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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--SESeptember 2006



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Title On Filter bank Based MIMO Frequency Multiplexing and

Demultiplexing

Master thesis in Electronics Systemsat 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



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Presentation Date2006-09-26

Publishing Date (Electronicversion) 2006-10-02

Division of ElectronicsSystems

Department of ElectricalEngineering

Abstract

The next generation satellite communication networks will provide multimedia services supporting high bit rate, mobility,

ATM, and TCP/IP. In these cases, the satel lite 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 asbetween 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 interferencesuppression.

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 existingapproaches 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 frequencymultiplexing 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-levelThesis D-levelReportOther (specify below)

Publication TitleOn Filter Bank Based MIMO Frequency Multiplexing and Demultiplexing

Author

Amir Eghbali

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

KeywordsFrequency Band Reallocation, Filter Bank, Multirate Signal Processing, MIMO, Satellite Communications



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Abstract

The next generation satellite communication networks will provide multimediaservices supporting high bit rate, mobility, ATM, and TCP/IP. In these cases, thesatellite technology will act as the inter-network infrastructure of future global systemsand assuming a global wireless system, no distinctions will exist between terrestrial andsatellite 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-compatibleservices for data applications, and point-to-point or point-to-multipoint on-demandcompressed video services. This calls for onboard processing payloads capable offrequency 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 certainsystem constraints, the system can handle all possible shifts of different user signals and

provide variable bandwidths to users. Furthermore, the aliasing signals are attenuated bythe stopband attenuation of the channel filter thus ensuring the approximation of the

perfect reconstruction property as close as desired. Study of the system efficientimplementation 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 bothSingle Input single Output (SISO) and Multiple Input Multiple Output (MIMO) cases.For the MIMO case, two different scenarios for frequency multiplexing anddemultiplexing 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 performfrequency multiplexing and demultiplexing and the stopband attenuation of the

prototype filter controls the aliasing signals since the filter coefficients resolution playsthe major role on the system performance. Hence, the system can approximate perfectreconstruction property by proper choice of resolution.



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Acknowledgments

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

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

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



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Foreword

The next generation information society will includetelecommunications, computing, video, TV, videoconferencing, andconsumer electronics in every building and requires wideband services to

provide multi-application networks at rates around 2 Mbps accessible toeverybody everywhere [1]. The terrestrial networks, even with the large bandwidth available due to optical fiber technology, cannot meet theserequirements. However, satellites play an important role since if asatellite is in orbit, the subscriber only has to install a satellite terminaland subscribe to the service. To solve the problem of the next generationnetworks, 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 asasynchronous digital subscriber line (ADSL) and high-rate DSL (HDSL)

[1]. One of the disadvantages of geostationary communicationssatellites, is the large delay for one up- and downlink, which is disturbingfor voice. However, the terrestrial copper, optical fiber, and the cellularradio networks carry most voice traffic. Thus, the satellites can be asuitable choice for interactive data services and delivery of a largeamount of data on request. In addition, low earth orbit (LEO) systemssuch as GLOBALSTAR and ICO are competing with the terrestrialnetworks for voice applications. Therefore, for wideband multimediaapplications, geostationary satellites with several high-gain spot-beamantennas, OnBoard Processing (OBP), and switching seem to be a logical

step in migration from pure TV broadcast to interactive multimediaservices. The functionality that the OBP system offers is suited to provide the services required by the information society. The elementsmaking up the OBP system are [1]:

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

• The Master Control Center (MCC): The MCC translates thesubscriber terminal’s protocols and algorithms into commands. Italso controls the communication flow inside the broadband

satellite communications network. This block is also responsible



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

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

This thesis focuses on digital signal processing of satellite payloadswhich has two major categories as [2]:1. Onboard regeneration and baseband processing: Examples

for this type are data buffering and multiple accessreformatting, data rate conversion, coding, and encryption.These systems decouple noise and interference on theuplink and downlink and are able to optimize access,modulation, and coding techniques for the uplink anddownlink.

2. Onboard non-regenerative processing: Here, signals aresampled with appropriate precision and sampling rate.

Subsequent processing is performed as arithmeticoperations on the signal samples. In particular, suchtechniques allow the digital demultiplexing of narrowbandchannels and processing of individual channels to includelevel control and beamforming. Hence, we need atransparent payload architecture where signals are notregenerated onboard. The system level advantages of thissystem are power efficiency, frequency reuse, flexibilityin response to changing traffic, reproducibility, and lackof 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 lowcomplexity, achieves high level of parallelism, and is easy to analyze anddesign. The system uses a new class of oversampled complex modulatedFilter Banks (FB), which brings superiority over previously proposedarchitectures. 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 offlexibility and complexity. Furthermore, the proposed systemoutperforms the overlap/save DFT/IDFT based networks if perfectreconstruction property is important.The key features of the proposed system are:



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• Use of oversampled filter banks: This choice makes thesuppression of aliasing easier and allows the combination ofsmaller subbands into wider subbands without introducing largealiasing distortion. This property brings full flexibility to thesystem.

• More FB channels than granularity bands: This feature brings the

ability to generate all possible frequency shifts and reduces thecomplexity of the system.• Complex modulated filter banks: These filter banks result in very

low complexity and simplicity in terms of analysis, design, andimplementation.

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 structureuses filter banks, building blocks of filter banks and their mathematicalrepresentations are derived. Based on the structure and parameters, themaximally decimated and oversampled filter banks are discussed. Next,

the concept of paraunitariness followed by DFT and cosine modulatedfilter 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 bankanalysis starts with the definition of frame theory followed by theexample on oversampled DFT modulated filter banks. Having discussedthe time invariant systems, basics and properties of the time varying filter banks are investigated. The chapter ends with common issues in designof filter banks from a system point of view described as constraints in ahierarchical 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 interblockinterference, and channel equalization are discussed. As special cases oftransmultiplexers, the multiple access schemes such as Code DivisionMultiple Access (CDMA), Time Division Multiple Access (TDMA), andFrequency Division Multiple Access (FDMA) are introduced. Havingdiscussed the transmultiplexers, different architectures of payloads i.e., bentpipe, partial processing, full processing, and hybrid systems used insatellite applications and their features are studied. The chapter ends witha review of filter bank applications in payload systems.

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



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multiplexing and demultiplexing is formulated followed by theintroduction to a new class of online variable oversampled complexmodulated filter banks. Based on the problem formulation and the filter bank definition, the constraints of the architecture are derived. Next, thecharacteristics of the filter bank blocks namely analysis/synthesis banksand channel switch are defined. In order to decrease the implementation

complexity, the polyphase decomposition is applied to derive the newsystem architecture. In reality, there are several users in the uplink whichmust be multiplexed to different downlink spot beams. This calls for aMIMO system capable of performing the multiplexing anddemultiplexing and satisfying the defined Perfect Reconstruction (PR) properties. The extension of the proposed system to a MIMO case iscovered in two scenarios.The last part of the chapter illustrates simulation results of the proposedarchitecture from the functionality and performance points of view. Todo so, the system test setup and the error measurement algorithm whichis Mean Square Error (MSE) are described. Next, examples on SISO andMIMO cases verifying the system functionality to multiplex anddemultiplex signals are presented. To evaluate the system performance,the finite word length effects are introduced and examples for a 64-QAMsignal with different resolutions are illustrated. The chapter ends withconclusion and topics for future research.



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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 wordlength effects.

5. Evaluation of the MIMO polyphase network from the BER pointof 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 ANDSATELLITE 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



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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 SIMULATIONRESULTS......................................................................................................................55

3. INTRODUCTION ................................................................................57

3.1. PROBLEM FORMULATION ............................................................................ 58 3.2. CLASS OF O NLINE 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 R ESULTS................................................................................. 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 R EMARKS AND FUTURE TOPICS ............................................ 77

REFERENCES .............................................................................................................79

APPENDIXES...............................................................................................................83

APPENDIX A: MATLAB PROGRAM TO DESIGN THIRD AND SIXTH BANDFILTERS.......................................................................................................................85

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

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

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

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



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LIST OF FIGURES

Figure 1: Effect of Aliasing and Imaging in Upsamplers andDownsamplers........................................................................................... 5Figure 2: Noble Identities in Multirate Systems. ...................................... 6Figure 3: Efficient Polyphase Decimator and InterpolatorImplementation. ........................................................................................ 7

Figure 4: Filer Realization Using Subband Decomposition. .................... 8Figure 5: Typical Analysis and Synthesis Banks...................................... 9Figure 6: Typical Frequency Responses of Analysis Filters. ................. 10Figure 7: General Filter Bank Architecture. ........................................... 12Figure 8: Realization of the Analysis and Synthesis Banks Based onPolyphase Matrices. ................................................................................ 13Figure 9: Simplified Realization of Filter Banks Using Noble Identities.................................................................................................................. 14Figure 10: Filter Characteristics for Uniform and Non-Uniform FilterBanks....................................................................................................... 18Figure 11: Analysis Bank Polyphase Realization of DFT Modulated

Filter Banks. ............................................................................................ 21Figure 12: Analysis Bank Polyphase Realization of DFT ModulatedFilter Banks. ............................................................................................ 22Figure 13: Simplest Case of the DFT Modulated Filter Banks.............. 23Figure 14: Analysis Filters for the Cosine Modulated Filter Banks. ...... 26Figure 15: Polyphase Realization Analysis Bank for the CosineModulated Filter Banks........................................................................... 26Figure 16: Architecture of Oversampled DFT Modulated Filter Bank. . 32Figure 17: Polyphase Realization of the Oversampled DFT ModulatedFilter Bank............................................................................................... 33Figure 18: General Architecture of Time Varying Filter Banks............. 35

Figure 19: Different Stages of a Time Varying Filter Bank. .................. 35Figure 20: General Architecture of a Transmultiplexer.......................... 44Figure 21: Architecture of Transmultiplexer with Transmit and receiveFilters. ..................................................................................................... 45Figure 22: Modeling the Channel to Cancel InterBlock Interference..... 46Figure 23: Time Frequency Tilde of a General Discrete Time Function.49Figure 24: CDMA System Based on Transmultiplexer.......................... 50Figure 25: Simple TDMA System Based on Transmultiplexer.............. 50Figure 26: Transmultiplexer Synthesis/Analysis Filter Characteristics forFDMA System. ....................................................................................... 51Figure 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 SynthesisBank. ....................................................................................................... 59Figure 29: FFBR system with Fixed Analysis/Synthesis Banks andChannel Switch. ...................................................................................... 63Figure 30: Polyphase Implementation of the FFBR Network. ............... 64Figure 31: K-Input K-Output MIMO FFBR with Fixed Analysis and

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



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List of Abbreviations

Abbreviation CommentsAFB Analysis Filter BankATM Asynchronous Transfer ModeBER Bit Error RateCDMA Code Division Multiple AccessDCT Discrete Cosine TransformDFT Discrete Fourier TransformDSL Digital Subscriber LineDSP Digital Signal ProcessingENS Electronic Network SystemsESA European Space AgencyFB Filter BankFDM Frequency Division MultiplexingFDMA Frequency Division Multiple AccessFFBR Flexible Frequency Band ReallocationFIR Finite Impulse ResponseGDFT Generalized Discrete Fourier TransformHDSL High bit rate Digital Subscriber LineIDFT Inverse Discrete Fourier TransformIIR Infinite Impulse Response

ISDN Integrated Services Digital NetworkISI Inter Symbol InterferenceISP Internet Service ProviderLEO Low Earth OrbitLP Low PassLTI Linear Time InvariantLTV Linear Time VariantMCC Master Control CenterMIMO Multiple Input Multiple OutputMSE Mean Square ErrorMUI Multi User InterferenceOBP OnBoard ProcessingPFBR Perfect Frequency Band ReallocationPR Perfect ReconstructionPU ParaUnitaryQAM Quadrature Amplitude Modulation

SFB Synthesis Filter BankSISO Single Input Single OutputSNR Signal to Noise RatioSS/TDMA Satellite-Switched Time Division Multiple AccessTCP/IP Transmission Control Protocol/Internet ProtocolTDMA Time Division Multiple AccessTM TransMultiplexerTVFB Time Varying Filter BanksUTS User StationVPN Virtual Private Network



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1

Chapter One: Overview of Multirate Systemsand Filter Banks



2



3

1. Introduction

Multirate digital filters and filter banks find wide application inareas such as speech processing, communications, analog voice privacysystems, image compression, antenna systems, and digital audio industry.

This applicability has excited immense amount of research leading to asubstantial progress in multirate systems including decimation andinterpolation filters, polyphase structures, and several types ofanalysis/synthesis filter banks with specific properties that suit someapplications. To analyze different systems mathematically, it is useful tohave some blocks that are common among the systems and furthermore,ease the analysis process. In the analysis of the multirate systems andfilter banks, which is the subject of this chapter, the basic building blocksare the interpolators and decimators, which used along with the conceptof 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 thiscontext, different types of maximally decimated and oversampled filter banks are discussed. Furthermore, a brief introduction to time varyingfilter banks is provided.

1.1. Basic Building Blocks of Multirate Systems

In the area of the multirate signal processing, interpolators anddecimators are the basic blocks that alter the sampling frequency atdifferent parts of the system leading to name “Multirate”. An interpolator

is a combination of an upsampler and a lowpass filter where theupsampler inserts 1− M zeros between consecutive samples of theoriginal signal. Doing so, the output signal spectrum is a compressedversion of the input signal spectrum. In the mathematical representationof an upsampler, we have [3]



4

)()()()( M j j M

e X eY or z X zY ω ω == , 1.1

where )(n y and )(n x are the output and input sequences, respectively. If

)( ω je X is periodic with π 2 , then )( ω jeY will be periodic with

M π 2 [3].

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

M j M

k

M k j

j M

k

k M eW e X M

eY or W z X M

zY π π ω

ω 21

0

)2(1

0

1

,)(1

)()(1

)(−−

=

−−

=

=== ∑∑ . 1.2

Hence, )( ω jeY is a sum of M uniformly shifted versions of an − M fold

stretched version of )( ω je X [3]. An important issue in the analysis of

these blocks is imaging and aliasing.Looking at Equation (1.2), one can conclude that if )(n x is band

limited to M M

π ω π <<− (more generally M

π α ω α 2+<< [3]), the

original signal can be recovered from )(n y 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 inputsignal spectrum, which must be removed by a lowpass filter followingthe upsampler.

Similarly, a decimator can cause aliasing due to the stretching ofthe input signal spectrum. To deal with this problem, a lowpass filtermust remove unnecessary signals before the downsampler. The imaging

and aliasing effects and the characteristics of the lowpass filters for asystem 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 berealized, so the filters should have transition bands. This can be solved byconsidering the fact that data signals are not also strictly band limitedwhich 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 nextsection.

1.1.1. Polyphase Decomposi tion

Polyphase decomposition realizes any lowpass filter as the sum of polyphase components [3]. Any finite or infinite length sequence { })(nh

with a z-transform )( z H can be written as [4]

[ ]

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

===

−

−−−−

=

−∞

−∞=

− ∑∑

)(

.

.

.

)(

)(

...1)()()(

1

1

0

)1(11

0

M

M

M

M

M M

k

M

k

k

n

n

z H

z H

z H

z z z H z znh z H . 1.3

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

Interpolation Filter

23π π 2

23π −π 2−

Aliasing in the absence of the filter

π 2 π 2− 2

π +2

π −

Decimation Filter)( ω je X

)( ω je X

π 2π + π −π 2−

32π

3π

3π −

Images to be removed



6

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

M π can be written as

∑−

=

−=1

0

)()( M

i

M

i

i z H z z H , 1.4

where )( z H i are the polyphase components. In the time domain, the

impulse responses of the polyphase components can be derived as10),()( −≤≤+= M i Mnihnhi. 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

)( z H 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

fold h zh zh zh zh zh zh

fold h zh zh zh zh zh zh

h zh zh zh zh zh zh z H

z E z E z E

z E z E

−++++++=

−++++++=

++++++=

−−−−−−

−−−−−−

−−−−−−

4 4 34 4 214 4 34 4 214 4 4 4 34 4 4 4 21

4 4 4 4 34 4 4 4 214 4 4 4 4 4 34 4 4 4 4 4 21

1.5

The second type of the polyphase decomposition can be derived as

∑−

=

−−−=1

0

)1( )()( M

i

M

i

i M z R z z H and is useful in the analysis of synthesis bank filters

[3]. The relationship between these types is )()( 1 z E z R i M i −−= [5]. The

advantage of polyphase components can be better understood by the useof 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 andinterpolation filter implementations as shown in Figure 3.

≡

≡

[ ]n x

[ ]n x [ ]m y

[ ]m y

[ ]mv1

[ ]mv1 L

M )( z H

)( L z H [ ]m y [ ]n x

[ ]m y [ ]n x

[ ]nv2

L )( z H

)( M z H M [ ]nv2



7

Figure 3: Efficient Polyphase Decimator and Interpolator Implementation.

In these structures, the filters run at lower sampling rates comparedto the input signal sampling rate i.e. T 1 . Since the samples across the

adders are phased by T seconds and hence they do not interact in theadder, some commutator models as described in [7] can be used to avoidthe adders for easier implementation. A generalization of the polyphasedecomposition is called the structural subband decomposition given by[4]

[ ]

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−

−−−

)(

.

.

.

)(

)(

...1)(

1

1

0

)1(1

M

M

M

M

M

zV

zV

zV

T z z z H , 1.6

where ijt T = is an M M × non-singular matrix. A non-singular squarematrix is one that has a matrix inverse. In other words, a square matrix isnonsingular if and only if its determinant is nonzero. The relationship between the polyphase components and the generalized polyphasecomponents is as

Polyphase Interpolator

.

.

.

L

L

)(0 z H

)(2 z H L−

)(1 z H L− 1− z

1− z [ ]m y+

+

L

[ ]n x

Polyphase Decimator

1− z

1− z

.

.

.

[ ]n x [ ]m y M

)(1 M

M z H −

)(1 M z H

)(0 M z H +



8

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

− )(

.

.

.

)(

)(

)(

.

.

.

)(

)(

1

1

0

1

1

1

0

M

M

M

M

M

M

M

M

z X

z X

z X

T

z H

z H

z H

. 1.7

As with the case for the polyphase decomposition, the structuraldecomposition can be used to realize an FIR filter. Suppose )( z H is an

FIR filter with an impulse response of length M P N ×= , where P and M are positive integers. One can apply the structural subbanddecomposition and write the filter as [4]

[ ]

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

,)()(

)(

.

.

.

)(

)(

...1)(

1

01,1

1

0

1

1

0

)1(1

−==

=

⎥⎥⎥⎥

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎢⎢⎢⎢

⎣

⎡

=

∑

∑

−

=

−++

−

=

−

−−−

M k zt z I

zV z I

zV

zV

zV

T z z z H

M

j

j

jk k

M

k

M

k k

M

M

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),()()( −== M i zV z I zF M

iii. It must be mentioned that by

choosing simple invertible transform matrices T , the complexity canfurther be reduced. Polyphase decomposition reduces the complexity ofthe filter realization and hence finds extensive use in the analysis and

implementation of filter banks, as discussed in the next sections.

)(1 zF

.

.

.

+)(n x )(n y

)(0 zF

)(1 zF M −



9

1.2. Digital Filter Banks

The idea of filter banks is to split the input signal )(n x into subband

signals )(n xk through the use of analysis filters )( z H k

. The subband

signals can then be processed which is usually called subband processing.

The last stage is the reconstruction to approximate the output signal )(

^

n xk by the use of synthesis filters )( zF k 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 banksand their properties for specific types of filter banks namely maximallydecimated, oversampled, and time varying filter banks which will bediscussed 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 nextsubsections.

1.2.1. Analysis Filter Bank

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

)(n x

)(1 z H

)(0 z H

)(1 z H M −

)(0 n x

)(1 n x M −

)(1 n x

.

.

.

Analysis Bank

.

.

.

)(0 zF

)(1 zF M −

)(1 zF

+

+

)(^

n x

)(1 n y

)(1 n y M −

)(0 n y

Synthesis Bank



10

Figure 6: Typical Frequency Responses of Analysis Filters.

1.2.2. Downsamplers

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

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

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

1.2.3. Subband Processing

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

Overlapping

Marginally Overlapping

Non-Overlapping

T ω

T ω

T ω



11

1.2.4. Upsamplers

In order to have the data at the original sampling rate, upsamplingwhich simply inserts a number of zeros in between every two samples isused.

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 collectionof M so called synthesis or interpolation filters with a summed outputwhich is simply a combination of the subband signals. In order to have perfect reconstruction, the frequency responses of the synthesis filtersmust be matched to frequency responses of the analysis filters. Thewaveform )(t h is said to be matched to the waveform )(t s if [8]

∆−∆− =−=−∆= f j f j e f jkS e f jkS f j H or kst h π π π π π τ 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 thespectrum of the filter to which it is matched. The use of a matched filtergives the maximum Signal to Noise Ratio (SNR). However, in mostcases, 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 bankarchitecture can be drawn as shown in Figure 7.



12

Figure 7: General Filter Bank Architecture.

In general, the decimation and interpolation ratios m R can be

different resulting in the aliased channel outputs as [9]

)()(1

)()

2(1

0

)2

(mm

m

mm R

k

R j

m

R

k

R

k

R j

m

j

m e H e X R

eY

π ω π ω

ω −−

=

−

∑= . 1.10

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

representation of the original signal. To construct the input signal and

assuming there is no processing, the signals { })(n ym must be upsampledand filtered through the synthesis filters )( zF m . The reconstructed signal

can be written as

)()()(1

)()()(

1

0

1

0

)2

(2

(

1

0

^^

ω

π ω

π ω

ω ω ω

j

m

M

m

R

k

R

k j

m

R

k j

m

M

m

j

m

R j

m

j

eF e H e X R

eF eY e X

m

mm

m

∑ ∑

∑−

=

−

=

−−

−

=

=

=

. 1.11

The drawback of this system is that the information about thealiased signals in one channel is available in the other channel signals.However, it is possible to design exactly reconstructing analysis andsynthesis systems despite existence of aliasing in every individualchannel [9]. A special case can be derived letting 10, −≤≤= M i M Ri

and

is called a maximally decimated filter bank where the number of samplesin the set of { })(n ym

and )(n x is equal. This type of filter bank will be

discussed in the next section.

0Y

1Y

1− M Y

.

.

.

)( z X

)(1 z H M −

)(1 z H

)(0 z H 0 R

1 R

1− M R Processing

Processing

Processing...

1

^

− M Y

0

^

Y

1

^

Y

1− M R

0 R

1 R

)(1 zF M −

)(1 zF

)(0 zF

+

)(^

z X



13

1.4. Maximally Decimated Filter Banks

As stated before, a simplification by setting 10, −≤≤= M i M Ri in

the general filter bank system of Figure 7 leads to maximally decimatedcase. In this system, the number of samples for full band and subbandsignals is equal. To analyze this system, the input-output relationship can

be written as [10]

)()}()({1

)(})()({1

)(1

1

1

0

1

0

l M

l

M

k

k

l

k

M

k

k k zW X zF zW H M

z X zF z H M

zY ∑ ∑∑ −

=

−

=

−

=

+= . 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

k

k k zF z H . 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

k

k

l

k zF W z H 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 )( M z E and )( M z R represent the

polyphase components of the analysis and synthesis filters in the sensethat, the ith row of )( M z E and the ith column of )( M z R have the

polyphase components of the )(),( zF z H ii respectively. In the

mathematical form, this can be shown as [10]

)1( +− M n y )(n x ...

.

.

.1

)1( −− M z

)2( −− M z

1

1− z

)1( −− M z

M

M

M

)( M z E )( M z R

M

M

M

+



14

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−−−

−

−−−−−

−

−

)(...)(

::

)(...)(

.

1

:

)(

:

)(

:

1

.

)(...)(

::

)(...)(

)(

:

)(

1,10,1

1,00,0)1(

1

0

)1(1,10,1

1,00,0

1

0

N

N N

N

N

N

N

N T

N T

N

N N

N N

N

N

N

N

N

N

z R z R

z R z R z

zF

zF

z z E z E

z E z E

z H

z H

. 1.13

Using the noble identities, this system can further be simplified toease the extraction of perfect reconstruction conditions as shown inFigure 9.

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

In this system, the only part affecting the PR is the product)()( z R z E 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 circulantmatrix. A pseudo circulant matrix is a circulant matrix i.e., a matrixwhose rows are cyclically shifted versions of a sequence, but theelements below the main diagonal are multiplied by 1− z . So, the matrix isof the form [10]

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−

−−

−−−

−

)(...)()(

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

)(...)()(

021

11

011

21

2011

110

z p z p z z p z

z p z p z z p z

z p z p z p z

z p z p z p

M M

M M

M

. 1.14

In this case, the first row is comprised of the polyphase componentsof distortion function )(...)()( 1

)1(1

10

M

M

M M M z p z z p z z p −−−− +++ which must be a

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

)(n x M

M

M M

M

M

+)( z E )( z R

1 1− z

)1( −− M z 1

)1( −− M z )2( −− M z )1( +− M n y.

.

.

.

.

.



15

10 ,0

0)()(

1 −≤≤⎥

⎦

⎤⎢⎣

⎡=

−−

−−

N r I z

I z z E z R

r

r N

δ

δ

1.15

where N I is the identity matrix with r being a constant. The PR

condition can be stated in another way. If we have the set of powercomplementary analysis filters, by a proper choice of the synthesis filters

[3], the subband signals can be combined in a way to produce the originalinput signal at the output. In general, a set of filters )( z H k is said to be

complementary of order p if we have [11]

.1)(1

0

=∑−

=

p M

k

j

k e H ω 1.16

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

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

.1)(,1)(21

0

1

0

== ∑∑ −

=

−

=

M

k

j

k

M

k

j

k e H e H ω ω 1.17

It can be shown [11] that the higher order complementary filterscan generate ordinary magnitude and power complementary filters whilemaintaining superior cut-off characteristics.The procedure to design amaximally decimated filter bank has the following steps [10]:

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

• Having designed the analysis filters, polyphase matrix )( z E can

be determined.• The polyphase matrix of the synthesis filters )( z R can be

determined by inverting )( z E .

In general, we prefer the Finite Impulse Response (FIR) solutionswhich are guaranteed to be stable despite having larger delays comparedto their Infinite Impulse Response (IIR) counterparts. However, theinverse of a FIR matrix generally leads to IIR solutions which necessitatestability checks. For a special case of FIR matrices, called unimodularmatrices, FIR inverse solutions exist. A polynomial matrix is calledunimodular, 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 bankswhich will be discussed in the next section.

1.5. Paraunitary Filter Banks

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

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

defined as )()( 1~

−∗= z H z H . Thus, to obtain the paracojugate, one

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

*1*

~

})({)()( ω ω ω

j j j

e ze ze z

z H z H z H ==

−

=

== . 1.18

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

defined as )()( 1*

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

transpose the matrix, replace z by 1− z , and replace eachcoefficient by its complex conjugate. On the unit circle, paraconjugation is equivalent to transpose conjugation since wehave

T

e ze z

T

e z

j j j

z H z H z H })({)()( 1*

~

ω ω ω

==

−

=

== . 1.19

Having these definitions, a matrix transfer function )( z H , is

defined to be paraunitary if I z H z H =)()(~

. In the case of a square matrix

function, and using the concept of the inverse matrix, we have

{ } 1~

)()( −= z H z H . So the paraconjugate can be derived from the inverse

matrix.



17

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

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

N I z E z R =)()( , and assuming )( z E to be paraunitary i.e. I z E z E =)()(~

, the

PR system can be obtained choosing )()(

~

z E z R = . In this case, If )( z E isFIR, then )( z R 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 propertiesas follows:

1. If the polyphase matrix )( z E is paraunitary, then )( N z E is

paraunitary also. Hence, assuming

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−

−

−)1(

1,10,1

1,00,0

1

0

:

1

)(...)(

::

)(...)(

)(

:

)(

N N

N N

N

N

N

N

N

N z z E z E

z E z E

z H

z H , 1.20

it can be shown that the vector transfer function )( z H composed of all

the analysis filters is paraunitary.2. If the vector transfer function )( z H is paraunitary and assuming

that its components are power complementary as

.)(1

0

2const e H

N

k

j

k =∑−

=

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 )( z H k

of order L , the

synthesis filter coefficients can be computed by conjugating theanalysis filter coefficients and reversing their order as

10],[][ * −≤≤−= N k n Lhn f k k . Using the Fourier series properties, it

can be shown that the magnitude response of the synthesis filter

)( zF k is the same as the magnitude response of the analysis filter



18

)( z H k 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 powercomplementary 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 inthe next section.

1.6. DFT Modulated Filter Banks

Before moving to the discussion of DFT modulated filter banks, wewill 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 toshift the analysis and synthesis filters uniformly or non-uniformly alongthe frequency axis. This leads to new classes of filter banks whosesample 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(

−=+=−

−+−

−+−

M iengaenganh L

ni M

j

i

Lni

M j

ii

π π

1.22

Uniform

Non-uniform

T ω

T ω



19

where subscript * denotes the complex conjugation. In this system, sincethe LP prototype has real coefficients, the channel filters are obtained bycosine modulation and will be discussed in Section 1.7. The complexmultiplying factors ia define the modulation phase. The synthesis filters

are similar to the analysis filters but with a different modulation phaseusually chosen so that the resulting filters are the time-reverse of the

analysis filters. In this case, the overall filter bank response will havelinear phase. By appropriate design of the prototype filter, the overallfrequency response can be made flat also.

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

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

2

)1()(5.0(

**)

2

)1()(5.0(

−=+=

−−+

−−+−

M iengaenganh

ii

i

ii

i

Lnk

M j

ii

Lnk

M j

iii

π π

1.23

Hence, the synthesis filters are given as

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

2

)1()(5.0(

**)

2

)1()(5.0(

−=+=

−−+

−−+−

M iengbengbn f

ii

i

ii

i

Lnk

M j

ii

Lnk

M j

iii

π π

1.24

Here, the term )5.0( +±i

i

k M

π defines the ith channel center frequency, ik is

an integer, andi M is the decimation factor. The coefficients ii ba , are

complex and define the modulation phase. The choice of differentdecimation factors

i M 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 synthesisfilters 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 z H z H z H N − in the time and z domain as [16]



20

)()(,][][22

N

k j

k N

nk j

k zeP z H en pnhπ π

−

== . 1.25

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

21− N

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

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

1[][

)2

1(

2

−=−=−

−=−

−

M k N ne N

n pnh N

n N

k j

k

π

. 1.26

The synthesis filters are identical to the analysis filters. It can beshown [16] that if the prototype filter has the zero phase property, thenall the analysis and synthesis filters will be linear-phase. In theimplementation phase, the polyphase decomposition can be used toreduce the implementation complexity. The polyphase components of the

prototype filter can be written as

∑−

=

−=1

0

)()( N

l

N

l

l z E z zP . 1.27

So, the analysis filters can be written as

)()()()(1

0

1

0

/2/2/2 ∑∑ −

=

−−−

=

−−− === N

l

N

l

kll N

l

N kN j N

l

N kl jl N k j

k z E W ze z E e z zeP z H π π π , 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

z X

z E z

z E z

z E z

z E

IDFT

W W W W

W W W W

W W W W

W W W W

z X

z H

z H

z H

z H

N

N

N

N

N

N

N N N

N

N

N ⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−+−

−

−

−−−−−−

−−−−

−−−−

−

4 4 4 4 4 4 4 84 4 4 4 4 4 4 76

1.29

where N jeW /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 filtersare 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 ofa transfer function. The relationship between signals )(n xi

and )(n yk can

be written as [3]

∑−

=

−=1

0

)(1

)( N

i

ik

ik W n x M

n y . 1.30

By defining the transfer function)(

)()( z X

zY z H k

k = , it can be verified that

)( z H k is a shifted version of a prototype response )( zP through Equation

1.26 namely )()( k

k zW P z H = [3].

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

)()( /2/2 N k j N k j

k zePe zF π π −= . 1.31

Again, using the polyphase representation, we have

)()(1

0∑

−

=

−= N

l

N

l

l z R z zP . 1.32

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

)()(1

0

)1(∑−

=

−−−= N

l

N

l

l N k l

k z RW z zF . 1.33

Thus, the reconstructed output signal )(n y in the matrix domain can be

written as

)(n x 1− z

1−

z

N

N

N

.

.

.

.

.

.

IDFT

)(0 n y

)(1 n y

)(1 n y N −

)(0 n x

)(1 n x N −

)(1 n x)(1 z E

)(0 z E

)(1 z E N −



22

[ ]

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−

−

)(

:

)(

)(

)(

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

1

2

1

0

1210

zY

zY

zY

zY

zF zF zF zF zY

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

zY

zY

zY

DFT

W W W W

W W W W

W W W W

W W W W

z R z R z z R z z R z

N N N N

N

N

N N N N

N

N

4 4 4 4 4 4 4 84 4 4 4 4 4 4 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 )()( 11 z E z R l N l

−−−= and is the second type of the polyphase

representation [3]. In this scheme, the first filter should be centered at0= but using a Generalized Discrete Fourier Transform (GDFT) [7],

this constraint can be removed. It must be mentioned that by appropriatechoices of GDFT matrix, one can obtain a filter bank with

2 M 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 )(n p that has even length L as [17]

N nk en pnhnnk k

M j

k ∈= ++

,,)()())((

200

π

. 1.36

DFT

)(0 n y

)(1 n y N −

)(1 n y

.

.

.

)(1 z R N −

)(2 z R N −

)(0 z R )(n y

1−

1− z N

N

N



23

Here, the offsets0k and

0n are introduced leading to the name GDFT.

Choosing a linear-phase prototype filter and setting0n 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 2 M 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 2 M

filters. The synthesis filters can be obtained by time reversion of theanalysis filter as )1()( * +−= n Lhn f k k

. Thus, all filters can be derived from

one single prototype. The procedure to design these filter banks can besummarized in the following steps [10]:

• The prototype filter must be designed according to the systemrequirements.

• Having the prototype filter, the polyphase components )( z E k can

be achieved.• Assuming that )( z E

k

can be inverted, the synthesis filters can be

chosen as )()( 11 z E z R k N k

−−−= .

It can be shown that the maximally decimated DFT filter banks at thesame time satisfy perfect reconstruction, have FIR analysis and synthesisfilters, and are paraunitary.

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

121 ...1)( +−−− ++++= N z z z zP , 1.37

which leads in polyphase components as delays reducing the filter bankstructure 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 matricesis equal to a unity matrix i.e.

N I DFT IDFT =−1. , one can assume the overall

)(n x )1( +− M n y

M

M

M M

M

M

+ IDFT DFT )2( −− M z

1

)1( −− M z1

1− z

)1( −− M z

.

.

.

.

.

.



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 andsynthesis filters is similar to the general case. It can be shown that theanalysis filters can be modeled as

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

−

−

−−−

−

−

−1

2

1

)1()1(210

)1(2420

1210

0000

1

2

1

0

:

1

...

::::

...

...

...

)(

:

)(

)(

)(

2 N

DFT

N N N

N

N

N z

z

z

W W W W

W W W W

W W W W

W W W W

z H

z H

z H

z H

4 4 4 4 4 4 84 4 4 4 4 4 76

. 1.38

In other words, we can assume the analysis filters as

)()( )/2( N k j j

k ePe H π ω ω −= , 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 )()(2

N

k j

k zeP z H

π −

= . Cosine modulated filter banks

are defined by the use of Discrete Cosine Transform (DCT) which hasfour 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

:

N or iif

N or iif

c

N k ii N

k N

IV Type DCT

N k i N

k ic

N III Type DCT

N k i N

ik c

N II Type DCT

N k i N

kicc

N I Type DCT

i

k

i

k i

π

π

π

π

1.40

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

modulated filter banks and there is a relationship between the filter banksdefined using these two types of modulation. To be specific, suppose wehave a type IV cosine modulated filter bank with ),(),( z H zP k

and )( zF k

being its prototype, analysis, and synthesis filters, respectively as

}4

.)1()2

)(12(2

cos{)(.2)(

}4

.)1()2

)(12(2

cos{)(.2)(

π π

π π

k

k

k

k

Lnk

N n pn f

Lnk

N n pnh

−−−+=

−+−+=, 1.41

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

}4

.)1()2

1)(12(

2cos{)(2)(

}4

.)1()2

1)(12(

2cos{)(2)(

^^

^^

π π

π π

k k

k k

Lnk

N n pnh

Lnk

N n pnh

−−+

−+=

−++

−+=. 1.42

Hence, if the prototype real-coefficient lowpass filter is )( zP with a cutoff

at N 2/π ± , the analysis filters are

)()()()5.0(

*)5.0(

N k j

k N

k j

k k zeP zeP z H π π

α α ++−

+= . 1.43



26

In the time domain, we have

})1(4

)2

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

k

Lnk

N n pnh −+−+=

π π . 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 inFigure 14 [10].

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

Exploiting the advantages of the polyphase decomposition, andassuming that )( zP has N 2 -fold polyphase, we have

∑∑∑ ∞

−∞=

−−

=

−

=

− +===k

k

l

N

l

N

l

l L

k

k l Nk p z z E z E zk p z zP ]2[)(,)(][)(12

0

2

0

. 1.45

If )( zP has a length of N m..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 N N T 2× is [10]

)(k x )( 20

N z E −

)( 21

N z E −

)( 212

N

N z E −−

)(0 z H

)(1 z H

)(1 z H N −

N N T 2×

1− z

1− z

N 2π

0 H

π 2

P

π 2 N 2π

1 H

π 2



27

[ ]

.

...00

:::

0...0

0...0

00...1

:::

01...0

10...0

,

1...00

:::

0...10

0...01

)()(

})5.0(cos{

})5.01(cos{

})5.0(cos{

22

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=Λ

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

+−−Λ=

−

+

××

m N

m

m

J I

J I J I C N T N N N N

π

π

π

1.46

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

N C ijπ are the elements of an

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

If the prototype filter has 1+ L real coefficients where mN L 21 =+ for some integer m and it is linear-phase, then it can be shown that theFIR cosine modulated analysis bank is paraunitary if and only if the polyphase components )(,)( z E z E N k k + of the prototype filter are power

complementary. This means that we have a lossless one input-two outputsystem. 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]. Toillustrate this, assume )(),( z R z E be the polyphase component matrices of

the analysis and synthesis filter banks and the linear-phase prototype

filter has a length of 12 mmN L += with a transfer function

∑−

=

−=12

0

2 )()( N

l

N

l

l zG z z H , 1.47

with )( 2 N

l zG being the polyphase components. It can be shown [18] that

)( zGk satisfies



28

⎪⎪⎩

⎪⎪⎨

⎧

>

−≤=

−−+−

−−

.),(

1),()(

1121

11~

1

1

mk zG z

mk zG

z zG

k m N

k mm

k . 1.48

Thus, the polyphase matrices )(),( z R z E can be written as

T

C zg zg z z R

zg z

zgC z E

~2

1

~2

0

~1

21

1

20

^

)()()(

)(

)()(

⎟ ⎠

⎞⎜⎝

⎛ −−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−=

−

−, 1.49

where

.4

)1()2

(2

)12(cos2

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

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

,

^

1211

1100

⎟ ⎠

⎞⎜⎝

⎛ −+−+=⎥⎦

⎤⎢⎣

⎡=

=

−+

−

π π k

lk

M M M

M

Ll

N k C

zG zG zGdiag zg

zG zG zGdiag zg

1.50

To derive the PR conditions, we define )()()( z E z R zP∆

= . It can be shown

[18] that for PR, we must have

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ =

+−

−

0

0)(

2)1(

1

I z

I z zP

v

v

, 1.51

where v is a positive integer and the dimensions of unity matrices21, I I

add to N . Further simplification can be made leading to the necessaryand sufficient conditions for PR as [18]

10,2

1)()()()(

~~

−≤≤=+ ++ N k N

zG zG zG zG k N k N k k . 1.52

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



29

1. First, the lossless systems )(,)( z E z E N k k + 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 polyphasecomponents satisfying the given specifications.

As a comparison between cosine modulated and DFT modulatedfilter banks, it must be mentioned that, in a maximally decimated cosinemodulated filter bank, two polyphase components of the prototype filterreplace only one polyphase component in the DFT modulated case. Insuch a system, in order to satisfy paraunitariness, or equivalently havingFIR system to be PR, each such pair of polyphase filters should form a power complementary pair. In other words, they must be a losslesssystem. On the other hand, for a DFT modulated system to be paraunitary, each polyphase component must separately be losslesswhich means that each polyphase component should be an allpasstransfer function. There is a problem here since allpass functions are

usually IIR and concerns about stability arise. To further simplify the PRconditions, the oversampled filter banks are used which will be discussedin 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. Asolution to this is the choice of oversampled FBs that easily suppressaliasing and allow the combination of smaller subbands into wider

subbands without introducing large aliasing distortion. To analyze theoversampled FBs, the concept of frame expansion needs to be defined.Signal decomposition in )(2 Z l , is the expansion of the signal through a

sliding window using a selected set of elementary waveforms as [12]

∑ ∑−

=

∞

−∞=

=1

0,, )()(

K

i j

ji ji ncn x ϕ , 1.53

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



30

K N jN nn i ji ≤−= ),()(, ϕ ϕ . 1.54

Any signal in )(2 Z l can be represented using such an expansion if and

only if the family Φ

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∈−=−==Φ Z jK i jN nn i ji ji ,1,...,1,0),()(: ,, ϕ ϕ ϕ , 1.55

forms a frame in )(2 Z l . A family of vectors Φ given in Equation 1.56 is

said to be a frame if for any )(2 Z l x ∈ ,

∞<>≤≤ ∑ ∑−

=

∞

−∞=

B A x B x x AK

i j

ji ,0,,2

1

0

2

,

2ϕ . 1.56

The constants B A, are called the frame bounds and the frame is tight if B A = . If the family Φ is a frame, there is another frame as

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∈−=−==Ψ Z jK i jN nn i ji ji ,1,...,1,0),()(: ,, ψ ψ ψ . 1.57

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

∑ ∑−

=

∞

−∞=

=1

0

,, )(,)(K

i j

ji ji n xn x ϕ ψ . 1.58

In addition, ji x ,,ψ stands for the inner product. Furthermore, frames

ΨΦ, can be interchanged in a way that any signal in )(2 Z l can also be

written as

∑ ∑−

=

∞

−∞=

=1

0,, )(,)(

K

i j

ji ji n xn x ψ ϕ . 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 j

ji ji n x A

n x ϕ ϕ . 1.60

The starting point to relate this expansion to filter banks is the fact thatthe inner product of a signal x with the vectors of the family Φ can be

obtained as the outputs of an analysis bank )(),...,(),( 110 z H z H z H K − followed by downsampling ratio of K N ≤ . As discussed before, the analysis filtersare complex conjugates of the time-reversed versions of a prototypefilter. In this case, the prototype filter becomes the elementarywaveforms i . So, the analysis filters are constructed as

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

It must be added that the case K N = 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 zG zG zG K − whose impulse responses are derived as )()( nng ii ψ = .

Hence, the synthesis bank can be expressed as

∑ ∑−

=

∞

−∞=

−=1

0, )()()(

K

i j

jii k k n yn x ψ , 1.62

where )(n yi is the input of the ith channel of upsampler and synthesis

filter. As with almost all the filter banks, the polyphase idea can be usedto reduce the implementation complexity. The polyphase matrix of theanalysis 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

N

ij

j

i

N K K

N

N

N

j

N

ij

j

i

N K K

N

N

zG z zG

zG zG

zG zG

zG zG

zG

z H z z H

z H z H

z H z H

z H z H

z H

. 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

N M × matrix, its rank is the largest number of columns of A forming a linearly independent set. This set of columns is notunique, but the number of elements of this set is unique. A matrix

is Full Rank if ),min()( N M Arank = .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 )( z H , its dual frame consists of finite length

vectors if and only if )()(~

z H z H is unimodular.

A special case is oversampled DFT modulated filter banks whichare FIR, PR, DFT modulated, and paraunitary as we shall see. To analyzethis system, we assume Figure 9 with the property

N I z E z R =)()( . As a

special case, one can choose different dimensions for )(),( z E z R leading tothe system in Figure 16 [10].

Figure 16: Architecture of Oversampled DFT Modulated Filter Bank.

.

.

.

.

.

. )1( +− M n y

)(n x 1

1− z

)1( −− M z

M

M

M

M N z E ×)(

1

)1( −− M z

)2( −− M z

M

M

M

N M z R ×)(+



33

In this system, the condition N I z E z R =)()( still guarantees the system to

be PR. It can be shown that the PR condition becomes easier to achieve if N M > . As an example, if N M 2= , we have [10]

[ ] .)(

)()()()()(

2

121 ⎥

⎦

⎤⎢⎣

⎡==

z E

z E z R z R I z E z R N

1.64

It is important to note that this equation does not necessarily imply)()( 1

11 z E z R −= , so inverses may be avoided and we can expect FIR PR

filter banks. In general, an analysis filter bank with M channels and thedecimation ratio of N can be realized based on an M -point IDFTcascaded with an N M × polyphase matrix containing the N -fold polyphase components of the prototype filter )( zP . It can easily be

verified that if N M = , 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

k

l lk N h z z E ][)( '0

, 1.65

with),gcd(

' N M

MN N = and ),gcd( N M representing the Greatest Common

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

1−

z

1+− N z

1

N M N z B ×)( IDFT

)(0 z H

)(1 z H

)(1 z H M −

)(n x



34

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−−

+−

−

−

)(...0

.0

.

.

..)(0

0)(

)(.

.

.0.

.....)(0

0...0)(

)(

124

81

1

1

0

M

N

N

N

M

N

N

M

N

M

M

N

z E z

z E z

z E z

z E

z E

z E

z B

. 1.66

It is shown that if )()( z B IDFT z E ×= is paraunitary, then with the choice

of DFT z B z R ×= )()(~

, the PR property can be satisfied. This means that

the analysis bank will be paraunitary if )( z B is paraunitary. The

paraunitariness of )( z B can be proved if and only if1,...,1,0),(),( −=+ N k z E z E N k k 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 timevarying systems to improve the efficiency. In the next section, we willdiscuss the time varying filter banks and their properties.

1.9. Time Varying Filter Banks

In Time Varying Filter Banks (TVFB), the analysis/synthesisfilters, the number of bands, the decimation ratios, and the frequencycoverage of the bands change in time. This is in contrary to the FBsdiscussed in the previous sections, where the system structure does notchange 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 analysissection according to the input signal properties and hence improve thesystem performance. In this case, the PR property is the same as the

regular FBs i.e. )()(^

∆−= n xn x where ∆ is an integer. The design

problem is to choose the system parameters in a way that the PR propertyholds for all times. TVFBs can be analyzed using the time-domainformulation [20] where the time varying impulse response of the entirefilter bank is derived in terms of the analysis and synthesis filtercoefficients. To do this, the filter bank is divided into three stages namely

the analysis filters, the down/up samplers, and the synthesis filters asshown in Figure 19.

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

The analysis filters’ output is [ ]T

n M 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 [ ]T

n M nwnwnwnw )(),...,(),()( 1)(10 −= .

Assuming the length )(n N input signal at time n to be

[ ]T

N n N n xn xn xn xn x )1)((),...,2(),1(),()( +−−−= , we have

)(0 nw

)(1 nw

)(1)( nwn M −

)(n x )(n x∧

.

.

.

)(nQ

SynthesisFilters

)(1)( nv n M −

)(1 nv

)(0 nv

)(nP

AnalysisFilters

)(nΛ

Down/UpSamplers

+

)(n x +...

),(0 zn H

),(1)( zn H n M −

),(1 zn H

)(n x∧

Processing

Processing

Processing

)(0 n R

)(1 n R

)(1)( n R n M −

)(0 n R

)(1 n R

)(1)( n R n M −

),(0 znG

),(1)( znG n M −

),(1 znG

.

.

.



36

)()()( n xnPnv N = , 1.67

where )(nP is an )()( n N n M × 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 )()( n M n M × with mth diagonal

element at time n being one if the input and output of the mth down/upsampler are identical. The last stage is the contribution of the synthesisfilters, 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

n N ngngngng

n N ngngngng

n N ngngngng

n N ngngngng

nQ

n N

n M n M n M n M

−

−−−−

=

⎥⎥⎥

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

−

−

= , 1.69

where [ ]T

n M i 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

^

)()()(n N

i

T i inwnqn x 1.70

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

[ ]T T

n N

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



37

⎥⎥

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−+−+−Λ

−−−Λ

−−−Λ

Λ

=

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

.

.

.

)2()2()2(

)1()1()1(

)()()(

)()(^

n N n xn N nPn N n

n xnPn

n xnPn

n xnPn

nsn x

N

N

N

N

T . 1.71

Using the fact that the last 1)( −n N elements of vector )( in x N − are

identical to the first 1)( −n N elements of vector )1( −− in x N , this equation

can be decomposed as

[ ]

[ ]

[ ]

[ ]

,

)1)(2(

.

.

.

)2(

)1(

)(

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

...

...

..

0...0)2()2(00

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

0...0)()(

)()(^

⎥⎥⎥⎥

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎢⎢⎢⎢⎢

⎣

⎡

+−

−

−

⎥⎥⎥⎥

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎢⎢⎢⎢⎢

⎣

⎡

+−+−Λ

−−Λ

−−Λ

Λ

=

n N n x

n x

n x

n x

n N nPn N n

nPn

nPn

nPn

nsn x T

1.72

where 0 is the zero column vector of length )(n M . In this case, the

input/output relationship of the TVFB becomes [20]

[ ]1)(2

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

−=

+−−−==

n N I

I n xn xn xn xn xn xn zn x T

I I

T

. 1.73

The time varying impulse response vector of the FB is defined as)()()( nsn An z = . In this case, the [ ] [ ])()(1)(2 n M n N n N ×− matrix )(n A is

defined as



38

[ ][ ]

[ ]⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−Λ+−

−Λ−

Λ

=

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

0..

...

..0

.)1()1(0

00)()(

)(

n N nn N nP

nnP

nnP

n A

T T T

T

T

T T

T T T

. 1.74

In order for the system to act as a delay of integer ∆ , it is necessaryand sufficient [20] that all elements except the th)1( +∆ in )(n z be zero at

all times. Having a desired impulse response )(nb , PR property holds

[20] if and only if )()()( nbnsn A = for all n .

1.10. Differences between Time Varying and LTIFilter Banks

Since the nature of LTI systems is different from that of the timevarying 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 inthe LTV case.

2. For an LTI PU system, the analysis and synthesis banks arelossless whereas in the LTV case, the losslessness of analysis bank does not imply losslessness of the synthesis bank. A systemwith a transfer function )( z H is lossless if it preserves signal

energy for all inputs. Mathematically, if the system input andoutput are )(),( n yn x respectively, we have

22

)()( ∑∑ ∞

−∞=

∞

−∞=

=nn

n yn x . 1.75

1. Replacing the delay 1− z in the implementation of an LTI PUsystem with L z− for some integer L does not change the PU property. However, this is usually not true for a LTV losslesssystem.



39

1.11. Filter Bank Design Issues

Regardless of the different types of filter banks, the general systemof Figure 7 can be viewed as a hierarchical system [9] having specificrequirements and issues at each level. In the next subsections, we willdiscuss 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 problemsand tradeoffs. In the analysis and synthesis side, the stop band, pass band,and transition band characteristics are important. Furthermore, for thesynthesis part, we must consider the reconstruction issues as well. Thereare some common issues such as the implementation complexity andnumerical sensitivity also.

1.11.2. Filter Bank Issues

In this context, the quality of the frequency coverage of the analysisfilter bank must be considered. In order to save in the realizationcomplexity, it must be noted that one can implement the whole filter bank requiring less operations than the sum of the operations for theindividual filters and must be checked thoroughly. As the last issue, thecapability of the system to reconstruct the data in the presence ofdistortion, 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 toreconstruct the input at the output. In this case, the analysis/synthesislevel distortions can be modeled as LTI distortions in the form ofmagnitude and phase as well as distortions caused by aliasing. Here,there is a problem whose goal is to minimize these distortions. Thisminimization 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 inthe analysis/synthesis filters to reject out of band and out of time processing distortion.



41

Chapter Two: Overview of Transmultiplexersand Satellite Payload Systems



42



43

2. Introduction

Digital filter banks find applications in subband coders for speechsignals, frequency domain speech scramblers, image coding, andfrequency multiplexing/demultiplexing. In this chapter, we will describe

the mathematical theory of the transmultiplexers as duals of filter banksfollowed by issues such as channel equalization and interferencecancellation. As a special case of transmultiplexers, TDMA, CDMA, andFDMA systems will be studied. Next, we will discuss and comparedifferent payload architectures for satellite applications. The chapter endswith introduction to applications of filter banks in payload systems. Thistopic will be studied in detail in the third chapter.

2.1. Transmultiplexers

By definition, Transmultiplexer (TM) converts the time multiplexedcomponents of a signal into a frequency multiplexed version and back[21]. A TM can also be used for applications such as channelequalization, channel identification, etc. In [22], it was shown that a FBand a TM are duals and the transposition of the analysis/synthesis banksgives the dual TM. Using the duality, at the transmitter side, M differentsource signals are multiplexed into one transmit signal by upsamplersand synthesis filters. On the receiver side, the received signal isdecomposed into M source signals by analysis filters and downsamplers.As it can be predicted, non-ideal synthesis/analysis filters result incrosstalk 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 Transmultip lexers

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

)( zF k to produce the signals [21]



44

∑ −=i

k k k iPn f isn x )()()( . 2.1

The term pulse shaping comes from the fact that the filters take eachsample of )(nsk and put a pulse )(n f k around it [21].

Figure 20: General Architecture of a Transmultiplexer.

Here we have M users transmitting through a channel described bya 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 )( z H k

separate the signals and only a downsampling by P is needed to get theoriginal symbol streams. In this system, M signals are multiplexed intoone channel which necessitates the constraint M P > giving the name ofredundant transmultiplexer [21] as opposed to minimal transmultiplexerswhere M P = . Ignoring the effects of the channel, the input-outputrelationship can be written as [23]

M j

i

M

l

k

M

k

N

k

N i eW N i zW H zW F zS

M

zS π 2

11

0

11

0

^

,10),()()(1

)( −

−−

=

−−

=

=−≤≤= ∑∑ . 2.2

The transfer function )()()( 11

0

1 −−

=

−∑= zW H zW F zT i

M

l

k

N

ki relates the

output signal )(^

N i 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, thereis always a difference between the transmitted and the received signals.In order to decrease the Bit Error Rate (BER) of the system, channelequalization is needed. For the case M P > , channel equalization can bedone by the use of transmit and receive filters. The idea is shown in

)(1 n x

)(1 n x M −

)(0 n x

P

P

P

)( zC

)(0 ns

)(1 ns

)(1 ns M −

P

P

P

)(0 zF

)(1 zF

)(1 zF M −

+

Noise

+)(1 z H

)(0 z H

)(1 z H M −

)(^

1 ns

)(1

^

ns M −

)(^

0 ns



45

Figure 21, where M different channels )( zC k 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 isequivalent to the system in Figure 20. Interestingly, in the case ofmultiuser communications over wireless channels, the new representation becomes useful. In the next section, we will discuss the perfectreconstruction 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 simplifythe analysis, we deploy the fact that, if an LTI filter )(ng is placed

between an upsampler and a downsampler of ratio P , the overall systemis equivalent to the decimated version of the filter impulse responsewhich becomes )(nPg [21]. In this case, designing the transmit/receive

filters in each branch, so that the decimated version of )()()( zF zC z H mmk

becomes a pure delay, the MUI can be cancelled and the system is a PRsystem. From the duality property mentioned in Section 2.1, andaccording to the PR condition discussed in Section 1.4, if )(),( zG z H PP

are

the polyphase matrices of the analysis and synthesis filters, then the FB isPR if and only if I z H zG PP

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

only if I zG z H PT

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

channels are the same, the PR conditions are identical. It must bementioned that, the PR properties are independent of filter lengths,causality of filters etc., and can be satisfied for both minimal andredundant transmultiplexers. However, for the minimal case, there maynot always exist FIR or stable IIR solutions. So, allowing some

redundancy will make the solutions feasible.

)(0 ns

)(1 ns

)(1 ns M −Additive Noise

+

)(1 z H

)(0 z H

)(1 z H M −

P

P

P

)(^

1 ns

)(1

^

ns M −

)(^

0 ns

+

P

P

P

)(0 zF

)(1 zF

)(1 zF M −

)(0 zC

)(1 zC

)(1 zC M −

Channels

Transmitter Filters

Receiver Filters



46

2.1.3. Canceling InterBlock Interference inTransmultiplexers

Using the polyphase realization of the transmit/receive filters, wecan derive the channel between the mth transmitter and kth receiver inmatrix form. The new system diagram is shown in Figure 22.

Figure 22: Modeling the Channel to Cancel InterBlock Interference.

In this figure, )(),( ,, z E z R ik mi are the polyphase components of thetransmitter and receiver filters, respectively. It is shown in [21] that theInterBlock Interference defined as the interference between input vectors

occurring at different times at the input of ∑=

−= L

n

n

mm znc zC 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 have0)(...)( ,1, === −− z R z R mPm LP .

• Zero Jamming: In this scheme, a block of L samples at the beginning of each block of P successive received symbols are setto zero. Mathematically, we have 0)(...)( 1,0, === − z E z E Lk k .

2.1.4. Canceling Multi User Interference inTransmultiplexers

In reality, the signal )(nsk

∧

is affected by both )(nsk and k mnsm ≠),( ,

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 z R m

)(,1 z R mP−

)(,0 z R m

P

P

P

+

+1− z

1− z

1− z

)( zcm

1− z

1− z

1− z z

z

z

)(1, z E k

)(1, z E pk −

)(0, z E k

P

P

P

+

+

)(^

nsk

Channel



47

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

)(...0

...

...

...)(0

)(

....

....

....

0...)0()1(

0...0)0(

Lc

Lc

Lc

cc

c

A

m

m

m

mm

m

m

. 2.3

This matrix is a banded Toeplitz matrix. An N N × matrix1,...1,0,;, −== N jk t T jk N

is Toeplitz if jk jk t t −=,. In other words, the matrix

has constant values along negative slope diagonals. The matrix is bandedif there is a finite m for which mk t k >= ,0 . A banded Toeplitz matrix has

a finite number of diagonals with nonzero entries and zeros everywhereelse [25]. Hence, for any nonzero numberk ρ we have

)1(1)1(1 ...1)(...1 −−−−−−− = LP

k k k mm

P

k k C A ρ ρ ρ ρ ρ . 2.4

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

)1(,1

2,2

1,1,0

)1()1(221

...)(

)...1()(−−−

−−−−

−−−−−

++++=

++++= LP

m LPmmmm

PP

k k k k k

zr zr zr r zF

z z za z H ρ ρ ρ , 2.5

The transfer function from )(nsm to )(nsk

∧ becomes

)()()( k mk mk km F C a zT ρ ρ = which in the ideal case, must equal )( mk −δ . If

the multipliers k a are chosen to be)(

1

k k C ρ , the constraint to satisfy the

MUI cancellation becomes

1,0),()( −≤≤−= M mk mk F k m δ ρ . 2.6

Some conclusions can be made as follows:• The values

k ρ can be arbitrary numbers but they must be distinct.



48

• The receive filter )( zF m must at least have an order of M leading

to L M P +≥ .• 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 multipleaccess schemes such as TDMA, FDMA, and CDMA. To describe these

architectures, we will define the time-frequency interpretation and linkthe definition to the transmultiplexer theory. Doing so, we canunderstand the relationship between the transmultiplexer theory and theseschemes.

2.1.5. Time Frequency interpretation

The “uncertainty principle” states that no function can at the sametime 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

nn

n nhn E

ndwe H nh E nhnn E

2

0

_ 2

0

2

0

2

0

2 _ 2 )(

1,)(

2

1)(,)()(

1 π

π π

σ , 2.7

where _

, n E are the energy and time centers respectively.

• Frequency SpreadThe frequency domain spread of a function { })(0 nh is defined as

∫∫−−

=−=π

π

π

π π π

σ dwe H w E

wdwe H ww E

jw jw

w

2

0

_ 2

02

_ 2 )(

2

1,)()(

2

1, 2.8

with _

w being the frequency center. The general frequency-time tilde ofthe signal is shown in Figure 23. The shape and the location of the tildecan 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 FDMAschemes as results of expanding the time frequency interpretation todesign 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 N

iii cccc ,...,,, 321 chosen from aset of orthogonal codes called spreading sequences. A set of codes aresaid to be orthogonal, if for any jicc i

m

j

m ≠,, we have [13]

ji

N

m

j

m

i

mcc ,1

δ =∑=

. 2.9

These codes are simultaneously spread in time and frequency whichmeans they are both allpass like and spread in time domain [26]. Thesecodes 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

][k ui is replaced with the sequence ][],...,[],[],[ 321 k uck uck uck uc ii

N

iiiiii . This

is equal to time domain multiplication of the user data with the spreadingsequence. By setting the filter coefficients of a TM equal to orthogonaluser codes, a CDMA transmitter and receiver can be achieved. In thissense, the transmitter code multiplication may be viewed as filteringoperation, with FIR transmit filter as i

N

N iii c zc zc zC 12

11 ...)( +−− +++= .On

the receiver side, the receiver code multiplication and summation can

also be treated as a filtering operation, whose receive filter is

wσ 2

nσ 2

_

n

_

w

n

w



50

)...()1

( 11

221 i N i N i

N

N ic zc zc z

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 Transmult iplexers

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

ω ω jk j

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

1,...,1,0,0,1)( −=≤≤= M ieF j

k π ω ω [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 ++ k uk uk uk uk u N

which can be retrieved after passing the

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

..

.

.

.

.

[ ] [ ]1, 11 +k uk u

[ ] [ ]1, 22 +k uk u

[ ] [ ]1, +k uk u N N

Channel

N

N

N)( 12 − zC

+

N

N

N

)(1 zC

)(2 zC

)( zC N

[ ] [ ]k uk u 22 ,1+

1

1− z

1+− N

z

N z−

1+− N z

1−

z

.

.

.

.

.

.

[ ] [ ]1, 11 +k uk u

[ ] [ ]1, 22 +k uk u

[ ] [ ]1, +k uk u N N

Channel

N

N

N

+

N

N

N



51

2.1.8. FDMA System Based on Transmult iplexers

In FDMA systems, each user is assigned a portion of the availablechannel frequency. To do this, we can choose the synthesis and analysisfilters as frequency selective filters that add up to cover the full channelfrequency 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 amongterminals with a wide range of data rates, wideband payloads will play animportant role in the next generation communications satellites. Thefundamental topics in this area are channelization and satellite routing. Inorder to combat the effects of low radiated power and receiver sensitivityat the mobiles, there is a need to increase the size and complexity of the

satellite antennas. The satellites must also generate a large number ofspot beams in order to cover the full field-of-view from the satellite [27].It is necessary to have onboard switching or equivalently FDMmultiplexing and demultiplexing to direct the received carriers to thedesired spotbeams. In literature, four types of payload architectures have been proposed namely Bentpipe, Full Processing, Partial Processing, andHybrid payloads [28]. The following sections will study and comparethese architectures.

π 0

M

π

ω

)( ω j

i eF

……



52

2.2.1. Bentpipe Payload

This type of payload is the simplest architecture and converts theuplink carrier frequency to another carrier frequency for downlink signalwithout 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, atransponder-hopping technique [28] must be used. However, thedrawback of transponder hopping solution is that, the number oftransponders is the square of the number of beams, which makes itimpractical for a large number of beams. Another solution is the onboard beam switching technique based on a so-called Satellite-Switched TimeDivision Multiple Access (SS/TDMA) [28]. However, the bentpipesystem 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 thecapacity is always wasted, reducing the payload efficiency.

The main drawback of the bentpipe payload is the transfer of theuplink noise to downlink, since there is no error correction algorithm inthe 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

N

E

N

E

N

E . 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 doneat the packet level according to the destination information provided inthe transmitted user data packets. To transmit the routed data in thedownlink, there is a need for encoding and modulation, which can bedone 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 anintelligent routing at the packet level, which increases the efficiency. Therouting subsystem for a full processing payload can be implementedusing a combination of analog and digital components reducing theweight 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 onthe overall end-to-end BER performance. On the other hand, a full processing system is complicated and there is a trade-off betweenreduction of weight and power consumption, complexity, and ability ofusing digital components.

2.2.3. Partial Processing Payload

As discussed before, the main features of bentpipe payload aresimplicity and degradation of overall SNR. On the other hand, a full

processing payload is complicated but it has better performance from theSNR point of view. According to the applications, a compromise onmemory size, onboard decoder, speed, and power consumption can bemade using a partial processing system. A partial processing systemincludes demodulator and modulator, but not decoder and encoder.Therefore, channel corruption in uplink and downlink are decoupled butthere is no coding gain. Furthermore, a hard decision has to be applied inthe uplink signal demodulation, which will further destroy the softinformation of the received uplink signal since a detection error madeduring the processing of the uplink signal cannot be corrected. From theend-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 addan uncoded header that contains the routing information. However, tomake the system less vulnerable to the uplink noise, we need a longheader to compensate for the lack of coding gain. Another approach can be to include a simpler full processing subsystem to process headers thusallowing 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. Thissystem has some of the disadvantages and advantages of the individualarchitectures as discussed before. The next sections will discuss filter

bank solutions for frequency multiplexing/demultiplexing.

2.3. Frequency Multiplexing/Demultip lexingUsing Filter Banks

Filter banks provide solutions to frequency multiplexing anddemultiplexing problems in the satellite communications. The mainsolutions reported in the literature are as [29]:

• Channel-individual digital filtering with single- or multi-stepdecimation: In this algorithm, center frequency and bandwidth of

each channel is independent of adjacent channels, which bringsthe highest flexibility. The drawback is that, this approach has themaximum computational load.

• Tree-structured filter bank: This architecture has cascadeddirectional filter cells, where the complex FDM input is firstdownsampled by two followed by a separation into two complexsubsignals with half bandwidth. Therefore, each directional filtercell is a four channel oversampled complex modulated uniformfilter bank. However, only two channels are used for subsequent processing. Before the whole filter bank, a Hilbert transformconverts 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 isused. This approach has the highest efficiency in arithmeticoperations and storage. The previous versions have no flexibilityto channel allocation and bandwidth, but it will be shown in thenext chapter that a new solution utilizing a channel switch can bring full flexibility to the system making it a suitable choice forfrequency multiplexing and demultiplexing.



55

Chapter Three: Proposed Bentpipe Systemand Simulation Results



56



57

3. Introduction

In order to provide a solution to the increasing demands onmultimedia services supporting high bit rates and mobility, the EuropeanSpace 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 bentpipearchitecture has simplicity and easy system evolution support.

• Meshed type of regenerative satellite network for professionalusers: This type of network will support different classes of professional users. A set of earth stations will support differentclasses 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 terrestrialnetworks is the capability to interconnect several Internet ServiceProvider (ISP) access points. This calls for a regenerative onboard processor to flexibly direct spot beams creating an add-on toterrestrial networks.

In all, the aim is to have a globally interconnected digital society, withmultimedia applications, information on demand, and low cost deliveryof advanced data services which is the user’s expectation and theoperator’s promise [31]. In these systems, satellites communicate withusers through multiple spot beams, which necessitate efficient use of thelimited available frequency spectrum. This calls for satellite onboard

signal processing to support frequency band reusage among the beamsand bring flexibility in bandwidth and transmission power allocated toeach user. To support services at different data rates and bandwidths, adynamic frequency reusage system is required. Consequently, there is aneed 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 implementationsimplicity.

In this chapter, a new class of FFBR networks based on variableoversampled 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 ofview. The proposed system can be deployed in any communicationsenvironment that requires transparent (bentpipe) reallocation ofinformation. 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 theMIMO case where we will discuss different scenarios of the MIMO case.We will illustrate the simulation results of the system to evaluate thearchitecture 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 (atmost Q ) of these granularity bands. Consequently, the input signal

contains a variable number of user subbands q where Qq ≤≤1 [32]. Inthe 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 betweendifferent user subbands, and ensure the realizability. In brief, the SISOFFBR network has three major tasks:

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

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

Q

πα 2

QQ

πα π 22+

QQ

πα π π

222 +−

Q

πα 2Q Granularity Bands

Granularity Band Guard Band



59

• Combine the frequency-shifted user subbands into the outputsignal: 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 bechosen. In the next section, we will discuss a class of filter banks used forthe proposed FFBR system.

3.2. Class of Online Variable OversampledComplex Modulated Filter Banks

This section introduces the proposed class of variable oversampledcomplex modulated FBs used in the proposed FFBR network. We willstart with constraints of the system followed by the structure of the proposed filter bank. Finally, we will discuss the implementation issuesof 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 theinput/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.

)(n x ...

)(0 z H

)(1 z H

)(1 z H N −

M

M

M

.

.

.

M

M

M

)(0 zG

)(1 zG

)(1 zG N −

+)(n y

)(0 n y

)(1 n y

)(1 n yq−

Fixed Analysis FB Adjustable Synthesis FB

Channel Combiner



60

In this system, the analysis filter bank splits the input signal intosubbands. Furthermore, the combination of downsamplers, upsamplers,and synthesis filter bank with adjustable synthesis filters generates therequired frequency shifts and recombination of FB subbands into the q

shifted user subbands 1,...,1,0),( −= qin yi. To satisfy the system

requirements, specific constraints on M N , must be posed, which will be

discussed in the next section.

3.2.2. Constraints on Sampling Rate Converters andNumber of Channels

As discussed in Section 3.2.1, the choices of M and N play animportant role for the system to satisfy its properties. For instance, if

Q N M == , the system becomes a maximally decimated FB and hence the

variable subband widths and zero aliasing cannot be achievedsimultaneously. As discussed in Section 1.8, letting a slight oversampling by choosing N M < , makes the PR conditions milder. To generate allinteger frequency shifts of the granularity frequency, decimation andinterpolation by M can be used. Thus,

.int,1, B B BQ M ≥= . 3.1

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

.int,, A B A B

AM AQ N >== . 3.2

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

.int,11, K AK K A B −≤≤−= , 3.3



61

where

KQ N M −= . 3.4

In addition, K is the smallest integer allowed without introducingaliasing. In practice, it is possible to make aliasing components arbitrarily

small by the stopband attenuation of the analysis filters [33]. Precisely, ifthe filter bandwidth and transition band are N

π 2 and ∆2 respectively, the

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

)(π +∆

∆≥

N

N

Q

N K . 3.5

The next sections will study the building blocks of the system includinganalysis filters, synthesis filters, channel switch, and the channelcombiner.

3.2.3. Analysis Filters

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

∑=

−= D

n

n zn p zP0

)()( , 3.6

where D is the filter order with a constraint that it must be linear-phaseand symmetric so that, )()( n D pn p −= . Hence, its frequency response

can be written in the form of real zero-phase frequency response as [15]

)()( 2 T PeeP R

T jDT j ω

ω ω

−

= . 3.7

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

1,...,1,0),()( −== + N k zW P z H k

N k k

α β , 3.8



62

where

2)(2

, Dk

N k N

j

N W eW α π

β +−

== . 3.9

In addition, α is a real valued constant to shift the filters to the desired

center frequencies and the constants k β compensate the phase rotationscaused by replacing the D th-order linear-phase filter )( z H with

)( α +k

N zW H . 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 areinterested 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 ofthe synthesis filters as

M

NDm

N kr r kr ckr k

r

kr W Ask c z H zG 2,),()( =+== µ µ , 3.10

with

⎪⎪

⎩

⎪⎪

⎨

⎧

<+

≥

=0,

0,

r r

r r

r

s Bs M

s Bs

m, 3.11

satisfies these constraints. Here, r s is the number of granularity band

shifts for each signal. In general, the synthesis filters are obtained byreplacing the D th-order linear-phase filters )( z H k

with )( r m

M k zW H . The

constants kr compensate the resulting phase rotations. For simplicity,

the constants can be made 1=kr µ if [33]



63

.int2

= M

Dmr . 3.12

As a result of this discussion, we are able to transform the adjustablesynthesis filter bank to the combination of a set of fixed filters, anadjustable switch, and a series of multipliers. In the next section, the

adjustable switch operation that shifts the signals will be discussed.

3.2.5. Appl ication of Switch in the FFBR Network

To decrease the complexity of the system and considering the factthat variable filter banks are expensive to implement, with an appropriatechoice of filters and parameters in the FFBR network [33], it is possibleto implement the same function using variable channel switch and fixedFBs 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 theinputs of the synthesis bank. In this way, the complexity can be reducedsince fixed filters are less complex to implement. Furthermore, the fixedanalysis/synthesis FB, can be implemented using only one filter and anIDFT/DFT block [3]. In the next section, the efficient implementation ofthe system making use of the polyphase decomposition will be discussed.

)(0 n y

)(1 n y

)(1 n yq−

+ )(n y

Channel Switch

M

M

M

Fixed Analysis FB

)(0 z H

)(1 z H

)(1 z H N −

)(n x

M

M

M

Fixed Synthesis FB

)(0 zG

)(1 zG

)(1 zG N −

Channel Combiner

kr µ

.

.

.



64

3.2.6. Efficient Implementation

The polyphase decomposition reduces the implementationcomplexity of filters and filter banks. The starting point in the analysis isthe construction of the polyphase components of the prototype filter as

∑−

=

−=1

0

)()( N

i

N i

i zP z zP , 3.13

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

)( z H k can be written as [32]

[ ] i

N i

ki

N

N

i

N

N

N

ii

i

k k W W W zP z z H α α α α β −−−

=

− == ∑ ,)()(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 -pointDFT as shown in Figure 30.

Figure 30: Polyphase Implementation of the FFBR Network.

Here k

N k k W β γ = compensate for the phase rotations and B

A L = is chosen

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

kr µ

)(0 N

N

LW zP α

)(1 N

N

LW zP α

)(1 N

N

L

N W zP α

−

Channel Switch

IDFT

0α

1α

1− N α

0 β

1 β

1− N β

0γ

1γ

1− N γ

1− N α

2− N α

0α

)(1 N

N

L

N W zP α

−

)(2 N

N

L

N W zP α

−

)(0 N

N

LW zP α

DFT

)(n y

+

+

M

M

M

1− z

1− z

.

.

....

)(n x

1− 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/outputssince the proposed system will be used in the satellite payloads for thenext generation communications networks. To deal with theserequirements, 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 systemwith equal number of inputs and outputs, the system in Figure 31 can beused [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 canredirect the outputs from one input beam to another output beam. If theSISO FFBR network is designed to satisfy the required BER, the overall performance for each output subband in the MIMO network will be thesame as in the SISO network. In general, the satellite payload may havedifferent number of inputs and outputs which is the topic of the nextsection.

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 systemshown in Figure 32 can be used [32].

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

In this system, RS K = and the output beam’s bandwidth is Rtimes narrower than that of the input beam [32]. However, this case can be generalized to allow outputs with different data rates requiringdifferent downsampling factors at the outputs. In the implementation,different instances of polyphase synthesis FBs must be used with some ofthe DFT inputs set to zero. This keeps the required signal branches onlyand the task of the channel combiners is to add the necessary SFBoutputs to form the desired signals. The channel switch can direct thesignals to baseband giving more flexibility in the FFBR network. Thenext section will illustrate the simulation results on system functionalityand performance.

3.4. Simulation Results

To test the system functionality and quality, some issues wereconsidered 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 to1K

Out 11 +K to2K

Out 1+r K to K

MIMO FFBR

Channel Combiners



67

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

3.4.1. System Parameters Select ion

As discussed in Section 3.2.2, and in order to control the aliasing bythe stopband attenuation of the filters, the system parameters must satisfyEquations 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 ) 44 Transition Band Width ( ∆ )

Qπ 125.0

5 Number of Subbands ( q ) 36 Prototype Filter Order ( D ) 134

7 Phase Rotation Factor ( L ) 28 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 M 2 will make

kr µ equal to unity

sincer m will in any case be an integer.

3.4.2. Transmitter/Receiver Filter Design

The purpose of the transmitter/receiver pair is to evaluate thequality of the FFBR system for different types of input data i.e. M-QAMor Gaussian signals. Transmit and receive filters design is a traditionalcommunication problem. In brief, the transmit filter constrains thetransmitted signal’s spectrum to a limited bandwidth while the receivefilter rejects out of band noise thus maximizing SNR. Furthermore, thecascade of the transmit and receive filter must minimize ISI, whichmeans that the convolution of the transmit and receive filters’ impulseresponses must satisfy [35]



68

⎪⎪

⎩

⎪⎪

⎨

⎧

−+

=±−=

−=

=

12

1,...,2,1,2

1,0

21,

1

)(

M

Lk kM Ln

Ln M

nh , 3.15

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

RF

M s= . 3.16

Furthermore, M

L2

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

Nyquist filter. Having designed the equiripple linear-phase Nyquist filterwith a nonnegative frequency response, one can use the standard spectralfactorization 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 thetransition band of the M-band transmit filter with a sharp transition bandand large attenuation in the stopband as shown in Figure 33. This leads tohigh order filters resulting in expensive receivers and a future researchtopic 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 filterswhose characteristics satisfied the constraints of the system weredesigned. The MATLAB program to design these filters can be found inAppendix 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 arethen upsampled and filtered with the third or sixth band filters,respectively. Finally, different users are modulated to appropriate

M

π T ω

)( ω je H

Dashed Line: Receive FilterSolid Line: Transmit Filter



69

positions in the frequency spectrum and summed to form a beam ofsignals. The MATLAB program constructing two test beams for thegeneral MIMO case can be found in Appendix B.Having constructed input beams, the transmit and receive filters withcharacteristics 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 asshown in Figure 29 and Figure 30. One aim in this thesis was to verifythe equivalence of these systems in the presence of the channel switch.To implement the system in Figure 29, the MATLAB program inappendix C is used. It must be mentioned that this program implements ageneral MIMO case. However, setting the number of inputs to one willresult in a SISO case. As mentioned before, the polyphase decompositioncan reduce the implementation complexity. To verify the system in

Figure 30, the MATLAB program in Appendix D is used. This programimplements the general MIMO case shown in Figure 31 and can easily beconverted to a SISO system by setting the number of inputs to one.Regarding the polyphase implementation of the FFBR network, someissues must be mentioned as follows:

• Prototype Filter Polyphase DecompositionSince the designed prototype filter in this thesis was linear-phase and ithad symmetry, the polyphase decomposition resulted in polyphasecomponents with different lengths. Otherwise, the prototype filter wouldlack 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

InputData

TransmitFilter

ReceiveFilterFFBR 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, ina sample-based implementation, zero padding can be avoided thusallowing vectors of different lengths.

• DFT/IDFT ImplementationThe DFT and IDFT operations can be modelled as the multiplication of a

column vector whose elements are the values of each branch in Figure 30at 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 valuesof branches in Figure 30 at time n .

An important issue in the design of filter banks is the procedure todesign the prototype filter. Generally, two techniques namely minimaxand 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 usersubbands was generated at the input of the FFBR Network. A channel

switch directs outputs of the analysis filters to different synthesis filtersaccording to the reallocation scheme. For the SISO case, redirectionoccurs between branches of one filter bank. An example switch schemeis shown in Figure 35.

Figure 35: Example Channel Switch for SISO Case.

The input, output, and the analysis/synthesis filters of the FFBRnetwork with this switch is shown in Figure 36.

Analysis BankOutputs

Synthesis BankInputs



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 differentlocations in the spectrum.

3.4.4. Implementation of the MIMO System

Extension of the SISO case to MIMO is done by increasing theinstances of the filter bank and adding a channel switch capable of

directing signals from one filter bank to another. An example of theswitch structure for the case with two inputs and two outputs is shown inFigure 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

1 X

1 X

2 X

2 X

3 X

3 X



72

In this scheme, two input beams each containing three differentusers are multiplexed into two output beams where four users share one beam and the remaining two users are present in the other beam. Theinputs and outputs of the FFBR network in Figure 31 with the switch inFigure 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, >= R RS K outputs. To havemore flexibility, it is desired to direct all the signals to baseband. Thisneeds modifications in the channel switch as well as setting some of theDFT inputs of the SFBs to zero. The latter removes the unnecessary branches.

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

1 X 2 X 3 X

4 X 5 X 6 X

2 X 4 X

1 X 3 X 5 X 6 X



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 examplechannel switch to shift the user signal 2 X 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 Figure41.

From AFB 2 From AFB 1

To SFB 2 To SFB 1

1 X

1 X

2 X

2 X



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 multiplexingflexibility of the network.

3.5. Finite Word Length Effects on the FFBRNetwork

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 thedesigner decide on the system resolution. To evaluate the FFBR network performance, the quantization effects were introduced in the filtercoefficients, filter outputs, and DFT/IDFT outputs as shown in Figure 42.

1 X

1 X

2 X

2 X



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 thefilter implementation structure. In the quantization scheme, we assumedifferent resolutions at different branches of the filter bank. This needs aninvestigation on the propagation of error in different branches of thesystem and is a future research topic. To illustrate the effects of finiteword length on the FFBR network performance, Figure 43 shows theconstellation for a 64-QAM data multiplexed according to the channelswitch in Figure 37 for three different filter coefficient lengths.

1− z

+

+

1− z

)(n y

)(n x

IDFT

0α

1α

1− N α

Q

Q

Q

Q

Q

Q

DFT

1− z

1−

z

M

M

M

)(0 N

N

LW zP α

)(1 N

N

LW zP α

)(1 N

N L

N W zP α −

.

.

.

)(1 N

N

L

N W zP α

−

)(2

N

N

L

N W zP α

−

)(0 N

N LW zP α

1− N α

2− N α

0α

Channel Switch

0 β

1 β

1− N β

0γ

1γ

1− N γ

kr µ

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 ofthe prototype filter suppresses aliasing and the designer can maketradeoffs according to the required BER. To compare the system MSEfor different stopband attenuations, we can utilize the fact that since thetransmit/receive pair has a MSE in the order of 1110− , it can detect largererrors. To illustrate the MSE trend for different stopband characteristics,the variance of the input/output difference for six different channels ofFigure 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 inaccordance with the fact that the attenuation of the stopband suppressesaliasing. It must be mentioned that to achieve lower BER, the prototypefilter must have larger attenuation and for higher attenuation, thedifference 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 variableoversampled complex modulated N -channel FBs, which has fixeddecimation and interpolation ratios M in the final implementation. Thenetwork 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 ofeasy suppression of aliasing allowing the combination of smaller

* 1 X

○ 2 X

+ 3 X

∆ 4 X □ 5 X

. 6 X



78

subbands into wider subbands without introducing large aliasingdistortion. 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 andsimplicity in terms of analysis, design, and implementation.

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

• Handle all possible frequency shifts and all possible user subbandwidths.

• 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 offlexibility.

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

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

Furthermore, both tree-structured FB and overlap/save DFT/IDFT basednetworks 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 willhelp 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 thisthesis, the purpose was to evaluate the FFBR network, thereforethe chosen transmit/receive filters had high orders making thereceiver expensive. Reduction of the orders using spectralfactorization methods is a future research topic.



79

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[13] Proakis, John G., Digital Communications, McGraw-Hill,1995.



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[23] Ramachandran R.P. and Kabal P., “Configuration and performance of modulated filter banks,” Proc. IEEE Int.Symp. Circuits Syst., pp. 1809-1812, 1-3 May 1990.

[24] R.A. Gopinath and C.S. Burrus, “A tutorial overview offilter banks, wavelets, and interrelations,” Proc. IEEE Int.Symp. Circuits Syst., pp. 104-107, May 1993.

[25] Robert M. Gray, Toeplitz and Circulant Matrices: AReview, Department of Electrical Engineering, StanfordUniversity, http://www-ee.stanford.edu/~gray/toeplitz.pdf .

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review,” IEEE Tran. SP, vol. 46, no. 4, pp. 979-995, Apr.1995.

[27] Andersen B.R., ”Digital filter bank designs for satellitetransponder payloads: implementation on VLSI circuits,”5th IEEE Int. Conf. Universal Personal Comm., vol. 2, pp.750-754, 29 Sept.-2 Oct. 1996.

[28] Tien Nguyen, Hant J., Taggart D., Chit-Sang Tsang,Johnson D.M., Jo-Chieh Chuang, ”Design concept andmethodology for the future advanced wideband satellitesystem,” Proc. IEEE Military Comm. Conf., MILCOM2002, vol. 1, pp. 189-194, 7-10 Oct. 2002.

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[36] Samueli H., “On the design of optimal equiripple FIRdigital filters for data transmission applications,” IEEE



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83

Appendixes



84



85

Appendix A: MATLAB Program to DesignThird 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));gridon find(h == 1/Mth_Band) find(h == 0)figure(3);plot(h);grid on



86



87

Appendix B: MATLAB Program to GenerateUser 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)) ;



88



89

Appendix C: MATLAB Program to Implementthe 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-1vtemp( (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-1vmat( (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-1ymat( (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-1FBR_Out( l+1 , : )=FBR_Out( l+1 , : )+ymat( (l*N)+k+1 , : );

end end for l=0:Num_Inputs-1



90

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



91

Appendix D: MATLAB Program to Implementthe 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_Bit

s,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-1Delayed_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


vtemp{(l*N)+k+1}=quant(conv(Decimated_Delayed_x((l*N)+k+1,:),P_Upsampled_Fli pped{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 = [] ;



92

for k=0:Num_Inputs*N-1if (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-1vtemp((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-1vtemp_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-1vtemp_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


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-1vtemp_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




93

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-1vtemp_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


vtmep_DFT_fil{(l*N)+k+1}=quant(conv(vtemp_DFT((l*N)+k+1,:),P_Upsampled_Flip ped{N-k}),2^(-FFT_Out_Bits((l*N)+k+1)+1));

end end new_vtemp=[];%*******************

for k=0:Num_Inputs*N-1if (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-1for 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-1y(l+1,:)=y(l+1,:)+Delayed_new_vtemp((l*N)+k+1,:);

end end




94

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



95

Appendix E: MATLAB Program to DesignPrototype 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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