the filter bank and adding a channel switch capable of directing signals from one filter bank to another. An example of the switch structure for the case with two inputs and two outputs is shown in Figure 37 .

Figure 37: Example Channel Switch for Two-Input Two-Output MIMO FFBR Network.

From AFB 2 From AFB 1

To SFB 2 To SFB 1

1X

1X

2X

2X

3X

3X

72

In this scheme, two input beams each containing three different users are multiplexed into two output beams where four users share one beam and the remaining two users are present in the other beam. The inputs and outputs of the FFBR network in Figure 31 with the switch in Figure 37 are depicted in Figure 38.

Figure 38: Inputs and Outputs for MIMO FFBR Network with two Inputs and two Outputs.

As discussed in Section 3.3.2, by increasing the number of SFBs, the FFBR Network can handle S Inputs and 1, >= RRSK outputs. To have more flexibility, it is desired to direct all the signals to baseband. This needs modifications in the channel switch as well as setting some of the DFT inputs of the SFBs to zero. The latter removes the unnecessary branches.

As an example, duplicating the AFB output, setting branches five to eight in the first SFB, and setting branches one to four in the second SFB to zero keeps the required user signals without shifting them to baseband. The input and outputs of the FFBR network in Figure 32, without any channel switch are shown in Figure 39.

1X 2X 3X

4X 5X 6X

2X 4X

1X 3X5X6X

73

Figure 39: Input and Outputs of the FFBR Network without Channel Switch.

The use of channel switch can direct the signals to baseband. An example channel switch to shift the user signal 2X to baseband is shown in Figure 40.

Figure 40: Example One-Input/Two-Output Channel Switch for MIMO FFBR Network.

The outputs of the system with this channel switch are shown in Figure 41.

From AFB 2 From AFB 1

To SFB 2 To SFB 1

1X

1X

2X

2X

74

Figure 41: Input and Outputs of the FFBR Network with Channel Switch of Figure 40.

As it can be seen, the outputs are at baseband increasing the multiplexing flexibility of the network.

3.5. Finite Word Length Effects on the FFBR Network

Any system designer deals with the tradeoffs of the system cost and

performance. The more bits we specify for the system, the better performance we get. However, for some applications, a specific performance (usually measured in BER) is required, which will help the designer decide on the system resolution. To evaluate the FFBR network performance, the quantization effects were introduced in the filter coefficients, filter outputs, and DFT/IDFT outputs as shown in Figure 42.

1X

1X

2X

2X

75

Figure 42: Quantization in the Polyphase FFBR Network.

Having the resolution of the data bits at the output of the filter

block will help us define the required number of bits according to the filter implementation structure. In the quantization scheme, we assume different resolutions at different branches of the filter bank. This needs an investigation on the propagation of error in different branches of the system and is a future research topic. To illustrate the effects of finite word length on the FFBR network performance, Figure 43 shows the constellation for a 64-QAM data multiplexed according to the channel switch in Figure 37 for three different filter coefficient lengths.

1−z

+

+

1−z

)(ny

)(nx

IDFT

0α

1α

1−Nα

Q

Q

Q

Q

Q

Q

DFT

1−z

1−z

M

M

M

)(0N

NLWzP α

)(1N

NLWzP α

)(1N

NL

N WzP α−

.

.

.

)(1N

NL

N WzP α−

)(2

NN

LN WzP α−

)(0N

NLWzP α

1−Nα

2−Nα

0α

Channel Switch

0β

1β

1−Nβ

0γ

1γ

1−Nγ

krµ

Q

Q

Q

M

M

M

.

.

.

76

Figure 43: Multiplexed 64-QAM Data Constellation for Three Filter Coefficient Lengths.

This figure further proves the fact that the stopband attenuation of

the prototype filter suppresses aliasing and the designer can make tradeoffs according to the required BER. To compare the system MSE for different stopband attenuations, we can utilize the fact that since the transmit/receive pair has a MSE in the order of 1110− , it can detect larger errors. To illustrate the MSE trend for different stopband characteristics, the variance of the input/output difference for six different channels of Figure 38 is shown in Figure 44.

77

Figure 44: FFBR Network Noise Variance for Channels in Figure 38.

The results show the same trend for all the channels and are in

accordance with the fact that the attenuation of the stopband suppresses aliasing. It must be mentioned that to achieve lower BER, the prototype filter must have larger attenuation and for higher attenuation, the difference between noise variance in different channels is negligible. Thus, the noise behaviour of system is stable for high attenuation values. It is important to note that the noise is white and Gaussian.

3.6. Concluding Remarks and Future Topics

The proposed FFBR network is based on a new class of variable oversampled complex modulated N -channel FBs, which has fixed decimation and interpolation ratios M in the final implementation. The network handles a variable q input and output user subbands where

Qq ≤≤1 . The proposed system architecture uses the following: • Oversampled FB: The oversampled FBs have the advantage of

easy suppression of aliasing allowing the combination of smaller

* 1X ○ 2X + 3X ∆ 4X □ 5X . 6X

78

subbands into wider subbands without introducing large aliasing distortion. This property brings full flexibility to the system.

• More FB channels than granularity bands: This helps generate all possible frequency shifts.

• Complex modulated FBs: This results in very low complexity and simplicity in terms of analysis, design, and implementation.

By properly selecting N , M , and analysis/synthesis filters with given a maximum value of Q , this new class of FBs can:

• Handle all possible frequency shifts and all possible user subband widths.

• Achieve as low complexity as in regular complex modulated FBs. • Achieve as much parallelism as in any of the previously existing

FFBR methods. • Approximate PR as close as desired via a proper design. • Easily be analyzed, designed, and implemented compared to

previously existing FFBR networks. In comparison with other FFBR systems, the proposed system can:

• Outperform the regular modulated FB based networks in terms of flexibility.

• Outperform the tree-structured FB based networks in terms of flexibility and complexity.

• Outperform the overlap/save DFT/IDFT based networks in terms of PR.

Furthermore, both tree-structured FB and overlap/save DFT/IDFT based networks are more complicated to analyze and design. The future research topics can be as follows:

• Analysis of the error propagation in the FB branches: This will help define different word lengths for different branches of the system leading to tradeoffs on complexity and performance.

• Application of the general polyphase decomposition: As discussed in Section 3.2.6, the variable L is an integer. However, if L is not an integer, the general polyphase decomposition must be used.

• The study on transmit/receive filters for the test setup. In this thesis, the purpose was to evaluate the FFBR network, therefore the chosen transmit/receive filters had high orders making the receiver expensive. Reduction of the orders using spectral factorization methods is a future research topic.

79

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[28] Tien Nguyen, Hant J., Taggart D., Chit-Sang Tsang, Johnson D.M., Jo-Chieh Chuang, ”Design concept and methodology for the future advanced wideband satellite system,” Proc. IEEE Military Comm. Conf., MILCOM 2002, vol. 1, pp. 189-194, 7-10 Oct. 2002.

[29] H. G. Göckler and B. Felbecker, “Digital onboard FDM-demultiplexing without restrictions on channel allocation and bandwidth,” 7th Int. Workshop on Digital Sign. Processing Techn. for Space Comm., 1-3 Oct. 2001, Sesimbra, Portugal.

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Appendixes

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Appendix A: MATLAB Program to Design Third and Sixth Band Filters warning off;close all;clc;clear all;format long wsT=0.21875*pi;Mth_Band=6;N=89; % 6th band %wsT=0.46875*pi;Mth_Band=3;N=25;% third band t=1;M=N/2+1;m=1:M;Ks=1000;wT=linspace(wsT,pi,Ks) ; D=zeros(1,Ks);W=ones(1,Ks) ;A=[trigMat(t,m,wT) -1./W'] ; A=[A' [-trigMat(t,m,wT) -1./W']']'; b=[D -D]';c=[zeros(1,M) 1]'; vlb=1/Mth_Band ; vub=1/Mth_Band ; vlb(2:M)=-1 ; vub(2:M)=1 ; for k=Mth_Band+1:Mth_Band:M vlb(k)=0 ; vub(k)=0 ; end x=linprog(c,A,b,[],[],vlb,vub) ; h=[0.5*fliplr(x(2:M)') x(1) 0.5*x(2:M)'] ; wT=linspace(0,pi,4000) ; H=freqz(h,1,wT) ;Mag=20*log10(abs(H)) ;plot(wT/pi,Mag);grid on figure(2);plot(Mag(1:790));grid on find(h == 1/Mth_Band) find(h == 0) figure(3);plot(h);grid on

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Appendix B: MATLAB Program to Generate User Signals function[y,x60,x30,x6,x3]=MIMO_Transmitter(QAM_Type,Sample_Number,h6,h3,Ch_Shift,Type,Data_Bits,Desired_Mean,Desired_Variance) if (Type == 1) x60(1,:)=Desired_Mean+sqrt(Desired_Variance)*randn(1,Sample_Number); x60(2,:)=Desired_Mean+sqrt(Desired_Variance)*randn(1,Sample_Number); x60(3,:)=Desired_Mean+sqrt(Desired_Variance)*randn(1,Sample_Number); x60(4,:)=Desired_Mean+sqrt(Desired_Variance)*randn(1,Sample_Number); Sample_Number = Sample_Number /2 ; x30(1,:)=Desired_Mean+sqrt(Desired_Variance)*randn(1,Sample_Number); x30(2,:)=Desired_Mean+sqrt(Desired_Variance)*randn(1,Sample_Number); else x60(1,:)=qam(Sample_Number,QAM_Type); x60(2,:)=qam(Sample_Number,QAM_Type); x60(3,:)=qam(Sample_Number,QAM_Type); x60(4,:)=qam(Sample_Number,QAM_Type); Sample_Number = Sample_Number /2 ; x30(1,:)=qam(Sample_Number,QAM_Type); x30(2,:)=qam(Sample_Number,QAM_Type); end x6(1,:)=6*conv(upsample(x60(1,:),6),h6);n=0:length(x6(1,:))-1;x6(1,:)=x6(1,:).*exp(j*Ch_Shift(1,1)*pi*n); x6(2,:)=6*conv(upsample(x60(2,:),6),h6);n=0:length(x6(2,:))-1;x6(2,:)=x6(2,:).*exp(j*Ch_Shift(1,3)*pi*n); x6(3,:)=6*conv(upsample(x60(3,:),6),h6);n=0:length(x6(3,:))-1;x6(3,:)=x6(3,:).*exp(j*Ch_Shift(2,2)*pi*n); x6(4,:)=6*conv(upsample(x60(4,:),6),h6);n=0:length(x6(4,:))-1;x6(4,:)=x6(4,:).*exp(j*Ch_Shift(2,3)*pi*n); x3(1,:)=3*conv(upsample(x30(1,:),3),h3);n=0:length(x3(1,:))-1;x3(1,:)=x3(1,:).*exp(j*Ch_Shift(1,2)*pi*n); x3(2,:)=3*conv(upsample(x30(2,:),3),h3);n=0:length(x3(2,:))-1;x3(2,:)=x3(2,:).*exp(j*Ch_Shift(2,1)*pi*n); y(1,:)= x6(1,:)+[x3(1,:) zeros(1,( length(x6(1,:))-length(x3(1,:)) ))]+x6(2,:); y(2,:)= x6(3,:)+[x3(2,:) zeros(1,( length(x6(3,:))-length(x3(2,:)) ))]+x6(4,:); y=quant(y,2^(-Data_Bits+1)) ;

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Appendix C: MATLAB Program to Implement the System in Figure 29

function FBR_Out = HL_MIMO(x,Shift,N,M,L,h0,alpha,wT,High_Level_FBR_Out_Data_Bits,Num_Inputs) %********************************************************** h=[];g=[];n=0:length(h0)-1;K=length(n)-1; for k=0:N-1 h(k+1,1:K+1)=h0.*exp(j*((k+0.5)*(n-K/2)*2*pi/N));g(k+1,1:K+1)=h0.*exp(j*((k+0.5)*(n-K/2)*2*pi/N));end %********************************************************** for k=0:N-1 for l=0:Num_Inputs-1 vtemp( (l*N)+k+1 , : )=conv( x(l+1,:) , h(k+1,:) ); vtemp_down( (l*N)+k+1 , : )=downsample( vtemp((l*N)+k+1,:) , M); end end %********************************************************** Channel_Switch_Out=[]; for k=0:N-1 for l=0:Num_Inputs-1 Channel_Switch_Out((l*N)+k+1+Shift((l*N)+k+1),:) = vtemp_down((l*N)+k+1,:) ; end end %********************************************************** vmat=[]; for k=0:N-1 for l=0:Num_Inputs-1 vmat( (l*N)+k+1 , : )= upsample( Channel_Switch_Out( (l*N)+k+1,: ) ,M); end end %********************************************************** ymat=[]; for k=0:N-1 for l=0:Num_Inputs-1 ymat( (l*N)+k+1 , : )=conv( vmat((l*N)+k+1,:) , g(k+1,:) ); end end FBR_Out=zeros(Num_Inputs,length(ymat(1,:))); for k=0:N-1 for l=0:Num_Inputs-1 FBR_Out( l+1 , : )=FBR_Out( l+1 , : )+ymat( (l*N)+k+1 , : ); end end for l=0:Num_Inputs-1

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FBR_Out(l+1,:)=M* (quant(FBR_Out(l+1,:),2^(-High_Level_FBR_Out_Data_Bits(l+1)+1))) ; end

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Appendix D: MATLAB Program to Implement the System in Figure 31 function [FBR_Out] = Poly_MIMO(x,Shift,N,M,L,h0,alpha,wT,Filter_Out_Bits,FFT_Out_Bits,Final_Out_Bits,Num_Inputs) D = length(h0) -1 ;n=0:length(h0)- 1 ;K=length(n) - 1 ;h=[];g=[]; for k=0:N-1 h(k+1,1:K+1)=h0.*exp(j*((k+alpha)*(n-K/2)*2*pi/N)); g(k+1,1:K+1)=h0.*exp(j*((k+alpha)*(n-K/2)*2*pi/N)); end; for k = 0:N-1 Poly(1,k+1) = {h0(k+1:N:end)};end for k = 0:N-1 Temp_P = Poly{k+1} ; for m = 1:length(Poly{k+1}) Temp_P(m) = ((-1)^(m+1)) .* Temp_P(m) ; end P(1,k+1) = {Temp_P}; end for k = 0:N-1 P_Up(k+1,:) = {upsample(P{1,k+1},L)} ;end;P_Upsampled_Flipped=P_Up; %****************delayed versions of the input signal Delayed_x=[]; for k=0:N-1 for l=0:Num_Inputs-1 Delayed_x((l*N)+k+1,:)= [zeros(1,k) x(l+1,1:end-k)]; end end Decimated_Delayed_x=[]; for k=0:N-1 for l=0:Num_Inputs-1 Decimated_Delayed_x((l*N)+k+1,:)=downsample(Delayed_x((l*N)+k+1,:),M); end end %****************filtering vtmep=[]; for k=0:N-1 for l=0:Num_Inputs-1 vtemp{(l*N)+k+1}=quant(conv(Decimated_Delayed_x((l*N)+k+1,:),P_Upsampled_Flipped{k+1}),2^(-Filter_Out_Bits((l*N)+k+1)+1)); end end % all the inputs should have the same length .zero padd if needed new_vtemp = [] ;

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for k=0:Num_Inputs*N-1 if (length(vtemp{k+1}) ~= length(vtemp{1}) ) new_vtemp(k+1,:) = [ vtemp{k+1} zeros(1,length(vtemp{1})-length(vtemp{k+1}))]; else new_vtemp(k+1,:) = vtemp{k+1} ; end end vtemp = [];vtemp = new_vtemp ; % multiplication by alpha for k=0:N-1 for l=0:Num_Inputs-1 vtemp((l*N)+k+1,:)=vtemp((l*N)+k+1,:) * exp(j*(2*pi/N)*alpha*k); end end %*********IDFT operation in matrix form DFT_Matrix=fft(eye(N)) ;IDFT_Matrix=DFT_Matrix';vtemp_IDFT=[]; for m=0:length(vtemp(1,:))-1 for l=0:Num_Inputs-1 vtemp_IDFT((l*N)+1:(l*N)+N,m+1)=quant((IDFT_Matrix * vtemp((l*N)+1:(l*N)+N,m+1)),2^(-FFT_Out_Bits((l*N)+k+1)+1)); end end %*********multiplication by beta for k=0:N-1 for l=0:Num_Inputs-1 vtemp_IDFT((l*N)+k+1,:)=vtemp_IDFT((l*N)+k+1,:) * exp(-j*(2*pi/N)*(k+alpha)*(D/2)); end end %*********channel switch Channel_Switch_Out=[]; for k=0:N-1 for l=0:Num_Inputs-1 Channel_Switch_Out((l*N)+k+1+Shift((l*N)+k+1),:) = vtemp_IDFT((l*N)+k+1,:) ; end end %******************** for k=0:N-1 for l=0:Num_Inputs-1 vtemp_IDFT_Mu_Gama((l*N)+k+1,:)=Channel_Switch_Out((l*N)+k+1,:) * exp((+j*(2*pi/N))*(((k+alpha)*D/2)-k)); end end %******************* for m=0:length(vtemp_IDFT_Mu_Gama(1,:))-1 for l=0:Num_Inputs-1

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vtemp_DFT((l*N)+1:(l*N)+N,m+1)=quant((DFT_Matrix * vtemp_IDFT_Mu_Gama((l*N)+1:(l*N)+N,m+1)),2^(-Filter_Out_Bits((l*N)+k+1)+1)); end end %*******************reverse alpha multiplication for k=0:N-1 for l=0:Num_Inputs-1 vtemp_DFT((l*N)+k+1,:)=vtemp_DFT((l*N)+k+1,:) * exp(j*(2*pi/N)*alpha*(N-1-k)); end end %*******************reverse filtering for k=0:N-1 for l=0:Num_Inputs-1 vtmep_DFT_fil{(l*N)+k+1}=quant(conv(vtemp_DFT((l*N)+k+1,:),P_Upsampled_Flipped{N-k}),2^(-FFT_Out_Bits((l*N)+k+1)+1)); end end new_vtemp=[]; %******************* for k=0:Num_Inputs*N-1 if (length(vtmep_DFT_fil{k+1}) ~= length(vtmep_DFT_fil{N}) ) new_vtemp(k+1,:)=[vtmep_DFT_fil{k+1} zeros(1,length(vtmep_DFT_fil{N})-length(vtmep_DFT_fil{k+1}))]; else new_vtemp(k+1,:)=vtmep_DFT_fil{k+1} ; end end %******************* for k=0:Num_Inputs*N-1 vtmep_DFT_fil_up(k+1,:)=upsample(new_vtemp(k+1,:),M); end new_vtemp=[];new_vtemp=vtmep_DFT_fil_up; %******************* for k=0:N-1 for l=0:Num_Inputs-1 Delayed_new_vtemp((l*N)+k+1,:)=[zeros(1,N-1-k) new_vtemp((l*N)+k+1,1:end-(N-1-k))]; end end; %******************* y = zeros(Num_Inputs,length(Delayed_new_vtemp(1,:))); for k=0:N-1 for l=0:Num_Inputs-1 y(l+1,:)=y(l+1,:)+Delayed_new_vtemp((l*N)+k+1,:); end end for l=0:Num_Inputs-1

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FBR_Out(l+1,:)=M* (quant(y(l+1,:),2^(-Final_Out_Bits(l+1)+1))) ; end

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Appendix E: MATLAB Program to Design Prototype Filters Using Minimax Algorithm clc;close all;clear all for iteration=0:70 M=4;N=8;Q=4;delta=0.125*pi/Q; PB_dB=0.0000025 SB_dB=2+iteration A=10^((PB_dB)/20);dc=(A-1)/(A+1) B=10^((SB_dB)/20);ds=(1+dc)/B [n0,f,m0,w]=remezord([pi/N-delta pi/N+delta],[1 0],[dc ds],2*pi); n0=8*(fix(n0/8)+1) h0=remez(2*round(n0/2),f,m0,w);n=0:length(h0)-1;K=length(n)-1; options=foptions;options(1)=1; wT=linspace(0,pi/N+delta,250);wsT=linspace(pi/N+delta,pi,200); x0=[0.1 h0(1:K/2+1)]; x=minimax('FIR_fun_new',x0,options,[],[],[],wT,wsT,N,K,n); h0=x(2:length(x));h0=[h0 fliplr(h0(1:length(h0)-1))]; h(iteration+1)={h0} end;save Prototypes_25eMinus7_0to70.mat h function [f,g]=FIR_fun_new(x,wT,wsT,N,K,n) f=x(1);h0=x(2:length(x));h0=[h0 fliplr(h0(1:length(h0)-1))]; H=freqz(h0,1,wT);Hc=freqz(h0,1,wT-2*pi/N); g1=abs(abs(H).*abs(H)+abs(Hc).*abs(Hc)-1)-10*x(1); H=freqz(h0,1,wsT);g2=abs(H)-x(1); g=[g1 g2];

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