Abstract

Performanceanalysisplaysa centralrole in thedesignof reliablecommunicationsys-

tems. It helpsto reducethe probability of error in the transmissionof digital dataor to

improve the quality of the recoveredsignalswhenanalogsignalsarecommunicated.In

this dissertation,we first addresstheproblemof performanceanalysisby presentingana-

lytical methodsto deriveupperandlowerboundsfor theerrorratesof varioussystems.We

thendesigna joint source-channelcodingsystemfor the transmissionof analogsources

overamultipleantennacommunicationlink.

We begin by consideringa single-antennasystemwith additive white Gaussiannoise

aswell asblock Rayleighfadingchannels.It is shown thatonecanreducethecomplexity

of someof the existing methodsand, at the sametime, obtain tighter results,which is

very favorableparticularlywhenthe complexity of the problem(i.e., the codebooksize)

is prohibitive. Closed-formformulasfor theblock pairwiseerrorprobabilitiesarederived

everywhere.

Wethenconsideramulti-antennacommunicationlink with space-timeorthogonalblock

codingandestablishtight upperandlowerboundson thesymbolandbit errorratesof the

system.Thesignalingschemeandthemappingbetweenthesignalsandbits canbearbi-

trary.

We next considerthe casewherethe input bit-streamhasa non-uniformdistribution,

which allows the useof maximuma posteriori (MAP) decoding. We derive the MAP

decodingrule, calculatea closed-formformula for the symbolpairwiseerrorprobability,

andestablishtight upperand lower boundsfor the symbolandbit error rates. Another

ii

contribution is to demonstratethat systemswith MAP decoding,which is a form of joint

source-channelcoding,canoutperformthosewith separatesourceandchannelcoding(i.e.,

tandemsystems)atall errorratesof interest.

Finally, we proposea joint source-channelcodingsystemfor the transmissionof ana-

log sourcesover multiple-antennalinks. The proposedsetupusessoft-decisiondecod-

ing channel-optimizedvectorquantization(COVQ) to achieve a largeportionof thesoft-

decodinggain with reducedcomplexity. Two COVQs with fixed encodersandadaptive

decodersarealsodesigned.Thesuperiorityof theproposedCOVQ over tandemsystems

is alsodemonstrated.

iii

Acknowledgments

I would like to expressmy sincerethanksto my supervisors,ProfessorFady Alajaji and

ProfessorTamasLinder, for their support,continuousencouragement,andthoughtfulsug-

gestionsandcommentsthroughoutthecompletionof this work. I amalsogratefulto my

committeemembersProfessorSteve Blostein,ProfessorLorneCampbell,ProfessorAmir

Khandani,andProfessorShahramYousefi,for spendingvaluabletimeto readmy thesis,at-

tendingmy defense,andproviding detailedandconstructivecomments.I amalsothankful

to ProfessorMartin Guayfor chairingmy defensesession.

I wouldalsoliketo gratefullyacknowledgethefinancialsupportreceivedfrom theNat-

uralSciencesandEngineeringResearchCouncil(NSERC),theOntarioPremier’sResearch

ExcellenceAward (PREA), the Departmentof ElectricalandComputerEngineering,the

Departmentof MathematicsandStatistics,andtheSchoolof GraduateStudies.

I amgreatlyindebtedto my bestfriend andloving wife, Arezou,whosepatience,un-

derstanding,andsacrificesenabledme to pursuemy studies.I cannotthankher enough.

I have beenfortunateto have wonderfulparents,brother, andsisters;I thankthemall for

their constantloveandstimulation.

iv

Contents

Abstract ii

Acknowledgments iv

List of Figures xi

1 Introduction 1

1.1 TheCommunicationSystemModel . . . . . . . . . . . . . . . . . . . . . 2

1.2 TheSource-ChannelSeparationTheorem . . . . . . . . . . . . . . . . . . 4

1.3 PerformanceEnhancement. . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Improvementvia JointSource-ChannelCoding . . . . . . . . . . . 6

1.3.2 Improvementvia Diversityin SpaceandTime . . . . . . . . . . . . 7

1.4 Summaryof theThesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.1 ProblemStatement. . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.2 ThesisOrganizationandContributions. . . . . . . . . . . . . . . . 14

v

1.5 LiteratureReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5.1 ErrorAnalysisof ChannelCodedSISOSystems . . . . . . . . . . 16

1.5.2 JointSource-ChannelCodingMethods . . . . . . . . . . . . . . . 17

1.5.3 Multi-antennaSystems. . . . . . . . . . . . . . . . . . . . . . . . 21

2 Bounds on the Probability of a Finite Union of Events 33

2.1 TheLowerandUpperBounds . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.2 TheHunterUpperBound. . . . . . . . . . . . . . . . . . . . . . . 34

2.1.3 TheKouniasLowerBound . . . . . . . . . . . . . . . . . . . . . . 35

2.1.4 ThedeCaenLowerbound . . . . . . . . . . . . . . . . . . . . . . 35

2.1.5 TheKAT LowerBound . . . . . . . . . . . . . . . . . . . . . . . 36

2.1.6 TheCohenandMerhav Lowerbound . . . . . . . . . . . . . . . . 36

2.2 Algorithmic Implementationof theHunterandKouniasBounds . . . . . . 37

2.2.1 TheHunterUpperBound. . . . . . . . . . . . . . . . . . . . . . . 37

2.2.2 TheKouniasLowerBound . . . . . . . . . . . . . . . . . . . . . . 38

2.3 Final Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 A Brief Review of Space-Time Orthogonal Block Codes 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Encodingin Alamouti’sSystem . . . . . . . . . . . . . . . . . . . 42

vi

3.1.2 Decodingin Alamouti’sSystem . . . . . . . . . . . . . . . . . . . 43

3.1.3 Extensionto ThreeandFour TransmitAntennas . . . . . . . . . . 45

3.2 GeneralTreatmentof Space-Time OrthogonalBlock Codes. . . . . . . . . 47

3.3 TheSymbolPEPfor GeneralSpace-Time OrthogonalBlock Codes . . . . 48

3.4 TheCodeword PEPof Arbitrary Space-Time Codes. . . . . . . . . . . . . 50

4 Bounds on the Block and Bit Error Rates of Coded AWGN and Block Rayleigh

Fading Channels 53

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 TheErrorRatesin Termsof Probabilitiesof aUnion . . . . . . . . . . . . 54

4.3 TheAWGN Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 TheBlock RayleighFadingChannel . . . . . . . . . . . . . . . . . . . . . 57

4.5 LinearBlock CodesandBPSKSignaling . . . . . . . . . . . . . . . . . . 60

4.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5.2 Evaluationof theAlgorithmic Boundsfor LargeBlocks . . . . . . 61

4.6 NumericalResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Bounds for the Symbol and Bit Error Rates of Space-Time Orthogonal Block

Codes 76

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2 Boundson the Codeword Pairwise Error Probability of Space-Time Or-

thogonalBlock Codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

vii

5.3 Boundson theErrorRatesof Space-Time OrthogonalBlock Codes. . . . . 79

5.3.1 TheSymbolErrorRate . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3.2 TheBit ErrorRate . . . . . . . . . . . . . . . . . . . . . . . . . . 81


6 Error Rates of STOB Coded Systems under MAP Decoding 94

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2 TheMAP DecodingRule . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.2.1 DecoupledSymbolDetection . . . . . . . . . . . . . . . . . . . . 96

6.2.2 TheMAP DecodingRule. . . . . . . . . . . . . . . . . . . . . . . 97

6.3 ExactSymbolPairwiseErrorProbabilitywith MAP Detection . . . . . . . 98

6.3.1 TheConditionalPEP . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3.2 Interpretationof thePEPasaLaplaceTransform . . . . . . . . . . 99

6.3.3 ThePEPin ClosedForm . . . . . . . . . . . . . . . . . . . . . . . 99

6.4 TheOptimalBinaryAntipodalSignaling. . . . . . . . . . . . . . . . . . . 102

6.5 Boundson theSERandBER of STOB CodedChannelsunderMAP De-

coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.5.1 TheSymbolErrorRate . . . . . . . . . . . . . . . . . . . . . . . . 104

6.5.2 TheBit ErrorRate . . . . . . . . . . . . . . . . . . . . . . . . . . 104


viii

6.6.1 BinaryAntipodalSignaling . . . . . . . . . . . . . . . . . . . . . 105

6.6.2 TandemversusJointSource-ChannelCoding . . . . . . . . . . . . 106

6.6.3 SERandBERBoundsfor � -arySignaling. . . . . . . . . . . . . 107

6.6.4 TheEffectof ConstellationMapping . . . . . . . . . . . . . . . . 108

7 Quantization with Soft-decision Decoding for MIMO Channels 125

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.2 SystemComponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.2.1 TheChannel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.2.2 Soft-DecisionDecodingandtheEquivalentDMC . . . . . . . . . . 130

7.2.3 TheStep-Sizeof theUniform Quantizerat theDecoder. . . . . . . 133

7.3 Quantizationwith Soft-DecisionDecoding. . . . . . . . . . . . . . . . . . 135

7.3.1 Soft-DecisionDecodingCOVQ . . . . . . . . . . . . . . . . . . . 135

7.3.2 Fixed-EncoderAdaptive-DecoderSoft-DecisionDecodingCOVQ . 136

7.3.3 On-lineFEAD Soft-DecisionDecodingCOVQ . . . . . . . . . . . 138

7.4 NumericalResultsandDiscussion . . . . . . . . . . . . . . . . . . . . . . 139

7.4.1 ImplementationIssues . . . . . . . . . . . . . . . . . . . . . . . . 139

7.4.2 COVQ for VariousMIMO Channels. . . . . . . . . . . . . . . . . 140

7.4.3 COVQ versusTandemCoding . . . . . . . . . . . . . . . . . . . . 141

7.4.4 COVQ, FEAD COVQ, andOn-lineFEAD COVQ . . . . . . . . . 142

ix

8 Summary 152

Bibliography 155

x

List of Figures

1.1 Block diagramof adigital communicationsystem. . . . . . . . . . . . . . 2

1.2 Theideaof thesource-channelseparationprinciplefor discretesources. . . 5

1.3 Improvementof channelcapacitywhenmultiple antennasareused. The

CSNRis 20dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Block diagramof acommunicationsystemwith MAP sequenceestimation. 11

1.5 Block diagramof a typical multi input-multi output system,where �� is encodedto �� ! , attenuatedby " , receivedas#

, anddecodedto $� whichcorrespondsto $�%� � $�� $�� . . . . . . . . . . 22

2.1 Formingtheoptimalspanningtreeto evaluateaBonferroni-typeupperbound. 38

3.1 Thetransmitterof Alamouti’sspace-timeorthogonalblock codingscheme. 42

4.1 Theideato form aspanningtreeandcomputeanupperboundfor theBlER.&('is theindex of codeword )+*�, for which -.*/,10324-.*50 & �76 ��8 . . . . . . . 66

4.2 Block andbit error ratecurvesfor BCH (15, 5) code,8-PSKmodulation,

andAWGN channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

xi

4.3 Block error rate curves for BCH (15, 7) code, BPSK modulation,and

AWGN channel.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4 Comparisonamongthreelowerboundson theBlER. Golay(23,12) code,

BPSKsignaling,andAWGN channel. . . . . . . . . . . . . . . . . . . . . 70

4.5 Block error rate boundsfor Golay (23, 12) code,BPSK signaling,and

AWGN channel.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.6 Block error rate boundsfor BCH (31, 16) code, BPSK signaling, and

AWGN channel.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.7 Block error rate boundsfor BCH (63, 24) code, BPSK signaling, and

AWGN channel.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.8 Block error rateboundsfor Hamming(7, 4) code,BPSK signaling,and

Rayleighfadingchannel. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.9 Blockerrorrateboundsfor BCH(15,5)code,BPSKsignaling,andRayleigh

fadingchannel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.1 Our upperboundandthelower boundson -:9;9 ��=< ��?>3@ 9 < *< * < ** >;> , Alam-

outi’scodewith Q-PSKmodulation,AB�DC , and EF�76 . . . . . . . . . . . 85

5.2 Resultsfor Q-PSKsignalingwith AB�DC and EF�76 (2 bits/s/Hz). . . . . . 86

5.3 Resultsfor 8-PSKsignalingwith AG�:C and EH�I6 (3 bits/s/Hz).. . . . . . 87

5.4 Resultsfor 16-QAM signalingwith AJ�:C and EH�:C (4 bits/s/Hz). . . . . 88

5.5 Resultsfor 32-PSKsignalingwith AJ�:C and EH�LK (5 bits/s/Hz). . . . . . 89

5.6 Resultsfor 64-QAM signalingwith AJ�:C and EH�:C (6 bits/s/Hz). . . . . 90

xii

5.7 Resultsfor 8-PSKsignalingwith AG�DM and EH�I6 C and K (1.5bits/s/Hz). 91

5.8 Thestar-QAM signalingschemewith quasi-Graymapping.N is anormal-

izing factorandequals6;OPC�Q 6SR and& � Q T 6 . . . . . . . . . . . . . . . . . 92

5.9 Resultsfor Star8-QAM signalingwith AB�4K and EF�U6 (1.5bits/s/Hz).. . 93

6.1 Resultsfor BPSKsignaling,AB�LC , and EF�U6 . . . . . . . . . . . . . . . . 110

6.2 Resultsfor theoptimumbinaryantipodalsignaling,AJ�:C , and EF�I6 . . . 111

6.3 ComparisonbetweenBPSKandoptimumsignalingschemes,AV�WC , andEF�I6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.4 ComparisonbetweentandemandMAP-decodedschemesfor BPSK sig-

naling, AB�:C , and EF�I6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.5 ComparisonbetweentandemandMAP-decodedschemesfor 16-QAM sig-

naling, AB�:C , and EF�I6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.6 SERcurvesfor 8-PointStarQAM modulationwith quasi-Graymapping,AB�:C , and EF�I6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.7 BER curvesfor 8-PointStarQAM modulationwith quasi-Graymapping,AB�:C , and EF�I6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.8 SERcurvesfor 16-QAM signaling,AB�LC , and EX�U6 . . . . . . . . . . . . 117

6.9 SERcurvesfor 32-PSKsignaling,AB�LC , and EF�U6 . . . . . . . . . . . . . 118

6.10 BER curvesfor 32-PSKsignaling,AB�LC , and EF�U6 . . . . . . . . . . . . . 119

6.11 SERcurvesfor 64-QAM signalingwith Graymapping,AJ�:C , and EH�I6 . 120

xiii

6.12 Thestar-QAM signalingschemewith M1 mapping. . . . . . . . . . . . . . 121

6.13 Comparisonbetweenthequasi-GrayandtheM1 mappingswith StarQAM

for a systemwith AB�:C , and EF�I6 . . . . . . . . . . . . . . . . . . . . . . 122

6.14 Comparisonbetweenthe M1 andGray mappingsfor 64-QAM signaling,AB�:C , and EF�I6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.15 ComparisonbetweentheBER performanceof theM1 andGraymappings

for 64-QAM signaling,AJ�DC , and EF�I6 . . . . . . . . . . . . . . . . . . . 124

7.1 Systemblock diagram,whereevery Y bits in a Z\[ -bit index8

is transmit-

tedvia a space-timecodeword �U�]�^� �� _ (with orthogonalcolumns��* & �`6 ��a ��b�Y ), receivedas# �c�ed �� df�_ , andspace-timesoft-

decodedas g# �� gd � �� gdfhi . For simplicity, wehaveassumedthat Yj�:Z\[ . 128

7.2 Linearcombinerwhich follows thespace-timesoftdecoderat thereceiver.

Left: problemstatement,right: solutionto theproblem. . . . . . . . . . . . 144

7.3 DMC capacityversusthestep-sizeof theuniformquantizer, AB�:C , EF�I6 . 146

7.4 SimulatedSDRin dB for azero-meanunit-varianceGauss-Markov source

( kl�mR �on ) vectorquantizedat rate [p�`6 � R bpsandsoft-decisiondecoded

with q bits. Quantizationdimensionis Zr�LC . . . . . . . . . . . . . . . . . 148

7.5 Jointlydesignedversustandemcodingschemesfor azero-meanunit-variance

Gauss-Markov source( ks�UR �tn ). Quantizationdimensionis Zu�vC andthe

overall rateis [w�LM � R bps. AJ�:C and EF�I6 . . . . . . . . . . . . . . . . . 149

xiv

7.6 SimulatedSDR in dB for an i.i.d. xy�eR 6; sourcevectorquantizedat rate[z�DC � R bpsandsoft-decodedwith q bits. Quantizationdimensionis Zr�LC .AB�:C and EF�I6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.7 Comparisonamongthe COVQ, FEAD VQ, andOn-line FEAD VQ for axy�eR 6; Gauss-Markov sourcevectorquantizedat rate [F� 6 � R bpsand

harddecoded.Quantizationdimensionis Zs�DC , AB�DC , and EF�76 . . . . . 151

xv

Chapter 1

Introduction

The next generationof wirelesscommunicationsystemsfacesthe demandfor increased

datarates,highermobility, larger carrierfrequencies,andmorelink reliability. Wireless

channelsarecharacterizedby fading,multipath,limited bandwidth,andfrequency andtime

selectivity which make systemdesigna challenge.It is thereforecrucialto have anunder-

standingof thebehavior of wirelesschannelsin orderto know theirperformancelimits and

to beableto designefficient communicationsystemsfor them.This dissertationconsiders

theanalysisof theperformanceof wirelesssystemswith singleor multipleantennasat the

transmitandthereceive sides.It addressesbothof thecaseswherethetransmitteddatais

or is not protectedagainstchannelnoise(i.e., channelcoded).An applicationof theanal-

ysisresultsis presentedby designinga robustcommunicationsystemfor thetransmission

of analogdatasuchasspeechovera multipleantennacommunicationlink.

This chapterbeginswith a shortdescriptionof conventionalcommunicationssystems.

Two methodsto improve the performanceof the generalsystemarethendescribed.The

1

inputsequence { encoder { modulator |

waveformchannel}demodulator}decoder}reconstructedsequence

discretechannel

Figure1.1: Block diagramof a digital communicationsystem.

two problemsconsideredin this thesis,i.e., performanceanalysisandsystemdesign,are

next introduced.Thechapterendswith a review of therelatedliterature.

1.1 The Communication System Model

A typical communicationsystemis depictedin Figure1.1. We considerthe transmission

of discrete-time(discreteor continuous-amplitude)sequencesover noisywaveformchan-

nels.Theassumptionthattheinput is discretein time is justifiedby thesamplingtheorem

whichstatesthatany band-limitedsignalcanberepresentedby its sampleswithout lossof

informationprovidedthatit is sampledataratewhichis at leasttwicethesignalbandwidth

[143, 162]. In practice,thesignalsto be transmittedby a communicationsystemarefirst

low-passfiltered,andhenceit is reasonableto considerband-limitedsignals.

Theencoderanddecoderaresignalprocessingblocks. Theencoderaimsto represent

the sourcein an efficient form (sourceencoding)and to protectthe informationagainst

channelnoise(channelencoding). The decoderexploits the error detectionstructureto

canceltheeffectof channelnoiseandreconstructtheinputmessage.

2

Theuseof sourcecodingis inevitabledueto restrictionsonchannelbandwidthor stor-

agespace. Sourceencodingaims to remove the redundancy of the input and represent

it with as low a bit rateand inaccuracy aspossible. In effect, the sourceencodermaps

the input sequenceto a string of binary symbols. For analogsources,this mappingin-

corporatesquantization,which causespartial lossof information.Theencoderalsoinserts

controlledredundancy to helperrordetectionandcorrectionatthereceiver, whichis known

aschannelcoding.Themodulatormapsblocksof certainlengthfrom theencoderoutputto

multidimensionalwaveforms,sothatthedatasequencecanbetransmittedover theanalog

channel.We assumethatpulseshapingis alsoincludedin themodulatorblock. By pulse

shaping,we meanusinga modifiedwaveforminsteadof therectangularpulse,suchasthe

raisedcosinepulse,in orderto reduceinter-symbolinterference(ISI) at thereceiver [150].

Amplitudeattenuationandnoiseadditionarethe two maineffectsof thewaveformchan-

nel. While bothof thesephenomenaoccurin channelswith afrequency responsenotequal

to unity, suchastheRayleighfadingchannel,thesoleeffectof theadditivewhiteGaussian

noise(AWGN) channelis theadditionof a constantpower densitynoiseto thewaveform

at thechannelinput. Thedemodulatorestimateswhatwaveformwastransmittedover the

analogchannel.Theoutputof thedemodulatoris a discrete-timestreamwhich is passed

throughthe decoderso that the errorscausedby the channelaredetectedandcorrected.

The decoderalsomapsits input sequenceto symbolsthat are in a subsetof the original

inputalphabetto reconstructthetransmittedmessage.Notethattheinputandoutputof the

concatenationof the modulator, the waveform channel,andthe demodulatorarediscrete

sequencesandhencethesethreeblocksmayberegardedasadiscrete(in termsof timeand

alphabet)channelasindicatedin Figure1.1.

3

1.2 The Source-Channel Separation Theorem

The designof the encoder-decoderpairs can be performedbasedon resultsfrom infor-

mationtheory. The resultsstatethat, for a givendistortion ~ , thereexistsa rate ��e~p –

calledtherate-distortionfunction– below which thesourcecannotbecompressedwith a

distortionlessthan ~ . In thepositivedirection,therate ��e~p is asymptoticallyachievable

at distortionlevel ~ by encodinglarge blocksof the sourcetogether. The resultsfurther

statethat,for everycommunicationchannel,thereexistsaquantitycalledchannelcapacity�, which is themaximumratethat informationcanbe transmittedover thechannelwith

arbitrarily small probability of error. The rate-distortionfunction dependson the source

statistics.Thecapacityof a single-antennawaveformchanneldependson thechannelfre-

quency responseandnoisedistribution. For discretechannels,channelcapacitydepends

on thechannelcrossoverprobabilities.

Letusassumethatasource-channelcodewith transmissionrate[ sourcesymbolsO channel

symbolsis usedat the encoder. From the lossy joint source-channelcoding theorem

[160, 161,63,31], it follows thataslongas��e~p �[=� �holds,communicationwithin distortion ~ is asymptoticallypossible.

Many communicationsystemsaredesignedaccordingto thefollowing interpretationof

thefirst two results:“communicationwith distortion ~ canbeachievedby designingtwo

independentcodes:a sourcecodewith rate [S� anda channelcodewith rate [ � . Theonly

constraintis that ��~p �[ ��O�[S�� ; thereis noperformancepenaltyassociatedwith thesep-

arateindependentdesign.” Thisresultis calledthesource-channelseparationprinciple.For

4

discretesource { sourceencoder { channelencoder |discretechannel}channeldecoder}sourcedecoder}decodedmessage

error-freechanneluniformi.i.d. source

Figure1.2: Theideaof thesource-channelseparationprinciplefor discretesources.

discretememorylesssources,if we requirelosslessbit errorrateperformance(this results

in the Hammingdistancedistortionmeasure),thenwe replace��e~p by sourceentropy� �� to require� �� [ ��O�[S��

. This resulthasgiven rise to the communicationsys-

temmodelof Figure1.2. Accordingto this model,channelcodingwill correctall channel

errors,no matterwhatthesourcestatisticsare,andhencethesourceencoder/decoderpair

shouldbedesignedasif thechannelwaserror-free,no matterwhatthechannelconditions

are.

1.3 Performance Enhancement

In this section,we describetwo methodswhich leadto performancegainsover thecom-

municationsystemof Figure1.2.Thefirst schemereconsiderstheseparationof thesource

andchannelcodesandthesecondmethodchangesthephysicalsetupby deploying multiple

antennasat thetransmitandreceivesidesto obtainperformancegains.

5

1.3.1 Improvement via Joint Source-Channel Coding

Thesource-channelseparationprinciplesimplified thetheoryandtechnologyby dividing

adifficult probleminto two lesschallengingproblemswhichmaybesolvedindependently.

In agreementwith the literature,we usethe term“tandemsystems”for systemsdesigned

basedon this principle. Thereare numeroussuccessfultheoreticalresultsandpractical

systemsavailabletodaywhich arebasedon tandemsystems.In fact, the performanceof

sometandemsystemsis becomingverycloseto thetheoreticallimits for simplechannels,

suchassingle-userGaussiannoiselinearchannels[29, 151]. However, thereareanumber

of factsabout the separationapproachwhich merit attention. The first point is that in

order for this result to hold, the sourceandchannelcodelengthsneedto grow without

bound.As a result,communicationsystemsbuilt from independentlydesignedsourceand

channelcodesmay requiregreatercomputationalresourcesand causemore delay than

jointly designedsystems.Separation-basedcodesmaybefar from theoptimaldesignfor

systemswith critical constraintsondelayandcomplexity, suchaswirelesscommunication

systems.Joint source-channelcodingwasshown in [123] to outperformtandemcoding

for systemshaving delayor complexity below a certainthreshold. It wasfurther proved

in [216] that givena probability of block error, optimal joint source-channelcodingmay

requirehalf of theencoding/decodingdelayof optimaltandemcoding.

Secondly, the separationprinciple ignoresthe imperfectionsin real systems.In par-

ticular, sourcecodesaredesignedassumingthat the channelcodeswill correctall errors

introducedby thechannel.Knowing thatthis is notalwaysthecaseevenfor themostpow-

erful channelcodes,oneis not informedby this theoremhow to modify thesourcecodes

6

to dealwith noisychannels.Similarly, channelcodesareusuallydesignedassumingthat

all bits createdby thesourcecodeareof equalimportance[31]. This assumption,which

resultsfrom theassumptionof a perfectsourcecode,leadsto thedesignof channelcodes

thatprotectall bits equally. In a largenumberof applicationsthesourceoutputbit-stream

is thebinaryexpansionof somedecimalnumbersor it is relatedto more“sensitive” parts

of thedata(asin thecasewheretransformcoefficientsareencoded)andhenceallocating

moreprotectionto the moresignificantbits than the lesssignificantbits is a betterway

to usethebit rate. Indeed,equalerrorprotectioncausesperformancedegradationin such

conditions.

The third point is that therearechannelsfor which the separationprinciple doesnot

hold. An exampleof suchchannelsis the multiuserchannelwhosecapacitydependson

thelengthof thesourcecode[31]. Clearly, joint designof thesourceandchannelcodesis

amoreviableway to designcodesfor suchsystems.

Severalmethodshave beenproposedfor joint source-channelcoding.They arebriefly

describedandreviewedin Subsection1.5.2.

1.3.2 Improvement via Diversity in Space and Time

Diversityprovidesthe receiver with several replicasof the transmittedsignal. Therefore,

it is a powerful techniqueto mitigate fading and interferencefrom other users,thereby

improving link reliability. Dependingon thedomainwherediversity is created,diversity

techniquesmaybe divided into threemain categories;i.e., temporal,frequency, andspa-

tial. Thefirst two techniquesnormally introduceredundancy in thetime and/orfrequency

7

domain,causinglossin transmissionefficiency. Spatialdiversitytechniquesemploy mul-

tiple antennasfor transmissionand/orreceptionwithout necessarilysacrificingbandwidth

or power. Therefore,it is naturalto considerchannelswith multiple input (transmit)and

multipleoutput(receive)antennas,oftenreferredto asMIMO channels,to obtainimproved

reliability and/orincreaseddatarateswhereverpossible.

HigherdataratesandmorereliablecommunicationoverMIMO channelsarepromised

by informationtheoreticresults.For thecoherentcase;i.e., whenthe receiver knows the

channelfadingcoefficientsperfectly, andfor independentRayleighfading,channelcapac-

ity at ahighchannelsignal-to-noiseratio (CSNR)is approximatelyequalto [196]�U�D�r�� eA E� ��P��\� bits/sec/Hz

where A and E arethenumbersof transmitandreceive antennasand �� is theCSNR.For

thenon-coherentcase,whichis thescenariothatthechannelis notknown at thetransmitter

or thereceiver, channelcapacityathighCSNRapproximatelyequals[215]�U� A '�� 6 T A '� �� +� �\� bits/sec/Hz

where A ' � �s�� eA E � ��OPCP and� � is the coherencetime of the channel,that is the

numberof symbolintervalsoverwhich thechannelis static.

Both of theabove resultsarevery significant,becausethey show that thecapacityin-

creaseslinearly in theminimumof thenumberof receive andtransmitantennas(provided

that� � is largeenoughfor thenon-coherentcase)andthereforeit canbemuchlargerthan

thesingle-antennacase.To increasechannelcapacityin thesingle-antennascenario,one

hastwo options: increasingthe bandwidth,for which the capacityis asymptoticallyup-

perboundedby a constant,or increasingthe transmitpower which leadsto a logarithmic

8

growth in capacity. Linearincreasein channelcapacityis thereforeveryattractivefor wire-

lessapplications.

For thecoherentcase,channelcapacityis plottedin Figure1.3for threecases;namely,

for a systemwith a singletransmitantennabut multiple receive antennas,a systemwith a

singlereceive elementbut multiple transmitantennas,anda systemwith thesamenumber

of transmitandreceive antennas.It is observedthat thegainof having multiple antennas

at bothsidesis muchlarger thanhaving a numberof antennasat only oneside. If oneis

constrainedto have a singleantennaat onesideof the channel,thenreceive diversity is

morepromising.Transmitdiversity is thechoicewhenthemobilehaslimited processing

resources.

One signal processingapproachto exploit the larger capacityof MIMO channelsis

spatialmultiplexing. In thisapproach,which is typically known astheBLAST architecture

[56], the input bit-streamis de-multiplexed to a numberof brancheswhereconventional

channelcodingmaybeemployed. Theinformationat eachbranchis thentransmittedin a

certainorderover the transmitantennas.At the receiver side,thesignalsareestimatedin

a step-by-stepfashionaccordingto theCSNRof the received signals.Anotherapproach

which combinescodingandmodulationis known asspace-timecoding.Space-timecodes

introducetemporalandspatialcorrelationinto thesignalstransmittedfrom differentanten-

naswithout increasingthe total transmitpower andoftenwithout extra bandwidth.There

is a diversity gain that resultsfrom multiple pathsbetweenthe transmitand the receive

terminals,andacodinggainthatresultsfrom thecorrelationamongthesymbolsacrossthe

transmitantennas1.

1In this work, we areinterestedin point-to-pointcommunications.Therefore,cooperativediversity[158,

103, 114], in which themobilesrelayeachother’smessagesto obtaindiversitygains,is not considered.

9

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

Number of antennas

Cha

nnel

Cap

acity

in n

ats

K = LK = 1L = 1

Figure1.3: Improvementof channelcapacitywhenmultipleantennasareused.TheCSNR

is 20 dB.

10

¡£¢�¤¦¥ ��§�¨©¨©¨©§ ¥ ��ª { channelencoder[ � « ¢�¤ ¬ ��§�¨©¨©¨©§ ¬® ª { modulatorC°¯ ± ¢X² ³ � ¤¦´ ��§�¨©¨©¨©§ ´ � ª|channel} µ ¢4¤ ¶ � §�¨©¨©¨©§ ¶ � ªMAP channeldecoder

·± ¢�¤ ·´ � §�¨©¨©¨©§ ·´ � ª·¡3¢�¤ ·¥ � §�¨©¨©¨©§ ·¥ � ª}Figure1.4: Block diagramof acommunicationsystemwith MAP sequenceestimation.

1.4 Summary of the Thesis

1.4.1 Problem Statement

Error Analysis of Communication Systems

Let us considerthe communicationsystemin Figure1.4, which encodesa groupof

databits �u� �� at a time andmapsthemto a sequenceof basebandsignals.The

above mappingmay includeany kind of channelcodingwith rate [� , suchaslinearblock

codes,for which the codedbits aredenotedby )�� 5¸S�� ¸ �with ¹º�¼»½O�[ � , or trellis

codedmodulation. Eachof the signal sequenceswill be called a “frame” and denoted

by ��*w� �o¾ * � ��¾ *� �� & �¿6 ÀSÀSÀ_�Á , whereÁ �ÂC � ,

a �Â»½Oi��[ ��ÃÄ is the frame length

in symbols,and C�¯ is the constellationsize to which¾ *Å belongs. We assumethat

¾ *Å is

selectedfrom a unit-energy constellationandweightedlaterat themodulator. When¾ *Å is

transmittedandfor a transmitpowerof ÆÇ� �È[ �1ÃÉÆËÊ , whereÆËÊ is thepowerperanuncoded

(information)bit, theinput-outputrelationshipof thechannelis representedby� Å ��Ì Æ£� � ¾ *Å�ÍÏÎ Å 11

where� Å and Î Å arethechanneloutputandadditivenoise,respectively, and�

is thechan-

nelgain.Weassumethat Î Å , � , and¾ *Å areindependent.Theadditivenoiseis assumedto be

complex Gaussianwith independentandidenticallydistributed(i.i.d.) realandimaginary

parts,eachhaving distribution x��R ÎzÐ OPC+ . We denotesucha distribution by Ñ�xy�eR ÎzÐ .We hereinconsiderthreecases:

� �V6Ò� ¸�Ó ¹ ¾ Ô�� , which indicatesthe AWGN channel,�beingi.i.d. Ñ�x��eR 6; amongframes,which is theblock Rayleighfadingmodel,and

�following an i.i.d. chi squaredistribution, which modelstheblock Rayleighfadingchan-

nel with diversityreceptionandalsothespace-timeorthogonalblock codedchannels(see

Chapter3). In thefadingcases,weassumecoherentdetection,which is thecasewherethe

receivercanaccuratelyestimatethechannelgain�

.

We are interestedin finding upperand lower boundson the error ratesof the above

system.Moreprecisely, wewantto deriveupperandlowerboundsontheframeerrorrate,

which is theprobabilitythata frameis detectedin error;i.e.,

FER �LÕ Ö.� $�Ø×�:�� where � and $� arethetransmittedanddecodedframes,respectively, aswell astheproba-

bility thata bit is detectedwrongly, which is givenby

BER � 6» ÙÚ ÛÜ � Ã.�^� Û Ú 0ÝÜ�Û ~�Þ3�eß áà �-j� $�=�D��0�â¦��D� Û where ~�Þ£�eß áà is thenumberof bit differencesbetweenthedatabits correspondingto � Ûand ��0 and -j� $�ã��0�â5�ã�y� Û is theconditionalprobability that $�ã�y��0 is detectedgiven

that ��L� Û is transmitted.

Weareinterestedin bothuniformandnon-uniformi.i.d. binaryinputsequences.There-

fore,weconsidermaximuma posteriori(MAP) decoding.

12

Quantization for MIMO Wireless Channels

Unlike the previous systemin which the input is a bit-stream,in many applicationsthe

problemat handis to transmitananalogsource,suchasthespeechor imagesignals,over

adigital communicationsystem.In suchcases,it is requiredto quantizethesource;i.e., to

approximateit with asourcewhich hasafinite alphabet.

The performanceof a quantizeris expressedby its signal-to-distortionratio (SDR).

Therefore,whenan analogsourceis transmitted,it is reasonableto chooseto minimize

theend-to-enddistortionof thesystemasthedesigngoal,asopposedto approacheswhich

opt to minimize the systembit, symbol,codeword, or frameerror rate. We measurethe

inaccuracy by a distortionmeasure,suchasthesquarederrorbetweentheoriginal andthe

decodedsequences.

Let ä and $ä be, respectively, the original and reconstructedsourcevectors,i.e., the

systeminput andoutput. We are interestedin this caseto transmitanalogsourcesover

communicationlinks with multiple transmitandpossiblymultiple receive antennas.Our

objective is to minimizethemean-squareerror;namely, wewantto minimize~m�DÆÈåçæ�ä T $ä�æ �+èsubjectto constraintson the averagetransmitpower andchannelbandwidth. The band-

width constraintcansimplybemetby fixing thequantizationrateanddimension,themod-

ulationscheme,andtheconstellationsizeasdescribedin [150,200].

13

1.4.2 Thesis Organization and Contributions

This dissertationis divided into eight chapters.Relevant upperandlower boundson the

probabilityof a unionof a finite numberof eventsarereviewedin Chapter2. We present

theseboundsin their generalform andemploy themthroughtoutthis thesisto establish

upperandlower boundson theerror ratesof variouscommunicationsystems.In Chapter

3, we briefly review the space-timeorthogonalblock codes. Thesecodesarefrequently

usedin the following chaptersof the thesis. The maximumlikelihood (ML) decoding

rule andthe symbolpairwiseerror probability of space-timeorthogonalblock codesare

alsoreviewed. As a contribution, an alternatederivation of the codeword pairwiseerror

probabilityof generalspace-timecodesis presented.

In Chapter4, we derive upperandlower boundson the error ratesof channelcoded

singleinput-singleouput(SISO)AWGN aswell asblock Rayleighfadingchannelswith

arbitrarysignalingschemesandunderML decoding.In this chapter, we show how to use

the HunterandKouniasupperand lower bounds,which areBonferroni-typebounds,to

boundthe error ratesof suchsystemsandalsoimprove an alreadyexisting lower bound,

known asthe KAT bound. We alsodevise a way to make the computationof the above

boundsfeasiblewhenthecodebooksizeis large. Furthermore,we find a closed-formex-

pressionfor theprobabilityof theintersectionof two pairwiseerroreventsunderRayleigh

fading.

MIMO systemswhich usespace-timeorthogonalblock codesandML decodingare

consideredin Chapter5. As suchMIMO channelsareequivalentto a numberof parallel

SISOchannelswith chi squarefading,a similar approachto that in Chapter4 is usedto

14

derive tight upperand lower boundson the symbolandbit error ratesof the system. A

closed-formexpressionfor theprobabilityof the intersectionof two pairwiseerrorevents

is alsoderived.

We considertheerror ratesfor orthogonalspace-timecodedchannelsunderMAP de-

coding in Chapter6. We derive the MAP decodingrule and show that, similar to ML

decoding,symboldetectionis decoupledin thiscase.Themaincontributionof thischapter

is thatwederiveaclosed-formexpressionfor thesymbolpairwiseerrorprobability. Signal

mappingandcomparisonwith tandemsystems,which performsourceandchannelcoding

independently, arealsopresented.

In Chapter7,wedesignacommunicationsystemfor thetransmissionof analogsources

overMIMO channels.Thecontributionis to show how to usespace-timesoft-decodingand

soft-decisiondecodingchannel-optimizedvectorquantization(COVQ) to achieve perfor-

mancegainsover tandemsystemscomprisingvector quantization,channelcoding, and

space-timecodingandalsoover systemswhich employ COVQ with harddecoding.Two

COVQs which do not assumethe knowledgeof the CSNRat the transmitterarealsode-

signed.

Chapter8 presentsasummaryof thethesis.

1.5 Literature Review

In what follows, we mentionthe major work on the subjectof this thesis;namely, per-

formanceanalysis,joint source-channelcoding,andspace-timecodes.For anoverview on

15

thefundamentalproblemsin informationtheory, onecanreferto Shannon’soriginalpapers

[160, 161], furtherdescriptionsonratedistortiontheoryin [163], andGallager’stutorialon

Shannon’scontributionsin [64].

1.5.1 Error Analysis of Channel Coded SISO Systems

Themostcommonestimateof theerrorrateof acommunicationsystemis theunionbound

whichonly dependsonthepairwiseerrorprobabilities(see,e.g.,[150]). Theunionbound,

which presentsan upperboundto the codeword error rate, is looseparticularly at low

CSNRvalues. Shannonpresenteda lower boundto the codeword error rateof systems

with AWGN channelandÁ

-ary modulationin [164] usinga spherepackingapproach.

Shannon’s lowerboundis tight at low CSNR,but it becomeslooseastheCSNRgrows. An

upperboundto thecodeword errorrateof BPSK-modulatedAWGN channelswasderived

in [149], extendedto PSK-modulatedchannelsin [84], andtightenedfor somelongbinary

blockcodesin [211]. A lowerboundto deCaen’s lowerboundontheprobabilityof aunion

[39] wasderived in [156] andappliedto BPSK-modulatedAWGN channelswith linear

blockcodes.Althoughthelowerboundin [156] is poorevenatmoderatelylow errorrates,

it is significantbecauseit is asymptoticallytight andconvergesto theunionupperbound

athighCSNR[156] andit only dependsontheweightenumerationof thecode.Thelower

boundin [39] wasimprovedin [109], wheretheKAT boundwasintroducedby solvingthe

sameoptimizationproblemasin [39], but usingstrongeranalysis.Sincethelowerbounds

in [39, 109] arestrongerthanthe boundin [156], they alsoconverge to the union upper

boundastheCSNRgrows to infinity whenappliedto thesamecommunicationsystemas

16

in [156]. The lower boundsin [39, 156, 109] areall in termsof sumsof ratios. Another

improvementof theboundin [39] is givenin [30], wherethe lower boundis expressedin

termsof aweightedsumof ratiosandtightenedvia carefulselectionof theweights(aswell

asexhaustivesearch).Thesemethodsaredescribedin detail in Chapter2.

1.5.2 Joint Source-Channel Coding Methods

Wecategorizejoint source-channelcodingapproachesasunequalerrorprotection,channel-

optimizedvectorquantization,optimizationof index assignment,andexploitation of the

(residual)redundancy at thesourceencoderoutputvia MAP decoding.COVQ andMAP

decodingarereviewedherewith moreemphasissincethesetwo methodsareusedin the

following chapters.

Unequalerrorprotection(UEP) tradesoff sourceresolutionandchannelerrorprotec-

tion to allocatetheavailablebit rate.Therateof thechannelcodedependson thechannel

conditions,suchasthebit error rate. As thechannelbecomesmorenoisy, thenumberof

bits usedfor channelcodingincreases.Theaim in UEPis to find theoptimalbalancebe-

tweenthesourceandchannelcoderates.Most UEPsystemsareimplementedusingrate-

compatiblepuncturedconvolutional (RCPC)codesfor error protection[76], often with

a measureof parity checking,suchascyclic redundancy check(CRC) [205], for detec-

tion of errorsin the channel-decodedsequence.Goodexamplesof UEP systemsinclude

thework in [187] – whereentropy-codedsubbandcodingtogetherwith packetizationand

RCPCchannelcodingwereusedfor imagetransmissionover memorylessnoisychannels

– andthework in [165] which follows [187], but insteadof adhocsourcecoding,it uses

17

the strongimagecoderSPIHT introducedin [152]. A differentapproachto UEP canbe

foundin [4], wherelargerenergy levelsandsmallersignalingschemeswereassignedto the

moresensitive transformcoefficientsin imagetransformcodingusingthediscretecosine

transform(DCT). A major sourceof complexity in UEP methodsis the allocationof the

availablebit ratebetweenthesourceandchannelcodes.An adaptive sourceandchannel

codingrateallocationschemefor finite statechannels,includingtheGilbertchannelmodel

[68, 48], wasexaminedin [94]. In [142], thelogarithmof thechannelcodeblockerrorrate

wasassumedto havea linearrelationwith thechannelcoderate.Rateallocationwasnext

doneusingiterativeoptimization.

Channeloptimized scalar/vector quantizationis anotheralternative to joint source-

channelcoding.In thisapproach,channelconditionsin theform of COVQ index crossover

probabilitiesareincorporatedin quantizerdesign.As aresult,thesourceis compressedde-

pendingon thechannelconditions.Theoptimality conditionsof a generaldigital commu-

nicationsystemwith noisychannelsandreal-valuedinputandoutputwerestudiedin [54].

Later, thenecessaryconditionsfor optimality of scalarquantizationfor binarysymmetric

channelsgivena fixedindex assignmentwerederivedin [113]. In [50], it wasshown that

the algorithmin [113] neednot converge anda realizability constrainton the codewords

wasimposedto guaranteeconvergence.ThegeneralizedLloyd algorithm(GLA) [66] for

joint source-channelcodingover noisychannelswasstudiedin [14] for trellis andpredic-

tive codingandit wasshown that theconsideredschemesoutperformedtandemsystems.

Theapproachestowardvectorquantizationfor noisychannelsbeganwith thework in [43]

wheretheoptimality conditionsof theencoderwerefound. Finally, in [111], COVQ op-

timality conditionswereformulated.Amongthevariousalgorithmsthatareproposedfor

18

COVQ design,we cite the modifiedGLA initialized by simulatedannealing[49], noisy

channelrelaxation[60, 61], stochasticrelaxation[214], deterministicannealing[136], and

COVQ designusingfuzzylogic [92]. Theoreticalstudiesonvectorquantizationweredone

in [125, 126, 213]. For example,it wasshown in [126] that, asexpected,the distortion

of a quantizerapproachesto what is predictedby theratedistortionfunctionasthequan-

tizer dimensionincreases.In [125], therateof convergenceof a quantizerin thepresence

of channelnoisewasstudied. In [147], COVQ designfor the Polyachannelwith finite

memory[5] wasaddressedandit wasshown thatsignificantgainscanbeachievedthrough

exploiting the memoryof the channel(insteadof interleaving). The work in [25] used

subbandcoding,all-passfiltering, andCOSQfor subbandimagetransmissionover mem-

orylessnoisy channels.This methodis similar in spirit to the approachin [20, 21, 22],

wherechannelmemoryis exploitedanddiscretewavelettransform(DWT)-basedsubband

coding, which is substantiallylesscomplex due to reducedfilter lengths,is used. Soft

decodingCOVQ wasintroducedin [200,179] andappliedto imagecodingin [177]. Soft-

decodingCOVQ is computationallydemandingdueto the needfor matrix inversionand

evaluationof trigonometricfunctions.In aneffort to reducethecomputationalcomplexity

of the decoder, COVQ with soft-decisiondecoderwasdevelopedin [6] for non-coherent

Rayleighfadingchannelsandusedin [146] for channelswith additiveGaussiannoisewith

inter-symbolinterference.

Anotherapproachto joint source-channelcodingassignsindicesto thequantizercode-

vectorsaccordingto channelconditions.Thesourceis codedfor a noiselesschanneland

noexplicit channelcodingis applied.Then,takingthechannelcrossoverprobabilitiesinto

account,indicesareassignedto source(scalaror vector)samplessuchthattheend-to-end

19

distortionis minimized.Theindex assignmentapproachto joint source-channelcodingwas

studiedin [212], whereclosercodevectorsin theEuclideandistancesensewereassigned

to closerindicesin theHammingdistancesensein aniterativemanner. Thesimulatedan-

nealingalgorithmwasappliedin [49] to find thebestassignmentof indicesvia perturbing

theindex assignmentandacceptingthelabelingwhichcouldincreasedistortionaccording

to aprobabilisticmeasure.In [106], thecodevectorswerewrittenasa linearsummationof

theindex bits andtheHadamardtransformwasusedto find thebestindex assignment.

The fourth methodof joint source-channelcodingconsiderssourcea priori probabil-

ities in channeldecoding. Ideally, a sourcecoderwould compressany discrete-alphabet

sequenceinto ani.i.d. bit-streamhaving a rateequalto thesourceentropy rate[31]. Nev-

ertheless,no sourceencoderis ideal.Theoutputof thesourceencoderwould thencontain

redundancy in theform of memoryand/ornon-uniformdistribution. A MAP detectormay

exploit this residualredundancy to gain performanceimprovementsover ML detection.

MAP decodingwasfirst usedin [154] for scalarquantization. This work was later ex-

tendedto the vectorcasein [148]. In [145], the characteristicsof the arithmeticsource

coderandthesourceweremodeledby a Markov sourceof orderoneandMAP decoding

wasemployed using the model statistics. A similar approachwas taken in [52], where

MELP-compressedspeechwasmodeledasa first-degreeMarkov chain. SequenceMAP

decodingfor varioustrellis codingsystemsandchannelconditions,includingtheRayleigh

fadingchannel,wasstudiedin [13, 75]. MAP decodingwasusedfor channelswith memory

in [7, 178] and,later, in [182], wheretheGilbert channelmodelwasused.Robustspeech

transmissionvia MAP decodingwasdevelopedin [8] andapplicationof MAP decodingto

imagecommunicationovernoisychannelswasstudiedin [210].

20

1.5.3 Multi-antenna Systems

Communicationsystemswith receive antennadiversity have attracteda lot of attention

sincerelatively longago.In fact,in currentcellularapplications,receivediversityis already

in usefor improving receptionfrom mobiles.A few generalideason receive diversityare

explainedin [150]. A moreadvancedstudy canbe found, for example,in [16]. In the

following, we cite themajor referenceson thestudieson channelswith multiple transmit

andmultiple receive antennas.This subsectionis divided into four partswhich address

the informationtheoreticissues,codedesign,performanceanalysis/evaluation,andother

issueswhichareimportantin MIMO channelsbut arenotdirectly relatedto this thesis.

Let us considera multiple-antennasystemwhich employs A transmitand E receive

antennasasdepictedin Figure1.5. The encoderconvertsa block of » input bits � into

matrix codewordsof size A é a symbolsdenotedby �c�¿�^� �� ! , where � Å �� ¾+�áê Å �¾ � ê Å ��¾Sëìê Å áí with¾ * ê Å beingthe signalassignedto transmitantenna

&at time indexÔ

and í denotingtransposition.At time indexÔ, the entriesof � Å are transmittedsimul-

taneouslyfrom the transmitantennas.At the receiver, the E receive antennascollect the

incomingsignalsattenuatedby fadingandperturbedby additive noise. Thereceivedsig-

nalsarethendemodulatedanddecoded.

In this thesis,we considerRayleighflat fading, so that the complex path gain from

transmitantenna&

to receive antennaà, denotedby

� 0 ê * , hasa zero-meanunit-variance

complex Gaussiandistribution, denotedby Ñix��R 6; , with i.i.d. realandimaginaryparts.

In many scenarios,it is assumedthat the channelis quasi-static,which meansthat the

pathgainsremainconstantduring a codeword transmission,but vary in an i.i.d. fashion

21

¡ { encoder î ï 2

ï 1ï ë... ðñ ññ ñ�òó ó 2 ô1 ôh ô ... õ decoder

·î {·¡Figure 1.5: Block diagramof a typical multi input-multi output system,where �`��5��

is encodedto �J� �^� �� ö� �� ! , attenuatedby " , received as#

, andde-

codedto $� which correspondsto $�÷� � $�� $�� .from onecodeword interval to theother. Theadditivenoiseat receiver

àat time

Ô, Î 0 ê Å , is

assumedto be ÑixW��R 6; distributedwith i.i.d. realandimaginaryparts.Weassumethatthe

input,fadingcoefficients,andchannelnoiseareindependentfrom eachother. Basedonthe

above,for a CSNRof �\� at eachreceive branchandat timeÔ, thesignalat receiveantennaà

canbewrittenas �ø0 ê Å � Ì ùûúëýü ë* Ü � � 0 ê * ¾ * ê Å ÍþÎ 0 ê Å , or in matrix form,d Å � ÿ �\�A "U� Å Í � Å (1.1)

whered Å ��e� �áê Å �� h ê Å áí , � Å �W� Î �áê Å �� Î h ê Å áí , and " � � � 0 ê * � .Information Theoretic Studies

Researchon systemswith multiple transmitantennasbegan in the late 1980sand early

1990s.Thepotentialfor lower error ratesat no extra bandwidthandtransmitpower with

multiple transmitantennaswasdemonstratedvia simulationsin [206, 207,208, 209]using

delayelements,transmit-antennadiversity, andequalization. It wasshown in [196, 57]

that the capacityof fast fading MIMO channelsscaleslinearly in the minimum of the

numberof the transmitandreceive antennasassumingindependentfadingbetweeneach

22

transmit-receive antennapair. This potentialincreasewasshown to hold also for block

Rayleighfadingchannelsin [134]. In [87], it wasshown that temporaldiversity canbe

replacedwith transmitantennadiversityto gainnon-zerocapacityin asinglecoherencein-

terval if thenumberof transmitantennasandthecoherencetime cangrow without bound.

Capacityreductiondueto spatialcorrelationwasinvestigatedthroughMonte-Carlosimu-

lationsin [169] andtheoreticallyin [26], whereit wasshown that thereductionin outage

andShannoncapacityis not significantwhenthecorrelationcoefficient betweenthe fad-

ing coefficients is lessthan0.5. In [27] somedegeneratechannelmodelsto accountfor

spatialcorrelationand keyhole (whereeachentry in the channelmatrix is a productof

two complex Gaussianrandomvariables)wereinvestigatedandvalidatedthroughphysical

measurements.Usingtheideaof [72], tight upperboundsfor thecapacityof spatiallycor-

relatedanddouble-scaterringMIMO channelsweredevelopedin [168] which canbeused

togetherwith the lower boundderivedin [144]. Thecapacityof keyholeMIMO channels

wascalculatedin closedform in [168]. Using randommatrix theory [93] andGaussian

approximationof theunaveragedchannelcapacity, asymptoticbehavior of MIMO outage

capacitywasstudiedin [72, 138, 159]. Finally, in [59], spherepackingboundswerede-

rived for codeword error rateof space-timecodesover block Rayleighfadingchannels.

It wasshown that thecodeword error ratesignificantlyimprovesasthecodewordsspana

largernumberof fadingblocks.

Code Design for Multi-Antenna Systems with Coherent Detection

Thestepnext to informationtheoreticstudiesis to designcodesto achieve a performance

ascloseaspossibleto theinformationtheoreticlimits. In thissubsection,weconsidercode

23

designmethodsfor coherentscenarios;namely, themethodsin which thechannelmatrix

is assumedto beknown at thereceiver. This assumptionis dueto thefact thatthechannel

fadingcoefficientsneedto beestimatedat thereceiver for synchronizationpurposes.

We divide codedesignapproachesinto threegroups:spatialmultiplexing, space-time

coding,andadhocmethods.Spatialmultiplexing containsmostlytheschemeswhich add

little (or no) redundancy at the transmitter, andusesignalprocessingtechniquesfor de-

tection.Spatialmultiplexing wasintroducedin [56], wherethediagonalBell Laboratories

layeredspace-time(D-BLAST) architecturewasdeveloped.In D-BLAST, the input data

bits arefirst de-multiplexed into A sub-sequences,eachof which may be channelcoded

by a conventional(SISO)encoder. The codedbits are thenmodulatedandsentvia an-

tennasin turn; for example,modulator1 sendsits first symbolvia antenna1, its second

symbolvia antenna2, andsoon. At the receiver, thesymbolsaredetectedin an iterative

mannervia “nulling andcancelling”which detectsandremovestheeffect of thesymbols

with thehighestsignal-to-noiseratioateachiteration.TheD-BLAST decodersuffersfrom

beingtoo computationallyintensive. A simplifiedversionof D-BLAST, calledthevertical

BLAST (V-BLAST) wasproposedin [58], wheremodulator&

sendsA symbolsover theA transmitantennasat the& th symbolperiod.Typicalvariationsof BLAST arethework in

[129, 157], whereinterleaving, BLAST, anditerative decodingwereusedto improve the

BLAST performance.Themajordrawbackof BLAST is its complexity asthenumberof

transmitand/orreceiveantennasgrows.

In orderto designspace-timecodes,thecodewordpairwiseerrorprobability(PEP)was

consideredin [194]. The codeword PEPis the probability of the event that when � was

sentandbetween� and $� , $� is detected.Weshalldenotethis probabilityby Õ ÖS�(� @ $�£ .24

UsingtheChernoff bound,it wasshown in [194] thatthecodeword PEPis upperbounded

by Õ Ö;�á� @ $�Ç � ��\��O�K� h�� where [ is the rank of the matrix � T $� and � , alsoknown asthe codingadvantage,is

givenby �þ�mâ©�á� T $�� (� T $�£ �� â � � where� denotescomplex conjugatetransposeand â�� â for a matrix � is thedeterminantof� . Theproduct E [ is known asthediversityorder. Thecodedesigncriteriaproposedin

[194] is to maximizetheworse-case(minimum) rank [ anddeterminantof thecodeword

differences.The maximumvaluethat [ cantake on is A . Codesfor which [ �JA are

calledfull-rank (or full diversity)codes.

The first coding schemefor multiple transmitantennaswas designedin [9], where

a rate-1full-diversity space-timecodewith orthogonalcolumnswasdesignedfor a dual

transmitantennasystem.Using a generalizationof the theoryof orthogonaldesigns,the

work of [9] wasextendedto any numberof transmitantennasin [192], wherecodesfor

two to eight transmitantennaswere given. The codewordsof suchcodeshad orthogo-

nal columnsandwerehenceknown asspace-timeorthogonalblock (STOB) codes.Such

codesdo not dependon thenumberof receive antennas.Thekey advantageof space-time

orthogonalblock codesis that decodingof the STOB codedsymbolsis decoupled. In

otherwords,onecanperformscalarML decodinginsteadof vectordecodingandhence

have polynomialasopposedto exponentialdecodingcomplexity. An importantresult in

[192] is that rate-1space-timeorthogonalblock codesexist only for two transmitanten-

25

naswith complex signalingschemes.Space-timeorthogonalcodeswerealsostudiedin

[82, 120, 121, 185]. An interestingresultin [121] is that themaximumrateof space-time

orthogonalcodeswith complex signalingand AJ�DC�» or AJ�DC�» T 6 transmitantennasis��» Í 6S ®OPC�» . Space-timeorthogonalblockcodesarereviewedin moredetailin Chapter3.

Space-timeorthogonalblock codesachieve full diversity gain, but they fail to have

a large codinggain. In order to achieve a codinggain (at the price of increaseddecod-

ing – andpossiblyencoding– complexity), onemayusea concatenationof conventional

channelcodingandeitherspace-timeorthogonalcodes(see,for example,[10]) or BLAST

(suchas the work in [132]). Otherchoicesarespace-timetrellis coding [194, 95] (and

their designvia computersearch[18, 73, 189]), usingiterative (turbo)codingideas(with

parallel [34, 98, 98, 184] andserial [23, 70, 124, 127, 183] encoderconcatenation),and

non-orthogonalblock coding.An importantclassof high-ratenon-orthogonalblock codes

wasintroducedin [80] andcalledthe lineardispersion(LD) code. Every entryof anLD

codeword is aweightedsumof thebasebandsignalswith theweightschosensuchthatthe

mutualinformationbetweenthechannelinput andoutputcodewordsis maximizedgiven

the numberof transmitandreceive antennas.Othermajor space-timecodedesignmeth-

odsincludethework in [35, 36], whereconstellationrotationandtheHadamardtransform

wereused,and in [46, 77] and their extensionin [47], wherea connectionbetweenthe

rankcriterionof [194] with binaryfieldsto designspace-timecodesfor PSKconstellations

usingAlgebraictoolswascreated.Amongadhocmethods,we cite thework in [118] that

usedconventionalchannelcodingto obtaincodinggainfollowedby randomconstellation

mappingof thesamedataat every transmitbranchto achievediversitygain.

26

Performance Evaluation/Analysis

Ourmainfocusin this thesis(exceptfor Chapter7) is performanceanalysisof digital com-

municationsystems. Many conventionalmethodsfor analysisof SISO communication

systemsbecomeinfeasiblein multi-antennasystems(e.g.,dueto their large codebooks).

Therefore,it is requiredto find waysto eitherextendtheexistingmethodsto MIMO chan-

nelsor to comeup with new analysisstrategies.

Early paperson space-timecodedesignderivedtheChernoff upperboundon thepair-

wiseerrorprobabilitybetweencodewords[194, 74]. Chernoff upperboundsfor channels

with correlatedRician fadingandcorrelatednoisecanbefound in [41]. Thepairwiseer-

ror probability underfastfadingwasconsideredin [173], wherean integral solutionwas

found,andin [199] wherethesolutionwasin termsof a derivative. In [190] a form of the

momentgeneratingfunctionof theML metricwasderivedwhich wassimplerto evaluate

for block fadingchannels,and the PEPswere found numerically. The exact expression

and a lower boundfor the pairwiseerror probability of arbitrary space-timecodewords

underquasi-staticfadingwerefoundin [131] (we will presentanalternative derivationin

Chapter3). For a land-mobileto satellitelink, thepairwiseerrorprobabilitywasfoundin

[197] for Rician-lognormalfadingandin [198] for Rician-Nakagami-» fading.A moment-

generatingfunctionapproachwasusedin [176], wherethepairwiseerrorprobabilityof the

so-calledsuper-orthogonalspace-timetrellis codeswasfound,andin [222], whereupper

boundson thesymbolerrorrateof D-BLAST systemswithout channelcodingwereeval-

uated. The transferfunction methodcanbe usedto derive upperboundson the bit error

rateof space-timetrellis codesaswasthecasein [173, 223],but theboundsareoftenover

27

anorderof magnitudelargerthansimulationresultsevenat highsignal-to-noiseratiosand

they becometoo complex to evaluatefor largediversityorders.

For standardÁ

-ary PSK and QAM constellationswith Gray signal mapping[150],

thestandardapproachfor performanceanalysisin [175], which is widely adoptedin many

references,is to usethe exact conditionalsymbolerror rateandapproximateconditional

bit error rateformulasof [175] which areconditionedon the channelfadingcoefficients.

For example,theconditionalsymbolerrorrate -ì�°�� of standardÁ

-ary PSKis givenby- �°�� £â � ��É � 6� �� Ù < �� ÙÐ �� T �\�! �� O Á "� � �� $# %�& # (1.2)

andtheconditionalSERof standardsquareÁ

-ary QAM is givenby- �� Çâ � ��É � KPq� � �� Ð �� T �\�áb QAM � � �� $# % & # T KPq �� ('Ð �)�*� � T ��áb QAM � � �� +# % & # (1.3)

where qp� 6 T 6;O Q Á and b QAM � M�OPCi� Á T 6; . Similar expressionshold for theBER,

which we do not reportherefor brevity. To derive the error rates,oneneedsto find the

distribution of theequivalentfadingcoefficientsbasedon which thesymbolerrorrateand

approximatebit error maybe evaluated.An exampleof suchan approachis thework in

[53], in which an approximateexpressionfor the bit error rateof space-timeorthogonal

block codedchannelswith correlatedNakagami-» fadingwasderived. In [19] and[166],

the symbolerror rate for space-timeorthogonalcodedchannelswith i.i.d. Rayleighand

keyhole Nakagami-» fading were found, respectively (althoughthe resultsof the latter

referenceare complex to evaluate). An interestingresult in the latter paperis that the

symbolerrorrateof keyholespace-timeorthogonalcodedchannelswith agrowing number

28

of receive antennasapproachesthat of the i.i.d. channelswith a single receive antenna,

which is expectedbecausekeyhole channelshave only onedegreeof freedom(i.e., their

diversitygainin one).Notethattheaboveresultsholdonly for standardsignalingschemes

andGraysignalmapping.

Miscellaneous Issues

We have left out many areaswhich areappliedin MIMO systems,becausethey arenot

directly relatedto this thesis.Many of theseissuesareimportantby themselves. An im-

portantissueis space-timecodedesignfor non-coherentdetection.Dif ferentialandnon-

coherenttechniquesin communicationsdo not requireanexplicit carrierphasereference.

Thesetechniquesarespeciallyneededfor wirelessradio communicationsystems,where

carrierphasesynchronizationis difficult dueto fading,high mobility, severeinterference,

andtheuseof shortdatapackets.Non-coherentdetectionis requiredalsowhentherearea

lot of antennasin thesystemandtrainingsequencesbecometoo largeto bepossible.The

ideasfor communicationovernon-coherentSISOchannelscanbeappliedto MIMO chan-

nelsaswell. Thesesolutionsincludefrequency shift keying (FSK) [117] anddifferential

transmission.

It wasshown in [86] thatfor asymptoticallylargechannelsignal-to-noiseratiosor when

thechannelcoherencetime� � andthenumberof transmitantennasA grow withoutbound

but theratioof� ��O+A is fixed,capacity-achieving codewordsaretheproductof anisotrop-

ically distributedunitary matrix with an independentreal non-negative diagonalmatrix.

A systematicmethodto designunitary constellationswasproposedin [88], but decoding

complexity of suchconstellationsremainedexponential.Designof differentialspace-time

29

codesbasedon the orthogonaldesignswasaddressedin [191, 96], but thesecodesexist

only for certainratesandcertainnumbersof transmitantennas.In orderto designnon-

coherentspace-timecodeswith somestructure(so thatdecodingcomplexity is reduced),

space-timecodeswhich form a groupundermatrix multiplicationwerestudiedanddevel-

opedin [85, 89,90,170, 171,170, 172,99, 81]. As groupcodesdo not exist for arbitrary

rates,otherstrategiessuchascodeconcatenation[12, 15, 155, 188] needto be devised.

An interestingcodedesignmethodproposedin [79] (andits extensionto non-squarecode-

words in [100]), consideredthe designof unitary space-timecodesbasedon the Cayley

transform. The key advantageof the Cayley transformis that it makes it possibleto do

codedesignin alinearvectorspaceovertherealsinsteadof thenon-linearandnon-concave

spaceof complex unitarymatrices,significantlyreducingencodinganddecodingcomplex-

ity.

Many algorithmshave beenproposedto reducethe complexity of encoding/decoding

in multi-antennacommunicationsystems.We categorizethesealgorithmsin threegroups:

algorithmswhichareaimedto reducethedecodingcomplexity throughsub-optimalmeth-

ods(suchas[37, 78]), thosethatusefewer antennasto reducecomputations,andcombi-

nationsof spatialmultiplexing andspace-timecoding. The algorithmin [37] appliedthe

spheredecodingalgorithmof [202] to multi-antennaapplications.The key advantageof

thisalgorithmis thatit doesnotdependon thetransmissionrate.Althoughit hasexponen-

tial complexity [97], it hasa smallerexponentascomparedwith ML decoding.Antenna

selectionfor space-timeorthogonalblock codeswasaddressedin [193] (andextendedin

[104]), where,amongthereceiveantennas,theonewith thehighestchannelsignal-to-noise

ratio was chosen. An optimal algorithm for receive antennaselectionfrom a capacity-

30

maximizingperspectivewasgivenin [69], where,at eachiteration,theantennawhich had

the leasteffect on channelcapacitywasremoveduntil thedesirednumberof antennasre-

mained.In [67], asub-optimalalgorithmwasgivenwhichhadareducedcomplexity by an

orderof magnitudeascomparedwith thealgorithmin [69] by addingtheantennawith the

largestincreasein channelcapacityat eachstep.Thethird approachis to combinespatial

multiplexing andspace-timecoding. In suchmethods,the transmitantennasaredivided

into a numberof groupsamongwhich theincomingbit-streamis de-multiplexed. A form

of space-timecodingis thenappliedto eachgroup; for example,in [38, 45] thediagonal

space-timecodesare usedwith this method,while in [195] space-timetrellis codes,in

[105] space-timeorthogonalblockcodes,andin [203] V-BLAST wereapplied.

An importantissueis theestimationof thechannelfadingcoefficientsat the receiver,

whichis madefor coherentdetection.Channelestimationmethodscanbefound,for exam-

ple, in [133,32, 153, 24]. As thetransmissionbandwidthincreasesbeyondthecoherence

bandwidthof the channel,equalizationbecomesindispensable.An overview of practical

equalizationtechniquesfor space-timecodedsystemscanbefoundin [11]. Whenthechan-

nel is frequency selective, it maybedivided into a numberof sub-channelswhich canbe

assumedfrequency flat andmultiple tonescanbeused.This methodis known asorthog-

onal frequency division multiplexing (OFDM). OFDM for MIMO channelswasstudied,

for example,in [17, 119,139]. Note thatasshown in [219], frequency selectivity canbe

exploited to obtaindiversity gains. If, at least,the channelcovariancematrix canbe fed

backto thetransmitter, optimalpowerallocationvia water-filling [31] andalsobeamform-

ing canbe employed to increasethe throughput.The main result in beamformingis that

theoptimalbeamformershouldhavemultiplebeamspointingalongtheeigenvectorsof the

31

channelcorrelationmatrixwith properpower loadingacrossbeams[218]. Otherissuesin-

cludemultiuserMIMO communication[130,141, 204], broadcastMIMO channels[115],

andMIMO channelmodeling[2, 28, 65,102,122, 137].

32

Chapter 2

Bounds on the Probability of a Finite

Union of Events

2.1 The Lower and Upper Bounds

2.1.1 Introduction

Let ,;N �� N£� �� N Ù�- beafinite setof eventsandlet -j��N£*� betheprobabilitythattheout-

comeof the experimentis N£* . In this section,we review a numberof upperand lower

boundson -r�(.ì*1NÇ*e , which is the probability of the union of N£* . The upperand lower

boundsthatwe considerin this chapterrequiretheknowledgeof theprobabilityof thein-

dividualevents-r�eN£*� & �76 ��Á , whichwehereinreferto asthefirst orderprobabilities,

andthesecondorderprobabilities-j��N£**/ N 0 & áà ��6 ��Á (which is theprobabilityof

theintersectionof apairof eventsN£* and N 0 ). Theseboundsholdfor any probabilityspace.

33

Weareinterestedin suchboundsbecause,asis demonstratedin thefollowing chapters,the

errorratesof communicationsystemscanbeexpressedin termsof sumsof probabilitiesof

theunionof errorevents.Therefore,establishingupperandlowerboundsontheprobability

of afinite unionhelpsto establishboundson theerrorrates.

We mentiontwo typesof bounds. The first two boundsareBonferroni-typebounds

[62] which areexpressedin termsof sumsof thefirst andsecondorderprobabilities.The

secondgroupof boundsarein termsof ratios.

2.1.2 The Hunter Upper Bound

A Bonferroni-typeupperbound,dueto Hunter, for theprobabilityof theunionof a finite

numberof eventsis givenby [91]-10 Ù2* Ü � NÇ*43 � ÙÚ * Ü � -j��N£*� T �65 �í87"9;: Ú� * ê 0 � 9°í87 -r�eN£*�/ãNø0° (2.1)

where < is the setof all spanningtreesof theÁ

indices,i.e., the treesthat includeall

indicesasnodes.

Thecomputationalcomplexity of findingtheoptimalspanningtreeviaexhaustivesearch

is exponentialmakingit infeasiblefor mostof theapplications.In [91,110], it is notedthat

theproblemof finding theoptimalspanningtreecanberecastinto determiningthemaxi-

mal spanningtreeof a completelyconnectedweightedgraphwhosenodesarethe indices

of theeventsandtheweightof theedgebetweena pair of nodes&

andà

is theprobability

of the joint event NÇ*=/lNø0 . This ideamakesit possibleto useKruskal’s greedyalgorithm

[3] whichfindstheoptimalspanningtreefor aweightedgraph.Thisalgorithmis described

in Subsection2.2.1.

34

2.1.3 The Kounias Lower Bound

For the set of events N �� N Ù , the Kouniaslower bound,which is a Bonferroni-type

bound,is givenby [108]

-10 Ù2* Ü � NÇ*43m2 �65 �> ?@A @B Ú * 9 > -r�eNÇ*� T ÚC�D EGFIHC�J)E -r�eNÇ*�/ Nø0� LK @M@N (2.2)

whereO is a subsetof thesetof indicesPRQ��,�6 C ��Á - . Thecomputationalcomplexity

of directsearchto performtheabovemaximizationis exponential.UnliketheHunterupper

bound,thereis no algorithmicsolution(or otherwise)to find theoptimal index setin this

problem. Nevertheless,a sub-optimalalgorithmwith complexity Su� Á � is proposedin

[110]. Thisalgorithmis describedin Subsection2.2.2.

2.1.4 The de Caen Lower bound

In [39], a lowerboundontheprobabilityof aunionof eventsis derived.This lowerbound,

which is dueto deCaen,is givenby- 0 Ù2* Ü � N£* 3 2 ÙÚ * Ü � -r�eNÇ*� �ü Ù0Ü � -r�eNÇ*�/ Nø0� À (2.3)

Noticethattheaboveboundis differentfrom thetwo previousonesin thatit is in termsof

a ratio. In the following, we review two otherlower boundswhich arealsoexpressedin

termsof asumof ratiosandimproveoverdeCaen’s lowerbound.

35

2.1.5 The KAT Lower Bound

A lower boundon the probability of the union of a finite numberof eventsis derived in

[109]. Thisbound,to whichwereferastheKAT lowerbound,is givenby- 0 Ù2* Ü � N£* 3 2 ÙÚ * Ü � 0 # *�-r�eNÇ*� �ü Ù0Ü � -r�eNÇ*�/½Nø0° Í ��6 T # *e û-r�eNÇ*� Í ��6 T # *e û-r�eNÇ*� �ü Ù0

Ü � -j��N£*�/½N 0 T # *1-j��N£*� 3 (2.4)

where # *!�UT *V * TXW T *V *�Y with Z4[]\ beingthelargestintegersmallerthan [ ,V *!�L-j��N£*1 and T *_� Ú0)^ 0ÝÜ * -j��N£*�/ãNø0° ÀNotethattheaboveboundreducesto deCaen’s lowerboundif weset # *!�DR for all

&. The

KAT boundis tighterthandeCaen’s lowerboundby a factorof at most9/8 [40].

2.1.6 The Cohen and Merhav Lower bound

Anotherlower boundfor theprobabilityof a finite unionis derivedin [30]. Similar to the

KAT bound,this lowerboundis animprovementoverthedeCaenlowerboundalthoughit

requiresmoreinformationthanjust theknowledgeof -j��N£*� and -r�eNÇ*_/ÒN 0 . For a finite

eventset,CohenandMerhav’s lowerboundis givenby-`0 Ù2* Ü � N£*43m2 ÙÚ * Ü � 9 üba 9dc C Ã.��[ �»u*��[ > �ü Ù0Ü � ü a 9;c Cfe c E Ã ��[ �» �* ��[ (2.5)

36

where»u*��[ ø24R is anarbitraryrealfunctionwhichcanbechosensothatthelowerbound

is tightened.It is notedin [30] thatequalityin (2.5) is achievedwhen »u*��[ ø��6;Ohg � �Ä��[ ,whereg � � ��[ is thenumberof eventsto which [ belongs;in otherwords,g � �Ä��[ iQ�Iâj, &lk P�mn[ k N£* - â �Finally, notethat for »u*á��[ w�c6 , CohenandMerhav’s lower boundreducesto de Caen’s

lower bound.Therefore,by including »u*á��[ ø�y6 in thepossiblechoicesfor »u*��[ , oneis

guaranteedto geta lowerboundwhich is at leastastight asdeCaen’s lowerbound.

2.2 Algorithmic Implementation of the Hunter and Kou-

nias Bounds

2.2.1 The Hunter Upper Bound

A greedyalgorithm, known as Kruskal’s algorithm, is given in [3] to find the optimal

spanningtree that maximizesthe sumon the right handsideof (2.1). The algorithmis

describedasfollows. First,all nodes&and

àareconnectedvia edgesof weight -r�eN£*�/�Nø0°

to form a fully connectedgraph. To form the tree� Ð , the algorithmstartsfrom the edge

with the largestweight � the � & (à with the largest -r�eNÇ*o/ýN 0 û . Then,at eachstep,the

edgewith thelargestweightis addedto� Ð , subjectto theconstraintthatnocycle is formed

in� Ð . This stepis executeduntil all of the nodesare in

� Ð . This algorithmis formally

statedasfollows [110]. Considera fully connectedgraphwithÁ

edges� & áà of weights-r�eNÇ*�/½Nø0° . Constructasetof edges� Ð accordingto thefollowing steps:

37

p p pp

1

2

3

4

q q q q q q q q qq q q q q q q q qq q q q q q q q q rrrrrrrrq q q q q q q q0.2 0.1

0.04

0.02

0.05

0.3

Figure2.1: Formingtheoptimalspanningtreeto evaluateaBonferroni-typeupperbound.

1. Set� Ð ��s .

2. Add theedgewith maximumweightto� Ð .

3. Fromtheremainingedges,addto� Ð theedgewith maximumweightsubjectto the

constraintthat� Ð remainscycle-free.

4. Gobackto step3 until� Ð contains

Á T 6 edges(i.e.,all nodesandacompletetree).

Figure 2.1 shows the result of applying the above algorithm to a set of eventswith-r�eN � / N�� LR � C , -r�eN � / Nut� �LR � M , -r�eN � / N ' �LR � R�C , -j��N��8/�Nvt� ��LR � 6 , -j��N��8/�N ' ��R � R+K , and -j��Nutw/lN ' � R � Ryx . Thefirst edgeto addto the treeis theonebetween1 and

3, becauseit is hasthe largestweight. The last edgeis the onebetween3 and4. Note

thatalthough-r�eN��o/pNvt� is largerthan -r�eNutz/uN ' , theedgecorrespondingto theformer

probabilityis not addedto thetree,becauseit resultsin a loop.

2.2.2 The Kounias Lower Bound

A stepwisealgorithmis proposedin [110] which findsanindex setto maximizetheright

handside of (2.2). The algorithm in [110] startsfrom two sets O � � s and Oç�ý� P ,

38

andoptimizesO � and Oç� in an iterative (stepwise)mannerby addingan index to O � and

removing anindex from Oç� andvice versa,until they converge. Thealgorithmis detailed

below [110]. In thefollowing, {j��Oø denotesthetermto bemaximizedin (2.2) for asubsetOR|}P . Also, ~�� denotesthesetwhichcontainstheelementsin ~ but not in � .

1. Begin from two setsO � �bs andOç� ��P . Set {u��O � �DR andcompute{u��Oç�� .2. a. If possible,augmentO � by addingto it the bestindex

&�k P��8O � ; that is, the

index for which {j��O � .�, & - is maximizedand {j��O � .�, & - ��{r��O � .b. If possible,shrink Oç� by removing from it the bestindex

&�k O!� ; that is, the

index for which {j��O!�� is maximizedand {u��Oç��, & - i�R{u��Oç�� .3. Repeatstep2 until thetermontheright-handsideof (2.2)cannolongerbeimproved.

4. a. If possible,shrink O � by removing from it the bestindex&�k O � ; that is, the

index for which {j��O � is maximizedand {u��O � ��, & - i�R{u��O � .b. If possible,augmentO!� by addingto it the bestindex

&�k P��8Oç� ; that is, the

index for which {j��O!�h.�, & - is maximizedand {j��Oç�h.�, & - ��{r��Oç�� .5. Repeatstep4 until thetermontheright-handsideof (2.2)cannolongerbeimproved.

6. Go backto step2 until themetric { canno longerbeimproved. In sucha case,the

outputof thestepwisealgorithmis�65 � ,�{j��O � {j��Oç�� - �Notethatthisalgorithmis sub-optimalanddependsonthewayit is initialized(in theabove

algorithm,the initial setsarethe emptysetandthe global set P ). Onecanusedifferent

39

initial sets(otherthan s and P ) or usemorethantwo initial setsto find a better(yet still

sub-optimal)setof indicesfor O .

2.3 Final Remarks

As is shown in the following chapters,theblock error rate,thesymbolerror rate,andthe

bit errorrateof single-andmulti-antennacommunicationsystemswith arbitrarysignaling

schemesandsymbolmappingscanbeexpressedin termsof sumsof probabilitiesof unions

of errorevents.Therefore,theaboveboundsarepowerful toolsto establishupperandlower

boundsfor theerrorratesof thesystemsunderstudy. Whenspecializedto theanalysisof

the communicationsystems,the overall error rateof the systemcanbe representedby a

finite unionof events,where-r�eNÇ*� is thepairwiseerrorprobability, and -r�eNÇ*y/uNø0� is the

secondorderpairwiseerrorprobability.

40

Chapter 3

A Brief Review of Space-Time

Orthogonal Block Codes

3.1 Introduction

Space-timeorthogonalblock codesarea sub-classof space-timecodeswhosecodewords

haveorthogonalcolumnsandhencethey arefull-rank. Thesecodesareimportantbecause

of their simpleencodingand,specially, becausetheir decodingcomplexity is polynomial

(asopposedto exponential)in the sizeof the signalingscheme.Space-timeorthogonal

block codeswereintroducedby Alamouti in [9] for a systemwith two transmitantennas

andany numberof receive antennas.A theoryto extendthesecodesfor morethantwo

transmitantennaswaspresentedin [192].

In thefollowing,wedescribeencodinganddecodingfor Alamouti’scode,whichis also

known asthe � � code,in detail for a dual-transmitandsingle-receiveantennasystem.ML

41

input bits¡ � § ¡ � { constellationmapper� �� ¬ � ¬ �� { space-timeblock coder� ¬ � ¬ � �o� �� ¬ �� ¬ '�¬ � ¬�' ��

Figure3.1: Thetransmitterof Alamouti’sspace-timeorthogonalblock codingscheme.

decodingandthesymbolpairwiseerrorprobability(PEP)of generalspace-timeorthogonal

block codesarenext explained.As a contribution, we presentanalternative derivationof

thecodeword PEPof multi-antennacodeswith arbitrarystructure.This PEPexpressionis

valid for bothspace-timecodesandBLAST.

3.1.1 Encoding in Alamouti’s System

Let us considera dual-transmitsingle-receive antennasystem. The encodingstrategy

of Alamouti’s schemeis as follows (seeFigure 3.1). First, every incoming Ã bits �4��5�� ¯ � (whereC ¯ is thesizeof thesignalingscheme)aremappedinto abasebandsignal

usingascalarfunction.For example,if � � and �+� aretwo consecutiveblocksof Ã bitseach,

they aremappedto¸ � � ��á� � and

¸ �Ç� ��á�+�� , respectively, where �� À is themodulation

functionand¸ �

and¸ � arebasebandsignals.At a givensymbolperiod,two signalsaresi-

multaneouslytransmittedfrom thetwo transmitantennas.At symbolperiodone,we have¾��áê¦� � ¸ �and

¾ � ê¦� � ¸ � . During the next symbolperiod,¾��áê �� T ¸ '� is transmittedfrom

antennaoneand¾ � ê �Ç� ¸ ' �

is transmittedfrom antennatwo, where'

denotescomplex con-

jugate. Notice thatno explicit channelcodingis performedandbecausetwo consecutive

42

symbolsaresentin two symbolintervals,therateof thischannelcodeis unity. Weassume

thatthefadingconditionsremainthesameduringthetwo consecutivesymbolperiodsused

to transmitsymbols �

and¸ � . Denotingthechannelfadingcoefficientsby

� �and

� � , at

symbolperiodone,we receive � � � � ��¸S� Í � � ¸ � ÍÏÎ �andatsymbolperiodtwo, weget�£�?� T � �û¸ '� Í � � ¸ ' � ÍÏÎ � �3.1.2 Decoding in Alamouti’s System

Defining ¡" Q� ¢£¤ � � � �� '� T � '�d¥;¦§ wenotethat

¡" is anorthogonalmatrix;namely,

¡" � ¡" �W��â � � â � Í â � �+â � I¨�� .For

¡d©Q�v�� '� í , )=�� ¸ �B¸ �� í , and

¡� �I� Î � Î '� í , wecanwrite¡du� ¡"U) Í ¡� � (3.1)

At this stage,ML decodingmay be performeddirectly using (3.1); i.e., choosing $)D�� $¸S� $¸ �� í which satisfies

$)=� argminª æ ¡d T ¡"U)çæ � (3.2)

where $[ denotesthe estimateof [ and minimization is doneover all two-tupleswhose

elementsarebasebandsignals.However, decodingmaybefurthersimplifiedby exploiting

43

theorthogonalityof thepathgainsmatrix

¡" . Multiplying (3.1) from the left by

¡" � , we

have gd Q� ¡" � ¡dã� ¡" � ¡"U) Í ¡" � ¡� �W��â � � â � Í â � �+â � ®) Í ¡" � ¡�� â � � â � Í â � ��â � û) Í g� � (3.3)

In otherwords, g� � ��â � � â � Í â � �+â � ¸ � Í gÎ � and g��?�I��â � � â � Í â � �+â � ¸ � Í gÎ � whereg� � Q� � '� � � Í � �� (3.4)g�£��Q� � '� � � T � � �£� (3.5)gÎ � Q� � '� Î � Í � � Î � , and gÎ ��Q� � '� Î � T � � Î � . As theentriesof g� arei.i.d. and� �

and� �

areknown to thereceiver, ML decodingof theentriesof ) canbedoneindependentlyvia$¸ � � argmin� âSg� � T ��â � � â � Í â � �+â � ¸ â � (3.6)$¸ � � argmin� â g�£� T ��â � � â � Í â � �+â � ¸ â � (3.7)

whereminimizationis doneover thesignalalphabet.Sincethemodulationfunction �� À is

scalar, theestimatedbits wouldsimplybe $� � �«� < � � $¸S� and $�+�ø�b� < � � $¸ �� .For asignalingschemeof size

Á �:C°¯ , while usingtheML detectionruleontheoutput

vector

¡d asoutlinedin (3.2) will requireÁ � ML costfunctioncomputations,thenumber

of thecomputationswill only be C Á if gd , definedin (3.4) and(3.5), is usedfor detection

asoutlinedin (3.6)and(3.7).Thiswill resultin muchshorterdecodingtimes,speciallyfor

largesignalalphabets.In a typical implementationof thesystemwith 16-QAM signaling,

decodingof 5 million bits with detectionrule (3.2) took 62918seconds,while employing

thedetectionrulesin (3.6)and(3.7) took about100seconds.Thetestswererun on a Sun

Ultra 60 computer.

44

3.1.3 Extension to Three and Four Transmit Antennas

Alamouti’s work wasgeneralizedto A transmitantennas,insteadof only 2 antennas,in

[192]. There,it wasstatedthat it is not possibleto constructspace-timeorthogonalblock

codesof rateonewith complex signalingschemeswhentherearemorethantwo transmit

antennas.Also, two examplesof space-timeorthogonalblock codeswereintroducedfor

threeandfour transmitantennasas

� t � ¢£££££¤¸ � T ¸ � T ¸ t T ¸�' ¸ ' � T ¸ '� T ¸ 't T ¸ ''¸ � ¸ � ¸�' T ¸ t ¸ '� ¸ ' � ¸ '' T ¸ 't¸ t T ¸)' ¸ � ¸ � ¸ 't T ¸ '' ¸ ' � ¸ '� ¥ ¦¦¦¦¦§ (3.8)

and

� ' � ¢££££££££¤¸S� T ¸ � T ¸ t T ¸)'¿¸ '� T ¸ '� T ¸ 't T ¸ ''¸ � ¸ � ¸)' T ¸ t ¸ '� ¸ ' � ¸ '' T ¸ 't¸ t T ¸�' ¸S� ¸ � ¸ 't T ¸ '' ¸ ' � ¸ '�¸)' ¸ t T ¸ � ¸ � ¸ '' ¸ 't T ¸ '� ¸ ' �

¥ ¦¦¦¦¦¦¦¦§ �(3.9)

Notethatbothof theabove codewordstransmitfour symbolsin eightsymbolperiodsand

hencethey havea rateof 6;OPC .For encoding,four blocksof Ã bits eacharemappedinto a block of four signals )l�� ¸S��¸ � �¸ t �¸)' (í . A codeword is thenformedaccordingto (3.8)or (3.9)andtransmittedover

thechannel,with onecolumnof thematrix transmittedat a time.

Thedecodingphaseis similar to thedualinputsystem.It is assumedthatthechannelis

slow fading(quasi-static),with thepathgainsremainingconstantduring thetransmission

of a codeword (eight symbol intervals). Therefore,we canusethe sameapproachas in

45

Subsection3.1.2. In orderto find theML detectionrules,we definetheoutputandnoise

vectorsas¡dã�W�e� �� ut � ' � '¬ � ' � '® � '¯ í and

¡� �W� Î �� Î � Î t Î '; Î '¬ Î ' Î '® Î '¯ í respectively, so that the input-outputrelationshipbecomesasin (3.1),with thepathgains

matrix

¡" givenby

¡" �¢£££££££££££££££££££££££¤

� � � � � t R� � T � � R T � t� t R T � � � �R � t T � � T � �� '� � '� � 't R� '� T � '� R T � 't� 't R T � '� � '�R � 't T � '� T � '�

¥ ¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦§for thethreetransmitantennasandby

¡" �¢£££££££££££££££££££££££¤

� � � � � t � '� � T � � � ' T � t� t T � ' T � � � �� ' � t T � � T � �� '� � '� � 't � ''� '� T � '� � '' T � 't� 't T � '' T � '� � '�� '' � 't T � '� T � '�

¥ ¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦§46

for systemswith four transmitantennas.Notethat theabove matricesareorthogonal,and

henceaftermultiplying (3.1) from theleft by

¡" � , wehave

gdp�:C�°=) Í g� (3.10)

where °J� ü ë* Ü � â � *�â � . It canbe shown that the elementsof g� arei.i.d. andhenceML

detectioncanbeperformedseparatelyoneachentryof gd .

3.2 General Treatment of Space-Time Orthogonal Block

Codes

Let usnow considerasystemwith A transmitand E receiveantennas.Let )=�� ¸S�� ¸)± áíbe a vectorof Y consecutive symbolsto encodeand �c� �� _ be the space-time

codewordcorrespondingto it. In thecaseof STOB codes,wehavea �4b�Y , whereb�� a O�Y

is thecodinggainand �3� � �Ib æ°) æ � I ë . As anexample,for thecode � t in (3.8),a �³² ,Yl�yK , and b �yC , andfor Alamouti’s code, b � 6 and

a �IY��yC . It canbeshown that

(1.1)canbere-writtenas[140]¡d 0 � ÿ �\�A¡" 0 ) Í ¡� 0 à �76 ÀSÀSÀ! E (3.11)

where

¡d 0 � � ¡� 0 � �� ¡� 0 � � í ,

¡� 0 � � ¡Î 0� �� ¡Î 0� � í , andwe have

¡� 0 Å �]� 0Å and

¡Î 0Å � Î 0Å for6 � Ô � � � , and

¡� 0 Å �� 0 'Å and

¡Î 0Å � Î 0 'Å for � � � Ô � aand

¡" 0 is derivedfrom theà

th

row of " via negationandcomplex conjugationof someof its entries.It is clearthat

¡Î 0Åarei.i.d. Ñix��R 6; .

47

We will hereafterconsiderthe set2 of orthogonalcodesfor which the corresponding

matrix

¡" 0 hasorthogonalcolumns,i.e.,

¡" 0 � ¡" 0 �Jb*°ö0�¨ ± , where °\0 � ü * â � 0�*�â � áà �6 �� E . Therefore,when(3.11)is multiplied from theleft by

¡" 0 � we obtain

gd 0 Q� ¡" 0 � ¡d 0 �4b ÿ �\�A °ö0)´ Í g� 0 (3.12)

where g� 0 Q� ¡" 0 � ¡� 0 � � gÎ 0� �� gÎ 0± � í � Notethateachentryof gd 0 � � g� 0 � �� g� 0 ± � í is associ-

atedwith only onesymbol. Since,given " , g# � � gd � �� gdÄh � is an invertible functionof# � � d �� di� � , ML decodingcanbebasedon g# insteadof#

in thefollowing way¸ *_� argmax� -j� ¸ âµ,Ëg�·¶* - h¶Ü � "7

whichcanbeshown to beequivalentto¸ * � argmin� hÚ ¶ Ü � âSg� ¶* T bL¸¹° ¶ ¸ â �° ¶ (3.13)

whereb ¸ �4b Ì ��O+A .

3.3 The Symbol PEP for General Space-Time Orthogonal

Block Codes

Basedon(3.13),it canbeverifiedthatthePEPbetweenapairof symbolstransmittedover

thespace-timeorthogonalblockcodedchannelconditionedon thepathgainsis givenby-r� ¸ * @ ¸ 0Pâ5"U �bº¼»�½�*50 Q °¿¾ (3.14)

2This classof orthogonalcodesincludesmostof suchcodespublishedin theliterature,includingAlam-

outi’scodeandthosein (3.8)and(3.9).

48

where¸ * @ ¸ 0 denotestheevent that

¸ 0 hasa smallerML metric than¸ * (in (3.13))when¸ * is sent(

¸ * and¸ 0 areapair of symbolsinput to theSTOB encoder),ºr� À is theGaussian

tail functiongivenby ºj��[ � 6Q C � �ÁÀa Â < ÅfÃ � & Ô�½�*50?� Ì Ä ù ú� ë â ¸ * T ¸ 0�â , and °7� hÚ ¶ Ü � ° ¶ �ëÚ Å Ü � hÚ ¶ Ü � â � ¶ ê Å â � � (3.15)

To find the PEP, oneshouldaverage(3.14) with respectto the distribution of ° , which

canbederivedasfollows. Let usdefinerandomvariable Æ�* as Æ � 0 < ��ëlÇ * �ÉÈÊ, � 0 ê * - andÆ ë h Ç � 0 < ��ëlÇ * �ÌËÍ, � 0�* - for& �76 �� A and

à �W6 �� E , where È and Ë aretherealand

imaginaryparts,respectively. Wenotethat Æ Å�Î i.i.d. x��R �� , andwecanwrite ° as°I� Ú * ê 0 â � 0 ê *�â � � Ú * ê 0 ÈÊ, � 0 ê * - � Í Ë·, � 0 ê * - � � Ú * Ü � Æ �* where¹Ò�LAlE . Usingthemomentgeneratingfunctionof normalrandomvariablesyields

theprobabilitydensityfunctionof ° as�ÐÏ ��Ñi � 6�e¹ T 6; )Ò Ñ < � Â <LÓ Ñ��ºR (3.16)

Hence,° hasa scaledchi-squareddistribution with C�¹ degreesof freedom.Theexpected

valueof (3.14)is thereforeequalto-r� ¸ * @ ¸ 0� �� 6�e¹ T 6; )Ò � ÀÐ Ñ < � Â <LÓ º��Ô½�*50 Q Ñ\ & Ñ � (3.17)

As noticedin [19], thePEPof diversityreceptionsystemswith maximumratio combining

hasa form similar to thatof (3.17)with theexceptionthat ½�*50 is givenby Ì �\�ûOPCÉâ ¸ * T ¸ 0Pâ .49

Therefore,theconditionalPEPfor thosesystemsis thesameas(3.14).MRC systemswere

previously analyzed,for example,in [150, 175], whoseresultscanbe usedto obtainthe

expectedvalueof (3.14)as-j� ¸ * @ ¸ 0° � 6C ¢¤ 6 T ½�*50Õ ½ �*50 Í C ë h < �Ú Å Ü Ð � C+ZZ÷� 6��Ci�½ �*50 Í C+ ® Å ¥§ �(3.18)

3.4 The Codeword PEP of Arbitrary Space-Time Codes

Here we presenta new simple derivation of the codeword PEPfor arbitraryspace-time

codes.Weshow thattheconditionalPEPof any two space-timecodewordsis in theform of

adoublesumof anexpressionsimilar to (3.14),andhenceits expectationcanbeevaluated

with thesameapproachasthatfor theSTOB codes.

For arbitraryspace-timecodingschemesthe receiver computesthe squareddistance

betweenany pair of fadedtransmittedcodedsequences� and $�Ö � � ê�×� � �\�A �Ú Å Ü � hÚ 0 Ü ��ØØØØØëÚ * Ü � � 0 ê * & * ê Å ØØØØØ

� � �\�A hÚ 0 Ü �oÙ �0)Ú í Ù 0 (3.19)

where &nÛ ê ÅzÜ ¾ Û ê Å+ÝßÞ¾ Û ê Å , à ÜÉá &yÛ ê â4ã , Ú Ü àäà � , and Ùoå is thetransposeof the æ th row of ç .

Let è6éëêbì Þêví denotetheprobabilitythatÞê hasa largermetricthan ê when ê is sent.It

is easyto verify that[190]è�éîêRì Þêví Ü�ïñðzòó=ôÔõóäö�÷1øñù úûýüÿþó�� ×ó ��(3.20)

Let � þ� Ü � ��þ�� , where � � is the � th non-zeroeigenvalue of U with multiplicity � � . To

computethe expectationof (3.20),we determinethe pdf of �þ ü þ ó�� ×ó andwrite (3.20)asa

weightedsumof expressionssimilar to (3.14).

50

SinceU is Hermitian(i.e.,U � = U) andnon-negativedefinite,it canbedecomposedas� Ü�� îà ��whereV is unitary(i.e.,V � V = I � ) andD is anon-negativedefinitediagonal

matrixhaving theeigenvaluesof U on its maindiagonal.V is a unitarymatrix (i.e.,V � V =

I � ) andits columnsaretheunit-normeigenvectorsof U. Substitutingthedecomposed�

in (3.19),wehaveúû ü þ ó�� ×ó Ü ��û � !" å Ü �$# �å � � à � # å Ü ��û%� !" å Ü �'& �å à & å Ü !" å Ü � �" Û Ü � ��û%� � Û)( * å � Û�( þ � (3.21)

where & å Ü+� # å , * å � Û is the , th elementof & å , and � Û Ü à Û � Û . We notethat theentriesof

& å arei.i.d. -/.Éé10 � ú í . Therefore,themomentgeneratingfunctionof �þ ü þ ó2� ×ó canbewritten

as 354ò ðoòó_ôÔõó é Ý76 í Ü �8� Ü � úé ú:9 � þ� 6 í !<; �>= (3.22)

In order to find the pdf of �þ ü þ ó2� ×ó , which equals ?A@ �CB 354ò ðoòó]ôÔõó é ÝD6 í<E (where ? is the

Laplacetransform),weconvert (3.22)into asumandusethelinearityof theLaplacetrans-

form. Letting F � Ü þ��G�� andusingpartial fraction expansion[112], we canwrite (3.22)

as 3 4ò ðoòó]ô õó é ÝD6 í Ü �" � Ü � ! ; � @ �" Û Ü2H I ÛKJ � � �é 6 9 � @ þ� í ÛKJ � � (3.23)

whereI ! ; � @ Û � � Ü ú,)L BNM ÛM 6 ÛPO é 6 9 F � í ! ; �354ò ðoòó*ô õó é Ý76 íRQ E ØØØØ � ÜTS � � , Ü 0 � �U�V� �XW � � Ý ú � (3.24)

TakingtheinverseLaplacetransformof (3.23),wehaveY 4ò ð ò ó]ô õó é[Z$í Ü �" � Ü � !<; � @ �" Û Ü2H I ÛKJ � � �,�L Z Û[\ @ ò^]_ ` �ba � Z�cd0 � (3.25)

51

Thenext stepis simply using(3.25)to evaluate(3.20).Thisyieldsè6éëêRì Þêuí Ü �" � Ü � ! ; �" Û Ü � I Û � �é1, Ý ú í�LfehgH Z Û @ � \ @ ò^]_ ` � a ÷ji�k Zml M Z � (3.26)

The integral in (3.26) is thesameastheonein (3.17)which wassolvedin (3.18). Using

(3.17)and(3.18)in (3.26)andsetting� þ� Ü � �þ�� , weobtaintheexpressionfor thecodeword

pairwiseerrorprobabilityof arbitraryspace-timecodesasè6éëê�ì Þêví Ü �" � Ü � ! ; �" Û Ü �An Û � �û ø ú Ý � �o � þ� 9 û Û @ �" å Ü2Hqp û ææsr úé û � þ� 9ut í å � �(3.27)

wheren Û � � Ü � þ Û� I Û � � . Theaboveresultagreeswith theresultin [131].

52

Chapter 4

Bounds on the Block and Bit Error

Rates of Coded AWGN and Block

Rayleigh Fading Channels

4.1 Introduction

Let usconsiderthecommunicationsystemof Figure1.4. This systemcanuseany typeof

channelcodingandany complex signalingscheme.In this chapter, we assumethe input

to thesystemto beuniform i.i.d. andhencewe considerML decodingat thereceiver. Our

goalis to establishupperandlowerboundsontheblockerrorrate(BlER) andbit errorrate

(BER) of thesystem.To this end,we first write theBlER andBER in termsof a sumof

theprobabilitiesof a unionof errorevents.We will thenusetheKouniasandKAT lower

boundsaswell asHunterupperbound,whosegeneralformsweredescribedin Chapter2,

53

to derive boundson the block and bit error ratesof SISO AWGN and block Rayleigh

fadingchannelsunderML decoding. Theoretically, theseapproachescanbe appliedto

any signalingschemesandchannelcodesof any typeor rate. For large v , where vxw � is

thechannelcoderate,computationof mostof theboundsin theliterature,includingthose

consideredhere,becomesinfeasible. Oneof the contributionsof this chapteris to usea

subsetof the codebookto evaluatethe bounds. In fact, it is shown that the performance

of thelower boundscanbeimprovedin that theboundsbecometightenedby considering

a subsetof the codebook.Anothercontribution of this chapteris deriving a closed-form

formula for the probability of the intersectionof two pairwiseerror eventsfor the block

Rayleighfadingchannel.

4.2 The Error Rates in Terms of Probabilities of a Union

As explainedis Subsection1.4.1,weconsideracommunicationsystemwhichmapsblocks

of inputbits yTz Ûf{ �}|�Ü 0 � �U�V� ��~ Ý ú into blocksof basebandsignalsy ó Ûf{ �}|�Ü 0 � �V�U� �X~ Ý úwhicharetransmittedover thechannel.Theblockerrorrateof thesystemis givenby

BlERÜ�� @ �" Û°Ü2H è�é�� ( ó Û í^F é ó Û í Ü�� @ �" Û°Ü2H F é ó Û í�è Û ø��Û[�Ü�Û � Û Û � �

(4.1)

whereF é ó Û í Ü F éRz Û í is theprobabilitythat z Û is emittedfrom thesource,è Û é = í��Ü è�é = ( ó Û íis the conditionalprobability given that

ó Ûwassent,and � Û Û indicatesthe pairwiseerror

eventbetweenó Û

andó Û ; i.e., theevent thatwhen

ó Ûis sentandbetween

ó Ûand

ó Û , ó Û is

detectedat thedecoder. Similarly, theBERcanbecomputedfrom

BERÜ � @ �" Û°Ü2H F é ó Û íIè Û é1��[�ëí � (4.2)

54

where è Û é��[�(í Ü úv � @ �" å Ü2HA�q� é æ �)| íIè6é Þó Ü ó å ( ó Ü ó Û íÜ úv � @ �" å Ü2H �q� é æ �)| í ø ú Ý è Û ø'�Û1�Ü å � å Û �� (4.3)

whereÞó

is thedecodedsignalsequenceand�5� é æ �)| í is theHammingdistancebetweenthe

databits correspondingtoó å and

ó Û. Notein theaboveequationsthatbecausewe assume

theinput bit-streamto beuniform i.i.d., all symbolsareequallylikely andF é ó Û í Ü ú w ~ .

Equations(4.1)and(4.3)expressboththeBlER andBERin termsof theprobabilityof

a union,which canbe boundedusingtheKAT, Kounias,andCohenandMerhav’s lower

boundsaswell astheHunterupperboundwhicharepresentedin Chapter2.

In thecomputationof theabove bounds,we needto computethefirst andthesecond

orderPEPs(see(2.1)-(2.5)). Clearly, thePEPexpressionsdependon thechannelmodel.

In thefollowing, wewill considertwo channelmodels:theAWGN andtheblockRayleigh

fadingchannel.

4.3 The AWGN Channel

For theAWGN channel,wehave � â Ü�6 â 9u� â �where

� âis the received signalat symbol interval � and � â�� -/.Éé10 � � H í is the additive

whiteGaussiannoiseat thereceiver. Letó Û Ü¼á�6 Û � � =�=�= ��6 Û� ã bethe

| th possiblesequenceof

55

signalsin ablock,andlet M Û � Ûâ Ü�6 Ûâ Ý�6 Ûâ. TheML decodingrule in this caseis givenbyÞó Ü

argmin� �� Ý ó � þ �where� Ü¼á � � � �U�V� �

� � ã.

Thepairwiseerrorprobabilitybetweenó Û

andó Û is givenbyè Û é�� Û Û í Ü è6é ó Û ì ó Û í Ü è�é �� Ý ó Û � þ c �� Ý ó Û � þ ØØ ó Ü ó Û íÜ è ø k û2� ó Û Ý ó Û �}� �� ó Û Ý ó Û � � H c úk û � ó Û Ý ó Û � � (4.4)Ü ÷ é ü Û � Û í � (4.5)

where��Ü¼á � � � �V�U� � � � ã

,� Z �)¡2� is thestandardinnerproductof Z and

¡, ü þÛ � Û Ü ��2¢ â ( M Û � Ûâ ( þ ,

and÷ é = í is theGaussiantail function. We alsohave �� Ü+£ � w � H

for BPSKsignalingand�� Ü+£ � w û � Hfor two dimensionalsignaling.Notethat theargumentof theexpressionon

theleft-handsideof (4.4) is aunit-variancerealGaussianrandomvariableandhence(4.5)

easilyfollows.

Theprobabilityof theintersectionof two pairwiseerroreventsis equaltoè Û é1� Û Û¥¤ � å Û í Ü è6é ó Û ì ó Û � ó Û ì ó å í Ü§¦ é^¨ Û Û å � ü Û � Û � ü å � Û í � (4.6)

where ¦ é1¨ � Z �)¡ í Ü úû<© o ú Ý ¨ þ ehga ehgª \ @C« ò�¬ ò® « `�¯<` òò�° 4 ¬ òR± MT²�M � (4.7)

is thebivariateGaussianfunctionand¨ Û Û å Ü ¢ �â Ü � � M Û � Ûâ � M å � Ûâ �i ¢ �â Ü � ( M Û � Ûâ ( þ l4ò i ¢ �â Ü � ( M å � Ûâ ( þ l

4ò =56

From(4.3),we needto evaluateè Û é�� å Û í and è Û é�� å Û ¤ � å � í for theBER bounds.These

quantitiesaregivenbyè Û é1� å Û í Ü è6é �� Ý ó å � þ c �� Ý ó Û � þ ØØ ó Ü ó Û íÜ ÷ p ü þÛ � Û Ý ü þå � Ûü Û � å rand è Û é1� å Û ¤ � å � í Ü�¦ p ¨ å Û � � ü þÛ � Û Ý ü þå � Ûü Û � å � ü þ� � Û Ý ü þå � Ûü � � å r =4.4 The Block Rayleigh Fading Channel

For theblock Rayleighfadingchannel,wehave� â Ü�³´6 â 9u� â �where

³, which is the only parameteraddedto the channelmodel, is a zero-meanunit-

variancecomplex Gaussianrandomvariable:³ � -/.Éé10 � ú í . We assumethat

³remains

unchangedduringthetransmissionof acodewordandchangesin ani.i.d. mannerto anew

valueafterwards.

Similar to theAWGN channelcase,we first needto evaluatethefirst andsecondorder

PEPsto calculatetheBlER andBER upperandlower bounds.Thepairwiseerrorproba-

bilities conditionedon thepathgain³

aresimilar to theAWGN case.More specifically,

wehave è Û é�� Û Û í Ü�£ � y ÷ é ü Û � Û ( ³ ( í { � (4.8)

57

and è Û é�� Û Ûµ¤ � å Û í Ü�£ � y ¦ é^¨ Û Û å � ü Û � Û ( ³ ( � ü å � Û ( ³ ( í { � (4.9)

We notethatequation(4.8) is thesameas(3.14)with ¶ Ü ( ³ ( þ . Hence,it equals(3.18)

with� Ü·W Ü ú . Therefore,thePEPis givenbyè Û é1� Û Û í Ü úûh¸¹ ú Ý ü Û � Ûº û 9 ü þÛ � Û�»¼ =

As for the expectationin (4.9), we can usethe result of [174] to write the¦ é = � = � = í

functionas¦ é^¨ Û Û å � ü Û � Û ( ³ ( � ü å � Û ( ³ ( í Ü úû<© eh½m¾2¿TÀÂÁ¿�Ã Á� Ä ÁÅÀ Ã�ÆH Ç�È�É BËÊ ü þÌ ÛûÎÍ�ÏÑÐ þÓÒ ( ³ ( þ E M Ò9 úû<© e ½�Ô ¿ Ã Á¿ ÀÂÁ� Ä ÁÅÀ ÃRÕH Ç�È�É B Ê ü þå Ûû�Í�ÏÑÐ þ Ò ( ³ ( þ E M Ò � (4.10)

where Öié[Z � ¨!íØ×ÚÙXÛ Ð @ �ÝÜ Z o ú Ê ¨ þ w*é ú Ê ¨TZ$í�Þ , and ÙXÛ Ð @ � é[Z$í is re-definedhereas© ÊÙ}Û Ð @ � é Ê Z$í for negative Z . We thenuse[175, Eq. 5A.35] (with ��× ú ) to finally derive the

expressionfor thesecondorderPEPin theblock Rayleighfadingcaseasè Û é1� Ì Ûµ¤ � å Û í �×NßÊé[¨ Û Ì å � ü Ì � Û � ü å � Û íà×·á$é ü Ì Û � ü å Û � ¨ Û Ì å í 9 á$é ü å Û � ü Ì Û � ¨ Û Ì å í � (4.11)

where á$é[Z �}¡m� ¨!íÎ× ú© pãâ Êåä Zk ú:9 Z þ r �with â × Ù}Û Ð @ � ø Z o ú Ê ¨ þ¡ Ê ¨TZ � �

ä × úû B Ù}Û Ð @ � � � 9 © O ú Ê p ú:9 sgné � íû r sgné � í Q E �58

� × û Z k ú:9 Z þ Í)ÏVÐuû â , and� × é úA9 û Z=þ)í/æèç Í+û â Ê ú (with the definition of ÙXÛ Ð @ � é[Z$í

beingthesameasthatin (4.10)).

For theBERbounds,onecansimilarly obtainthefirst andsecondordererrorprobabil-

itiesasfollows è Û é1� Ì å íà× úû ¸¹ ú Ê ü þÌ � Û Ê ü þå � Ûº û ü þÌ � å 9 é ü þÌ � Û Ê ü þå � Û í þ »¼and è Û é�� Ì å ¤ � � å íà×�ß p ¨ å Ì � � ü þÌ � Û Ê ü þå � Ûü Ì � å � ü þ� � Û Ê ü þå � Ûü � � å r �whereßÊé = � = � = í is definedin (4.11)andwhenthesecondandthethird argumentsof

¦ é = � = � = íarenon-negative. If at leastoneof the two argumentsarenegative, the

¦ é = � = � = í function

canbe written assumof a numberof¦ é = � = � = í functionswith non-negative argumentsas

follows

¦ é1¨ � I � n í�×éêêêêêêêêë êêêêêêêêì

¦ é[¨ � I � 0ní 9 ¦ é Ê ¨ � I � 0yí Ê ¦ é Ê ¨ � I � Ê n í I cd0 � nîí 0 �¦ é[¨ � 0 � n í 9 ¦ é Ê ¨ � 0 � n í Ê ¦ é Ê ¨ � Ê I � n í I í 0 � n c·0 �ú Ê ¦ é1¨ � 0 � Ê n í Ê ¦ é Ê ¨ � 0 � Ê n íÊ ¦ é1¨ � Ê I � 0yí Ê ¦ é Ê ¨ � Ê I � 0yí 9 ¦ é[¨ � Ê I � Ê n í I í 0 � nîí 0 �(4.12)

Therefore,theclosed-formexpressionfor theprobabilityof theintersectionof twopairwise

erroreventscanalwaysbecomputedfrom (4.11).

59

4.5 Linear Block Codes and BPSK Signaling

4.5.1 Preliminaries

Sincethe numberof the block PEPsgrows exponentiallyin v , the computationalcom-

plexity of theaboveboundsbecomesprohibitivefor largedatablocks.In importantspecial

cases,suchaswhenlinear block codesandBPSK modulationareusedor for geometri-

cally uniform codes[55] with matchedsignaling[128], it is possibleto consideronly one

codeword insteadof thesumsin (4.1)and(4.2). In otherwords,onecanuse

BlER ×�è Û ø �Ì �Ü�Û � Û Ì � �and BER × úv � @ �" å Ü2H �q� é æ �}| í ø ú Ê è Û ø �Ì �Ü å � å Ì ��

(4.13)

where ï Û is an arbitrary codeword. Furthersimplification canbe madeby considering

the all-zerocodeword (| ×ð0 ) whenbinary modulationis used,becausethis will allow

to benefitfrom informationsuchasthe weight distribution of codesto simplify or speed

up analysis.We illustratethis in thefollowing via reviewing Seguin’s lower boundon the

BlER of a codedBPSKmodulatedsystemwith AWGN channelbasedon deCaen’s lower

boundon theprobabilityof aunionwhich is givenin (2.3).

deCaen’s lowerboundcanbeusedto derivealowerboundfor theblockerrorratefrom

BlER ×bè H ø$�Ì �Ü2H � H Ì � c � @ �" Ì Ü � è þ é1� H Ì í¢ � @ �å Ü � è�é�� H Ì ¤ � H å í � (4.14)

where,from [156], è6é1� H Ì íà× ÷ ø ù û £ �Rñóò� Hõô é�ï Ì í � �and è�é�� H Ì ¤ � H å íà× ¦ ø ¨ Ì å � ù û £ � ñóò� H ô é�ï Ì í � ù û £ �Gñóò� H ô éGï å í � �

60

whereô é�ï Ì í is theweight(i.e., thenumberof ones)of codeword ï å and¨ Ì å × ô éGï Ì í 9 ô é�ï å í Ê ô é�ï Ì 9 ï å íû o ô éGï Ì í ô éGï å í =

It is shown in [156] that¦ é[¨ � Z �}¡ í is strictly increasingin thecorrelationcoefficient ¨ . This

allows to computeanupperboundon ¨ Ì å andhenceon è6é1� H Ì ¤ � H å í on thedenominatorof

(4.14)which dependsonly on thecodeword weightsandis thereforevery fastto compute.

Theupperboundon ¨ Ì å is asfollows¨ Ì åDö ô éGï Ì í 9 ô éGï å í Ê ��÷ Ì ;û o ô éGï Ì í ô é�ï å í �where

�ø÷ Ì ; is theminimumweightof thecode.

Finally, wepresentCohenandMerhav’s dot-productlowerbound[30]. Assumingthatï H is sent,this boundreducesto

BlER c \ �Âùûú @ þ ù��Kü�ý ÷ � @ �" Ì Ü � ÷ þ é�þ é I � ï Ì � � H í"í¢ � @ �å Ü � ¦ é1¨ Ì å � þ é I$ÿ � ï Ì � � H í � þ é ICÿ � ï å � � H í�í � (4.15)

whereI × ú:9�� H w û , I ÿ × ú:9�� H, n ×³é ú Ê I þ í�w � H

, n ÿ ×¼é ú Ê é ú:9�� H í þ í�w � H, andþ é I � ï Ì � � H í�× û I £ � ñóò ô é�ï Ì í� H =

Parameter� canbechosento tightenthebound.Thebestvalueof � is foundin [30] through

anexhaustivesearch.

4.5.2 Evaluation of the Algorithmic Bounds for Large Blocks

It followsfrom (2.4)thatevaluatingtheKAT boundonasubsetof thecodebookwill result

in a lowerboundto theoriginalKAT lowerbound.This follows fromè ø ��Ì Ü � � Ì � c�è ø �Ì�� Ì � c KAT éÔè6é � Ì í � ,��í �61

where �jy ú � �U�V� ��~ {. Theabovecanbefurthertightenedbyè ø��Ì Ü � � Ì � c � Û È� KAT éè�é � Ì í��f,��í �

Theevaluationof thealgorithmicHunterandKouniasboundsmaystill betediousfor

large v . It would be desirableto useonly a subsetof the codebookto computethese

bounds.An examinationof thelower boundin (2.2)directly indicatesthatcomputingthis

boundon a subsetof the codebookwill result in a lower boundfor (2.2). We shall refer

to this boundasthe “Kouniaslower bound,subset”. It is exponentiallyexpensive to find

theoptimalsubsetfor theKouniaslowerbound.To find a sub-optimalsolutionwith a low

complexity, oneoptionis to employ thestep-wisealgorithmusedin Subsection2.1.3to find

a good(yet still sub-optimal)subset.A typical behavior of theapplicationof thestepwise

algorithmto theKAT boundis describedin thefollowing for acodewith asmallcodebook

sizefor clarificationpurposes.Wehaveobservedthesamebehavior for many othercodes,

suchasGolayandBCH codes.Hamming(7, 4) codehas15 non-zerocodewordsandits

generatormatrix is givenby

� × ¸��¹ú 0T0T0 ú 0 ú0 ú 0T0 ú�únú0T0 ú 0 ú�ú 00T0T0 ú 0 únú

»��¼ �Table4.1shows theindicesof thecodewordsfor variouscodewordweights.Eachindex is

thedecimalequivalentof thedatabitswhichcorrespondto acodeword(e.g.,databits0001

areencodedinto codeword 0001011,which hasa weight of 3). Table4.2 reportswhich

codewordsarechosenby the stepwisealgorithmto computethe KAT lower bound. The

62

initial index setsare � ×�� and þ ×��× y ú � �V�U� � ú�� { . We noticethatat low£ � w � H

(e.g.,

-5 dB), only the closestcodewordsto codeword 0 areused. As the signal-to-noiseratio

increases(e.g.,at 4 dB), morecodewordsareaddedto the index set. Eventually, at hight£ � w � H(e.g.,at 10 dB), theentirecodebookis usedto calculatetheKAT lowerbound.For

theGolay(23,12)code,usingthecodewordswith weight7 allowsusto computetheKAT

boundusing253codewordsinsteadof using2047codewords.For theBCH (63,24) code,

weonly considercodewordsof weight15. Thereareonly 651codewordswith this weight

andcomputationof theKAT boundtakesa very smallamountof time andmemory, while

usingtheentirecodebookof size16777216codewordsis computationallynot feasible.

If oneevaluatestheexpressionon theright handsideof (2.1) for a subsetof thecode-

book, onewill obtainneitheran uppernor a lower bound. We will refer to this quantity

asthe“Hunter-basedestimate”.Restrictingthemaximizationof thesecondsum(e.g.,via

Kruskal’s algorithm)on the right handsideof (2.1) will leadto a looseupperbound. To

tightenthisbound,morenodesneedto beaddedto the“sub-tree”whichcontainsthenodes

correspondingto the subset. Figure4.1 demonstratesan idea to derive an upperbound

basedonKruskal’salgorithm.First,weconsiderasubsetthatcontainsthecodewordswith

minimumweightandapplyKruskal’s algorithm. This will maximizethesumof thesec-

ondorderPEPsfor thosecodewords. We next considerthe remainingcodewords(whose

weightsare larger than��÷ Ì ; ). Let ï Ì � , × ú � �U�V� � á be the subsetof the codewordswith

minimum weight and let ï å be a codeword not in this subset. For each æ , we computeè Ì å �× è H é1� H Ì ¤ � H å í for , × ú � �V�U� � á . To includeevery nodeæ in thespanningtree,we add

theedgewith maximum è Ì å .63

4.6 Numerical Results

We considera systemwith a uniform i.i.d.binary source,variouschannelcodes,and2-

and1-dimensionalsignaling. A typical plot for a two dimensionalsignalingschemeis

shown in Figure4.2, wherethe incomingbits areBCH (15, 5) coded,8-PSKmodulated,

andsentover theAWGN channel.Theentirecodebookis usedto calculatetheboundsfor

thisfigure.WeconsidertheHunterupperboundandtheKounias,KAT anddeCaenlower

boundsandobserve thattheKouniaslower boundis thetightestamongthelower bounds.

A similar behavior is observed for othercodesandcoderates,suchasthat in Figure4.3,

which shows theboundsfor a systemwith BCH (15,7) code,AWGN channel,andBPSK

modulation.

Figure 4.4 comparesthe result of computingthe KAT lower boundusing the entire

codebookwith thatof usingonly thecodewordswith theminimumHammingweight for

theGolay(23,12) code.For thesubset,weonly usetheclosestcodewordsto codeword ï Hin theHammingdistancesensebeingmotivatedby theresultof Table4.2,which suggests

thatat low£ � w � H

, it is enoughto considerthecodewordswith theminimumweight. Inter-

estingly, wenoticethattheKAT boundcomputedonthesubset,givesa tighter lowerbound

at low to moderate£ � w � H

ascomparedwith the KAT boundcomputedon the codebook

andalsothebestlower boundin [30]. As predictedby Table4.2,at high£ � w � H

, theKAT

boundusedon theentirecodebookis slightly tighter; but this tightnessis achievedat the

priceof having amuchhighercomplexity andtheimprovementis almostnegligible.

Figures4.5,4.6,and4.7comparethetightnessof theBlER boundsfor thecasewhere

only a subsetof thecodebookis usedto computetheboundsfor theGolay(23,12),BCH

64

(31, 16), and BCH (63, 24) codes,respectively. BPSK modulationis used,hencethe

lowerboundof [156] canalsobecomputed.TheKouniasandKAT lowerbounds(usinga

subsetof theclosestcodewords)arestill tight, but theHunterupperboundis looseat low£ � w � H. Poltyrev’s upperbound[149] is superiorto theHunterupperboundoptimizedon

thesubset,althoughthey areboththesameastheunionboundat BlER valuesof interest

(BlER í ú 0 @�� ). Notefirst thattheHunterupperboundis easierto computethanPoltyrev’s;

therefore,it is thechoiceat BlER valuesof interest.Second,theKAT andKouniaslower

boundsaretighter thanSeguin’s bound;hencethey arealsotighter thanShannon’s lower

bound[164] at leastat mediumto high£ � w � H

. Also noticethat theKAT boundis always

tighterthantheCohenandMerhav (dot-product)lowerboundof [30].

Figures4.8and4.9demonstratetheperformanceof theunion,Hunter, Kounias,KAT,

andSeguinboundsfor theRayleighfadingchannelfor Hamming(7, 4) andBCH (15, 5)

codes.TheKouniaslowerboundis still tight, but theHunterupperboundis notastight for

theRayleighfadingchannel,suggestingthatthehigherorderprobabilitiesaresignificantin

theblock fadingcase.As for theBER,weobserve in ourcalculationsthattheupperbound

(resultingfrom the Kouniaslower bound)is tight, but the lower bound(derived from the

Hunterupperbound)is too looseto beuseful.

65

��

!"# $

%'&)( *+-,�./102436587:9<;>=@?BADC65FEHGIGJ;K5�L'M)N OP>QSRTPVUXWZY>W\[]@^_PKUXP`UFa6PVU_bBP>P

Figure4.1: Theideato form aspanningtreeandcomputeanupperboundfor theBlER. ,dcis theindex of codeword ï Ìfe for which è Ìfe å c�è Ì å � , × ú � �U�V� � á .

66

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

10−2

10−1

Eb/N

o in dB

Pro

babi

lity

of E

rror

Block Error Rate

Bit Error Rate

union boundHunter upper boundsimulationKounias lower boundKAT lower boundde Caen’s lower bound

Figure4.2: Block andbit error ratecurvesfor BCH (15, 5) code,8-PSKmodulation,and

AWGN channel.

67

−2 −1 0 1 2 3 4 5 6 6.510

−4

10−3

10−2

10−1

100

Eb/N

o in dB

Blo

ck E

rror

Rat

e

union boundHunter upper bound, subsetHunter−based estimatesimulationKAT lower boundKounias lower bound, subsetSeguin lower bound

Figure4.3: Block errorratecurvesfor BCH (15, 7) code,BPSKmodulation,andAWGN

channel.

68

codeword weight databits (in decimal)

3 1, 2, 5, 6, 8, 11,12

4 3, 4, 7, 9, 10,13,14

7 15

Table4.1: Thedatabits (in decimal)correspondingto variouscodewordweightsfor Ham-

ming (7, 4) code.

£ � w � Hcodeword indices

-5 dB 1, 2, 5, 6, 8, 11,12

4 dB 1, 2, 5, 6, 8, 9, 10,11,12,14

10dB 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11,12,13,14,15

Table4.2: Codewordsusedin the optimizationof the KAT boundvia the stepwisealgo-

rithm. Only theclosestcodewordsareselectedat low£ � w � H

, while thewholecodebookis

usedat high£ � w � H

.

69

0 1 2 3 4 5 610

−5

10−4

10−3

10−2

10−1

Eb/N

o in dB

Blo

ck E

rror

Rat

e

KAT lower bound, subsetCohen and Merhav’s lower boundKAT lower bound

Figure 4.4: Comparisonamongthreelower boundson the BlER. Golay (23, 12) code,

BPSKsignaling,andAWGN channel.

70

−1 0 1 2 3 4 5 5.510

−4

10−3

10−2

10−1

100

Eb/N

o in dB

Blo

ck E

rror

Rat

e

union boundHunter upper bound, subsetHunter−based estimatePoltyrev upper boundsimulationKAT lower bound, subsetCohen and Merhav lower boundKounias lower bound, subsetSeguin lower bound

Figure4.5: Block errorrateboundsfor Golay(23,12) code,BPSKsignaling,andAWGN

channel.

71

−1 0 1 2 3 4 5 5.510

−4

10−3

10−2

10−1

100

Eb/N

o in dB

Blo

ck E

rror

Rat

e

union boundHunter upper bound, subsetHunter−based estimatePoltyrev upper boundKAT lower bound, subsetKounias lower bound, subsetSeguin lower bound

Figure4.6: Block error rateboundsfor BCH (31, 16) code,BPSKsignaling,andAWGN

channel.

72

−1 0 1 2 3 410

−4

10−3

10−2

10−1

100

Eb/N

o in dB

Blo

ck E

rror

Rat

e

union boundPoltyrev upper boundHunter−based estimateKAT lower bound, subsetKounias lower bound, subsetCohen and Merhav lower bound

Figure4.7: Block error rateboundsfor BCH (63, 24) code,BPSKsignaling,andAWGN

channel.

73

10 15 20 25 30 35 40 45 50 55 6010

−6

10−5

10−4

10−3

10−2

10−1

Eb/N

o in dB

Blo

ck E

rror

Rat

e

union boundHunter upper boundKounias lower boundKAT lower boundSeguin lower bound

Figure 4.8: Block error rate boundsfor Hamming (7, 4) code, BPSK signaling, and

Rayleighfadingchannel.

74

10 15 20 25 30 35 40 45 50 55 6010

−6

10−5

10−4

10−3

10−2

10−1

Eb/N

o in dB

Blo

ck E

rror

Rat

e

union boundHunter upper boundKounias lower boundKAT lower boundSeguin lower bound

Figure4.9: Block errorrateboundsfor BCH (15, 5) code,BPSKsignaling,andRayleigh

fadingchannel.

75

Chapter 5

Bounds for the Symbol and Bit Error

Rates of Space-Time Orthogonal Block

Codes

5.1 Introduction

The original paperson space-timetrellis codes[194] and space-timeorthogonalblock

codes[192] adopt the Chernoff upperboundto estimatethe pairwiseerror probability

of codewordsandto establishcodedesigncriteria. Although the Chernoff boundyields

successfulcodeconstructions,it is quitelooseevenathighvaluesof thechannelsignal-to-

noiseratio. Furthermore,it is commonpracticeto usetheunionboundto approximatethe

symbolerror rateor bit error rate. However, theunionboundis loose,particularlyat low

CSNRs.Therefore,usingtheChernoff boundtogetherwith theunionboundmayresultin

76

poorapproximationsto systemperformance(see,e.g.,Section5.4).

With respectto theSER/BERanalysisof space-timeorthogonalblock codes,onecan

usethe closed-formconditionalSERformulasin [175] (see(1.2) and(1.3)) andaverage

themover the fadingcoefficientsdistribution to derive exact SERformulasfor standard

PSK andsquareû þ ÷ -ary QAM, where v is a positive integer; this approachis taken in

[167]. However, the resultingSERformulasdo not hold for arbitraryconstellationssuch

asstar-QAM andit is not clearhow to extendthis approachto evaluatetheBER,specially

if Graymappingis not used.ThesymbolPEPscanbeusedto derive anupperboundon

theBER; however, aswill beshown in thesequel,theresultingBER upperboundis very

looseevenat mediumCSNRs.

In this chapter, we show how to usethe symbolPEPresultsto derive boundsfor the

SERandBERof STOB codesvia establishingtheHunterandKouniasbounds.Oneimpor-

tantfeatureof theboundsis thatthey hold for arbitraryconstellationsandsignalmappings

sincethey donotdependonthegeometryof thesystemathand.Numericalresultsindicate

that the boundsfor STOB codesoften coincidewith the true error probabilitiesobtained

via simulationsevenat low CSNRs.Furthermore,thecomputationalcomplexity of these

algorithmicboundsis very modestevenfor largeconstellationsandlargernumberof an-

tennas.

Theapproachpresentedin thepreviouschaptercanalsobe appliedto studytheerror

rateof space-timeorthogonalblockcodedMIMO channelsusedin conjunctionwith error-

controlblockcodes;wehoweverdo notconsiderthis for thesakeof simplicity.

77

5.2 Bounds on the Codeword Pairwise Error Probability

of Space-Time Orthogonal Block Codes

Let usdenotetwo STOB codewordsby ê and gê , anddefinematrix h ×ji M Ì � kml with M Ì � k ×6 Ì � k Ê g6 Ì � k andmatrix� ×nhoh � . For STOB codes,

�hasonly oneeigenvalue � with

multiplicity�

. To verify this point, we notethata STOB codeword hasthepropertythatê·ê � × � ê � þ I � , hencewehave� ×phqh � ×¼éëê Ê gê�í;éîê Ê gêví � ×³é � ê � þ 9 � gê � þ í I � �which indicatesthat � � × =�=�= ×§� � �×h�Ø× � ê � þ 9 � gê � þ �It is shown in [131] thatthePEPbetweenê and gê is givenbyè�éîêRì gêuíÎ× ú© e r òH p Í�ÏÑÐ þÓÒ�s �� 9 Í)ÏÑÐ þÓÒ r ; M Ò � (5.1)

A lowerboundis givenin [131] which is derivedby replacingÍ)ÏVÐ þ Ò at thedenominatorof

(5.1)by 1, resultingin è6éëêRì gêuí<t þ û p ú:9 út �� r @ ; �where þ × t @ ; i þ ;; l . A simpleupperboundcanbederivedby minimizing thedenominator

of (5.1),i.e.,by replacingÍ�ÏÑÐ þ Ò by zero,to getè6éëê�ì gêví í þ û p út �� r @ ; � (5.2)

Note that as the CSNRgrows, the above upperand lower boundsconverge. Figure5.1

shows a typical behavior of the above boundsfor the codewords ê × i �� @ �� l and gê ×78

i @ Ì@ Ì @ ÌÌ l for Alamouti’sschemewith onereceiveantennaandquadraturephase-shiftkeying

(Q-PSK)signaling.Thetightnessof thenew upperboundandthelower boundin [131] is

obvious.Ourupperboundis closerto theactualPEPatmediumto highCSNRandcanbe

usedfor space-timecodedesignasin [131].

5.3 Bounds on the Error Rates of Space-Time Orthogonal

Block Codes

In thissection,weshow how theapproachof thepreviouschapterfor theHunterupperand

Kouniaslowerboundscanbeusedfor space-timeorthogonalblockcodes.

5.3.1 The Symbol Error Rate

Following theapproachin (4.1),for aconstellationyvu Ì � , × ú � �V�U� ��~ {, wherev ×xwVçzy þ é ~ í

is apositive integer, anda uniform i.i.d. bit-stream,theSERis givenby

SER × ú~ �" ÛÜ � è Û ø �Ì �Ü�Û � Û Ì �d�(5.3)è Û é�� Û Ì í is thePEPandis given in (3.18). In orderto find lower andupperboundson the

probabilityof eachunionin (5.3)via (2.2)and(2.1),respectively, we alsoneedto find the

probability of the intersectionof � Û Ì and � Û { , the secondorderPEPof symbols u Ì and u {with u Û . Thisprobabilitycanbederivedfromè Û é�� Ì Û¥¤ � { Û íÎ× ï|q}'¦ Ü ¨ Ì~{ Û � � Ì Û k ¶ � � { Û k ¶ Þ<� �

(5.4)

79

where ¨ Ì~{ Û × � u Ì Ê u Û � u { Ê u Û �( u Ì Ê u Û (Ñ( u { Ê u Û ( �� Z �)¡2� ×�� y�Z { � y ¡ { 9�� y�Z { � y ¡ { ( �sy = { and � y = { arethe real andimaginaryparts,re-

spectively), ¶ is definedin (3.15),and¦ é1¨ � â Ì é ¡ í � â { é ¡ í"í is thebivariateGaussianfunction

(givenin (4.7))withâ Ì é ¡ íµ×+� Ì Û k ¡

. Asâ Ì é ¡ í and

â { é ¡ í arenon-negative,we canusethe

resultof [174] to write¦ é1¨ � â Ì é ¡ í � â { é ¡ í�í as¦ é[¨ � � Ì Û k ¡Ó� � { Û k ¡ í × úû<© e ½ ¾ � À�Á� Ã Á

� Ä ÆH Ç�È�É p Ê � þÌ ÛûÎÍ�ÏÑÐ þÓÒ ¡ r M Ò9 úûb© e ½ Ô � Ã Á� À�Á� Ä ÕH Ç�È�É p Ê � þ{ Ûû�Í�ÏÑÐ þ Ò ¡ r�M Ò � (5.5)

where M Ì Û × ( u Ì Ê u Û ( , Öié^Z � ¨!íØ×ðÙ}Û Ð @ �:Ü Z o ú Ê ¨ þ w_é ú Ê ¨TZ$í�Þ , and ÙXÛ Ð @ � é[Z$í is as re-

definedbelow (4.10). Usingthepdf of ¶ in (3.16),we find theexpectedvalueof eachof

theintegralsin (5.5)asfollows.£�| O e ½H \ @�� _�� | M Ò Q × úé[� Ê ú í�L ehgH ¡ ; @ � \ @ ª e ½H \ @�� 8�� ª M Ò M ¡× úé[� Ê ú í�LTe ½H e gH ¡ ; @ � \ @ � � J � �8��U� ª M ¡ M Ò (5.6)× e ½H p úú:9�� é Ò í r ; M Ò � (5.7)

where � é Ò í is a givennon-negative functionof Ò . Thestepfrom (5.6) to (5.7) follows by

writing theinnerintegralin (5.6)in termsof the é1� Ê ú í st derivativeof theLaplacetransform

of the unit stepfunction. Using (5.7) with � é Ò í�× � þ w û�Í)ÏÑÐ þ Ò and[175, Eq. 5A.35], we

80

have á$éGÖ � ��í �× úû<© eh½H p û�Í�ÏÑÐ þ Ò� þ 9 û�Í)ÏÑÐ þ Ò r ; M Ò× Öû<© Ê nû<© �k û 9 � þ ; @ �" � Ü2H p û �� r úé û é û 9 � þ í�í �Ê �© k û 9 � þ ; @ �" � Ü2H � @ �"÷ Ü2H p û �v r é Ê ú í ÷ J �é û é û 9 � þ í"í � Í)ÏVÐ i û n éG� Ê v í lû éG� Ê v í �(5.8)

where

n × úû Ù}Û Ð @ � �� 9 © û O ú Ê p ú:9 sgné � íû r sgné � í Q �with

� × � k û 9 � þ Í)ÏVÐ�û Ö ,� ×Ué úµ9 � þ í/æèç Í=û Ö Ê ú , andsgné[Z$í �× ( Z ( w<Z if Zx�× 0 and 0

otherwise.From(5.4) and(5.5),we obtainthe following expressionfor thesecondorder

PEP è Û é�� Ì Û¥¤ � { Û íÎ×�á p Ö p � Ì Û� { Û � ¨ Ì~{ Û r � � Ì Û r 9 á p Ö p � { Û� Ì Û � ¨ Ì~{ Û r � � { Û r �(5.9)

whereáýé = � = í is definedin (5.8).

5.3.2 The Bit Error Rate

Thesamealgorithmcanbeusedasoutlinedin thepreviouschapterto estimatetheBER.

Wehave

BER × ú~ �" Û°Ü � è Û é�� [� í �whereè Û é�� [� í is thebit errorprobabilitywhen u Û is sentandis givenbyè Û é1� � �[� íà× úv �" { Ü � �q� é æ �}| íIè6é guµ×pu { ( u�×xu Û íà× úv �" { Ü � �5� é æ �)| í ø ú Ê è Û ø �Ì �Ü { � Ì~{ �>� �

(5.10)

81

where�q� éfæ �}| í is theHammingdistancebetweenthebit assignmentsof u { and u Û . From

theabove, it is clearthatfinding upperandlower boundson theBER requiresevaluating

lower and upperboundson the probability of the union in (5.10), respectively. These

bounds,in turn, requirethecomputationofè Û é1� Ì~{ í × £�| B ÷ p �ÐþÌ Û Ê � þ{ Û� Ì~{ k ¶ r E× úû ø ú Ê � þÌ Û Ê � þ{ Ûé�� þÌ Û Ê � þ{ Û í þ Ê û � þÌ~{ � � ! @ �" � Ü2H p û ��Pr p � þÌ~{û é�é�� þÌ Û Ê � þ{ Û í þ 9 û � þÌ�{ íbr � �aswell as è Û é�� Ì~{ ¤ � � { íð× £�| B ¦ p ¨ Ì � { � �ÐþÌ Û Ê � þ{ Û� Ì~{ k ¶ � �Ðþ� Û Ê � þ{ Û� � { k ¶ r E× á p Ö p � � { é1� þÌ Û Ê � þ{ Û í� Ì�{ é1� þ� Û Ê � þ{ Û í � ¨ Ì � { r � � þÌ Û Ê � þ{ Û� Ì~{ r9 á p Ö p � Ì~{ é��Ðþ� Û Ê � þ{ Û í� � { é1� þÌ Û Ê � þ{ Û í � ¨ Ì � { r � � þ� Û Ê �Ðþ{ Û� � { r �

(5.11)

whereáýé = � = í is definedin (5.8). Equation(5.11)holdswhenthesecondandthethird argu-

mentsof¦ é = � = � = í arenon-negative. Negativeargumentsshouldbedealtwith asexplained

below (4.12),showing that,in any case,anexpressionexistsfor thesecondorderPEP.


For this part, the lengthof the input bit-sequenceis max(100000,100 � BER@ � ). The

computationtime of the algorithmicBonferroniboundsis negligible for our codethat is

written in C andrunson aSUN Ultra 60 machine.TheChernoff andtheunionboundsare

alsogivenfor theSER.

82

The SERandBER versusCSNRcurvesarepresentedin Figures5.5 through5.9 for

various�

,W

, space-timecodes,and~

-ary PSKandQAM constellationswith Graymap-

ping. Figures5.5 and 5.6 show the performanceof Alamouti’s � þ code[9], while the

performanceof thecodes� � and � s of [192] arepresentedin Figures5.7and5.9,respec-

tively. TheHunterandKouniasboundsfor bothSERandBER curvesarevery tight and

canhardlybedistinguishedfrom eachotherandfrom theperformancecurve obtainedvia

simulation. It canbe seenthatasthe constellationsizegrows, the union boundbecomes

lessreliableandtheChernoff boundgetsfartherfrom theothercurves,while thelowerand

upperboundsremainverytight. Figure5.7showstheBERupperandlowerboundsaswell

assimulationresultsfor a systemwith threetransmitandW × ú � û � and t receiveantennas

for a wide rangeof CSNRs.Theboundsaretight evenat negativeCSNRvaluesandhigh

diversityorders.NoticethattheStar8-QAM constellationof Figure5.9(with signalpoints

andmappingsasdepictedin Figure5.8) is not a regular squareQAM nor 8-PSK,but the

boundsarestill very tight.

An upperboundon theBER canbecomputedby upperboundingè6é gu ×�u { ( u ×�u Û í in

(5.10)by thePEPto get

BER ö ú~ v "PÛË" { �q� é |$� æ!íIè6é�u Û ì u { í �Thisboundis alsoplottedin all figures.It is clearlyobservedthatourupperboundis much

tighter, particularlyfor largerconstellations.

AsGraymappingisusedin theabovesystems,it isexpectedthatathighenoughCSNRs

theBER of PSK-modulatedsystemsconvergesto [150] BER � �÷ SER�where

û ÷is the

constellationsize. Thecurveslabeledwith “SER/m” in Figure5.5 show this approxima-

83

tion. We alsoobserve that asthe constellationsizeshrinksor the numberof the receive

antennasgrows, this estimatebecomestighter. Therefore,at high enoughCSNRvalues,it

would beenoughto find only oneof theSERboundsto obtaina goodestimateof system

SERandBER.For example,for 16-PSKsignalingandataCSNRof 15dB, thecoinciding

SERlowerandupperboundsareequalto 0.195682.TheaboveapproximationyieldsBER� 0.048921,while thelower andupperboundson theBER coincideandequal0.051838.

It is alsoimportantto notethatalthoughwe have presentedtheBER resultsfor theGray

signalmapping(exceptfor Figure5.9), our boundsalsoapply to any othermappingand

show thesamebehavior.

84

0 2 4 6 8 10 12 14 16 18 20 2210

−4

10−3

10−2

10−1

100

CSNR in dB

Cod

ewor

d P

airw

ise

Err

or P

roba

bilit

y

upper bound in (5.2)upper bound of Tarokh et al. [194]lower bound of Lu et al. [131]

Figure5.1: Our upperboundandthelowerboundson � i<i �� @ �� l�� i @ Ì@ Ì @ ÌÌ lbl , Alamouti’s

codewith Q-PSKmodulation,� × û

, andW × ú .

85

−5 0 5 10 15 2010

−3

10−2

10−1

100

SER

BER

CSNR in dB

Pro

babi

lity

of E

rror

Chernoff−based union boundunion boundupper boundsimulationlower boundPEP−based BER estimateSER/2

Figure5.2: Resultsfor Q-PSKsignalingwith � ×�� andW ×�� (2 bits/s/Hz).

86

−5 0 5 10 15 20 2510

−3

10−2

10−1

100

CSNR in dB

Pro

babi

lity

of E

rror

SER

BER


Figure5.3: Resultsfor 8-PSKsignalingwith �� andW �� (3 bits/s/Hz).

87

4 6 8 10 12 14 16 1810

−3

10−2

10−1

100

CSNR in dB

Pro

babi

lity

of E

rror

SER

BER

Chernoff−based union boundunion boundupper boundsimulationlower boundPEP−based BER estimate

Figure5.4: Resultsfor 16-QAM signalingwith ��x� andW �x� (4 bits/s/Hz).

88

−5 0 5 10 15 20 2510

−3

10−2

10−1

100

CSNR in dB

Pro

babi

lity

of E

rror

SER

BER


Figure5.5: Resultsfor 32-PSKsignalingwith �� andW �x� (5 bits/s/Hz).

89

8 10 12 14 16 18 20 22 24 26 2810

−4

10−3

10−2

10−1

100

CSNR in dB

Pro

babi

lity

of E

rror

SERBER


Figure5.6: Resultsfor 64-QAM signalingwith ��x� andW �x� (6 bits/s/Hz).

90

−5 0 5 10 15 2010

−4

10−3

10−2

10−1

100

CSNR in dB

Bit

Err

or E

rror

L = 1

L = 2

L = 4

upper boundsimulationlower bound

Figure5.7: Resultsfor 8-PSKsignalingwith ��x� andW ��z�\�� and � (1.5bits/s/Hz).

91

�

�

� �¡[011]

� £¢6¡[111]

�¡[000]

�¢6¡[100]

� ¡¥¤[001]

� ¢6¡£¤[101]

� £¡£¤[010]

� £¢6¡¥¤[110]

Figure5.8: Thestar-QAM signalingschemewith quasi-Graymapping.¦ is anormalizing

factorandequals�@§z�©¨ �6ª and «¬� ¨ Ê � .

92

2 4 6 8 10 12 14 16 18 20 22 2410

−4

10−3

10−2

10−1

100

CSNR in dB

Pro

babi

lity

of E

rror

SER

BER


Figure5.9: Resultsfor Star8-QAM signalingwith ��x� andW �� (1.5bits/s/Hz).

93

Chapter 6

Error Rates of STOB Coded Systems

under MAP Decoding

6.1 Introduction

Whentheinput to thecommunicationsystemis uniformi.i.d., theoptimaldecodingrule in

theminimumcodeworderrorratesense,is theML decodingrule. Thiscasewasconsidered

in the previous chapter, wherewe derived tight upperand lower boundson the symbol

and bit error ratesof space-timeorthogonalblock codedchannelswith arbitrary signal

geometryandsignalmappings.

In this chapter, we study how exploiting the non-uniformsourcedistribution at the

transmitterand/orthe receiver can improve the performanceof MIMO systems.In par-

ticular, the effects of symbol MAP decodingas well as signal designand mappingare

addressedfor systemswhich usespace-timeorthogonalblock coding. As will beseenin

94

thesequel,obtainingthesymbolpairwiseerrorprobabilityof theMAP decodedsymbols

requiresfinding theexpectedvalueof �®B¯±°�²³�´ , where µ is a realnumber. This is more

involvedthantherelatedderivation in theML decodingcase,wheretheproblemis com-

puting theexpectedvalueof �®B¯ ´ . TheMAP decodingrule, which we alsoderive here,

andthepairwiseerrorprobabilityexpressionunderMAP decodingshow thattherecanbe

a largegainin performingMAP decodingascomparedwith ML decoding.

MAP decodingfor sourceswith redundancy (due to non-uniformdistribution and/or

memory)is a form of joint source-channelcoding.It would thenbeinterestingto compare

theperformanceof MAP-detectedschemeswith thatof separate,independentsourceand

channelcoded(tandem)systems.For BPSK signaling,it is observed that at low CSNR

values,MAP decodingis superiorto tandemcoding. As thesizeof thesignalingscheme

grows, this resultholdsfor all practicalerrorratesof interest.

Anothercontribution of this chapteris the establishmentof tight Hunter (upper)and

Kounias(lower) boundson the SERandBER of STOB codedchannelsunderMAP de-

coding.Similar to thepreviouschapter, wedeterminetheprobabilityof theintersectionof

two pairwiseerroreventsanduseit togetherwith our closed-formPEPformula to derive

HunterandKouniasbounds. We demonstratethat the boundsprovide an excellentesti-

mateof the error rateseven at low (negative) CSNRvalues. For moderatevaluesof the

CSNR,the upperandlower boundsareoften the sameup to four significantdigits. The

SERboundsaremuchtighterthantheunionbound,andtheBER boundsaresignificantly

betterthantwo computationallylessexpensiveestimates.

95

6.2 The MAP Decoding Rule

6.2.1 Decoupled Symbol Detection

As outlinedin Chapter3, theinput-outputrelationfor a MIMO channelwhichusesspace-

timeorthogonalblockcodingcanbeconvertedto theformgivenby (3.12),whichwerepeat

herefor simplicity ¶·�¸º¹� »¼ ¸4½ »·�¸ � ¾'¿ ÀÂÁ� Ã ¸TÄ ° ¶Å£¸@ÆWe recall in theabove equationthateachentryof

¶·�¸ is associatedwith only onesymbol.

Therefore,in thenew case(i.e.,underMAP decoding),if weshow thatthenoisevector

¶Å£¸is composedof i.i.d. randomvariables,thenwecandetectsymbol « by only consideringthe« th entryof thevectors

¶· ¸ , �ÈÇÊÉ�ÇÌË , whichmeansthatdetectionof symbolsis decoupled

underMAP decoding.

In order to find the distribution of the noisevector

¶Å£¸ , we considertwo noisesam-

ples

¶ÍÏÎÐÒÑ and

¶ÍÏÓÐ)Ô at two arbitrary symbol intervals ÕvÖ and ÕS× , and for two arbitrary re-

ceive antennasØ and Ù . Since

¶Í ÎÐ Ñ and

¶Í ÓÐ Ô are weightedsumsof independentnormal

randomvariables,they haveGaussiandistribution. Also, it is straightforwardto verify thatÚÜÛ ¶ÍÏÎÐÒÑ\Ý � ÚÞÛ ¶ÍoÓÐßÔTÝ �pª . Hence,thecorrelationof thesenoisesamplesisàâá ¶Í ÎÐ Ñ ¶Í ÓdãÐ Ôåä � Úçæéèxêë Ì Ü Ö »ì Î�ãÌ�í Ð Ñ »Í ÎÌïî èpêë ¸ Ü Ö »ì ÓÒã¸ í Ð Ô »Í Ó¸ î ãñð� ë Ì ë ¸ »ì Î\ãÌ�í Ð Ñ »ì Ó¸ í Ð Ô Úxò »Í ÎÌ »Í ÓÒã¸�óSince »Í ¸ô �6�ÜÇ ÉÏÇ�Ë , arezero-meani.i.d., theabove sumis zerounlessØq��Ù and «£�xÉ ,

96

in which caseit equalsêë Ì Ü Ö »ì Î\ãÌõí ÐKÑ »ì ÎÌ�í Ð)Ô Ú ò'ö »Í ÎÌ ö × ó � ë Ì »ì Î\ãÌ�í ÐÒÑ »ì ÎÌ�í Ð)Ô (6.1)

� ÷øù øúû Ì ö »ì ÎÌ�í ÐÒÑ ö × �Ì¾ Ã Î if Õ�Ö£� Õ�×ª otherwise,

(6.2)

where(6.1) follows from the fact that »Í ¸ô is unit-varianceand(6.2) follows from the or-

thogonalityof »¼ . Thus,weobtainthat¶Í ÎÌýü i.i.d. þïÿ ®)ªÂ�>¾ Ã Î ´ Æ (6.3)

As we mentionedin thefirst paragraphof this section,the above result,i.e., the fact that

theentriesof thenoisevector

¶Å Î � � ¶Í ÎÖ � ÆõÆfÆ � ¶Í ÎÐ�� arei.i.d., provesthatsymboldetection

is decoupled,similar to theML decodingcase.

6.2.2 The MAP Decoding Rule

For MAP decoding,

¶� � � ¶· Ö � ÆõÆfÆ � ¶·�� canbeusedinsteadof� � � · ÖD� ÆfÆfÆ � · ê � because

¶�is

generatedfrom�

throughinvertibleoperations.Thedetectionrule is givenby Ì � argmax ��® ö Û ¶�� Ì Ý � Ü Ö � ¼ ´� argmax � ® Û ¶�� Ì Ý � Ü Ö ö � ¼ ´�� ® ´� argmax �� Ü Ö �� ® Û ¶�� Ì�� ¾�� Ã Ý � Ü Ö ´�� ® ´ (6.4)

� argmax æ"!$# Û � ® ´ Ý � �ë Ü Ö ö¶� Ì%� ¾ � Ã ö ×¾ Ã ð � (6.5)

where¾ � � ¾'& ( )* , and(6.4)and(6.5)holdbecause

¶Í ÎÌ arei.i.d.andGaussian,respectively,

asindicatedin (6.3).

97

6.3 Exact Symbol Pairwise Error Probability with MAP

Detection

6.3.1 The Conditional PEP

Without loss of generality, we considerMAP decodingfor the Ø th symbol period. The

errorprobabilitiesmaybedeterminedusingtheMAP detectionmetricgivenin (6.5). The

receivershouldevaluatethismetricfor symbols Ì and ¸ giventhat Ì is transmitted(hence¶� Î �Ì¾ � Ã Ì ° ¶Í Î ) anddecidein favor of theonewhichyieldsa largermetric.

From(6.5), theprobability that ¸ is preferredover Ì when Ì is sent, +-,@® Ì-. ¸ ´ , is

theprobabilityof theevent!$#`Û � ® Ì ´ Ý � /0 ¾ �ë Ü Ö ö¶� Î � ¾ � Ã Ì ö ×Ã Ç !1# Û � ® ¸ ´ Ý � /0 ¾ �ë Ü Ö ö

¶� Î � ¾ � Ã ¸ ö ×Ã �which is equivalentto�ë Ü Ö ¨ 0�2 ¸ � Ì � ¶Í Î43ö ¸ � Ì ö 5 /ö ¸ � Ì ö 60 À�Á !$# � ® Ì ´� ® ¸ ´ ° ¾ ö ¸ � Ì ö ¿ ÀÂÁ0 6 �ë Ü Ö Ã Æ (6.6)

From(6.3),it followsthat 7 × 8 :9�;< � í �� =?>@ :9�;< � @ arei.i.d. þïÿ ®)ªÂ�>¾ Ã ´ . Hencethesumontheleft hand

sideof (6.6) is þïÿ ®)ªÂ�4¾ û � Ü Ö Ã ´ . Therefore,theprobabilityof theeventin (6.6),which is

thepairwiseerrorprobabilityconditionedon thepathgains,is givenby�ý® ÌA. ¸ ö ¼ ´ �p è ¿ ¾ À�Á0 6 ö Ì � ¸ ö ¨ Ã ° 60 ¾ À�Á /ö Ì � ¸ ö !$# � ® Ì ´� ® ¸ ´ /¨ Ã î � (6.7)

where,asmentionedin theChapter3,

Ã � �ë Ü Ö Ã � *ë Î Ü Ö �ë Ü Ö ö ì Î ö × Æ98

6.3.2 Interpretation of the PEP as a Laplace Transform

Basedon theprobabilitydensityfunctionof Ã , which wasderived in (3.16),andusinga

changeof variables,theaverageof (6.7)canbewrittenas�ý® ÌA. ¸ ´ � /®CB � / ´EDGF ×IHÌ ¸ JLKM N H ; ÖPO ;�Q RIS Ô� 9 UT ¨ N ° µ Ì ¸¨ N�VXW N � (6.8)

where F Ì ¸ � & Y (?)× * ö Ì � ¸ ö and µ Ì ¸ � Ö× !$#[Z]\ �1^Z]\ :9 ^ . We notethat the integral in (6.8) is the

Laplacetransformof N H ; Ö `_ ¨ N ° ² � 97 Qba evaluatedat cÜ� F ; ×Ì ¸ . We know that if � ®ed ´ andf ®gc ´ are Laplacetransformpairs (f ®hc ´ �ji Û � ®kd ´ Ý ), so are d H � ®ed ´ and ® � / ´ H�lPml Á m f ®gc ´ .

Therefore,weneedto find the B � / st derivativeof i á _ ¨ N ° ² � 97 Qba ä . Usingintegration

by parts,wehavederivedtheLaplacetransformof `_ ¨ N ° ² � 97 Qba asf�n�o�p ®gc ´ � irq T ¨ N ° µ Ì ¸¨ N Vts� / � sgn®)µ Ì ¸ ´0 c � /0 T /c ¨ 0 c£° / ° sgn®)µ Ì ¸ ´c V O ; \ ² � 9 u @ ² � 9 @ 7 × Á u Ö ^ Æ (6.9)

We now needto evaluatethe B th derivativeof (6.9)which is calculatedin thenext subsec-

tion.

6.3.3 The PEP in Closed Form

Herewederivethe B th derivativeof (6.9)with respectto c . First,wepresentalemmawhich

canbeprovedeasilyvia induction.

Lemma–Thefollowing hold

a) lvml Á mxw ÖÁ]y � \ ; Ö ^ m H{zÁ m | Ñb) l ml Á m ® 0 c<° / ´]} Ô � ® 0 c<° / ´]} Ô ; H H ; Ö� Ì Ü M ® � � 0 « ´

99

c) l ml Q m O ; \�~ u��Q ^ ��® � / ´ H��H O ; \G~ u��Q ^ ÆThederivativeof thefirst termin (6.9) maybefoundvia (a). As for theproductterm,

weusetheLeibniz’s formula[1] to treatthetwo termsseparately. Theformulais

W HW�� HA�%� � Hë Ì�� M T B « V W Ì �Wb� Ì W H; Ì �W�� H ; Ì � (6.10)

Weapply(6.10)with � � ~ Á ° ÖÁ 7 × Á u Ö and � � O ; ® ~ u�� 7 × Á u Ö ´ . Using(6.10)againto find the« th derivativeof thesecondtermin � with � Ö � ÖÁ and � Ö�� Ö7 × Á u Ö , andapplying(a)and(b)

with � � � / resultsin

W ÌW c Ì T�� c ° /c ¨ 0 c¥° / V � ® � / ´ Ì « D �c Ì u Ö ° Ìë ¸ � M ® � / ´ Ì « D®�« � É ´ED � Ì ; ¸ � Ö ® 0�� / ´c ¸ u Ö ® 0 c¥° / ´ Ì ; ¸ u ÑÔ� ® � / ´ Ì « D �c Ì u Ö ° ® � / ´ Ì « Dc Ì u Ö Ìë Î � M T 0 ØØ V c Î0 Î ® 0 c<° / ´ Î u ÑÔ � (6.11)

As for theexponentialterm,weuseaformulafor the B th derivativeof acompositefunction

from [71] which statesthatfor � ® � ´ � f ® N ´ , N ��Z® � ´ , wehave

W HW�� H � ® � ´ � Hë Ì�� Ö�� Ì« D f \ Ì ^ ® N ´ � where � Ì � Ì ; Öë Î � M T «Ø V ® � / ´ Î N Î W HW�� H N Ì ; Î Æ (6.12)

Lettingf ® N ´ � O ; \�~ u��Q ^

and N � ¨ 0 c¥° / , weuse(6.12),(b), and(c) to get

W H ; ÌW c H ; Ì O ; ® ~ u�� 7 × Á u Ö ´ � O ; ® ~ u�� 7 × Á u Ö ´ H ; Ìë ¸ � Ö ® 0 c¥° / ´ 9 Ô ; H u Ì � ¸É D ¸ ; Öë Î � M T É Ø V ® � / ´ ¸ u Î H ; Ì ; Ö� � M ®õÉ � Ø � 0�� ´ Æ(6.13)

Using (6.11)and(6.13) in (6.10)with � � µ Ì ¸ and � � ö µ Ì ¸ ö yields the B th derivative of

100

(6.9)as

W HW c H fMAP ®gc ´ � ® / � sgn®ßµ Ì ¸ ´>´ ® � / ´ H B D0 c H u Ö � /0 O ; ® ² � 9vu @ ² � 9 @ 7 × Á u Ö ´� Hë Î � M T B Ø V ® � / ´ Î Ø Dc Î u Ö è

sgn®ßµ Ì ¸ ´ ° Îë� � M T 0�� V c �0 � ® 0 c¥° / ´ � u ÑÔ î� H ; Îë � Ö ö µ Ì ¸ ö � D ® 0 c<° / ´ H ; Î ; �Ô ; Öë Z � M T �� V ® � / ´ u Z H ; Î ; Ö� Ó � M ® � � � � 0 Ù ´ Æ (6.14)

Using(6.14)with (6.8) resultsin the following expressionfor theexactPEPof MAP de-

codedspace-timeorthogonalblockcodes��® ÌA. ¸ ´ � / � sgn®)µ Ì ¸ ´0 � /0 O ;�� ² � 9?u @ ² � 9 @�� × Sv� Ô� 9 u Öe�� H ; Öë Î � M ® � / ´ H u Î ; Ö® 0 ° F ×Ì ¸ ´ H ; Î ; Ö�� sgn®ßµ Ì ¸ ´ ° F Ì ¸� 0 ° F ×Ì ¸ Îë� � M T 0�� V /® 0 F ×Ì ¸ °�� ´ �� H ; Î ; Öë � Ö ö µ Ì ¸ ö ® F ×Ì ¸ ° 0 ´ R ×� D¡F Ì ¸ ; Öë Z � M T �� V ® � / ´ u Z H ; Î ; ×�Ó � M ® � � � � 0 Ù ´ � (6.15)

wherewehavesetû£¢Ì¤� �"¥ Ì � / for ËL¦ � .

A specialcaseis ML decodingfor which µ Ì ¸ � Ö× !$# Z]\ �:^Z]\ :9 ^ � ª . Hence,thefirst sumin

(6.15)is non-zeroonly for ØÞ�§B � / andwehave�ý® ÌA. ¸ ´ � /0 �� / � F Ì ¸� 0 ° F ×Ì ¸ H ; Öë Î � M T 0 ØØ V /® 0 F ×Ì ¸ °�� ´ Î � �whichagreeswith theresultin (3.18).

101

6.4 The Optimal Binary Antipodal Signaling

In this subsectionwe considerbinary antipodalsignalingandoptimizeit in the senseof

minimizing theBERgivenby

BER �¨�ý®ª© � × ö � Ö ´�� ® Ö ´ °«�ý®ª© � Ö ö � × ´�� ® × ´ Æ (6.16)

Normally, oneshouldusetheaveragedPEPin (6.16)with Ö�� and ×<� � � , differentiate

theresult,andfind theoptimal � and � . However, this canbea tediousjob consideringthe

PEPgivenin (6.15).Therefore,weusethePEPsat thereceiverside,i.e.,given¼

, to find

thesolutionin aneasierway. Theoptimalconstellationderivedin thiswaywill notdepend

on¼

, thusjustifying our approach.

Let usassumethat � ® Ö ´ � � , andthebits 0 and1 aremappedto ×º� � � and Ö<� � ,

respectively. Letting ¬ � � × Y (?)* , ¨ ¦ �¬ \G~ u�� ^× ¨ N � and ® � Ö× !$# Ö ; ZZ , we canwrite the

BERconditionedon Ã as

BERé� � °T ¨ ¦ � ®¨ ¦ V °x® / � �V´ °T ¨ ¦�° ®¨ ¦ V Æ (6.17)

It is easyto verify that the BER is a strictly decreasingfunction of ¦ (regardlessof ® ).

Hence,given À�Á and� , in orderto minimizetheBER,onehasto maximize¦ . Notethat ¦is a scaleddistancebetweenthesignalpoints,therefore,signalingschemeswith thesame

distancebetweentheir signalshave identicalperformance.It is clearthattheconstellation

with constantaveragesignalenergy ( )* which maximizes¦ is thezero-meanconstellation,

becausea constellationwith a non-zeromeancansimply be shiftedto reduceits energy

withoutperformanceloss.

102

Fromthezero-meancondition,wehave� � �/ � � � �andtheaverageenergy conditionrequiresthat� � × °x® / � �V´ � × � À�Á6 �Theabovetwo equalitiesresultin

® � � � � ´ � ¿ ÀÂÁ6 T � ¿ �/ � � � ¿ / � �� V �which is thereforethe optimal binary antipodalconstellation.The above constellationis

identicalto theantipodalsignalingresultin [107] for thecaseof theAWGN channel.

6.5 Bounds on the SER and BER of STOB Coded Chan-

nels under MAP Decoding

In this section,we show how theapproachof thepreviouschapterin deriving Bonferroni-

typeupperandlower boundscanbe appliedto space-timeorthogonalblock codesunder

MAP decoding.Two lessexpensive upperanda lower boundson the BER arealsopre-

sentedin Section6.6andcomparedwith theBonferronibounds.

103

6.5.1 The Symbol Error Rate

As seenin thepreviouschapter, for a constellationof size ± � 0{�, where

�is a positive

integer, andani.i.d. bit-stream,theSERis givenby

SER �¨² ; Öë ³ � M �ý®h´ ö ³ ´k� ® ³ ´ �¨² ; Öë ³ � M � ® ³ ´ � ³ è-µÌC¶� ³ ´ Ì ³ î Æ (6.18)

Notethat � ³ ®h´ ³ Ì ´ is thePEPandis givenin (6.15).We now needto find theprobabilityof

theintersectionof ´ ³ Ì and ´ ³ ¸ which is givenby� ³ ®C´ ³ Ì�· ´ ³ ¸ ´ � à ¯ æ¹¸ èAº Ì ¸ ³ � F ³ Ì ¨ Ã ° µ ³ ÌF ³ Ì ¨ Ã � F ³ ¸ ¨ Ã ° µ ³ ¸F ³ ¸ ¨ Ã î ð � (6.19)

whereº Ì ¸ ³ � 8 � ;<e» í 19 ;<e» >l � » l 9 » , W Ì ³ � ö Ì � ³ ö , Ã is definedin (3.15), and

¸ ® º � � � � ´ is the

bivariateGaussianfunction definedin (4.7). After trying a few numericalmethods,we

choseto compute(6.19)usingacombinationof 96-pointGaussianintegration[1] together

with Donnelly’s algorithm[42] for each Ã � N . This selectionled to the fastestandthe

mostaccurateresults.

6.5.2 The Bit Error Rate

Similar to thepreviouschapter, wewrite theBER as

BER �¨² ; Öë ³ � M � ® ³ ´ � ³ ®C´ \ � ^ ´ � (6.20)

where � ³ ®h´ \ � ^ ´ is givenin (5.10). Now, we needto computethepairwiseerrorprobability

between Ì and ¸ giventhat ³ is sent.i.e.,� ³ ®h´ Ì ¸ ´ � Ú ¯ æ è F ×³ Ì%� F ×³ ¸F Ì ¸ ¨ Ã ° µ Ì ¸F Ì ¸ ¨ Ã î ð �104

whoseclosed-formsolutionis givenin (6.15). We alsoneedto find theprobabilityof the

intersectionof ´ ¸ Ì and ´ ¸ Î giventhat ³ is sent,which is givenby� ³ ®h´ Ì ¸ · ´ Î ¸ ´ � Ú ¯ æ¼¸ èAº Ì Î ¸ � F ×Ì ³ � F ×¸ ³F Ì ¸ ¨ Ã ° µ Ì ¸F Ì ¸ ¨ Ã � F ×Î³ � F ×¸ ³F Î ¸ ¨ Ã ° µ Î ¸F Î ¸ ¨ Ã î ð �

Therefore,theresultsof Section6.5.1canbeusedhereto computelowerandupperbounds

on theBER.


6.6.1 Binary Antipodal Signaling

We first study the BER of binary antipodalsignalingto show the exactnessof our PEP

formulas.We simulatethetransmissionof an i.i.d. bit sequenceover theMIMO channel.

Thelengthof thebit-sequenceis max ® / ª�½6� / ªzª � BER; Ö ´ bits. Wealsolet �ý®H¯ �xª ´ � � .

We considera systemwith two transmitandonereceive antennasusingthe ¾ × (Alam-

outi) space-timecode in Figure 6.1. It is seenthat the analysisand simulationcurves

coincideeverywhere.This simulationalsoindicateshow MAP decodingcanimprove the

BER performance.TheMAP decodinggainover ML decodingis 0.63,1.57,and5.98dB

for � � ª ÆG¿ ��ª ÆGÀ , and ª ÆGÀbÀ , respectively, at BER = / ª ;<Á . Figure6.2 presentstheresultsof

Section6.4. It showsthata largegaincanbeobtainedthroughmodifying theconstellation

accordingto theprior probabilities.At aBERof / ª ;<Á , thegainof usingtheoptimalbinary

signalingandMAP detectionis respectively 2.7, 6.1, and20.1 dB for � ��ª Æ¡¿ �\ª Æ¡À , andª Æ¡ÀbÀ overanML decodedsystemwith BPSKmodulation.

105

In Figure6.3, we comparefour systems,i.e., two systemswith BPSK signalingand

ML or MAP decoding,andtwo systemswith optimalsignalingandML or MAP decoding.

Thesesystemsareindicatedby ML symmetric,MAP symmetric,ML optimum,andMAP

optimum,respectively. Weseethatif theCSNRis highenough,optimumsignalingandML

detectionoutperformsBPSKandMAP detection.In otherwords,usingadaptivesignaling

is superiorto MAP detection.Whenreceiver noiseis strong(i.e., at low À�Á ), the second

term in the argumentof the ý® �8´ function in (6.7) hasthe dominanteffect; henceMAP

decodingwith BPSKsignalsis moreeffective thanML decodingwith optimumsignals.In

lessnoisychannelconditions(high À�Á ), thefirst termin theargumentof the ý® �8´ function

becomesdominantand henceML decodingwith optimum signalingoutperformsMAP

decodingwith symmetricsignals.As previouslymentioned,MAP decodingwith optimum

signalingis alwaysbetterthanothersystems.

6.6.2 Tandem versus Joint Source-Channel Coding

Figures6.4 and6.5 comparea MAP decodedsystemwith two tandemsystemsfor a dual

transmit-singlereceivechannelwith BPSKand16-QAM signaling,respectively. Theinput

bit-streamis i.i.d. with ��®B¯ �nª ´ � ª ÆG¿�À so that the sourceentropy isì ®H¯ ´ � ª ÆGÂ .

The tandemsystemsarecomposedof 4th orderHuffmancodingfollowedby eithera 16-

stateor a 64-staterate-1/2convolutionalcodingblock. Theconvolutionalcodesarenon-

systematicandchosenfrom [101]. The lengthof the input bit-streamis 10000bits. The

testis repeated1000timesandtheaverageBER is reported.It is observedthatthetandem

systembreaksdown (dueto errorpropagationin Huffmandecoding)forÚ � § Í MÄÃ / � dB

106

for thesystemwith BPSKsignalingandamuchhigherÚ � § Í M of � ª dB for thesystemwith

16-QAM modulation.For thesystemwith 16-QAM signaling,theMAP-decodedsystem

outperformsthe16-state(64-state)tandemcodedsystemfor BER ¦ � � / ª ;ÆÅ ( ¦ 0 � / ª ;ÆÅ ),which is therangeof theerrorratesof interestin mostpracticalapplications.This is avery

interestingresultwhich shows thatMAP decoding,which is far lesscomplex thantandem

coding,shouldbethechoicefor applicationsin whichtheinputbit-streamis notuniformly

distributed (suchas FAX). Notice that as the constellationsize (which translatesto the

systembit rate)grows, theMAP decodedsystembecomesmoresuperior.

6.6.3 SER and BER Bounds for Ç -ary Signaling

Figures6.8 to 6.14show theSERandBER boundsandthesimulationresultsfor various

PSKandQAM signalingschemesfor adualtransmit-singlereceivesystem.They compare

the curvesfor two systems:onewith an equiprobablei.i.d. bit-streamandML decoding

andanotherwith ani.i.d. bit-streamwith ��®B¯��pª ´ �pª Æ¡À andMAP decoding.Thelength

of thebit-sequenceis 2000000bits. For theSER,theunionboundgivenby

Unionbound � ë ³ � ® ³ ´ ë Ì �ý® ³ . Ì ´is alsoplotted. For theBER, we alsoplot SER§ � , where

0 Zis theconstellationsize. This

approximationis goodonly athighCSNRvaluesandwhenGraymappingis usedfor signal

mapping.TheothercurveasaBERestimateis basedon thesymbolPEPs.TheexactBER

is givenin (6.20).At highCSNR,onecanreplace�ý®ª© � ¸ ö � Ì ´ with �ý® ÌA. ¸ ´ . This

will resultin anupperboundontheBER,becauseclearly ��® ÌA. ¸ ´ ¦È�ý®ª© � ¸ ö � Ì ´ .107

Figure6.8demonstratestheSERboundsfor 16-QAM signalingandGraymapping(the

mappingis shown in [186, Figure8]). The upperandlower boundsprovide an excellent

approximationto the SERfor all (even negative) CSNRvalues. It is interestingto note

thatasthesourcebecomesmorebiased,theboundsbecometighter. Thecurvesalsoshow

that the union boundbecomeslooseras the CSNR decreases.Finally, we observe that

theMAP-decodedsystemis considerablysuperiorto theML-decodedsystem.TheMAP

decodinggainat BER � / ª ;<Á is 5.5dB.

TheSERandBERcurvesof 32-PSKsignalingareshown in Figure6.9andFigure6.10,

respectively. We observe from Figure6.10that theBER upperandlower boundsarealso

very tight for the entire CSNR range. This figure also shows that the PEP-basedBER

estimateis oftenvery looseascomparedwith theotherbounds.TheMAP decodinggain

for BER (SER)= / ª ;<Á is 1.85dB (1.68dB). Figure6.11presentstheSERcurvesfor 64-

QAM andGray mapping.The MAP decodinggainat SER= / ª ;<Á is aslarge as6.7 dB.

Theupperboundfor theML-decodedsystemis up to 10%larger thanthe lower boundat

low CSNRvalues,but it becomestight astheCSNRgrows. Both boundsaretight for the

MAP-decodedcase.Theboundsfor smallerconstellationsaretighterandothervaluesof�ý®H¯��xª ´ areat leastastight asthosereportedabove.

6.6.4 The Effect of Constellation Mapping

We next demonstratethat a large gain canbe achieved via signalmappingsdesignedac-

cordingto thesourcenon-uniformdistribution over Grayandquasi-Graymappings.The

M1 mappingwasintroducedin [186] andwasdesignedfor thetransmissionof non-uniform

108

binarysourcesover singleantennaadditive white Gaussiannoisechannels.It minimizes

the SERunion boundfor singleantennaRayleighfadingchannelswith ± -ary PSK and

squareQAM signaling[217].

Basedonthedesigncriteriaproposedin [186],weconstructanM1-typesignalmapping

for thestar-QAM signalingschemeasshown in Figure6.12.TheSERperformanceof the

M1 andquasi-Graymappingsarecomparedin Figure6.13. It is observed that even for

this small signalingscheme,theCSNRgainof M1 mappingover quasi-Graymappingis

nearly2 dB with MAP decodingat SER= / ª ;<Á . Thegaindueto sourceredundancy when

the sourcedistribution variesbetween0.5 to 0.9 is 7 dB with M1 mapping. Figure6.14

comparesthe SER curves for the Gray andM1 mappingsfor 64-QAM signaling. This

figureshowsthattheM1 mapperformsverywell for MIMO channels.Thegainof theM1

mappingoverGraymappingis 3.7dB at SER= / ª ;<Á . Thegaindueto sourceredundancy

is 10.4dB.

Note that the specificM1 mappingsusedin the above scenariosarenot designedto

minimizetheBERat all � ; they outperformtheGraymappingfor � �pª Æ¡À . This is demon-

stratedfor 64-QAM signalingin Figure6.15, wherethe M1 mappingis shown to have

superiorperformanceto the Gray mappingfor � � ª Æ¡À , but it is outperformedby Gray

mappingfor � �pª ÆGÂ .

109

−10 −5 0 5 10 1510

−5

10−4

10−3

10−2

10−1

100

CSNR in dB

Bit

Err

or R

ate

ML, p = 0.5, AnalysisML, p = 0.5, SimulationMAP, p = 0.8, AnalysisMAP, p = 0.8, SimulationMAP, p = 0.9, AnalysisMAP, p = 0.9, SimulationMAP, p = 0.99, AnalysisMAP, p = 0.99, Simulation

Figure6.1: Resultsfor BPSKsignaling,6 � 0, and Ë � / .

110

−10 −5 0 5 10 1510

−5

10−4

10−3

10−2

10−1

100

CSNR in dB

Bit

Err

or R

ate

ML, p = 0.5, AnalysisML, p = 0.5, SimulationOptimum, p = 0.8, AnalysisOptimum, p = 0.8, SimulationOptimum, p = 0.9, AnalysisOptimum, p = 0.9, SimulationOptimum, p = 0.99, AnalysisOptimum, p = 0.99, Simulation

Figure6.2: Resultsfor theoptimumbinaryantipodalsignaling,6 � 0, and Ë � / .

111

−10 −5 0 5 10 15 20 2510

−6

10−5

10−4

10−3

10−2

10−1

100

CSNR in dB

Bit

Err

or R

ate

ML symmetric, p = 0.5ML optimum, p = 0.9MAP symmetric, p = 0.9MAP optimum, p = 0.9ML optimum, p = 0.99MAP symmetric, p = 0.99MAP optimum, p = 0.99

Figure 6.3: ComparisonbetweenBPSK and optimum signalingschemes,6 � 0, andË � / .

112

−5 0 5 10 15 20

10−4

10−3

10−2

10−1

100

CSNR in dB

Bit

Err

or R

ate

Tandem, 16 statesTandem, 64 statesMAP Decoded

Figure6.4: ComparisonbetweentandemandMAP-decodedschemesfor BPSKsignaling,6ÊÉ 0, and Ë É / .

113

0 5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N

0 in dB

Bit

Err

or R

ate

Tandem, 16−stateTandem, 64−stateMAP decoded

Figure6.5: ComparisonbetweentandemandMAP-decodedschemesfor 16-QAM signal-

ing, 6ËÉ 0, and Ì É / .

114

0 5 10 15 20 2510

−3

10−2

10−1

100

CSNR in dB

Sym

bol E

rror

Rat

e ML

MAP

union BoundHunter upper boundsimulationKounias lower bound

Figure6.6: SERcurvesfor 8-PointStarQAM modulationwith quasi-Graymapping,Í ÉÎ, and Ì ÉÐÏ .

115

0 5 10 15 20 2510

−3

10−2

10−1

100

CSNR in dB

Bit

Err

or R

ate

ML

MAP

PEP−based estimateupper boundsimulationlower bound

Figure6.7: BERcurvesfor 8-PointStarQAM modulationwith quasi-Graymapping,ÍËÑÎ, and ÌÒÑ Ï .

116

−5 0 5 10 15 20 2510

−3

10−2

10−1

100

CSNR in dB

Sym

bol E

rror

Rat

e

ML

MAP

union boundHunter upper boundsimulationKounias lower bound

Figure6.8: SERcurvesfor 16-QAM signaling,ÍÊÑ Î, and ÌÒÑ Ï .

117

−5 0 5 10 15 20 25 30 3510

−3

10−2

10−1

100

CSNR in dB

Sym

bol E

rror

Rat

e

MLMAP


Figure6.9: SERcurvesfor 32-PSKsignaling,ÍËÑ Î, and ÌÒÑ Ï .

118

0 5 10 15 20 25 3010

−3

10−2

10−1

CSNR in dB

Bit

Err

or R

ate

ML

MAP

PEP−based estimateupper boundsimulationlower boundSER/5

Figure6.10:BER curvesfor 32-PSKsignaling,ÍËÑ Î, and ÌÒÑ Ï .

119

−5 0 5 10 15 20 25 3010

−3

10−2

10−1

100

CSNR in dB

Sym

bol E

rror

Rat

e

ML

MAP


Figure6.11:SERcurvesfor 64-QAM signalingwith Graymapping,ÍËÑ Î, and ÌÒÑ Ï .

120

Ó

Ô

ÕÖØ×[100]

ÕÖ�Ù]×[101]

Õ×[000]

ÕÙ]×[111]

Õ ×ÛÚ[001]

Õ Ù]×�Ú[011]

Õ Ö�×�Ú[010]

Õ Ö�Ù]×ÛÚ[110]

Figure6.12:Thestar-QAM signalingschemewith M1 mapping.

121

0 5 10 15 20 25 3010

−4

10−3

10−2

10−1

100

CSNR in dB

Sym

bol E

rror

Rat

e

quasi−Gray mapping, P(X = 0) = 0.5M1 mapping, P(X = 0) = 0.5quasi−Gray mapping, P(X = 0) = 0.9M1 mapping, P(X = 0) = 0.9

Figure6.13: Comparisonbetweenthe quasi-Grayandthe M1 mappingswith StarQAM

for asystemwith ÍÊÑ Î, and ÌÒÑ Ï .

122

0 5 10 15 20 25 30 35 4010

−4

10−3

10−2

10−1

100

CSNR in dB

Sym

bol E

rror

Rat

e

Gray mapping, P(X = 0) = 0.5M1 mapping, P(X = 0) = 0.5Gray mapping, P(X = 0) = 0.9M1 mapping, P(X = 0) = 0.9

Figure 6.14: Comparisonbetweenthe M1 and Gray mappingsfor 64-QAM signaling,ÍÊÑ Î, and ÌÜÑ Ï .

123

−5 0 5 10 15 20 25 3010

−3

10−2

10−1

100

CSNR in dB

Bit

Err

or R

ate

Gray mapping, P(X = 0) = 0.5M1 mapping, P(X = 0) = 0.5Gray mapping, P(X = 0) = 0.9M1 mapping, P(X = 0) = 0.9

Figure6.15:ComparisonbetweentheBERperformanceof theM1 andGraymappingsfor

64-QAM signaling,ÍÊÑ Î, and ÌÜÑ Ï .

124

Chapter 7

Quantization with Soft-decision

Decoding for MIMO Channels

7.1 Introduction

Likemany othererrorprotectionschemesthataredesignedin thespirit of Shannon’s sep-

arationtheorem[160,161],space-timecodesaredesignedto operateon uniform indepen-

dentandidenticallydistributedbit-streams.As explainedin Subsection1.3.1,Shannon’s

separationtheoremdoesnot take into considerationconstraintson systemcomplexity and

delay. Sincereal-world communicationsystemsareconstrained,systemswith independent

sourceandchannelcodes,known as tandemcoding systems,may have inferior perfor-

mancecomparedwith systemswhichperformsourceandchannelcodingjointly.

In this chapter, we considerthe transmissionof continuous-alphabet(analog)sources

over MIMO channels.Our proposedsystemaddsonly two blocks,with modestcompu-

125

tationalneeds,to a conventionalspace-timecodedsystem.As thesystemmayhave mul-

tiple receive antennas,an importanttask is the properprocessingof the received signals

from thedifferentantennas.Weproposeto addressthisproblemby performingspace-time

soft-decodingfollowedby linear combiningat the receiver. The linear combinerhasthe

following key advantages:i) it hasaverysimplestructure,ii) its designcriterionallowsthe

COVQ index transitionprobabilitiesto bedeterminedin closedform, andiii) its outputis

continuous,makingsoft-decisiondecodingpossible.Inspiredby thework in [6, 146], we

usesoft-decisiondecodingasopposedto soft-decodingmethodssuchasthework in [179],

to exploit the soft informationavailableat the outputof the linear combiner. Our choice

is motivatedby two factors.First, soft-decisiondecodingmaybeimplementedvia a Ý -bit

uniformquantizerat thereceiver(not to beconfusedwith theCOVQ blocksatthetransmit-

terandthereceiver)whichmakesthetaskof decodingcomputationallysimple.In contrast,

thefirst versionof soft-decodingin [179] needsthecomputationof trigonometricfunctions

andmatrix multiplicationandthesecondversionalsorequiresmatrix inversion. Second,

asdiscussedin [6], theperformanceof soft-decisiondecodingcanbecloseto thatof soft-

decodingandrequireslesscomputationalcomplexity (althoughits storagecomplexity may

behigher).

We show that the concatenationof the space-timeencoder, the MIMO channel,the

space-timesoft-decoder, thecombiner, andtheuniform scalarquantizeris equivalentto a

binary-input,Î�Þ

-outputdiscretememorylesschannel(DMC) usedßÆà times,where ß and àarethequantizerdimensionandrate,respectively. Thestep-sizeof theuniform quantizer

usedfor soft-decisiondecodingis numericallyselected,sothatthecapacityof theequiva-

lentDMC is maximizedfor eachvalueof thechannelsignal-to-noiseratio(CSNR).This is

126

aheuristic(sub-optimal)criterion,but asthesimulationresultsof [146] demonstrate,there

is a substantialcorrelationbetweenmaximizingchannelcapacityandminimizing distor-

tion. We show thatthetransitionprobabilitiesof this equivalentDMC canbeexpressedin

termsof thesymbolpairwiseerrorprobabilityof theML-decodedspace-timeorthogonal

blockcodedchannel.Hence,theseprobabilitiescanbedeterminedusing(3.18).

We designthree soft-decisiondecodingCOVQs for the equivalent DMC. The first

COVQ is the classicalCOVQ which assumesthat the index transitionprobabilitiesare

known at both the transmitterandthe receiver. The COVQ codebookis determinediter-

atively usingthemodifiedgeneralizedLloyd algorithm[66]. As theCSNRis not always

availableat thetransmitter, we considerthedesignof two fixed-encoderadaptive-decoder

(FEAD) COVQs. In a FEAD COVQ, the encoderis designedfor a fixed CSNRandthe

decoder, which canestimatethechannelfadingcoefficientsandtheCSNR,adaptsitself to

thechannelconditions.Ourfirst FEAD COVQ usesonly theknowledgeof theCSNRatthe

receiver (asin [180]), while thesecondone,whichwe call theOn-lineFEAD COVQ, em-

ploys alsotheknowledgeof thechannelfadingcoefficientsat thereceiver and,asa result,

it outperformstheFEAD COVQ. An importantfeatureof FEAD COVQ is thatits decoder

codebooksarecomputedin termsof thetransmitterparameters,andnot througha training

processasfor classicalCOVQ. Therefore,this methoddoesnot needa large memoryat

thereceiver to storeadifferentcodebookfor eachvalueof theCSNR.Wedemonstratethat

with a properchoiceof the designCSNR,the performancelossof FEAD COVQ canbe

significantlyreducedascomparedwith theclassicalCOVQ.

127

Figure7.1: Systemblock diagram,whereevery á bits in a ßÆà -bit index â is transmitted

via a space-timecodeword ã°ÑËägå�æªç]è:è1è:çEåbéëê (with orthogonalcolumns å�ìIç�í¼Ñ Ï ç]è:è:è1ç�îUÑï á ), receivedas ðñÑÊäCòóæEç4è1è:è1ç ò%éóê , andspace-timesoft-decodedas ôðjÑÊä ôò æ ç]è:è1è:ç ôò%õ�ê . For

simplicity, wehaveassumedthat áöÑ�ß<à .

7.2 System Components

Thesystemblock diagramis shown in Figure7.1. A COVQ encoderformsa vector ÷ of

dimensionß from theincomingscalarsourcesamples.It thenencodes÷ at a rateof à bits

per sample(bps) into a binary index â of length ßÆà bits, whosebits areeachmappedto

a signal in the BPSK signalingschemeconsistingof øúù æhû and øúùGü û . Encodingis specified

by thedecisionregions ý�þAìgÿ�� ì�� (�� Ñ Î��

), which form a partitionof the ß -dimensional

space,using the rule that â¨Ññí if ÷� þAì . Index â is then sentover the channeland

receivedasindex � . TheCOVQ decodersimply uses�� , the � th elementin thecodebook,

to reconstruct÷ as �÷ Ñ�� . Thegoal in COVQ designis to minimizetheexpectedvalue

of �ª÷��÷�� ü throughfinding theoptimalpartitionandcodebook.

Our objective is to modeltheconcatenationof theblocksbetweentheencoderandthe

decoderof thevectorquantizerby adiscretechannelandthendesignefficientvectorquan-

tization systemsfor this channel.The systemis designedso that the channeljudiciously

128

incorporatesthe soft-informationof the STOB-codedchannelandadmitsa closed-form

expressionfor its transitionprobabilities. This is achieved by designinga soft-decision

decodingspace-timereceiver which consistsof a space-timesoft detector, followedby a

linear combiner(with optimizedweight coefficients), anda simple Ý -bit scalaruniform

quantizer(whosecell sizeis chosenso the capacityof the equivalentdiscretechannelis

maximized). As a result of the simplememorylessstructureof our receiver anddue to

theorthogonalityof thespace-timecode,we obtainthat theequivalentdiscretechannelis

indeeda binary-inputÎ Þ

-outputmemorylesschannelwhosedistribution canbeeasilyde-

terminedanalyticallyin termsof thesystemparameters.In thefollowing, we describethe

systemcomponentsin detail.

7.2.1 The Channel

From(3.12),theoutputof theSTOB soft-decoderis givenby

ôò��Ñ ï�� ! #" ô$ � çwhere ï � Ñ ï&% ')(+* Í . We rememberthateachentry ô, � - of ôò�� is associatedwith only one

transmittedsymbolin . From(6.3),weknow thatô� �-. i.i.d. / ä10Æç ï2� � êEç 3ÄÑ Ï ç]è:è:è1ç54�ç�6-Ñ Ï ç4è1è:è1ç?á�ç (7.1)

andhencesymbol í canbe detectedby only consideringthe í th entry of the vectors ôò�� ,Ï87 3 7 4 . For our application,this will imply that the bits correspondingto a COVQ

index canbedetectedindependently. Thepairwiseerrorprobabilitybetweena pair of ML

129

decodedSTOB codedBPSKsymbolsø ù ì�û and ø ù � û wasgivenin (3.18)as9 ä1:�ê<;Ñ>=tähø]ù ì�û@? ø]ù � û ê Ñ ACBEDGF�HI:KJ �LNMÑ ÏÎPO Ï � :J Î " : ü<Q õ�R æS ��UT Î ßßWV Ïä Î : ü "YX ê 2Z ç (7.2)

where: Ñ % Î ï�'2([* Í .

7.2.2 Soft-Decision Decoding and the Equivalent DMC

Soft-Decision Decoding: Linear Combining and Scalar Quantization

Many communicationsystemsemploy harddecodingin processingthereceivedsignalsof

space-timecodedsystemsandsodonotexploit thesoft informationavailableat thespace-

time soft-decoderoutputs ôò æ ç]è:è:è1ç ôò õ . Methodsthat exploit soft information can provide

significantperformancegains. In our case,in additionto usingthe soft informationeffi-

ciently, thesolutionshouldallow theCOVQ index transitionprobabilitiesto bedetermined

in closedform, sincethis is requiredfor the COVQ designandencodingphases.As we

illustratebelow andin Section7.4,linearcombininghasbothof theaboveproperties.

This designproblemis a variationof the classicalmaximumratio combining(MRC)

setup[150] in whichthesignalsto becombined(thespace-timesoft-decodedsignals)have

differentnoisevariances(see(7.1)). Letting ô\ � ì ;Ñ ô, � ì * ä ï � � � ê , wecanwrite theoutputof the

linearcombineras(seeFigure7.2)

ô\ ì Ñ õS � �Aæ@] � ô, � ìï � � � Ñ õS � �Aæ^] � ähø ì " ô_ �ì êEç í Ñ Ï ç]è:è1è:ç á�ç (7.3)

130

where ø ì is the í th transmittedsymbolandthe ] � ’s areweightingcoefficientsto bedeter-

mined. From (7.1), we know that the distribution of ô_ �ì , the noisecomponentof ô\ � ì , is/ ä10�ç ï�* ä ï�` ü � � ê?ê . Therefore,theSNRat theoutputof thelinearcombineris

SNRCO Ñ HIa õ� �Aæ ] � L üa õ� �Aæ bb `dc B+e ] ü�gf (7.4)

In linear combining,the objective is thento choosethe weights ý ] � ÿ õ� �Aæ so that SNRCO

is maximizedby enforcingthe constraint a õ� �Aæ ] � Ñ Ï . Therefore,the problemis to

minimizethedenominatorof (7.4), i.e.,findinghjilk õS � �Aæ ïï ` üm� � ] ü� çsubjectto a õ� �Aæ ] � Ñ Ï è Minimizing theLagrangiannpo Ñ S � ïï ` üm� � ] ü� ">q O S � ] � � Ï Z çwe obtainthat q Ñr� Î ï)* ä ï ` ü � ê , and ] � Ñ � � *)� . Therefore,theoutputof thecombiner

canbewritten from (7.3)as ô\ ì Ñ¨ø ì " ô_ ìIç (7.5)

where ô_ ì Ñ a õ� �Aæ ] � ô_ �ì . Wenotethat Ats ô_ üìvu Ñ ä ï � � ê R ü a õ� �Aæ A�w ô_ � üìyx , andtherefore,ô_ ì . / T 0Æç Íï�'2(+� V ç í Ñ Ï ç]è1è:è:ç á�è (7.6)

It is easyto verify that when Ý Ñ Ï (i.e., harddecoding),the concatenationof the linear

combininganddecisionblocksis equivalentto ML decoding.

The linear combineroutput, ô\ ì , is next fed into a “uniform” scalarquantizerwhich

actsasthe soft-decisiondecoder. Let us indicatethe decisionlevels of this quantizerby

131

ýIz ÿ��{R æ � R æ andits codepointsby ý î ÿ��|R æ �� , where� Ñ Î{Þ

is thenumberof thecodewords.

As ô\ ì cantake any realvalue,thequantizershouldhave two unboundeddecisionregions.

Thedecisionregionsof theuniformquantizeraregivenby

z Ñ }~~~~~� ~~~~~��ç if ß Ñ�� Ïägß " Ï � � * Î ê�� ç if ß Ñ�0�ç f�f�f ç � � Î" ��ç if ß Ñ � � Ï ç

andthequantizationrule ��ä f ê is simply

��ä ô\ ê-Ñ ß�ç if ô\ Òä�z R æ ç�z u ç ßxÑ�0�ç f�f�f ç � � Ï èTransition Probabilities of the Equivalent DMC

For COVQ design,weneedto derivethetransitionprobabilitiesof theÎ��

-inputÎ Þ ��

-output

discretechannelrepresentedby the concatenationof the space-timeencoder, the MIMO

channel,thespace-timesoft-decoder, thelinearcombiner, andtheuniformquantizer. Since

thedetectionof bits which correspondto eachquantizerindex is decoupled,we notethat

thetransmissionanddecodingof COVQ indicesareindependentof oneanother, andalso

thediscretechannelis equivalentto abinary-inputÎ�Þ

-outputDMC usedß<à times.Weshall

referto this discretechannelasthe“equivalentDMC”.

Therequiredsetof thetransitionprobabilitiesare =täCî �� ø]ù æhû ê and =täCî �� øúùGü û ê for all î ,where ø ù æhû and ø ù�ü û are the BPSK constellationpointswhich, without lossof generality,

areassumedto correspondto 1 and0, respectively. Decisionis madein favor of the ß th

codepointif theoutputof thelinearcombinerfalls into the ä�z R æEç�z u interval of length � .

132

Using(7.5)and(7.6)wecanwrite

=täCî �� ø]ù ì¤û çm� ê Ñ =tä�z R æ 7 ø]ù ì¤û " ô_ ì��>z �� êÑ F H ä�z R æ��rø ù ì�ûhê�: J � L �PF H ä�z �rø ù ì¤ûgê+: J � L è (7.7)

Theexpectationover � of eachof theabove Ftä f ê functionscanbedeterminedusing(7.2)

to yield

=täCî �� ø ù ì�ûCê�Ñ 9Y�� z R æ��Üø ù ì�û��:�� 9Y�� z �rø ù ì�û��:��Øç (7.8)

where9 ä f ê is definedin (7.2).NotethattheDMC transitionprobabilitymatrixis symmetric

in thesenseof [63].

For our ß -dimensionalCOVQ with rate à shown in Figure7.1, we denotethe natural

binaryrepresentationof the index of decisionregion þAì by ý��5�kÿ ��d�Aæ andthatof codevectorî � by ý��#�kÿ ��d�Aæ , where�#� is abinary Ý -tuple.As theDMC is memoryless,theCOVQ index

transitionprobabilitiescaneasilybecomputedas

=�� úäl3 � íIê�Ñ �� Aæ = � îC�� ø ùGü R�� û � è (7.9)

7.2.3 The Step-Size of the Uniform Quantizer at the Decoder

Thefinal designparameterof thesystemis � , thestep-sizeof theuniformquantizerat the

receiver. For agivenCSNR,wenumericallyselectthe � whichmaximizesthecapacityof

theequivalentDMC. This is a sub-optimalcriterionsinceour ultimategoal is minimizing

theMSE,not maximizingthecapacity, but asthesimulationresultsof [146] demonstrate,

thereis a strongcorrelationbetweenhaving a high channelcapacityandreducedmean-

squarederrordistortion.

133

Foragivensoft-decisionresolutionÝ andCSNR'2( , wedeterminethestep-size� which

maximizesthe capacityof the DMC by maximizingthe mutual informationbetweenthe

DMC input andoutput. Becausethechanneltransitionprobabilitymatrix is symmetric,a

uniform input distributionachieveschannelcapacity[63]. Notethatthestep-sizedoesnot

dependon therateor dimensionof theCOVQ (usedto quantizethesource),andis only a

functionof Ý andtheCSNR.As atypicalsetof results,we list, in Table7.1,thecapacityof

theequivalentDMC versusthe“optimal” step-sizeof theuniformquantizerfor Alamouti’s

[9] dual transmitsinglereceive setup. Similar resultscanbe derived for systemswith a

differentnumberof transmitantennas,receive antennas,or space-timecodes.As shown

in Figure7.3,whenthestep-sizeis very small (closeto zero)or very large,soft-decision

decodingdoesnotsignificantlyincreasechannelcapacity. Wealsonotethatif theCSNRis

high,soft-decisiondecodingis not verybeneficialin termsof improving channelcapacity.

However, for moderateandlow CSNRs,andwith the optimal choiceof � , soft-decision

decodingsignificantly increaseschannelcapacity. For example,at CSNR Ñ � ÎdB in

Table7.1, thereis a 15% benefitin usingsoft-decisiondecodingwith Ý Ñ� bits. Also

notethat thechannelcapacityincreaseslessthan1% from ÝxÑU¡ to ÝxÑ� evenfor severe

channelconditions. This shows that typically Ý Ñ¢¡ achievesmostof the capacitygain

offeredby soft-decisiondecoding.

134

7.3 Quantization with Soft-Decision Decoding

7.3.1 Soft-Decision Decoding COVQ

Thetransitionprobabilitygivenin (7.9)canbeusedin themodifiedgeneralizedGLA algo-

rithm [66] to designasoft-decisiondecodingCOVQ for space-timecodedMIMO channels

asexplainedbelow. Every input ß -tuple is encodedat a rateof à bits persample.There-

fore, the input spaceis partitionedinto�� Ñ Î��

subsets.As we useBPSK modulation,

a vectorof ß<à real-valuedsignalsis received for every transmittedindex. This vector is

soft-decisiondecodedat a rate of Ý bits per dimension. Therefore,each ß -dimensional

sourcevectoris decodedto oneof the�£ Ñ Î Þ ��

codevectors.Theinputspacepartitioning

andthe codebookareoptimizedbasedon two necessaryconditionsfor optimality using

trainingdata ý{÷��eç5¤ëÑ¥0�ç]è:è1è:çm¦ � Ï ÿ asfollows.

§ Thenearestneighborcondition: for a fixed codebookand í�Ñ¨0�ç f�f�f ç �� Ï , the

optimalpartition ©Wª Ñ ý�þ|ªì ÿ isþ ªì Ñ¬«�÷> �¯®5R æS � �� =�� ä°3 � í ê+±�äh÷"ç�� ê 7 �¯®5R æS � �� =�� ä°3 � ¤Cê�±�äh÷"ç�� êEç+²N¤m³ (7.10)

where ´ is a training vector, ýI� � çµ3 Ñ¶0�ç]è:è:è1ç �·£ �¹¸�ÿ is the codebook,±�ä1´"ç��ê is

thesquaredEuclideandistancebetween and � , andtiesarebrokenaccordingto a

presetrule.

135

§ Thecentroid condition: givena partition ©ËÑ ý�þAìgç�í Ñº0�ç]è:è1è:ç �� »¸�ÿ , theoptimal

codebook¼ ª Ñ ýI� ª� ÿ is

� ª� Ñ � � R æS ì�� =�� {ä°3 � í ê S��½ ¾ �À¿vÁÃÂ ´�� R æS ì�� =�� úäl3 � íIê � þAì � ç 3 Ñ�0�ç f�f�f ç �·£ �t¸bè (7.11)

where � þAì � is thenumberof thetrainingvectorsin þAì .Trainingconsistsof usingtheaboveconditionsiteratively to updatethecodebookuntil the

averagetrainingdistortiongivenbyn Ñ ¸ß�¦ Ä R æS �� ® R æS � �� =�� ä°3 � Å ä1´|�1ê ê�±�ä�´��Cç�� êconverges,whereÅ ä1´�ê is theindex of thepartitioncell to which ´ belongs.

7.3.2 Fixed-Encoder Adaptive-Decoder Soft-Decision Decoding COVQ

Equations(7.2), (7.7)-(7.9)show that the CSNRshouldbe known at both the transmit-

ter andthe receiver to computetheCOVQ index transitionprobabilities.TheCSNRcan

be estimatedat the receiver and then fed back to the transmitter. As the feedbackpath

may not alwaysbe available,it is of particularinterestto considerthe casewhereno in-

formationaboutthechannelstateis availableto thetransmitter. In [180], a fixed-encoder

adaptive-decoder(FEAD) COVQ is proposedwhich addressesthis issuefor a different

setupinvolving hybrid digital-analogSISO transmissionsystems. In the following, we

show how to designa FEAD COVQ for thesoft-decisiondecodedSTOB codedchannel.

In additionto having multiple antennas,our FEAD COVQ differs from the onein [180]

136

in that we have a digital channel(insteadof a hybrid digital-analogchannel).The block

diagramof the FEAD COVQ looks the sameasin Figure7.1. The key differenceis that

heretheencoderpartitionmatchesa “designCSNR”andthereceivercodebookis adapted

to theactualCSNR.Theencoderusesadesigncodebooký�ÆAìhÿ � � R æì�� to find its partitionviaþ ªì Ñ « ´> � � R æS � �� =�� ä°3 � í ê+±Aä�´"ç5Æ � ê 7 � � R æS � �� =�� äl3 � ¤hê+±�ä1´"ç5Æ � êªç[²N¤ ³ èBecauseweassumecompletelack of informationat thetransmitter, theencodercodebooký�ÆAìhÿ is designedassumingthechannelis harddecoded(i.e, that Ý�ÑU¸ ).

Theproblemis thento adaptthedecodercodebookaccordingto theactualCSNRvalue,

while theencodercodebookremainsfixed. Let usdenotetheencoderindex by â andthe

decoderindex by � . Theaveragedistortionperdimensionis givenbyn Ñ ¸ß AÇwÈ�!´É�P�Ê� ü x Ñ ¸ß ��®mR æS � �� A�wË�!´É�P�Ê� ü � ��Ñ>3 x =��äl3<êÑ ¸ß � ® R æS � �� A w �!´É�Ì� � � ü � ��ÑÍ3 x =��ä°3<êªè (7.12)

The goal in quantizerdesignis to derive the � � which minimize (7.12). This gives the

optimal � � in theminimumMSE senseas� ª� Ñ AÎsÏ´ � ��Ñ>3 u Ñ � � R æS ì�� AÎsÏ´ � â Ñ¨í?çÐ��Ñ>3 u =��!� �ÆäCí � 3<êÑ � � R æS ì��ÒÑ ìÓ=��m� ��äCí � 3<êªç 3ÄÑÔ0�ç f�f�f ç �·£ �t¸�è (7.13)

whereÑ ì Ñ�Õ ÁÃÂ ´@Ö�ä1´Øê+±�´ denotesthemeanof thesamplesin þAì and(7.13)followsbecauseAts×´ � âÄÑ¨í çÐ� ÑY3 u Ñ AÎsÏ´ � âxÑ í u Ñ Ñ ì . Note that the decodercan simply determine=��!� �ÆäCí � 3<ê as = �m� � äCí � 3<ê-Ñ = �� ä°3 � í ê+=�� äeí êa � � R æ�d�� = �� ä°3 � ¤Cê�=��úä1¤Cê ç (7.14)

137

where =��{äCíIê ÑØ=öäeâ Ñ�í êÛÑU=tä1´Ù þAìkê . Also notethat =�� äeí ê and Ñ ì canbeapproximated

in thetrainingphaseof theencoderasfollows

=��{äeí ê{Ú � þAì �¦ ç Ñ ì@Ú ¸� þAì � S��½ ¾ �l¿vÁÃÂ ´|�eè (7.15)

From (7.13), we observe that unlike COVQ, given the encodermeans ý Ñ ìhÿ � � R æì�� , the

FEAD-COVQ doesnot requirea training phaseto determineýI� � ÿ . In otherwords, the

decodercodebookis computedfrom the channeltransitionprobabilitiesandthe encoder

meansvia (7.13)and(7.14).

7.3.3 On-line FEAD Soft-Decision Decoding COVQ

Oneof our assumptionsis that the receiver hasperfectknowledgeof the channelfading

coefficientswithout error. In light of this assumption,we observe in equation(7.5) that

theoutputof the linearcombinerhasan identicalform to theoutputof anadditive white

Gaussiannoisechannelwith the varianceof the additive noise ô_ ì known at the receiver

andgivenby (7.6). This motivatesusto applysoft-decisiondecodingdirectly on ô\ ì using

thestep-size� derived in [146, SectionII]) for AWGN channels.Thechanneltransition

probabilities,giventhepathgainsmatrix � , aregivenby

=öäeî �� ø ù � û çm� ê Ñ =tä�z R æ 7 ø ù � û " ô_ ì��Ùz �� êÑ F H äÛz R æ��rø ù � ûCê+: J � L �EF H ä�z �rø ù � ûCê�: J � L çNote that the above derivation is valid if the channelfadingcoefficientsremainconstant

duringthetransmissiontimeof anindex.

138

7.4 Numerical Results and Discussion

7.4.1 Implementation Issues

Weconsiderthetransmissionof zero-meanunit-variancei.i.d. GaussianandGauss-Markov

sourcesover MIMO channels.500,000trainingvectorsand850,000testvectorsareem-

ployed. Eachtestis performed5 timesandtheaveragesignal-to-distortionratio (SDR)in

dB is reported.MIMO systemswith Ü transmitand 4 receive antennasarereferredto asä�Ü - 4�ê systems.Alamouti’scode[9] is usedfor thedualtransmit-antennasystems.Thereal

(rate1) codeof [192] is employedfor thequad-transmitsystembecauseour constellation

is real.

Several training strategieswere examined,and the bestone in termsof having con-

sistentresultsandhigh training SDR wasusedasfollows. For any given COVQ rate à ,

dimensionß , andnumberof soft-decisiondecodingbits Ý , we first train a ß -dimensional

rate-Ý�à VQ with theSplit algorithm[66]. We next usetheSimulatedAnnealingalgorithm

[49] which aimsto minimize the averageend-to-enddistortionfor a givenVQ codebook

throughoptimizingtheassignmentof indicesof theVQ codevectors.Theaverageoverall

distortioncanbewrittenasn Ñ ¸ß AÎs×±�ä1´"ç��ê u Ñ ¸ß � � R æS ì�� ®5R æS � �� AÎsÏ±�ä1´"ç��ê � â Ñ§í çÐ��Ñ>3 u =��Ý ��äCâÄÑ¨í çÃ� ÑY3<êÑ ¸ß S ì =��{äCí ê S � =�� äl3 � í ê¯Þ ÁÃÂ �!´É�ß�&à ù � û�� ü Ö�ä1´�ê+±�´Ñ ¸ß A�wÈ�!´�� ü x " ¸ß S ì =��{äeí ê S � =�� úäl3 � íIêÐá1�@à ù � ûPçúä��@à ù � ûâ�Yã Ñ ìkê�ä ç (7.16)

where áÈå[ç�æ�äLÑ a ì ] ìÓç�ì is the standardinner productand èé ý�0�ç f�f�f ç �£ �ê¸�ÿ ?139

ý�0�ç f�f�f ç �£ �»¸�ÿ is the one-to-onemappingfunction to be optimized. It follows that the

costfunctionto beminimizedequals� � R æS ì�� =��{äeí ê � ® R æS � �� =�� {äl3 � íIêÐá1�ëà ù � ûPç ä��Nà ù � ûN�Eã Ñ ìeê�ä èSimulatedAnnealingis usedonly at thehighestCSNR.We thenuseanapproachsimilar

to the one in [50]; namely, we usethe modified generalizedLloyd algorithm to derive

theCOVQ codebooksstartingfrom thehighestCSNRto the lowestandvice-versa.This

methodis referredto asthe “decrease-increase”(DI) method.Anotherway to obtainthe

codebookscouldbestartingfrom thehighestCSNRdown to thelowest,whichwereferto

asthe“decreasing”method.

Table7.2 comparesthe resultsof the DI anddescendingmethodsfor variousSTOB

codedsystems.It is observedthattheDI methodis mostlybeneficialat low CSNRvalues.

This is becauseat low CSNRsomeencodercells areempty. Empirical resultsshow that

thesecellsareoptimizedmoreefficiently throughtheCSNR-increasingloop.

7.4.2 COVQ for Various MIMO Channels

Figure7.4 plots the SDR curvesof variousCOVQ-basedspace-timecodedsystemsasa

function of the CSNR.Even at the low COVQ dimensionand rateconsideredhere,the

gainof usingMIMO channelsover theSISOchannelis obvious. For example,at SDR=

5 dB, the (2-1) systemoutperformsthe SISOsystemby 6 dB (for hard-decoding)andis

outperformedby the(2-2)systemby 4.3dB.Thisfigurealsodemonstratestheeffectiveness

of our linearcombiner. Notethatasthesignalpowercollectedby the(2-2)systemis twice

140

thatof the(4-1)system,theformerhasabetterperformance,althoughthediversitygainof

bothof thesystemsis thesameandequalsÜ84ÒÑ X .We observe in Table 7.1 that soft-decisiondecodingbecomeslessbeneficialas the

CSNRgrows. Increasingthe numberof transmitor receive antennasresultsin a CSNR

gain dueto spacediversity. Therefore,we expectthat increasingthe numberof transmit

or receive antennaswould leave little roomfor furtherenhancementthroughsoft-decision

decoding. It follows from (7.6) that betweentwo systemswith the samediversity gain,

codinggain ï , andCSNR '2( , theonewith fewer transmitantennashasa higherSNRat its

linearcombineroutput.In otherwords,giventwo systemswith thesamediversitygain,the

onewith moretransmitantennasobtainsa largersoft-decisiondecodinggain. This result

canbe statedmore intuitively: systemswith more receive branchescollect more signal

power. Hence,the SNR at their linear combinerwould be higher, makingsoft-decision

decodinglesseffective. This observationis supportedby thesimulationsof Figure7.4: at

CSNR= 4 dB, thesoft-decisiondecodinggainin SDRis 0.29dB for bothof the(2-1)and

(4-1) systems;this gainreducesto 0.12dB for the(2-2) system.

7.4.3 COVQ versus Tandem Coding

Wenext compareour COVQ-basedsystemwith traditionaltandemcodingschemeswhich

useseparatesourcecodingandchannelcodingblockswith VQ andconvolutionalcoding

(CC), respectively. We consider, in Figure7.5, a (2-1) systemandquantizationwith di-

mensionß Ñìã . The overall rateis 3.0 bpsandhencetherearesix choicesfor the (VQ,

CC) coderates;namely, (0.5,1* 6), (1.0 , 1* 3), (1.5,1* 2), (2.0,2* 3), (2.5 , 5* 6), and(3.0,

141

0). Thefirst four convolutionalcodeshave 64 statesandarenon-systematicwith freedis-

tancesof 27 [116], 14, 10, and5 [101]. For rate5* 6, we usea rate-compatiblepunctured

convolutional (RCPC)codewith a rate-1* 2 mothercode. The generatorpolynomialsof

therate-1* 6, 1* 3, 1* 2, and2* 3 convolutionalcodesaregivenby (754,644,564,564,714,

574),(574,664,744),(634,564),and(3, 4, 5; 4 3 7). They arethestrongestcodesgiven

in [101, 150] for thegivennumberof states.Thegeneratorpolynomialsaredefinedasin

[101]. Thegeneratorpolynomialsof themothercodefor therate-5* 6 codeare(554,744)

andits puncturingmatrix is givenby� æ æ�� æ��æ�� æ�� æ � [150]. Figure7.5 shows a typical behavior:

the jointly designedCOVQ outperformsthe substantiallymorecomplex tandemsystems

almosteverywhere. Furthertestswith i.i.d. sourcesyield even more supportive results

towardsCOVQ.

Notethatif oneaimsto designanunequalerrorprotection(UEP)joint source-channel

coderwith the above separatecoders(i.e., selectthe besttandemcoderat eachCSNR),

oneneedsto designan algorithmto allocatethe sourceandchannelcoderatesfor each

givenCSNR,thusincreasingthecomplexity of theUEPsystem.COVQ doesnothave this

problemsinceerrorprotectionin COVQ is built-in.

7.4.4 COVQ, FEAD COVQ, and On-line FEAD COVQ

Figure 7.6 demonstratesthe performanceof a (2-1) systemquantizingan i.i.d. / ä10�çv¸úêsourcewith variousrate-2bpsFEAD COVQs with dimensionß Ñrã . A FEADí COVQ

is onewhosedesignCSNRis ' dB. TheFEAD VQ assumesthatthechannelis noiseless;

henceit hasalowercomputationalcomplexity attheencoder. Thefigureshowsthatsuchan

142

assumptionwill leadto a significantSDRlossat low to mediumCSNRs.FEAD� COVQ

alsosuffers from a considerableperformancedegradationat high CSNRs. It seemsthat

assumingamid-rangeCSNRof 8 dB (for thegivenMIMO system)will leadto reasonable

performanceeverywhere.

Thethreequantizerspresentedin thischapterarecomparedin Figure7.7,whereaunit-

varianceGauss-Markov sourceis quantizedwith dimension2 andrate1 bpsandsentover

a(2-1)system.TheFEAD VQsaredesignedassuminganoiselesschannel,hencetheSDR

of the threesystemsbecomecloserastheCSNRgrows becausethechannelmismatchof

the VQs decreases.The On-line FEAD VQ maintainsits gain over the FEAD VQ when

soft-decisiondecodingis employed. It is alsoobserved that althoughthe On-line FEAD

VQ encoderassumesthe channelis noiseless(CSNR ? � ), it slightly outperformsthe

COVQ, which is designedfor theexactCSNR,at highCSNRs.

143

îï æì Ó�ðÏñ òIólô ��õ æ+ö R æ Óî÷ æì ðøñ ò5ù æîï üì Ó�ð ñ ò ólô ��õ ü5ö R æ Óî÷ üì ð ñ ò ù ü...îï õì Ó�ð ñ ò ólô ��ú õ ö R æ Óî÷ õì ð ñ ò ù õ

ð û òÓ Ô î÷ ìýüEþ õ� �Aæ ù � óGÿ ì û î� �ì ö Óîï æì

òîï üì Ó...îï õì Ôð û Ó�ð ñ òólô � õ ö R æî÷ ì¯ü ÿ ì û î� ì Ó

Figure 7.2: Linear combinerwhich follows the space-timesoft decoderat the receiver.

Left: problemstatement,right: solutionto theproblem.

144

CSNR � ü�� ü�� ü Ù � ü�� ü�(dB) � � � �

16.0 0.9945 0.9969 0.414 0.9973 0.219 0.9974 0.183 0.9974 0.108

14.0 0.9881 0.9929 0.403 0.9937 0.214 0.9939 0.109 0.9940 0.109

12.0 0.9752 0.9842 0.394 0.9858 0.209 0.9862 0.107 0.9864 0.111

10.0 0.9506 0.9660 0.390 0.9695 0.207 0.9702 0.107 0.9705 0.113

8.0 0.9070 0.9333 0.393 0.9381 0.208 0.9392 0.108 0.9397 0.116

6.0 0.8374 0.8760 0.405 0.8833 0.214 0.8849 0.111 0.8855 0.120

4.0 0.7386 0.7891 0.430 0.7987 0.226 0.8009 0.118 0.8015 0.128

2.0 0.6164 0.6741 0.472 0.6852 0.248 0.6877 0.129 0.6884 0.139

0.0 0.4849 0.5427 0.536 0.5540 0.280 0.5565 0.146 0.5572 0.157

-2.0 0.3608 0.4142 0.627 0.4223 0.327 0.4246 0.170 0.4252 0.183

-4.0 0.2560 0.2974 0.751 0.3057 0.390 0.3076 0.203 0.3081 0.219

Table 7.1: Capacity(in bits/channeluse)of the DMC derived from � -bit soft-decision

decodingof BPSK-modulatedspace-timecodedMIMO channelwith Ü �ã and �� ¸ .� is thestep-sizewhichmaximizesthecapacityand � is numberof soft-decisiondecoding

bits.

145

0 0.1 0.2 0.3 0.4 0.6 0.8 1 1.40.4

0.5

0.6

0.7

0.8

0.9

1

∆

Cap

acity

(bi

ts/c

hann

el u

se)

CSNR = 0 dB

CSNR = 4 dB

CSNR = 10 dB

q = 4q = 3q = 2q = 1

Figure7.3: DMC capacityversusthestep-sizeof theuniformquantizer, �� , � �� .

146

CSNR (2-1) (4-1) (2-2)

in dB DI decreasing DI decreasing DI decreasing

10 7.385 7.368 7.809 7.795 7.899 7.876

8 6.886 6.845 7.551 7.536 7.867 7.839

6 6.119 6.026 6.949 6.908 7.712 7.691

4 5.167 4.968 5.948 5.832 7.288 7.272

2 4.178 3.830 4.781 4.508 6.477 6.416

0 3.315 2.783 3.714 3.245 5.353 5.179

-1 2.916 2.329 3.266 2.700 4.772 4.508

-2 2.546 1.930 2.830 2.224 4.217 3.853

-3 2.202 1.578 2.436 1.818 3.713 3.245

Table7.2: ComparisonbetweenthetrainingSDR(in dB) of two COVQ trainingmethods

for a unit-varianceGauss-Markov source( �� ) channel-optimizedvectorquantizedat

rate �� !�� bps.Quantizationdimensionis "# $� . ThreeMIMO systemsareconsidered

with %&� - �(') *%&� - �+' , %-, - �.' , and %&� - �!' .

147

−3 −2 0 2 4 6 8 10

1

2

3

4

5

6

7

8

CSNR in dB

SD

R in

dB

K=2, L=2

K=4, L=1K=2, L=1

K=1, L=1

COVQ, q = 3COVQ, q = 2COVQ, q = 1

Figure 7.4: SimulatedSDR in dB for a zero-meanunit-varianceGauss-Markov source

( �� ) vector quantizedat rate �/ �!�� bps and soft-decisiondecodedwith � bits.

Quantizationdimensionis "0 �� .

148

−2 0 2 4 6 8 10 12 14 16 18 20−5

0

5

10

15

20

CSNR in dB

SD

R in

dB

COVQVQ onlyTandem, 5/6Tandem, 2/3Tandem, 1/2Tandem 1/3Tandem 1/6

Figure7.5: Jointly designedversustandemcodingschemesfor a zero-meanunit-variance

Gauss-Markov source( �# 1��2� ). Quantizationdimensionis "3 � andtheoverall rateis

�4 65��2� bps. �7 �� and � �� .

149

−3 −2 0 2 4 6 8 10

1

2

3

4

5

6

7

8

9

CSNR in dB

SD

R in

dB

FEAD0 COVQ

FEAD VQ

FEAD8 COVQ

FEAD (CO)VQ, q = 3FEAD (CO)VQ, q = 2FEAD (CO)VQ, q = 1

Figure 7.6: SimulatedSDR in dB for an i.i.d. 81%9��:;�.' sourcevector quantizedat rate

�< �=�� bpsandsoft-decodedwith � bits. Quantizationdimensionis "> � . � � and

� �� .

150

−3 −2 −1 0 1 2 3 4 5 6 7 8

2

3

4

5

6

7

CSNR in dB

SD

R in

dB

COVQ, q = 3On−line FEAD VQ, q = 3FEAD VQ, q = 3On−line FEAD VQ (q = 1)FEAD VQ (q = 1)

Figure 7.7: Comparisonamongthe COVQ, FEAD VQ, and On-line FEAD VQ for a

8*%&��:?�.' Gauss-Markov sourcevector quantizedat rate �@ �!�2� bps and hard decoded.

Quantizationdimensionis "0 �� , �A �� , and � $� .

151

Chapter 8

Summary

In thisthesis,westudiedtheanalysisof theerrorratesof codedSISOchannelsanduncoded

MIMO channelswith both uniform andnon-uniformsources.We alsodesigneda soft-

decisiondecodingCOVQ for the transmissionof analogsourcesover MIMO channels.

Thecontributionsof this thesisaresummarizedin thefollowing.

B Wepresentedanalternativederivationof thepairwiseerrorprobabilityof space-time

codesof arbitrarystructureunderquasi-staticfading.Ourmethod,which is basedon

a momentgeneratingfunctionapproach,is very simpleandcanbeappliedto many

scenariosin singleor multiple antennasystems.It canalsobespecializedto exploit

thestructureof certaincodesto simplify thederivation,aswasdonein [176].

B We establishedupperandlower Bonferroni-typeboundson the block andbit error

ratesof single-antennacommunicationsystems. The KAT lower boundwas also

studied.TheAWGN aswell astheblockRayleighfadingchannelswereconsidered.

152

We derivedclosed-formformulasfor thepairwiseerrorprobabilityandalsofor the

probability of the intersectionof two pairwiseerror events. Thesequantitiesare

requiredfor thecomputationof theBonferroniandKAT bounds.We alsoproposed

to usea subsetof thecodebookandsuggesteda goodinitial setfor thecomputation

of theboundsfor thecasewherethenumberof bits inputto thechannelcoderis large

(thatis, for large " , where�;CD 6"FEHG is thechannelcoderate).Theboundswereoften

identicalup to four significantdigits for theAWGN channel.

B For MIMO channelswith auniformi.i.d. binaryinput,weconsideredthecasewhere

space-timeorthogonalblock coding is usedand establishedtight Bonferroni-type

upperandlowerboundson thesymbolandbit errorratesof thesystem.We derived

theprobabilityof the intersectionof two pairwiseerroreventsin closedform. The

bounds,whichcanbeappliedto any kind of signalinggeometryandarbitrarysymbol

mapping,areoftenthesameupto six significantdigits. In thecaseof PSKsignaling

(with any size),theboundsareidenticalup to tensignificantdigits evenat negative

channelsignal-to-noiseratios.

B We developeda new approachto derive the symbol pairwiseerror probability of

space-timeorthogonalblock codesunderMAP detectionfor non-uniforminputs.

With the pairwiseerror probability in hand,we computedtight upperand lower

boundson the symbol and bit error ratesof the system. It was shown that gains

up to 10 dB canbeobtainedvia MAP decodingandusingsymbolmappingsbetter

thanGraycoding.Furthermore,wederivedtheoptimalbinarysignalingschemeand

demonstratedthat very large gainsexist in usingthis schemeinsteadof BPSK sig-

153

naling. Moreover, it wasshown thatMAP decodingcansignificantlyoutperforma

typical tandemcodingsystemwhich is muchmorecomplex. In fact, it wasshown

thatMAP decodingis superiorto tandemcodingfor thebit errorratesof interest.

B In additionto MAP decoding,which is a form of joint source-channelcoding,we

consideredthe casewherethe input to the MIMO systemis a continuous-alphabet

(analog)signal. We designeda soft-decisiondecodingCOVQ for robust compres-

sionandcommunicationof theanalogsource.Wealsoconsideredthecasewherethe

transmitterdoesnothave thechannelsignal-to-noiseratioanddesignedtwo COVQs

for the systemoneof which exploits the knowledgeof the channelgainsat the re-

ceiver to gain betterperformance.Comparisonswith typical tandemsystemswere

alsopresentedand it wasshown that the COVQ cannot only outperformtandem

systems,but also it can beatunequalerror protectionsystemswhich useanother

approachto joint source-channelcoding.

154

Bibliography

[1] M. Abramowitz andI. Stegun,Handbookof MathematicalFunctionswith Formulas,

Graphs,and MathematicalTables. New York, NY: Dover Publications,fifth ed.,

1966.

[2] A. Abdi and M. Kaveh, “A space-timecorrelationmodel for multielementan-

tennasystemsin mobilefadingchannels,” IEEE J. Select.AreasCommun., vol. 20,

pp.550-560,Apr. 2002.

[3] R. K. Ahuja,T. L. Magnanti,andJ.B. Orlin, NetworkFlows.EnglewoodClif fs, NJ:

PrenticeHall, 1993.

[4] F. Alajaji, S.A. Al-Semari,andP. Burlina,“Visualcommunicationvia trellis coding

andtransmissionenergy allocation,” IEEETrans.Commun., vol. 47,pp.1722-1728,

Nov. 1999.

[5] F. Alajaji and T. Fuja, “A communicationchannelmodeledon contagion,” IEEE

Trans.Inform.Theory, vol. 40,pp.2035-2041,Nov. 1994.

155

[6] F. Alajaji andN. C. Phamdo,“Soft-decisionCOVQ for Rayleigh-fadingchannels,”

IEEE Commun.Let., vol. 2, pp.162-164,June1998.

[7] F. Alajaji, N. Phamdo,N. Farvardin, and T. Fuja, “Detection of binary Markov

sourcesover channelswith additive Markov noise,” IEEE Trans. Inform. Theory,

vol. 42,pp.230-239,Jan.1996.

[8] F. Alajaji, N. PhamdoandT. Fuja, “Channelcodesthatexploit the residualredun-

dancy in CELP-encodedspeech,” IEEE Trans.Speech Audio Processing, vol. 4,

pp.325-336,Sept.1996.

[9] S. M. Alamouti, “A simple transmitdiversity techniquefor wirelesscommunica-

tions,” IEEE J. Select.AreasCommun., vol. 16,pp.1451-1458,Oct.1998.

[10] S. M. Alamouti, V. Tarokh, and P. Poon,“Trellis-codedmodulationand transmit

diversity: designcriteria and performanceevaluation,” in Proc. IEEE Int’l Conf.

UniversalPersonalCommun., Florence,Italy, Oct.5-9,1998,pp.703-707.

[11] N. Al-Dhahir, “Overview andcomparisonof equalizationschemesfor space-time-

codedsignalswith applicationto EDGE,” IEEE Trans.SignalProcessing, vol. 50,

pp.2477-2488,Oct.2002.

[12] N. Al-Dhahir, “A new high-ratedifferentialspace-timeblockcodingscheme,” IEEE

Commun.Let., vol. 7, pp.540-542,Nov. 2003.

[13] S. Al-Semari,F. Alajaji, andT. Fuja,“SequenceMAP decodingof trellis codesfor

Gaussianand Rayleighchannels,” IEEE Trans.Veh. Technol., vol. 48, pp. 1130-

1140,July1999.

156

[14] E. AyanogluandR. M. Gray, “The designof joint sourceandchanneltrellis wave-

form coders,” IEEE Trans.Inform.Theory, vol. 33,pp.855-865,Nov. 1987.

[15] I. Bahceci,andT. M. Duman,“Combinedturbocodingandunitaryspace-timemod-

ulation,” IEEE Trans.Commun., vol. 50,pp.1244-1249,Aug. 2002.

[16] P. BalabanandJ. Salz, “Optimum diversity combiningandequalizationin digital

datatransmissionwith applicationto cellularmobileradio,” IEEETrans.Veh.Tech-

nol., vol. 40,pp.342-354,May. 1991.

[17] I. Barhumi,G. Leus,andM. Moonen,“Optimal trainingdesignfor MIMO OFDM

systemsin mobile wirelesschannels,” IEEE Trans. Signal Processing, vol. 51,

pp.1615-1624,June2003.

[18] S.Baro,G. Bauch,andA. Hansmann,“Improvedcodesfor space-timetrellis-coded

modulation,” IEEE Commun.Let., vol. 4, pp.20-22,Jan.2000.

[19] G. Bauch,J. Hagenauer, andN. Seshadri,“Turbo processingin transmitantenna

diversitysystems,” AnnalsTelecommun., vol. 56,pp.455-471,Aug. 2001.

[20] F. Behnamfar, F. Alajaji and T. Linder, “Progressive imagecommunicationover

binarychannelswith additiveburstynoise,” in Proc.IEEEDataCompressionConf.,

Snowbird, UT, Apr. 2002.

[21] F. Behnamfar, F. Alajaji andT. Linder, “Imagecodingfor binaryburstynoisechan-

nels,” in Proc.21st BiennialSymp.Commun., Kingston,ON, June2002.

157

[22] F. Behnamfar, F. Alajaji, andT. Linder, “Imagetransmissionover thePolyachannel

via channel-optimizedquantization,” IEEE Trans.SignalProcessing, to appear.

[23] R. S. Blum, “Some analytical tools for the designof space-timeconvolutional

codes,” IEEE Trans.Commun., vol. 50,pp.1593-1599,Oct.2002.

[24] C. Budianu and L. Tong, “Channel estimationfor space-timeorthogonalblock

codes,” IEEE Trans.SignalProcessing, vol. 50,pp.2515-2528,Oct.2002.

[25] Q. ChenandT. R. Fisher, “Imagecodingusingrobustquantizationfor noisydigital

transmission,” IEEE Trans.ImageProcessing, vol. 7, pp.496-505,Apr. 1998.

[26] M. Chiani, M. Z. Win, and A. Zanella, “On the capacityof spatially correlated

MIMO Rayleigh-fadingchannels,” IEEE Trans.Inform. Theory, vol. 49, pp. 2363-

2371,Oct.2003.

[27] D. Chizhik, G. J. Foschini,M. J. Gans,andR. A. Valenzuela,“K eyholes,correla-

tions, andcapacitiesof multielementtransmitandreceive antennas,” IEEE Trans.

WirelessCommun., vol. 1, pp.361-368,Apr. 2002.

[28] D. Chizhik, J. Ling, P. W. Wolniansky, R. A. Valenzuela,N. Costa,andK. Huber,

“Multiple-input-multiple-outputmeasurementsandmodelingin Manhattan,” IEEE

J. Select.AreasCommun., vol. 21,pp.321-331,Apr. 2003.

[29] S.-Y. Chung,J.G.D. Forney, T. Richardson,andR.Urbanke,“On thedesignof low-

densityparity-checkcodeswithin 0.0045dB of theShannonlimit,” IEEE Commun.

Lett., vol. 5, pp.58-60,Feb. 2001.

158

[30] A. CohenandN. Merhav, “Lower boundson the error probability of block codes

basedon improvementson de Caen’s inequality,” IEEE Trans. Inform. Theory,

vol. 50,pp.290-310,Feb. 2004.

[31] T. M. Cover andJ. A. Thomas,Elementsof InformationTheory. New York: Wi-

ley, 1991.

[32] C. CozzoandB. L. Hughes,“Joint channelestimationanddatadetectionin space-

timecommunications,” IEEE Trans.Commun., vol. 51,pp.1266-1270,Aug. 2003.

[33] J.W. Craig,“A new, simple,andexactresultfor calculatingtheprobabilityof error

for two-dimensionalsignalconstellations,” in Proc. IEEE Military Commun.Conf.

(MilCom), McLean,VA, Nov. 1991,pp.571-575.

[34] D. Cui andA. M. Haimovich, “A new bandwidthefficient antennadiversityscheme

using turbo codes,” in Proc. 34th Annual Conf. Inform. Sci. Syst., Princeton,NJ,

Mar. 2000,vol. 1, pp.24-29.

[35] M. O. Damen,K. Abed-Meraim,andJ.C. Belfiore,“Diagonalalgebraicspace-time

block codes,” IEEETrans.Inform.Theory, vol. 48,pp.628-636,Mar. 2002.

[36] M. O. DamenandN. C. Beaulieu,“On two high-ratealgebraicspace-timecodes,”

IEEE Trans.Inform.Theory, vol. 49,pp.1059-1063,Apr. 2003.

[37] M. O. Damen,A. Chkeif, andJ. C. Belfiore, “Lattice codedecoderfor space-time

codes,” IEEE Commun.Let., vol. 4, pp.161-163,May 2000.

159

[38] M. O. Damen,H. El Gamal,andN. C. Beaulieu,“Linear threadedalgebraicspace-

timeconstellations,” IEEETrans.Inform.Theory, vol. 49,pp.2372-2388,Oct.2003.

[39] D. deCaen,“A lower boundon theprobabilityof a union,” Discr. Math., vol. 169,

pp.217-220,1997.

[40] A. Dembo,unpublishednotes,communicatedby I. Sason,2000.

[41] A. Dogandzic, “Chernoff boundson pairwise error probabilities of space-time

codes,” IEEE Trans.Inform.Theory, vol. 49,pp.1327-1336,May 2003.

[42] T. G. Donnelly, “Bi variatenormaldistribution,” Commun.ACM, vol. 16, pp. 636,

1973,Algorithm 462.

[43] J. G. DunhamandR. M. Gray, “Joint sourceandnoisy channelencoding,” IEEE

Trans.Inform.Theory, vol. 27,pp.516-519,July1981.

[44] H. El GamalandM. O.Damen,“Universalspace-timecoding,” IEEETrans.Inform.

Theory, vol. 49,pp.1097-1119,May 2003.

[45] H. El Gamaland A. R. Hammons,“A new approachto layeredspace-timecod-

ing and signal processing,” IEEE Trans. Inform. Theory, vol. 47, pp. 2321-2334,

Sept.2001.

[46] H. El GamalandA. R. Hammons,“On the designandperformanceof algebraic

space-timecodesfor BPSKandQPSKmodulation,” IEEETrans.Commun., vol. 50,

pp.907-913,June2002.

160

[47] H. El GamalandA. R. Hammons,“On thedesignof algebraicspace-timecodesfor

MIMO block-fadingchannels,” IEEE Trans.Inform. Theory, vol. 49, pp. 151-163,

Jan.2003.

[48] E. O. Elliott, “Estimatesof errorratesfor codeson burst-noisechannels,” Bell Syst.

Tech. J., vol. 42,pp.1977-1997,Sept.1963.

[49] N. Farvardin, “A studyof vectorquantizationfor noisychannels,” IEEE Trans.In-

form.Theory, vol. 36,pp.799-809,July1990.

[50] N. Farvardin andV. Vaishampayan,“Optimal quantizerdesignfor noisy channels:

an approachto combinedsource-channelcoding,” IEEE Trans. Inform. Theory,

vol. 33,pp.827-838,Nov. 1987.

[51] N. FarvardinandV. Vaishampayan,“On theperformanceandcomplexity of channel

optimizedvector quantizers,” IEEE Trans. Inform. Theory, vol. 37, pp. 155-160,

Jan.1991.

[52] T. FazelandT. Fuja, “Robust transmissionof MELP-compressedspeech:an illus-

trative exampleof joint source-channeldecoding,” IEEE Trans.Commun., vol. 51,

pp.973-982,June2003.

[53] G. Femenias,“BER performanceof linear STBC from orthogonaldesignsover

MIMO correlatedNakagami-mfadingchannels,” IEEETrans.Veh.Technol., vol. 53,

pp.307-317,Mar. 2004.

[54] T. Fine, “Propertiesof an optimal digital systemand applications,” IEEE Trans.

Inform.Theory, vol. 10,pp.443-457,Oct.1964.

161

[55] G. D. Forney, Jr., “Geometricallyuniform codes,” IEEE Trans. Inform. Theory,

vol. 37,pp.1241-1260,Sept.1991.

[56] G. J.Foschini,“Layeredspace-timearchitecturefor wirelesscommunicationovera

fadingenvironmentwhenusingmultipleantennas,” Bell LabsTech.J., vol. 1, pp.41-

59,Autumn1996.

[57] G. J. Foschini and M. J. Gans,“On limits of wirelesscommunicationsin a fad-

ing environmentwhenusingmultiple antennas,” WirelessPers. Commun., vol. 6,

pp.311-335,Mar. 1998.

[58] G. J. Foschini,G. D. Golden,R. A. Valenzuela,andP. W. Wolniansky, “Simplified

processingfor high spectralefficiency wirelesscommunicationemploying multi-

elementarrays,” IEEEJ. Select.AreasCommun., vol. 17,pp.1841-1852,Nov. 1999.

[59] M. Fozunbal,S. W. McLaughlin, and R. W. Schafer, “On performancelimits of

space-timecodes:a sphere-packingboundapproach,” IEEE Trans.Inform.Theory,

vol. 49,pp.2681-2687,Oct.2003.

[60] S. GadkariandK. Rose,“Robust vectorquantizerdesignby noisy channelrelax-

ation,” IEEETrans.Commun., vol. 47,pp.1113-1116,Aug. 1999.

[61] S. GadkariandK. Rose,“Correctionsto “robust vectorquantizerdesignby noisy

channelrelaxation”,” IEEETrans.Commun., vol. 48,pp.176-176,Jan.2000.

[62] J. GalambosandI. Simonelli,Bonferroni-TypeInequalitieswith Appications.New

York: Springer-Verlag,1996.

162

[63] R.G.Gallager, InformationTheoryandReliableCommunication.New York: Wiley,

1968.

[64] R. G. Gallager, “ClaudeE. Shannon:a retrospectiveon his life, work, andimpact,”

IEEE Trans.Inform.Theory, vol. 47,pp.2681-2695,Nov. 2001.

[65] D. Gesbert,H. Bolcskei, D. A. Gore, and A. J. Paulraj, “Outdoor MIMO wire-

lesschannels:modelsandperformanceprediction,” IEEETrans.Commun., vol. 50,

pp.1926-1934,Dec.2002.

[66] A. Gersho,andR. M. Gray, Vector Quantizationand signal compression. Kluwer

AcademicPublishers,1992.

[67] M. Gharavi-Alkhansari and A. B. Gershman,“Fast antennasubsetselectionin

MIMO systems,” IEEE Trans.SignalProcessing, vol. 52,pp.339-347,Feb. 2004.

[68] E. N. Gilbert, “Capacity of a burst-noisechannel,” Bell Syst.Tech. J., vol. 39,

pp.1253-1265,Sept.1960.

[69] A. Gorokhov, D. Gore, and A. J. Paulraj, “Receive antennaselectionfor MIMO

flat-fadingchannels:theoryandalgorithms,” IEEE Trans.Inform. Theory, vol. 49,

pp.2687-2696,Oct.2003.

[70] L. GouletandH. Leib, “Serially concatenatedspace-timecodeswith iterativedecod-

ing andperformancelimits of block-fadingchannels,” IEEE J. Select.AreasCom-

mun., vol. 21,pp.765-773,June2003.

163

[71] I. GradsteinandI. Ryshik,Tablesof Integrals,Series,andProducts. SanDiego,CA:

AcademicPress,fifth ed.,1994.

[72] A. Grant,“Rayleighfadingmultiple-antennachannels,” EURASIPJ. AppliedSignal

Processing, vol. 2002,pp.316-322,Mar. 2002.

[73] J.Grimm,M. P. Fitz, andJ.V. Krogmeier, “Furtherresultsin space-timecodingfor

Rayleighfading,” IEEETrans.Commun., vol. 51,pp.1093-1101,July2003.

[74] J. C. Guey, M. P. Fitz, M. R. Bell, andW. Y. Kuo, “Signal designfor transmitter

diversity wirelesscommunicationsystemsover Rayleighfading channels,” IEEE

Trans.Commun., vol. 47,pp.527-537,Apr. 1999.

[75] A. Guyader, E. Fabre,C. Guillemot, and M. Robert,“Joint source-channelturbo

decodingof entropy-coded sources,” IEEE J. Select.Areas Commun., vol. 19,

pp.1680-1696,Sept.2001.

[76] J. Hagenauer, “Rate-compatiblepuncturedconvolutional codes(RCPCcodes)and

their applications,” IEEE Trans.Commun., vol. 36,pp.389-400,Apr. 1988.

[77] A. R. Hammonsand H. El Gamal, “On the theory of space-timecodesfor PSK

modulation,” IEEE Trans.Inform.Theory, vol. 46,pp.524-542,Mar. 2000.

[78] B. Hassibi,“A fastsquare-rootimplementationfor BLAST,” in Proc.34th Asilomar

Conf. SignalsSyst.Computers, Pacific Grove,CA, Oct.-Nov. 2000,pp.1255-1259.

[79] B. Hassibiand B. M. Hochwald, “Cayley differential unitary space-timecodes,”

IEEE Trans.Inform.Theory, vol. 48,pp.1485-1503,June2002.

164

[80] B. HassibiandB. M. Hochwald,“High-ratecodesthatarelinearin spaceandtime,”

IEEE Trans.Inform.Theory, vol. 48,pp.1804-1824,June2002.

[81] B. HassibiandM. Khorrami, “Fully-diversemultiple-antennasignalconstellations

andfixed-point-freeLie groups,” Proc. IEEE Int’l Symp.Inform. Theory, Washing-

ton,DC, June2001,p. 199.

[82] L. HeandH. Ge,“A new full-rate full-diversityorthogonalspace-timeblockcoding

scheme,” IEEE Commun.Let., vol. 7, pp.590-592,Dec.2003.

[83] R.W. Heath,Jr. andA. J.Paulraj,“Linear dispersioncodesfor MIMO systemsbased

onframetheory,” IEEETrans.SignalProcessing, vol. 50,pp.2429-2441,Oct.2002.

[84] H. Herzberg and G. Poltyrev, “The error probability of I -ary PSK block coded

modulationschemes,” IEEETrans.Commun., vol. 44,pp.427-433,Apr. 1996.

[85] B. M. Hochwald andW. Sweldens,“Dif ferentialunitary space-timemodulation,”

IEEE Trans.Commun., vol. 48,pp.2041-2052,Dec.2000.

[86] B. M. Hochwald andT. L. Marzetta,“Unitary space-timemodulationfor multiple-

antennacommunicationsin Rayleigh flat fading,” IEEE Trans. Inform. Theory,

vol. 46,pp.543-564,Mar. 2000.

[87] B. M. Hochwald, T. L. Marzetta,andB. Hassibi,“Space-timeautocoding,” IEEE

Trans.Inform.Theory, vol. 48,pp.2761-2781,Nov. 2001.

165

[88] B. M. Hochwald, T. L. Marzetta,T. J. Richardson,W. Sweldens,andR. Urbanke,

“Systematicdesignof unitaryspace-timeconstellations,” IEEE Trans.Inform.The-

ory, vol. 46,pp.1962-1973,sept.2000.

[89] B. L. Hughes,“Dif ferentialspace-timemodulation,” IEEE Trans.Inform. Theory,

vol. 46,pp.2567-2578,Nov. 2000.

[90] B. L. Hughes,“Optimal space-timeconstellationsfrom groups,” IEEE Trans. In-

form.Theory, vol. 49,pp.401-410,Feb. 2003.

[91] D. Hunter, “An upperboundfor theprobabilityof aunion,” J. Appl.Probab., vol. 13,

pp.597-603,1976.

[92] W. J. Hwang,F. J. Lin, andC. T. Lin, “Fuzzy channel-optimizedvectorquantiza-

tion,” IEEE Commun.Let., vol. 4, pp.408-410,Dec.2000.

[93] T. M. Ivrlac,W. Utschick,andJ.A. Nossek,“FadingCorrelationsin WirelessMIMO

CommunicationSystems,” IEEE Trans. WirelessCommun., vol. 2, pp. 819-824,

May 2003.

[94] H. Jafarkhani, P. Ligdas and N. Farvardin, “Adaptive rate allocation in a joint

source/channelcodingframework for wirelesschannels,” in Proc. IEEE Veh.Tech-

nol. Conf., Atlanta,GA, pp.492-296,1996.

[95] H. JafarkhaniandN. Seshadri,“Super-orthogonalspace-timetrellis codes,” IEEE

Trans.Inform.Theory, vol. 49,pp.937-950,Apr. 2003.

166

[96] H. JafarkhaniandV. Tarokh,“Multiple transmitantennadifferentialdetectionfrom

generalizedorthogonaldesigns,” IEEE Trans. Inform. Theory, vol. 47, pp. 2626-

2631,Sept.2001.

[97] J.JaldenandB. Ottersten,“An exponentiallowerboundontheexpectedcomplexity

of spheredecoding,” in Proc.IEEEInt. Conf. Acoustics,Speech andSignalProcess-

ing (ICASSP), Montreal,QC,May 2004.

[98] S.JiangandR.Kohno,“A new space-timemultipletrellis codedmodulationscheme

for flat fadingchannels,” Inst.Electron.Inf. Commun.Eng. (IEICE) Trans., vol. E87-

A, pp.640-647,Mar. 2004.

[99] Y. Jing andB. Hassibi,“Fully-diverse JLK (2) codedesign,” Proc. IEEE Int’l Symp.

Inform.Theory, Yokohama,Japan,June-July2003,p. 299.

[100] Y. JingandB. Hassibi,“Unitary space-timemodulationvia Cayley transform,” IEEE

Trans.SignalProcessing, vol. 51,pp.2891-2904,Nov. 2003.

[101] R. JohannessonandK. Zigangirov, Fundamentalsof Convolutional Coding. New

York, NY: IEEE Press,1999.

[102] J. P. Kermoal,L. Schumacher, K. I. Pedersen,P. E. Mogensen,andF. Frederiksen,

“A stochasticMIMO radio channelmodelwith experimentalvalidation,” IEEE J.

Select.AreasCommun., vol. 20,pp.1211-1226,Aug. 2002.

[103] M. A. Khojastepour, A. Sabharwal, and B. Aazhang,“Improved achievable rates

for usercooperationandrelaychannels,” in Proc. IEEE Int’l Symp.Inform. Theory

(ISIT), Chicago,IL, June-July2004.

167

[104] Y. J.Kim andH. S.Lee,“Generalizedprincipalratiocombiningfor space-timecodes

in slowly fadingchannels,” IEEE Commun.Let., vol. 4, pp.343-345,Nov. 2000.

[105] I. M. Kim andV. Tarokh,“Variable-ratespace-timeblock codesin M-ary PSKsys-

tems,” IEEEJ. Select.AreasCommun., vol. 21,pp.362-373,Apr. 2003.

[106] P. KnagenhjelmandE. Agrell, “The Hadamardtransform– a tool for index assign-

ment,” IEEETrans.Inform.Theory., vol. 42,pp.1139-1151,July1996.

[107] I. Korn,J.P. Fonseka,andS.Xing, “Optimal binarycommunicationwith nonequal

probabilities,” IEEETrans.Commun., vol. 51,pp.1435-1438,Sept.2003.

[108] E.G.Kounias,“Boundsontheprobabilityof aunion,with applications,” Ann.Math.

Statist., vol. 39,pp.2154-2158,1968.

[109] H. Kuai, F. Alajaji, andG. Takahara,“A lower boundon theprobabilityof a finite

unionof events,” Discr. Math., vol. 215,pp.147-158,2000.

[110] H. Kuai, F. Alajaji, andG. Takahara,“Tight error boundsfor nonuniformsignal-

ing over AWGN channels,” IEEE Trans.Inform. Theory, vol. 46, pp. 2712-2718,

Nov. 2000.

[111] H. Kumazawa,M. Kasahara,andT. Namekawa,“A constructionof vectorquantizers

for noisychannels,” ElectronicsEng. in Japan, vol. 67-B,pp.39-47,Jan.1984.

[112] B. C.Kuo,AutomaticControl Systems. EnglewoodClif fs,NJ:PrenticeHall, seventh

ed.,1995.

168

[113] A. KurtenbachandP. Wintz, “Quantizingfor noisychannels,” IEEETrans.Commun.

Technol., vol. 17,pp.291-302,Apr. 1969.

[114] J. N. Laneman,G. W. Wornell, andD. N. C. Tse,“An efficient protocolfor realiz-

ing cooperative diversity in wirelessnetworks,” in Proc. IEEE Int’l Symp.Inform.

Theory(ISIT), Washington,DC, June2001.

[115] E. G. Larsson,“Unitary nonuniform space-timeconstellationsfor the broadcast

channel,” IEEE Commun.Let., vol. 7, pp.21-23,Jan.2003.

[116] L. H. C. Lee, Convolutional Coding: Fundamentalsand Applications, Norwood:

ArtechHouse,Inc., 1997.

[117] G. Leus,W. Zhao,G. B. Giannakis,andH. Delic, “Space-timefrequency-shift key-

ing,” IEEETrans.Commun., vol. 52,pp.346-349,Mar. 2004.

[118] Y. Li, C. N. Georghiades,andG.Huang,“Transmitdiversityoverquasi-staticfading

channelsusingmultiple antennasandrandomsignalmapping,” IEEE Trans.Com-

mun., vol. 51,pp.1918-1926,Nov. 2003.

[119] Y. Li, J.H. Winter, andN. R. Sollenberger, “MIMO-OFDM for wirelesscommuni-

cations:signaldetectionandenhancedchannelestimation,” IEEE Trans.Commun.,

vol. 50,pp.1471-1477,Sept.2002.

[120] X. B. Liang, “A high-rateorthogonalspace-timeblock code,” IEEE Commun.Let.,

vol. 7, pp.222-223,May 2003.

169

[121] X. B. Liang,“Orthogonaldesignswith maximalrates,” IEEETrans.Inform.Theory,

vol. 49,pp.2468-2503,Oct.2003.

[122] M. Lienard, P. Degauque,J. Baudet,and D. Degardin, “Investigationon MIMO

channelsin subway tunnels,” IEEE J. Select.AreasCommun., vol. 21,pp.332-339,

Apr. 2003.

[123] J.Lim andD. L. Neuhoff, “Joint andtandemsource-channelcodingwith complexity

anddelayconstraints,” IEEETrans.Commun., vol. 51,pp.757-766,May 2003.

[124] X. Lin andR. S. Blum, “Improved space-timecodesusingserial concatenation,”

IEEE Commun.Let., vol. 4, pp.221-223,July2000.

[125] T. Linder, G. Lugosi, and K. Zeger, “Ratesof convergencein the sourcecoding

theorem,in empiricalquantizerdesign,andin universallossysourcecoding,” IEEE

Trans.Inform.Theory, vol. 40,pp.1728-1740,Nov 1994.

[126] T. Linder, G. Lugosi,andK. Zeger, “Empirical quantizerdesignin thepresenceof

sourcenoiseor channelnoise,” IEEE Trans.Inform. Theory, vol. 43, pp. 612-623,

Mar. 1997.

[127] Y. Liu, M. P. Fitz, andO. Y. Takeshita,“Full ratespace-timeturbocodes,” IEEE J.

Select.AreasCommun., vol. 19,pp.969-980,May 2001.

[128] H. A. Loeliger, “Signal setsmatchedto groups,” IEEE Trans. Inform. Theory,

vol. 37,pp.1675-1682,Nov. 1991.

170

[129] A. LozanoandC. Papadias,“Layeredspace-timereceiversfor frequency-selective

wirelesschannels,” IEEE Trans.Commun., vol. 50,pp.65-73,Jan.2002.

[130] B. Lu andX. Wang,“Iterative receiversfor multiuserspace-timecodingsystems,”

IEEE J. Select.AreasCommun., vol. 18,pp.2322-2335,Nov. 2000.

[131] H. F. Lu, Y. Wang,P. V. Kumar, andK. M. Chugg,“Remarkson space-timecodes

includinga new lower boundandanimprovedcode,” IEEE Trans.Inform. Theory,

vol. 49,pp.2752-2757,Oct.2003.

[132] X. Ma andG. B. Giannakis,“Full-diversityfull-rate complex-field space-timecod-

ing,” IEEETrans.SignalProcessing, vol. 51,pp.2917-2930,Nov. 2003.

[133] A. ManikasandM. Sethi,“A space-timechannelestimatorandsingle-userreceiver

for code-reuseDS-CDMA systems,” IEEETrans.SignalProcessing, vol. 51,pp.39-

51,Jan.2003.

[134] T. L. MarzettaandB. M. Hochwald, “Capacityof a mobilemultiple-antennacom-

munication link in Rayleigh flat fading,” IEEE Trans. Inform. Theory, vol. 45,

pp.139-157,Jan.1999.

[135] J. L. Massey, “Joint sourceand channelcoding,” in Commun.Syst.and random

processtheory, pp.279-293,TheNetherlands:Noordhoff, 1978.

[136] D. Miller and K. Rose, “Combined source-channelvector quantization us-

ing deterministic annealing,” IEEE Trans. Commun., vol. 42, pp. 347-356,

Feb./Mar./Apr. 1994.

171

[137] A. F. Molisch,“A genericmodelfor MIMO wirelesspropagationchannelsin macro-

andmicrocells,” IEEE Trans.SignalProcessing, vol. 52,pp.61-71,Jan.2004.

[138] A. L. Moustakas,S. H. Simon,andA. M. Sengupta,“MIMO capacitythroughcor-

relatedchannelsin thepresenceof correlatedinterferersandnoise:a (not so) large

Manalysis,” IEEE Trans.Inform.Theory, vol. 49,pp.2545-2561,Oct.2003.

[139] S.H. Muller-Weinfurtner, “Coding approachesfor multipleantennatransmissionin

fast fadingand OFDM,” IEEE Trans.Signal Processing, vol. 50, pp. 2442-2450,

Oct.2002.

[140] A. Naguib,N. Seshadri,andA. Calderbank,“Increasingdatarateoverwirelesschan-

nels,” IEEE SignalProcessingMagazine, vol. 46,pp.76-92,May 2000.

[141] B. K. Ng andE. S. Sousa,“On bandwidth-efficient multiuser-space-timesignalde-

signanddetection,” IEEEJ. Select.AreasCommun., vol. 20,pp.320-329,Feb. 2002.

[142] A. Nosratinia,J. Lu, and B. Aazhang,“Source-channelrate allocation for pro-

gressive transmissionof images,” IEEE Trans. Commun., vol. 51, pp. 186-196,

Feb. 2003.

[143] H. Nyquist,“Certaintopicsin telegraphtransmissiontheory,” Trans.AIEE, vol. 47,

p. 617,1928.

[144] O. Oyman,R. U. Nabar, H. Bolcskei, and A. J. Paulraj, “Characterizingthe sta-

tistical propertiesof mutual information in MIMO channels,” IEEE Trans.Signal

Processing, vol. 51,pp.2784-2795,Nov. 2003.

172

[145] B. D. Pettijohn,M. W. Hoffman,andK. Sayood,“Joint source/channelcodingusing

arithmeticcodes,” IEEE Trans.Commun., vol. 49,pp.826-836,May 2001.

[146] N. Phamdoand F. Alajaji, “Soft-decisiondemodulationdesignfor COVQ over

white, colored, and ISI Gaussianchannels,” IEEE Trans. Commun., vol. 48,

pp.1499-1506,Sept.2000.

[147] N. Phamdo,F. Alajaji, andN. Farvardin, “Quantizationof memorylessandGauss-

Markov sourcesover binary Markov channels,” IEEE Trans. Commun., vol. 45,

pp.668-675,June1997.

[148] N. PhamdoandN. Farvardin, “Optimal detectionof discreteMarkov sourcesover

discretememorylesschannels- Applicationsto combinedsource-channelcoding,”

IEEE Trans.Inform.Theory, vol. 40,pp.186-193,Jan.1994.

[149] G. Poltyrev, “Boundson the decodingerror probability of binary linear codesvia

their spectra,” IEEE Trans.Inform.Theory, vol. 40,pp.1284-1292,July1994.

[150] J.G. Proakis,Digital Communications. New York: McGraw-Hill, 4thed.,2001.

[151] T. J. Richardson,M. A. Shokrollahi, and R. L. Urbanke, “Design of capacity-

approachingirregularlow-densityparity-checkcodes,” IEEETrans.Inform.Theory,

vol. 47,pp.619-637,Feb. 2001.

[152] A. SaidandW. A. Pearlman,“A new, fastandefficient imagecodecbasedon set

partitioningin hierarchicaltrees,” IEEE Trans.CircuitsSyst.VideoTechnol., vol. 6,

pp.243-250,June1996.

173

[153] D. SamardzijaandN. Mandayam,“Pilot-assistedestimationof MIMO fadingchan-

nel responseand achievable datarates,” IEEE Trans.Signal Processing, vol. 51,

pp.2882-2890,Nov. 2003.

[154] K. SayoodandJ.C.Borkenhagen,“Useof residualredundancy in thedesignof joint

source/channelcoders,” IEEE Trans.Commun., vol. 39,pp.838-846,June1991.

[155] C. Schlegel andA. Grant, “Dif ferential space-timeturbo codes,” IEEE Trans.In-

form.Theory, vol. 49,pp.2298-2306,Sept.2003.

[156] G. Seguin, “A lower boundon the error probability for signalsin white Gaussian

noise,” IEEETrans.Inform.Theory, vol. 44,pp.3168-3175,Nov. 1998.

[157] M. SellathuraiandS. Haykin, “Turbo-BLAST for wirelesscommunications:the-

ory and experiments,” IEEE Trans. Signal Processing, vol. 50, pp. 2538-2546,

Oct.2002.

[158] A. Sendonaris,E. Erkip, andB. Aazhang,“Increasinguplink capacityvia userco-

operationdiversity,” in Proc. IEEE Int’l Symp.Inform. Theory(ISIT), Camberidge,

MA, Aug. 1998.

[159] M. ShafiandP. J. Smith, “On a Gaussianapproximationto the capacityof wire-

lessMIMO systems,” in Proc. IEEE Int’l Conf. Commun., New York, NY, Apr.-

May 2002,pp.406-410.

[160] C. E. Shannon,“A mathematicaltheoryof communication(Part1),” Bell Syst.Tech.

J., vol. 27,pp.379-423,1948.

174

[161] C. E. Shannon,“A mathematicaltheoryof communication(Part2),” Bell Syst.Tech.

J., vol. 27,pp.623-656,1948.

[162] C. E. Shannon,“Communicationin the presenceof noise,” in Proc. IRE, vol. 37,

pp.10-21,1949.

[163] C. E. Shannon,“Coding theoremsfor a discretesourcewith a fidelity criterion,” in

IRE Nat.Conv. Rec., pp.142-163,1959.

[164] C. E. Shannon,“Probability of error for optimalcodesin a Gaussianchannel,” Bell

Syst.Tech. J., vol. 38,pp.611-656,May 1959.

[165] P. G. SherwoodandK. Zeger, “Error protectionfor progressive imagetransmission

over memorylessandfadingchannels,” IEEE Trans.Commun., vol. 46, pp. 1555-

1559,Dec.1998.

[166] H. ShinandJ.H. Lee,“Performanceanalysisof space-timeblockcodesoverkeyhole

Nakagami-N fading channels,” IEEE Trans.Veh. Technol., vol. 53, pp. 351-362,

Mar. 2004.

[167] H. Shin andJ. H. Lee, “Exact symbolerror probability of orthogonalspace-time

block codes,” Proc. IEEEGlobal Telecom.Conf., Taipei,Taiwan,Nov. 2002,vol. 2,

pp.1197-1201.

[168] H. ShinandJ.H. Lee,“Capacityof multiple-antennafadingchannels:spatialfading

correlation,doublescattering,andkeyhole,” IEEE Trans.Inform. Theory, vol. 49,

pp.2636-2647,Oct.2003.

175

[169] D. S. Shiu, G .J. Foschini, M. J. Gans,andJ. M. Kahn, “Fading correlationand

its effect on thecapacityof multielementantennasystems,” IEEE Trans.Commun.,

vol. 48,pp.502-513,Mar. 2000.

[170] A. Shokrollahi, “Computing the performanceof unitary space-timegroup codes

from their charactertable,” IEEE Trans. Inform. Theory, vol. 48, pp. 1355-1371,

June2002.

[171] A. Shokrollahi,B. Hassibi,B. M. Hochwald, and W. Sweldens,“Representation

theory for high-ratemultiple-antennacodedesign,” IEEE Trans. Inform. Theory,

vol. 47,pp.2335-2367,Sept.2001.

[172] M. A. Shokrollahi, “Design of unitary space-timecodesfrom representationsof

J(O (2),” in Proc. IEEE Int’l Symp.Inform. Theory, Washington,DC, May 2001,

p. 241.

[173] M. K. Simon,“Evaluationof theaveragebit errorprobabilityfor space-timecoding

basedonasimplerexactevaluationof pairwiseerrorprobabillity,” J. Commun.Net.,

vol. 3, pp.257-264,Sept.2001.

[174] M. K. Simon,“A simplerform of theCraigrepresentationfor thetwo-dimensional

joint GaussianQ-Function,” IEEECommun.Let., vol. 6, pp.49-51,Feb. 2002.

[175] M. K. SimonandM. S.Alouini, Digital Communicationsover FadingChannels:a

UnifiedApproach. New York, NY: JohnWiley, 2000.

176

[176] M. K. SimonandH. Jafarkhani,“Performanceevaluationof super-orthogonalspace-

timetrellis codesusingamomentgeneratingfunction-basedapproach,” IEEETrans.

SignalProcessing, vol. 51,pp.2739-2751,Nov. 2003.

[177] M. Skoglund, “A soft decodervector quantizerfor a Rayleigh fading channel-

Applicationto imagetransmission,” in Proc. IEEE Int. Conf. Acoustics,Speech and

SignalProcessing(ICASSP), pp.2507-2510,May 1995.

[178] M. Skoglund,“Soft decodingfor vectorquantizationovernoisychannelswith mem-

ory,” IEEETrans.Inform.Theory, vol. 45,pp.1293-1307,May 1999.

[179] M. SkoglundandP. Hedelin,“Hadamard-basedsoftdecodingfor vectorquantization

overnoisychannels,” IEEETrans.Inform.Theory, vol. 45,pp.515-532,Mar. 1999.

[180] M. Skoglund,N. Phamdo,andF. Alajaji, “Designandperformanceof VQ-basedhy-

brid digital-analogjoint source-channelcodes,” IEEETrans.Inform.Theory, vol. 48,

pp.708-720,Mar. 2002.

[181] F. K. SoongandB. H. Juang,“Line spectrumpair (LSP)andspeechdatacompres-

sion,” in Proc. IEEE Int’l Conf. Acoustics,Speech, and Signal Processing, 1984,

pp.1.10.1-1.10.4.

[182] B. Srinivas, R. Ladner, M. Azizoglu and E. Riskin, “Progressive transmissionof

imagesusingMAP detectionoverchannelswith memory,” IEEE Trans.Image Pro-

cessing, vol. 8, pp.462-475,Apr. 1999.

[183] A. Stefanov andT. M. Duman,“Turbo-codedmodulationfor systemswith transmit

andreceive antennadiversityover block fadingchannels:systemmodel,decoding

177

approaches,andpracticalconsiderations,” IEEE J. Select.AreasCommun., vol. 19,

pp.958-968,May 2001.

[184] H. J. Su andE. Geraniotis,“Space-timeturbo codeswith full antennadiversity,”

IEEE Trans.Commun., vol. 49,pp.47-57,Jan.2001.

[185] W. SuandX. G. Xia, “Two generalizedcomplex orthogonalspace-timeblockcodes

of rates7/11 and3/5 for 5 and6 transmitantennas,” IEEE Trans.Inform. Theory,

vol. 49,pp.313-316,Jan.2003.

[186] G. Takahara,F. Alajaji, N. C. Beaulieu,andH. Kuai, “Constellationmappingsfor

two-dimensionalsignalingof nonuniformsources,” IEEE Trans.Commun., vol. 51,

pp.400-408,Mar. 2003.

[187] N. TanabeandN. Farvardin, “Subbandimagecodingusingentropy-codedquanti-

zationover noisychannels,” IEEE J. Select.AreasCommun., vol. 10, pp. 926-943,

June1992.

[188] M. TaoandR. S. Cheng,“Trellis-codeddifferentialunitaryspace-timemodulation

overflat fadingchannels,” IEEETrans.Commun., vol. 51,pp.587-596,Apr. 2003.

[189] M. TaoandR. S.Cheng,“Improveddesigncriteriaandnew trellis codesfor space-

time codedmodulationin slow flat fadingchannels,” IEEE Commun.Let., vol. 5,

pp.313-315,July2001.

[190] G. TariccoandE. Biglieri, “Exact pairwiseerror probability of space-timecodes,”

IEEE Trans.Inform.Theory, vol. 48,pp.510-513,Feb. 2002.

178

[191] V. TarokhandH. Jafarkhani,“A differentialdetectionschemefor transmitdiversity,”

IEEE J. Select.AreasCommun., vol. 18,pp.1169-1174,July2000.

[192] V. Tarokh,H. Jafarkhani,andA. R. Calderbank,“Space-timeblock codesfrom or-

thogonaldesigns,” IEEE Trans.Inform.Theory, vol. 45,pp.1456-1467,July1999.

[193] V. TarokhandT. K. Y. Lo, “Principal ratiocombiningfor fixedwirelessapplications

whentransmitterdiversity is employed,” IEEE Commun.Let., vol. 2, pp. 223-225,

Aug. 1998.

[194] V. Tarokh,N. Seshadri,andA. R. Calderbank,“Space-timecodesfor high datarate

wirelesscommunication:performancecriterionandcodeconstruction,” IEEETrans.

Inform.Theory, vol. 44,pp.744-765,Mar. 1998.

[195] V. Tarokh, N. Seshadri,and A. R. Calderbank,“Combinedarray processingand

space-timecoding,” IEEETrans.Inform.Theory, vol. 45,pp.1121-1128,May 1999.

[196] E. Telatar, “Capacityof multi-antennaGaussianchannels,” EuropeanTrans.Tele-

com., vol. 10,pp.585-595,Nov.-Dec.1999.

[197] M. Uysal, “Effect of shadowing on the performanceof space-timetrellis codes,”

IEEE Trans.WirelessCommun., to appear.

[198] M. Uysal,“Pairwiseerrorprobabilityof space-timecodesin Rician-Nakagamichan-

nels,” IEEE Commun.Let., vol. 8, pp.132-134,Mar. 2004.

179

[199] M. Uysal and C. N. Georghiades, “Error performanceanalysis of space-time

codesover Rayleighfadingchannels,” J. Commun.Networks, vol. 2, pp. 351-356,

Dec.2000.

[200] V. Vaishampayan,“Combined source-channelcoding for bandlimitedwaveform

channels,” Ph.D.Thesis,Universityof Maryland,CollegePark,MD, 1989.

[201] V. VaishampayanandN. Farvardin,“Joint designof blocksourcecodesandmodu-

lationsignalsets,” IEEETrans.Inform.Theory, vol. 36,pp.1230-1248,July1992.

[202] E. Viterbo and J. Boutros,“A universallattice codedecoderfor fading channel,”

IEEE Trans.Inform.Theory, vol. 45,pp.1639-1642,July1999.

[203] U. Wachsmann,J. Thielecke, and H. Schotten,“Capacity-achieving coding for

MIMO channels: multi-stratum space-timecoding,” IEEE Veh. Technol. Conf.,

Rhodes,Greece,May 2001,vol. 1, pp.199-203.

[204] X. WangandH. V. Poor, “Space-timemultiuserdetectionin multipathCDMA chan-

nels,” IEEE Trans.SignalProcessing, vol. 47,pp.2356-2374,Sept.1999.

[205] S.Wicker, Error Control Systemsfor Digital CommunicationandStorage. Prentice

Hall, 1995.

[206] J. H. Winters,“On the capacityof radio communicationsystemswith diversity in

Rayleighfadingenvironment,” IEEE J. Select.AreasCommun., vol. 5, pp.871-878,

June1987.

180

[207] J. H. Winters, “The diversity gain of transmitdiversity in wirelesssystemswith

Rayleighfading,” IEEETrans.Veh.Technol., vol. 47,pp.119-123,Feb. 1998.

[208] J. H. Winters, J. Salz, and R. D. Gitlin, “The impact of antennadiversity on

the capacityof wirelesscommunicationsystems,” IEEE Trans.Commun., vol. 42,

pp.1740-1751,Feb.-Apr. 1994.

[209] A. Wittneben,“A new bandwidthefficient transmitantennamodulationdiversity

schemefor linear digital modulation,” Proc. IEEE Int’l Conf. Commun., vol. 3,

pp.1630-1634,May 1993.

[210] W. Xu, J.HagenauerandJ.Hollmann,“Joint source-channelcodingusingtheresid-

ual redundancy in compressedimages,” in Proc.IEEEInt’l COnf. Commun., Dallas,

TX., June1996.

[211] S. Yousefi,“Bounds on the performanceof maximum-likelihooddecodedbinary

block codesin AWGN interference,” Ph.D.Dissertation,Departmentof Electrical

andComputerEngineering,Universityof Waterloo,Ontario,Canada.

[212] K. Zeger andA. Gersho,“Pseudo-Graycoding,” IEEE Trans.Commun., vol. 38,

pp.2147-2158,Dec.1990.

[213] K. Zeger and V. Manzella“Asymptotic boundson optimal noisy channelquanti-

zation via randomcoding,” IEEE Trans. Inform. Theory, vol. 40, pp. 1926-1938,

Nov. 1994.

181

[214] K. Zeger, J. Vaisey, and A. Gersho,“Globally optimal vector quantizerdesign

by stochasticrelaxation,” IEEE Trans. Signal Processing, vol. 40, pp. 310-322,

Feb. 1992.

[215] L. ZhengandD. N. C. Tse,“Communicationon the Grassmannmanifold: a geo-

metricapproachto thenoncoherentmultiple-antennachannel,” IEEETrans.Inform.

Theory, vol. 48,pp.359-383,Feb. 2002.

[216] Y. Zhong,F. Alajaji, andL. Campbell,“When is joint source-channelcodingworth-

while: aninformationtheoreticperspective,” in Proc.22nd BiennialSymp.Commun.,

Kingston,ON, June2004.

[217] L. Zhong,F. Alajaji, andG.Takahara,“Error analysisfor nonuniformsignalingover

Rayleighfadingchannels,” IEEE Trans.Commun., accepted.

[218] S. ZhouandG. B. Giannakis,“Optimal transmittereigen-beamformingandspace-

time block codingbasedon channelmeanfeadback,” IEEE Trans.SignalProcess-

ing, vol. 50,pp.2599-2613,Oct.2002.

[219] S. Zhou andG. B. Giannakis,“Space-timecodingwith maximumdiversity gains

over frequency-selective fading channels,” IEEE Signal ProcessingLet., vol. 8,

pp.269-272,Oct.2001.

[220] G. C. Zhu andF. Alajaji, “Turbo codesfor nonuniformmemorylesssourcesover

noisychannels,” IEEE Commun.Let., vol. 6, pp.64-66,Feb. 2002.

182

[221] G.-C.Zhu,F. Alajaji, J.BajcsyandP. Mitran, “Transmissionof non-uniformmem-

orylesssourcesvia non-systematicturbo codes,” IEEE Trans.Commun., vol. 52,

Aug. 2004.

[222] X. ZhuandR.D. Murch,“Performanceanalysisof maximumlikelihooddetectionin

aMIMO antennasystem,” IEEE Trans.Commun., vol. 50,pp.187-191,Feb. 2002.

[223] S.A. ZummoandS.A. Al-Semari,“A tight boundontheerrorprobabilityof space-

time codesfor rapid fadingchannels,” Proc. IEEE WirelessCommun.Networking

Conf., Chicago,IL, Sept.2000,vol. 3, pp.1086-1089.

183

Performance Analysis and Joint Source-Channel Coding

Documents

Transcript of Performance Analysis and Joint Source-Channel Coding