bidirectional decision feedback equalization and...

BIDIRECTIONAL DECISIONFEEDBACK

EQUALIZATION AND MIMO CHANNEL TRAINING

A Dissertation

Presentedto theFacultyof theGraduateSchool

of CornellUniversity

in Partial Fulfillment of theRequirementsfor theDegreeof

Doctorof Philosophy

by

JaiganeshBalakrishnan

August2002

c�

JaiganeshBalakrishnan2002

ALL RIGHTSRESERVED

BIDIRECTIONAL DECISIONFEEDBACK EQUALIZATION AND MIMO

CHANNEL TRAINING

JaiganeshBalakrishnan,Ph.D.

CornellUniversity2002

A majorobstaclein reliabledigital communicationis inter-symbolinterference(ISI),

which is encounteredin transmissionover frequency-selective channels. A decision

feedbackequalizer(DFE) offersaneffective anda low-complexity solutionto combat

ISI. However, theDFEis suboptimalandhasaperformancegapfrom thematchedfilter

bound. In addition,theDFE suffers from errorpropagationcausedby the feedbackof

incorrectdecisions.

Theincreasein popularityof packetbasedtransmissionsystemslikeGSMor EDGE

offers the possibility of block processingof the received signal. With block process-

ing comesthe freedomto processthesignalin eithera causalor a non-causalfashion.

A novel bidirectionaldecisionfeedbackequalizer(BiDFE) architecturethat employs

time-reversalof thereceivedblock of datais proposedin this dissertation.TheBiDFE

consistsof two parallelDFEstructures,oneto equalizethereceivedsignalandtheother

to equalizethetime-reversedversionof thereceivedsignal.TheBiDFE architectureis

shown to provideasignificantperformanceimprovementoveraconventionalDFEwith

little additionalcomplexity.

To gaininsightinto theperformancelimitationsof theBiDFE, theasymptotic(asthe

noisevarianceapproacheszero)mean-squarederror (MSE) performanceof an infinite

length designis evaluated. In an attemptto further improve performance,the filter

coefficientsof theBiDFE areoptimizedto minimize theoverall MSE. However, when

theidealfeedbackassumptionis relaxed,thesymbol-error-rate(SER)performancedoes

not show an improvement. To overcomethis problem,two approachesthat offer an

additionalimprovementin SERperformance,albeitmarginal,areproposed.

TheBiDFE architectureis extendedto themultiple-inputmultiple-output(MIMO)

channelequalization.Thedesignof theBiDFE assumesknowledgeof thechannelim-

pulseresponse,which is typically estimatedat the receiver. In training basedMIMO

channelestimation,thechoiceof trainingsequenceaffectsperformance.Thecriteriaof

optimalityof MIMO trainingsequencesis derivedanddesigntrade-offs in thechoiceof

traininglengtharediscussed.

Biographical Sketch

JaiganeshBalakrishnanwasbornin thecity of Madras(now renamedChennai),Indiain

theyear1976.Thefirst17yearsof hislife werespentin Chromepet,asuburbof Madras.

He did his schoolingat N.S.NMatriculationHigherSecondarySchool,Chromepet.In

1993,he joined the Bachelorof Technologyprogramin ElectricalEngineeringat the

IndianInstituteof Technology, Madras.

He enteredgraduateschoolat Cornell University, Ithaca,NY, in Fall 1997andre-

ceivedhisM.S degreein electricalengineeringin August1999.In theSummerof 1998

and2000,heinternedwith theWirelessTechnologiesResearchDepartment,Bell Labs,

LucentTechnologiesatCrawford Hill, NJ.After graduation,heplansto join theMobile

WirelessResearchLab,TexasInstrumentsat Dallas,TX. His researchinterestsinclude

equalization,detectionandestimation,adaptivesignalprocessingandwirelesscommu-

nications.

Heis anavid birderandspendsmostof hissparetimewatchingbirds.In theSummer

of 2001,hespentafew weeksin anisland,off thecoastof Maine,workingasaresearch

intern for theSeabirdRestorationProgramwith the NationalAudubonSociety. He is

thewinnerof the2001McIlroy birding competition,heldin Ithaca,NY.

iii

To my sister, Santhi

andmy parents,

VasanthaandBalakrishnan

iv

Acknowledgements

I would like to expressmy gratitudeto Rick Johnsonfor his interestandinvolvement

with my research.His constantinput hasgonetowardsimproving my technicalwriting

style. I havegreatlybenefitedfrom theemphasisthatRick placeson collaborationwith

otheracademicandindustrialresearchers.He encourageshis studentsto pursuetheir

interests,bothacademicandnon-academic,andthishasbeenaprimaryfactorin making

my stayatCornellenjoyable.

I amgratefulto Dr. LangTongandDr. Toby Berger for servingin my committee

andfor their feedbackon my dissertation.I thankLucentTechnologiesfor giving me

an opportunityto do an internshipwith their wirelesstechnologyresearchgroupand

for their generousgift to my researchgroup,C. U. BERG.I would like to thankHarish

ViswanathanandMarkusRupp,both from LucentTechnologies,for their insight and

contributionon theproblemof trainingsequencedesignfor MIMO channelestimation.

Thanksalsofor assistancefrom NSFGrantECS-9811297,AppliedSignalTechnology,

andNxtwaveCommunications.I would like to acknowledgeWonzooChung(currently

at Dotcast),Rick Martin andAndy Klein for many interestingdiscussionsandthe re-

laxedatmospherein theoffice.

Thanksto all of my friendsfor makingmefeelathomein Ithaca.Vinayakhasbeena

goodfriend for thelastfiveyearsandI thankhim for themany illuminatingdiscussions

v

ontopicsasvariedaspolitics,sports,societyandculture.Ashokhasbeenaninspiration

to meandconversationswith him arealwaysenlightening.Ganesh,with his dedication

to researchandTamil cinema,hasbeenan invaluablefriend. Many thanksto Sowmya

for beinga caringfriend andfor providing a differentperspective on life. Venkat,with

hisgoodsenseof humorandquickwit, is adelightto bewith. Thanksto Anuragfor the

delicioushome-cookedNorth Indianfood.

A significantportion of my sparetime during the pastthreeyearshasbeenspent

on birding. Antony, an ardentlover of parrots,hasbeeninstrumentalin arousingmy

enthusiasmfor birds. I thankall my birding friendsfrom Ithacafor sharingtheirknowl-

edgeonbirdsandbirding localities.Thenumerousspeciesof birdsthatI haveseenand

enjoyedin theCayugalakebasinarelistedin AppendixC.

I thankDr. S. Sundaramfor his encouragementandadviceall throughmy life and

Dilip for beingmy friend, philosopherandguide. I utilize this opportunityto express

my deepgratitudeto my parentsandsister, to whomthis dissertationis dedicated,for

their loveandaffection.They havesacrificeda lot for my sake.

vi

Tableof Contents

1 Intr oduction 11.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Organization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Bidir ectional DFE 142.1 SystemModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 MatchedFilter Bound . . . . . . . . . . . . . . . . . . . . . . 192.2 DFEReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Finite lengthMMSE-DFE . . . . . . . . . . . . . . . . . . . . 232.2.2 Infinite lengthMMSE-DFE . . . . . . . . . . . . . . . . . . . 252.2.3 Gapfrom theMFB . . . . . . . . . . . . . . . . . . . . . . . . 262.2.4 Error Propagation. . . . . . . . . . . . . . . . . . . . . . . . . 272.2.5 NumericalExample . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 DFEEnhancements. . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.1 Non-causalDFE . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.2 DecoderFeedback . . . . . . . . . . . . . . . . . . . . . . . . 322.3.3 DecisionDeviceOptimization . . . . . . . . . . . . . . . . . . 33

2.4 Time-reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4.1 SelectiveTime-ReversalDFE . . . . . . . . . . . . . . . . . . 35

2.5 BidirectionalDFE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.5.1 Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5.2 BAD: BidirectionalArbitratedDFE . . . . . . . . . . . . . . . 382.5.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.5.4 DiversityCombining . . . . . . . . . . . . . . . . . . . . . . . 412.5.5 Time-ReversalDiversity . . . . . . . . . . . . . . . . . . . . . 432.5.6 SimulationResults . . . . . . . . . . . . . . . . . . . . . . . . 442.5.7 ImplementationIssues . . . . . . . . . . . . . . . . . . . . . . 50

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3 Infinite Length BiDFE 583.1 SystemModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2 Performanceof anInfinite LengthLC-BiDFE . . . . . . . . . . . . . . 60

3.2.1 NumericalResults . . . . . . . . . . . . . . . . . . . . . . . . 65

vii

3.3 LC-BiDFE TapOptimization . . . . . . . . . . . . . . . . . . . . . . . 673.3.1 Relationto MSE optimizedNCDFE . . . . . . . . . . . . . . . 713.3.2 Uniquenessof MMSE-BiDFE . . . . . . . . . . . . . . . . . . 71

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4 Finite Length BiDFE 744.1 MMSE-BiDFEfor a SymmetricChannel. . . . . . . . . . . . . . . . . 75

4.1.1 NumericalResults . . . . . . . . . . . . . . . . . . . . . . . . 784.2 LC-BiDFE tapoptimizationwith modifiedcost . . . . . . . . . . . . . 804.3 LC-BiDFE with Iteration . . . . . . . . . . . . . . . . . . . . . . . . . 844.4 MMSE-BiDFEfor anAsymmetricChannel . . . . . . . . . . . . . . . 844.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 BiDFE for MIMO ChannelEqualization 885.1 SystemModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.1.1 MultichannelMatchedFilter Bound . . . . . . . . . . . . . . . 915.2 MIMO Equalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.3 BiDFE Extensionto MIMO DFE . . . . . . . . . . . . . . . . . . . . . 945.4 NumericalResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6 Training SequenceDesignfor MIMO Channel Estimation 1006.1 SystemModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.2 MIMO ChannelEstimation. . . . . . . . . . . . . . . . . . . . . . . . 1026.3 TrainingSequenceDesign . . . . . . . . . . . . . . . . . . . . . . . . 104

6.3.1 TrainingSequenceLengthDesign . . . . . . . . . . . . . . . . 1066.4 Searchfor goodTrainingSequences. . . . . . . . . . . . . . . . . . . 108

6.4.1 Full search . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.4.2 RandomSearch. . . . . . . . . . . . . . . . . . . . . . . . . . 1106.4.3 Cyclic Shift Search. . . . . . . . . . . . . . . . . . . . . . . . 110

6.5 TrainingSequencefor Delay-diversityScheme . . . . . . . . . . . . . 1116.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7 Conclusions 1177.1 Summaryof Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177.2 FutureDirections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

A Additional Simulation Results 121A.1 BroadbandWirelessChannels . . . . . . . . . . . . . . . . . . . . . . 121A.2 BiDFE Performancefor aChannelwith DeepNulls . . . . . . . . . . . 123A.3 SimulationExamplefor LC-BiDFE with Iteration . . . . . . . . . . . . 125A.4 FadingChannelSimulationfor MIMO LC-BiDFE . . . . . . . . . . . . 126

B Training Sequencesfor EDGE 129

viii

C Birding the CayugaLake Basin 134

Bibliography 139

ix

List of Tables

2.1 Known symbolinformationof equalizers . . . . . . . . . . . . . . . . 392.2 Complexity of equalizerstructures. . . . . . . . . . . . . . . . . . . . 542.3 Performancedegradationdueto channelestimation. . . . . . . . . . . 562.4 Performancecomparisonof equalizeretructuresfor �� . . . . . . . . . 57

4.1 � vs. SNRfor theLC-BiDFE tapoptimizationwith themodifiedcostfunctionfor �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

B.1 Trainingsequencesfor anEDGEsystemwith � � . . . . . . . . . 130B.2 Trainingsequencesfor anEDGEsystemwith � �� . . . . . . . . . 132

C.1 List of bird speciesseenin theCayugalakebasin . . . . . . . . . . . . 134

x

List of Figures

1.1 Structureof a decisionfeedbackequalizer. . . . . . . . . . . . . . . . 31.2 Theequalizationproblem . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Thesisorganization:a roadmap . . . . . . . . . . . . . . . . . . . . . 12

2.1 Basebandsystemmodel . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Digital basebandequivalentof system. . . . . . . . . . . . . . . . . . 162.3 Structureof a GSMpacket . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Block diagramof aDFE . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Block diagramof aDFEwith idealfeedbackassumption. . . . . . . . 212.6 Error propagationin DFE . . . . . . . . . . . . . . . . . . . . . . . . 282.7 Gapfrom thematchedfilter boundandidealDFE . . . . . . . . . . . . 302.8 Structureof a Non-causalDFE . . . . . . . . . . . . . . . . . . . . . . 312.9 Structureof a BidirectionalDFE . . . . . . . . . . . . . . . . . . . . . 372.10 Structureof a BidirectionalArbitratedDFE . . . . . . . . . . . . . . . 402.11 MSEperformancefor anasymmetricchannelwith animpulseresponse�� !"��#�� $%!"��&�$��(' . . . . . . . 462.12 SERperformancefor anasymmetricchannelwith animpulseresponse�� !"��#�� $%!"��&�$��(' . . . . . . . 472.13 SERperformancecomparisonwith MLSE for anasymmetricchannel

with animpulseresponse�� )!*��$+!�� &' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.14 MSE performancecurvesfor a symmetricchannelwith animpulsere-

sponse�$,-�.��/&��$0��1&�� $01' . . . . . . . . . . . . . . . . . . 492.15 SERperformancecurvesfor a symmetricchannelwith animpulsere-

sponse�$,-�.��/&��$0��1&�� $01' . . . . . . . . . . . . . . . . . . 502.16 Performanceof aBidirectionalArbitratedDFE(BAD) for asymmetric

channelwith animpulseresponse�$,-�� $0��1 �� 0(' . . . . 512.17 Comparative MSE performancewith sametotal numberof tapsfor a

BiDFE anda DFE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.18 Effect of channelestimationon SERperformancefor an asymmetric

channelwith an impulseresponseof �� 2� �� /&�� !��#�� $�!"�� &' . . . . . . . . . . . . . . . . . . . . . . 55

3.1 Structureof anLC-BiDFE. . . . . . . . . . . . . . . . . . . . . . . . . 59

xi

3.2 Performancegapfrom thematchedfilter boundfor a symmetric3-tap

channelwith root locationsat 3�465 798 . . . . . . . . . . . . . . . . . . 65

3.3 Performancegapfromthematchedfilter boundfor anasymmetric3-tapchannelwith root locationsat :;4<5 �1= . . . . . . . . . . . . . . . . . . . 66

3.4 LC-BiDFE structurefor tapoptimizationin aninfinite lengthscenario. 67

4.1 MSE performanceof a finite lengthMMSE-BiDFE . . . . . . . . . . . 794.2 SERperformanceof afinite lengthMMSE-BiDFE . . . . . . . . . . . 804.3 SERperformanceof afinite length“modified” MMSE-BiDFE . . . . . 824.4 Comparisonof feedbackfilter tapweights. . . . . . . . . . . . . . . . 834.5 SERperformanceof aniterativefinite lengthLC-BiDFE . . . . . . . . 85

5.1 BLAST schemefor amulti-elementantennasystem . . . . . . . . . . 905.2 Structureof a MIMO DFE . . . . . . . . . . . . . . . . . . . . . . . . 935.3 MSE performanceof user1 for the MIMO testchannelC with � �>5@?A�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.4 MSE performanceof user2 for the MIMO testchannelC with � �>5@?A�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.5 SERperformancecurvesof user1 for theMIMO testchannelC with� ��5B?A�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.6 SERperformancecurvesof user2 for theMIMO testchannelC with� ��5B?A�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

A.1 Powerprofileof fadingchannelfor a typicalurbanenvironment . . . . 122A.2 SERperformancecomparisonfor a fadingchannelenvironment . . . . 123A.3 SERperformancecomparisonfor afew samplechannelswith theurban

powerprofile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124A.4 Channelzerosfor ��CD�E�F!-��$ ��G��$&��1G� !"��$&�&' . 125A.5 SERperformancecomparisonfor adeepnull channelwith impulsere-

sponse��CB��H!I��$&��1G�� $ ��1G !"�� $ �&' . . . . . . . 126A.6 SERperformanceof an iteratedLC-BiDFE for the channelwith im-

pulseresponse� JK��H!-��/&�$G �� L�� !M�� $G&�&' . . 127A.7 SERperformancecomparisonof users1-4 for aMIMO fadingchannel

environmentwith � ��956?A�� . . . . . . . . . . . . . . . . . . . . 128

B.1 Lossdueto channelestimation,� � . . . . . . . . . . . . . . . . . 131B.2 Lossdueto channelestimation,� �� . . . . . . . . . . . . . . . . . 133

xii

Chapter 1

Intr oduction

In recentyears,therehasbeenan increasein demandon the dataratecapabilitiesof

wirelesssystems.Thishasnecessitatedanincreasein bandwidthandsignalingrate.As

the bandwidthincreases,themultipathdistortionor frequency-selective fadingcaused

by thephysicalmediumbecomesworse.Themultipathchannelcausesatimedispersion

of thetransmittedsignal,resultingin theoverlapof thevarioustransmittedsymbolsat

thereceiver. This is referredto asintersymbolinterference(ISI), which, if left uncom-

pensated,causeshigh error rates.A solutionto the ISI problemis to designa receiver

thatemploys a meansfor compensatingor reducingtheISI in thereceivedsignalprior

to detection.Suchacompensatorfor theISI is calledanequalizer.

Thedelayspreads(in symboldurations)of wirelesschannelscritically dependson

thesignalingbandwidthandphysicalenvironment. For instance,a GSM system,with

a symbolrateof 0 �� Kbaud,in anurbanenvironmenthasa typical delayspreadofG&NPO . A digital terrestrialTV broadcastchannel,on theotherhand,usesasymbolrateof��0 � Mbaudandcanhaveworstcasedelayspreadsin theorderof G �&NPO . Thebaseband

digital impulseresponsefor a DTV channelhasa spanof about �� symboltapsand,

hence,theequalizeris averycritical componentof thereceiver.

1

2

A significantnumberof wirelesscommunicationsystemsare packet based. In a

GSM system,the information to be transmittedis encoded,interleaved, and thendi-

videdinto smallbursts,prior to transmission.At thereceiver, eachreceivedsignalburst

is equalizedandconcatenated,prior to deinterleaving anddecoding.The useof such

packet basedcommunicationsystemshasmadeblock processingof the received data

possible.In this scenario,non-causalprocessingof thereceivedsignalbecomesa pos-

sibility andin somecasesmaybeadvantageous.

1.1 Background

A powerful techniquefor combatingISI is Maximum-LikelihoodSequenceEstimation

(MLSE) [41], which minimizesprobabilityof errorevents.This joint equalizationand

detectiontechniquecanbe implementedwith the trellis-basedViterbi algorithm[42].

However, thecomplexity of theViterbi algorithmis proportionalto thenumberof states

in the trellis, which grows exponentiallywith the numberof symbol intervals of the

channeltime dispersion.If thesizeof thesymbolalphabetis Q andthenumberof in-

terferingsymbolscontributing to ISI is R , theViterbi algorithmcomputesQTS�U branch

metricsfor eachnew receivedsymbol.Whenthechannelspanis large,or if thealphabet

sizeis big, thentheMLSE approachbecomesimpractical.A numberof schemes,e.g.,

[35, 57], generallyknown asreduced-statesequenceestimation(RSSE),wereproposed

in the1970sand1980sin anattemptto approachtheperformanceof MLSE atareduced

complexity.

Onesuboptimalschemethatis widely usedin practice,dueto its simpleimplemen-

tation,is thelinearequalizer[61]. This approachemploysa lineartransversalfilter and

hasa computationalcomplexity that is (in a sense)a linearfunctionof thechanneldis-

3

persionlength R . For a channelintroducingonly mild interference,the performance

achievableby a conventionallinearequalizeris oftenadequate.But, in suppressingthe

ISI, the linear equalizerinevitably enhancesthe noise. Therefore,asthe channeldis-

tortion becomesseveresuchthat thereappearspectralnulls in the Nyquist band,the

applicability of a linear equalizeris eventually limited by the noiseenhancementor

noisegain.

Thebasiclimitation of a linearequalizerin copingwith severeISI hasmotivateda

considerableamountof researchinto suboptimalnonlinearequalizerswith low compu-

tationalcomplexity, suchasthedecisionfeedback equalizer(DFE).Theclassicstructure

of this receiver is shown in Figure1.1. TheDFE consistsof a linear feedforwardfilter

(FFF)andafeedbackfilter (FBF).TheFFFsuppressesthecontributionof thepre-cursor

ISI, namelytheinterferencecausedby thesymbolstransmittedafterthesymbolof inter-

est. TheFBF cancelsthepost-cursorISI by subtractinga weightedlinearcombination

of theprevioussymboldecisions,assumedto becorrect.Theresultis thenappliedto a

thresholddevice to determinethesymbolof interest.TheFFFenhancesthenoise,but

thenoisegainis notassevereasin thecaseof a linearequalizer.

^

FeedForwardFilter

FeedbackFilter

y(n)Received

Signal

Output data

detectorsymbol

Symbol−by−

s(n)

Figure1.1: Structureof adecisionfeedbackequalizer

The ideaof usingprevious decisionsto copewith the ISI problemwasfirst intro-

ducedin 1967by Austin [9], only two yearsafterthedevelopmentof thedigital (adap-

4

tive) linearequalizerby Lucky [61]. Thedecisionfeedbackreceiver thatminimizesthe

MSEbetweentheinput to thethresholddeviceandthetransmittedsymbolwasfirst ob-

tainedby Monsen[70]. In [74], the joint optimizationof the transmitterandreceiver,

to maximizethe output signal-to-noiseratio (SNR) of a zero-forcingDFE, wascon-

sideredby Price. The jointly optimumMMSE transmitterandreceiver wasobtained

by Salz[80]. Using linearspacegeometricargumentsMesserschmitt[67] showedthe

equivalenceof zero-forcing(ZF) decisionfeedbackto MMSE predictionof a random

process(and also the equivalenceof a linear ZF receiver to linear interpolationof a

randomprocess),andthusprovidedsimplederivationsof theoptimumfilters andcon-

ditions for their existence.In [17], BelfioreandPark introduceda new DFE structure,

called the noisepredictive DFE andshowed its equivalenceto the conventionalDFE

for infinite-lengthfilters. Infinite-lengthresultson the MMSE-DFE wereextendedto

thefinite-lengthcasein [1]. Thus,developmentsin DFEdesignhavecontinuedover30

yearssinceits inception.

TheDFE designhastypically beencarriedout assumingthat thepastdecisionsare

error-free,thussimplifying themathematicsinvolved.However, whenanerror is made

by thereceiver theoutputof theFBF is no longerthedesiredvalueandtheprobability

of subsequenterrorsis increasedresultingin errorsthat tend to occur in bursts. The

first residual-induceddecisionerror, called a primary error, is fed back by the FBF

causingsecondaryerrorsandcreatinganerrorburst.Thisphenomenon,known aserror

propagation, is moreseverewhenthe tap weightsand/orthe numberof the feedback

tapsarelarge.

The performancelossof the DFE dueto error propagationcanbe evaluatedusing

Markov chainanalysis.This technique,however, is not feasiblewhentheFBF lengthis

largeasthenumberof statesin theMarkov chainincreasesexponentiallywith theFBF

5

length. Someresearch,which beganin the early 1970sshortly after the inventionof

theDFE,hasbeendoneto simplify theanalysisby clumpingerrorstatesandproviding

upperboundson theerrorprobabilities.[4, 25, 53] usethis approachto providebounds

onerrorpropagation.[16,52,28, 73] arebasedonMarkov chaintechniquesandprovide

othererrormeasuressuchasmeanerrorburst length. It is alsopossibleto studyerror

propagationfrom adynamicalsystemsperspective [54, 55].

Althoughthephenomenonof errorpropagationhasbeeninvestigatedfor alongtime,

only amoderateamountof work hasbeendoneonmitigatingerrorpropagationin deci-

sion feedbackequalizers.Several techniques,basedon transmitterprecoding[87, 46],

wereproposedin theearly1970sto remedytheerrorpropagationproblemin theDFE.

Thesetechniquesrequireknowledgeof thechannelresponseat thetransmitter, andthus

areinapplicableif thetransmissionis broadcastor if thechannelis time-varying. Dog-

nancay[29] proposedtechniquesfor detectingdecisionerrorsin equalizationschemes

including DFEs. In [36], Fertnerintroduceda schemewhich attemptsto prevent pri-

maryerrorsandhenceerrorpropagation.However, theperformanceof this algorithm

critically dependsonthevalueof thefirst post-cursorandtheaccompanying noiselevel.

Ariyavisitakulproposedtheuseof acombinationof soft decisionsanddelayeddecoder

decisionsto cancelISI for a joint convolutionalcodingandDFE schemein [8]. In [8],

the soft decisiondevice for the DFE wasobtainedby usinga simplified maximuma-

posterioriprobability (MAP) algorithm. In [10, 14] the useof a soft decisiondevice

to mitigate error propagationwas consideredand the decisiondevice was optimized

for minimizing the mean-squarederror (MSE) andthe bit-error rate(BER). In [31], a

techniquethatcombinesa DFE with a Viterbi algorithmto provide a trade-off between

complexity andperformancewasproposed.

Even in the presenceof ideal feedback,i.e., no decisionerrors, the performance

6

of the DFE is suboptimalwhencomparedto the matchedfilter bound. The matched

filter bound(MFB) is definedastheoutputsignal-to-noiseratio (SNR)whenamatched

filter is usedat the receiver andonly onesymbol is transmitted(i.e., no ISI). This is

equivalent to the performancethat can be obtainedthrougha channelwhereall the

energy is concentratedin asingletap.ThegapfromtheMFB is agoodmetricto evaluate

MSEperformanceof equalizerstructures.Theperformancegapfrom thematchedfilter

boundis anotherdrawbackof theDFE.

Theprincipleof non-causaldecisionfeedbackequalization(NCDFE)wasfirst pro-

posedby Proakis[75]. The NCDFE usesboth pastandfuture decisionsto cancelall

the ISI. TheMMSE optimizationof theNCDFEwasderivedby GershoandLim [43].

An estimateof the future transmittedsymbolsis obtainedby the useof a preliminary

equalizerandthe introductionof an appropriatedelayprior to the NCDFE. Whenno

decisionerrorsaremade,theNCDFEattainstheMFB andhenceperformsbetterthan

a DFE. However, thepresenceof decisionerrorscancausea performancedegradation

andonly a fractionof theachievableperformanceimprovement(whencomparedto the

DFE) canberealized.Thefeedbackfilter of theNCDFEis effectively twice aslong as

comparedto theDFEandresultsin increasederrorpropagation.

A burstmodeunbiasedMMSE versionof theNCDFEwasproposedby Slockandde

Carvalhoin [81]. Later, they alsoproposedtheuseof softdecisionsto decreasetheeffect

of errorpropagationin [27]. In 2001,ChanandWornellproposedablock-iterativeDFE

[21] whichattemptsto attaintheMFB. TheblockiterativeDFEis similarto theNCDFE,

but unliketheNCDFE,thetapcoefficientsof theblockiterativeDFEarerecomputedfor

eachiterationto incorporatetheeffect of thereliability of thedecisionsthatareusedto

cancelthepre-cursorandpost-cursorISI. TheoptimalDFE tapcoefficientsarederived

undertheassumptionthatthefrequency responseof thechannelis i.i.d (independentand

7

identicallydistributed)anddrawn from acomplex Gaussiansource.Thisassumptionis

satisfiedonly whenthenumberof channeltapstendsto infinity. Numericalresultsshow

thatwhenthechanneldelayspreadsarelarge(about100taps),theperformanceof the

block-iterativeDFEapproachestheMFB aftera few iterations.

1.2 Moti vation

Equalization

DFELinear MLSE/MAP

PropagationError

FeedbackDecoderDecision Device

OptimizationBidirectional

DFENon−causal

DFE

from MFBGap

Low HighLowComplexity:

Impairments:

ModeratePoorPerformance: Good

Figure 1.2: Theequalizationproblem

The motivation for this dissertationhasbeenpictorially representedin Figure1.2.

The low complexity andmoderateperformanceof theDFE make it a preferredchoice

asanequalizerstructure.However, theDFE hasbeenshown to possesstwo significant

drawbacks- error propagationanda gapfrom the matchedfilter bound. Someof the

earlierwork onDFEshaveattemptedto addressthesetwo issuesonanindividualbasis.

Recently, it hasbeenshown [11, 64, 12] that the employmentof time-reversalof the

8

receivedsignal,alongwith bidirectionalprocessingusingaDFEstructure,is aneffective

meansof simultaneouslyaddressingthesetwo drawbacksandimproving performance.

A time-reversaloperationis doneby reversingthesequentialorderof the received

samples,in time,prior to equalization.Thelinearconvolutionstructureimposedby the

channelis conserved by the time-reversaloperation.However, the equivalentchannel

impulseresponse,asseenby theequalizer, becomesatime-reverseof theactualchannel

impulseresponse.Theideaof bidirectionalequalizationof thereceivedsignalemploy-

ing aDFEwasindependentlyproposedin theearly90sby Ariyavisitakul [5] andSuzuki

[83].

Ariyavisitakul [5, 6] proposedtheuseof aselectivetime-reversalstructurein aDFE.

He considereda packet transmissionsystemcommunicatingthrougha quasi-staticfre-

quency selective fadingchannel.Thetime-reversalstructurewasoriginally proposedto

improve performancewhena finite lengthconstrainton thenumberof DFE filter taps

is imposed. Undersucha constraint,the performanceof a normalmodeDFE anda

time-reversalmodeDFEis different.Ariyavisitakulproposedtheuseof eitheranormal

modeDFE or a time-reversalmodeDFE, basedon the performanceof the two DFE

structuresover thechannelrealizationencounteredby thepacket. It wasfurthershown

that if the lengthconstrainton theDFE is relaxed,boththenormalmodeDFE andthe

time-reversalmodeDFE have thesameperformance.A similar ideawasalsoconsid-

eredin [50] and[60]. Although the term bidirectionalDFE is usedin someof these

publications,it refersto theselective time-reversalapproach.

Recently, bothBalakrishnanetal. [14] andMcGahey etal. [64] have independently

proposeda truly bidirectionalDFE architecture.In a bidirectionalDFE, henceforthre-

ferredto asBiDFE, the received signal is processedusingboth a normalmodeanda

time-reversalmodeDFE andthe outputof the two streamsarecombinedto improve

9

performance. In [64], the combiningof the two output streamsis doneusing a re-

constructionbasedarbitrationtechnique.The arbitrationis performedto select,on a

symbol-by-symbolbasis,betweentheoutputsof thenormalmodeandthetime-reversal

modeDFEs.As errorpropagationis a causalphenomenon,theerrorburstsof a normal

modeDFE anda time-reversalmodeDFE proceedin oppositedirectionsin time. On

the basisof numericalresults,McGahey et al. reportedthat the improvementin their

bidirectionalarbitratedDFE(alsoknown asBAD) is obtainedby exploiting thelow cor-

relationbetweentheerrorburstscausedby thenormalmodeDFEandthetime-reversal

modeDFE.

However, the advantageof combiningthe outputsof the normalmodeDFE with

that of the time-reversalmodeDFE is not limited to mitigating error propagation.In

fact, even in the presenceof ideal feedback,namelyno decisionerrors,the noiseat

the outputsof the normal modeDFE and the time-reversalmodeDFE exhibit a low

correlation. A diversity combiningschemeemploying a weightedlinear combination

of theoutputsof thenormalmodeDFE andthetime-reversalmodeDFE wasproposed

in [11] andit wasshown that theLC-BiDFE (linearcombiningbidirectionalDFE) has

a smallervalueof noise-enhancement,whencomparedto eitherof thetwo constituent

DFEs. In fact, if thefinite lengthconstraintis relaxed, theLC-BiDFE, underthe ideal

feedbackassumption,actuallyattainsthe matchedfilter bound[12]. This “diversity”

arisesfrom the assumptionthat the pastsymbolsareknown to the normalmodeDFE

andthefuturetransmittedsymbolsareknown to thetime-reversalmodeDFE.Thenon-

causalprocessingof the receivedsignalalongwith thenonlinearstructureof theDFE

make this knowledge,althoughimperfectin thepresenceof decisionerrors,possible.

The ideaof time-reversalhasbeenemployedin diverseareasin thepast.Raphaeli

[77] employstime-reversalto improveperformanceof areducedstateViterbi algorithm

10

appliedto demodulationof trellis codes. [78] extendsthis idea to reduced-statese-

quenceestimatorsfor combinedequalizationand decoding. Similar approachesem-

ploying time-reversalanda bidirectionalstructure[65, 58] have alsobeenproposedfor

RSSEs. In [59] Lindskog employs time-reversalto extend the Alamouti scheme[3],

that achievestransmitdiversity for two transmitantennasover flat fadingchannels,to

frequency-selectivechannels.

Thedemandfor higherdataratesin packet basedcommunicationsystemsnecessi-

tatesimprovedsignalprocessingatthereceiver, albeit,atalow complexity. Thepurpose

of this dissertationis to demonstratethat thebidirectionaldecisionfeedback equalizer

(BiDFE) architectureproposedhereis asuitablecandidatefor suchanimprovedequal-

izer structure. The idea of using time-reversalin a DFE hasreceived little attention

from the researchcommunityandinvestigationof this topic would contribute towards

promotingour understandingof theDFEandits unrealizedpotential.

The threeBiDFE architectures,namely, theselective time-reversalDFE [5, 6], the

LC-BiDFE [11, 12] andthe BAD [64], eachoffer a performance/complexity trade-off

that is distinct from that of the others. The primary focusof this dissertationwill be

the LC-BiDFE, the structureof which is particularlyamenableto theoreticalanalysis;

the intuition resultingfrom this studywill help us betterunderstandthe otherBiDFE

architectures.The LC-BiDFE doesnot necessarilyoffer the bestperformancewhen

comparedto every otherBiDFE architecture(e.g.,BAD), but in mostscenariosit pro-

videsagoodperformanceat asignificantlylowercomplexity.

11

1.3 Organization

A pictorial depictionof theorganization1 of this dissertationis presentedin Figure1.3.

Chapter2 describesthe systemmodel, notationand the variousassumptionsusedin

this dissertation.A brief review of the DFE is followed by a discussionon its limita-

tions. Someof theconventionalmethodsof addressingthedrawbacksof theDFE are

discussed.A generalizedbidirectionalDFE architecturethatemploys time-reversalof

thereceivedsignalis proposedasanappealingalternativeto simultaneouslyaddressthe

limitationsof theDFE.Theperformanceimprovementofferedby theBiDFE is demon-

strated,bothanalyticallyandnumerically. TheLC-BiDFE is shown to offer significant

performanceimprovementover the conventionalDFE for only a modestincreasein

complexity. Variousissuesrelatedto the implementationof the BiDFE arediscussed.

on this

Thelengthconstraintof anLC-BiDFE is relaxedandtheperformancefor aninfinite

lengthscenariois analyzedin Chapter3. Theinfinite lengthLC-BiDFE suffersfrom a

smaller, but non-zero,gapfrom theMFB whencomparedwith theMSEperformanceof

aninfinite lengthDFE.Thismotivatesthetapoptimizationproblemfor theLC-BiDFE,

wherethe coefficientsof the constituentDFEsareoptimizedto minimize the overall

MSE of thesystem,ratherthantheoutputMSE of theindividual streams.TheMMSE-

BiDFE solutionthatminimizestheoverallMSEof theLC-BiDFE is derived.It is shown

thattheinfinite lengthMMSE-BiDFE,undertheidealfeedbackassumption,attainsthe

matchedfilter bound.

In Chapter4, thetapoptimizationproblemis extendedto themorepracticalscenario

of a finite lengthLC-BiDFE. First, thetapcoefficientsof theLC-BiDFE areoptimized

1The materialin this dissertationis constitutedfrom partsor whole of the papers[11, 12, 13,89] writtenby theauthorin collaborationwith others

12

improvementBiDFE offers Evaluate BiDFE

performance

proposedBiDFE

Relax lengthconstraint

BiDFE tapsOptimize MFB is

attained

Known channelassumption BiDFE taps

Optimize Impose lengthconstraint

Estimatechannel

training sequencesDesign good Iterated

BiDFE

Relax idealfeedback

No gains fromtap optimization

modified costOptimize with

Only marginalimprovementdelay diversity

Training for

DFE haslimitations

Employtime-reversal

EqualizationMIMO

BiDFEExtension

Chapter 2 Chapter 3

Chapter 4Chapter 6

Chapter 5

Figure 1.3: Thesisorganization:a roadmap

for thespecialcaseof asymmetricchannel.It is shown thattheadditionalMSEperfor-

manceimprovementobtainedvia tapoptimization,usingtheidealfeedbackassumption,

doesnot readilytranslateinto animprovementin symbolerrorrate(SER)performance

whendecisionfeedbackis employed. Two solutions,onebasedon tap optimization

with a modifiedcostandanotheron an iterative approach,aresuggestedto overcome

this drawback in the absenceof ideal feedback. The BiDFE tap optimizationis also

extendedto theasymmetricchannelcase.

The ideaof employing time-reversalis extendedin Chapter5 to themultiple-input

multiple-output(MIMO) channelequalizationproblem. The MIMO DFE affords the

13

possibility of user re-orderingin addition to time-reversal. A MIMO BiDFE struc-

turethat incorporatesbothtime-reversalanduserre-orderingis proposedandis shown

to provide performanceimprovementssimilar to thesingle-inputsingle-output(SISO)

BiDFE.

Chapters2-5assumethatthechannelimpulseresponseis perfectlyknown at there-

ceiver. In practicethisassumptionis almostalwaysviolatedandthechannelresponseis

typically estimatedwith theaidof atrainingsequence.Thedesignof trainingsequences

for theSISOchannelestimationproblemhasbeencloselystudiedfor overthreedecades.

However, verylittle is known onthedesignof optimaltrainingsequencesfor theMIMO

channelestimationproblemandthis is consideredin Chapter6. Theoptimalitycriterion

for theMIMO trainingsequencesis derived,andthedesigntrade-offs in thechoiceof

training sequencelengtharediscussed.A few reducedcomplexity techniques,for the

implementationof a searchfor goodtrainingsequences,areproposedandnear-optimal

trainingsequencesdeterminedusingthesemethodsaretabulated. For thespecialcase

of a delay-diversityscheme,it is shown that transmittingthesameoptimal trainingse-

quencefrom eachof the two transmitantennasis optimal for training basedchannel

estimation.

Finally, Chapter7 providesa summaryof resultsanddirectionsfor futureresearch.

MATLAB scriptfiles for reproducingthesimulationplots in this dissertationareavail-

ableat “http://bard.ece.cornell.edu/matlab/balakrishnan/index.html”.

Chapter 2

Bidir ectional DFE

Thischapterreviewssomewell known resultsonDFEsandafew earlierattemptsto ad-

dressits drawbacks.Section2.1describesthesystemmodelandstatestheassumptions

madein thisdissertation.Thematchedfilter bound(MFB), asuitablemetricto evaluate

mean-squarederror (MSE) performanceof equalizerstructures,is definedin this sec-

tion. Section2.2 introducesthestructureof a DFE, tapoptimizationof the filter taps,

thegapfrom thematchedfilter bound,andtheerrorpropagationphenomenonobserved

in DFEs.Thenon-causalDFEstructureproposedby GershoandLim [43], to attainthe

matchedfilter bound,is describedin Section2.3. A few techniquesthatmitigateerror

propagationin DFEsarealsoreviewed. The conceptandimplication of time-reversal

of the received signalis introducedin Section2.4. The bidirectionalDFE structureis

proposedin Section2.5 asa suitablesolutionto addressthe limitationsof a DFE. The

rationalebehindtheBiDFE architecture,theBAD architectureproposedby McGahey

et al. in [64] andtheLC-BiDFE proposedin [11], is discussed.Theperformanceim-

provementofferedby theBiDFE is investigated,bothanalyticallyandnumerically, and

variousimplementationissuesarediscussed.Section2.6 providesa summaryof the

contentspresentedin this chapter.

14

15

2.1 SystemModel

Information

Symbols

noise

T

Sampled

sequence

Transmitpulse

shapingChannel

Whitenedmatched

filter

Figure2.1: Basebandsystemmodel

Datacommunicationthroughafrequency-selectivechannelis considered.Typically,

adigital communicationsystemconsistsof aninformationsource,asourceencoderand

a channelencoder. The output of the channelencoderis mappedonto a signal con-

stellation. The mappedsymbolsare then pulseshaped,modulatedby a carrier, and

transmittedthrougha frequency-selective channel.At the receiver, the receivedsignal

is down-converted,filtered(with anappropriatereceive filter), sampledandprocessed.

This systemcanbe modeledusingthe basebandequivalentdepictedin Figure2.1. In

thebasebandequivalentsystemshown in Figure2.1,theinformationsymbolsaretrans-

mitted througha frequency-selective, baseband,analogchannelandarecorruptedby

additive noiseat the receiver. A whitenedmatchedfilter front-endis employed at the

receiver andthe outputof the matchedfilter is sampledat baud-rate(symbol rate) to

obtaina sampledsequence.Forney [41] hasshown that the sampledsequenceat the

receiver constitutesa sufficient statisticfor detectingthe informationsymbols.Hence,

any optimalreceiverstructurecanbeprecededby thewhitenedmatchedfilter [41]. Fur-

thermore,thepresenceof a whiteningfront-endfilter ensuresthatthenoisecomponent

of thesampledsequenceis white.

Thedigital basebandequivalentof theanalogchannelmodelis illustratedin Figure

2.2. The channelV representsthe digital equivalentof the combinedtransmitpulse-

shapingfilter, analogchannelandfront-endwhitenedmatchedfilter. However, thechan-

16

w (k)

r (k)s (k) ChannelC

Figure 2.2: Digital basebandequivalentof system

nel impulseresponseis not known apriori at thereceiverandis usuallyestimatedusing

a trainingsequenceembeddedin thedata.Hence,it is not feasibleto have a front-end

analogfilter that is matchedto thechannelimpulseresponse.Typically, a front-endre-

ceivefilter thatis matchedto thetransmitpulseshapingfunctionis usedandtheoutput

of the receive front-endfilter is oversampledto obtainsufficient statistics.For exam-

ple,whena square-root-raised-cosine(SRRC)functionis usedasa pulseshapingfilter

this samefilter is thetypical front-endarchitectureof thereceiver andit hastheadded

advantageof aidingtiming synchronization.Yet,weassumethedigital basebandequiv-

alentmodelshown in Figure2.2; the intuition andresultsderived with sucha model

aregeneralenoughandcanbeeasilyextrapolatedto accommodateoversamplingat the

receiver.

Training DataData

Tail bits

Guard Period

Tail bits

Figure 2.3: Structureof aGSMpacket

We considera packet basedcommunicationsystemmodel,where W is thenumber

of symbolstransmittedin eachdatapacket. The structureof a typical packet for a

Global SystemsMobile (GSM) transmissionstandardis illustratedin Figure 2.3. A

17

GSM packet hasa lengthof about156 symbolswith the encodeddatasplit into two

chunksof 58 symbolseach. A 26 symbol long training sequenceis embeddedin the

centerof thepacketasamid-ambleandis usedfor channelestimation.Theremainderof

thepacket consistsof tail bits andguardsymbols.Thepacketizationof thetransmitted

dataresultsin someedgeeffects, i.e., the symbolsat the edgeexperiencea truncated

versionof thechannelimpulseresponseasopposedto thesymbolsat thecenterof the

packet. Kaleh[49] anddeCarvalho[81, 27] have discussedtheimplicationof theedge

effects in packet basedsystemsandshown that this canbe exploited to decreasethe

symbol-error-rate (SER) for the symbolsat the edges(or) the symbolsadjoining the

training sequences.However, as this aspectof the packet basedsystemis limited to

theedgesymbols,it will not bediscussedhere. Unlessstatedotherwise,we make the

following assumptions.

Assumption2.1 Thechannel V canbemodeledasa linear time-invariant (LTI) filter

for thedurationof each packet.

Most wirelesscommunicationchannelsaretime-varying,albeit slowly. Although, the

time variationscanbe significantbetweennon-contiguouspackets,the channelvaria-

tionsoverthedurationof apacketcanbeignoredundercertainconditions.For instance,

in a wirelessenvironmentunderlow mobility scenarios,this assumptionis valid andis

referredto asaquasi-staticfadingchannel[6]. In awirelineenvironment,thetimevari-

ationsare typically slow, and if the durationof the packet is small, comparedto the

time-constantof channelvariations,this assumptionis valid.

Assumption2.2 Thechannelcanbemodeledasa finite impulseresponse(FIR) filter.

This is a commonassumptionusedin theliterature.Although,thedelayspreadsof the

channelimpulseresponsedependson the transmissionenvironment,the FIR approxi-

mationis typically valid. For aGSMtransmissionsystem,thechanneldelayspreadcan

18

beasmuchas20 NXO (equivalentto 20 symbolperiodsfor a datarateof 1 Mbps) for a

hilly terrain.

For thecausallinearchannelmodelwith a finite span,i.e., finite-impulse-response

(FIR), thereceivedsequenceY9Z\[9] canbeexpressedas,

Y�Z_^`]<� S&acb d egf9h-i Z\[9]jOkZ_^l!+[�]Xm+nTZo^X] (2.1)

where �p�q� i Z\�$] i Zr��]s�t�u� i Z\R�vB!A��]c'Fw is the channelimpulseresponsewith R�v taps,OkZ_^`] representsthetransmittedsourcesymbolsand nTZ_^`] is theadditivenoisesequence.

Fromequation(2.1), we noticethatat any samplinginstantthe receivedsequenceis a

functionof notonly thecurrenttransmittedsymbol,but alsocontainscomponentsfrom

theearliertransmittedsymbols.Thisphenomenonis known asintersymbolinterference

(ISI).

Assumption2.3 ThenoisesequencenxZ_^`] is additivewhiteGaussian(AWGN)with a

variancey �z andis independentfromthesourcesequenceOkZ_^`] .This is a standardassumptionmadein the literature. The digitization of the received

signalresultsin thetruncationof theprobabilitydistribution function(pdf) of thenoise

process. Although, the pdf of the noiseprocesshasonly a finite support,it will be

approximatelyGaussian.

Assumption2.4 Thechannelimpulseresponse� is knownat thereceiver.

In typical communicationsystems,the channelimpulseresponseis not known at the

receiver. Most packet basedcommunicationsystemsprovidea trainingsegment,which

couldbein theorderof 10-20%of thepacketsize,in thedataburst.A priori knowledge

of thetrainingsequenceis usedat thereceiver to estimatethechannelimpulseresponse.

Theaccuracy of thechannelestimateis dependentonthelengthof thetrainingsequence

19

relative to the delayspreadof the channelimpulseresponse,and the noisevariance.

However, we assumeperfectknowledgeof thechannelat thereceiver.

2.1.1 Matched Filter Bound

Thematchedfilter receiverwasoriginally designedasanoptimaldetectorfor messages

transmittedusinga setof orthonormalbasisfunctions[76, 47]. Turin [88] providesa

review of thematchedfilter andits properties.In thecontext of digital communication

througha frequency-selective channel,the sampledoutputof a matchedfilter receiver

provides sufficient statisticsfor detection. Here, the sufficient statisticbearsall the

information(aboutthe transmittedsymbols)containedin the received signal. For the

systemillustratedin Figure2.2, the matchedfilter consistsof a digital FIR front-end

with animpulseresponsegivenby {|$} , where ~�&��9�<��&�r�I�9� . Let � �_�`� denotetheoutput

of sucha matchedfilter receiver. Then � �o�X� canbeexpressedas,

� �_�`�� g��>�&�\�� } �r�I�9�r��_��"�9�� g��>�&�\��$� �\�9�j�k�_�l�M�9�X� ��g��>�(�� } �r�I�9�r�x�_�l�+�9� (2.2)

where

�� |�� ~| } (2.3)

is theauto-correlationfunctionof thedigital channelimpulseresponse.Whenonly one

symbol is transmittedin a packet, i.e., � � � , the interfering symbolsare all zero.

Hence,thereis nocontribution from theISI termsin equation(2.2),andit reducesto,

� �o�X�� \�$� �k�_�l�+�9�X� ��g��>�&�\�� } �c��r�T�o��"�9�D¡ (2.4)

In suchascenario,thematchedfilter receivermaximizesthesignal-to-noiseratio(SNR)

at theoutput.TheoutputSNRof suchamatchedfilter receiver is commonlyreferredto

20

asthematchedfilter bound(MFB) andis givenby,

¢mfb

�¤£`¥¦ � �_�$�£ ¥§ (2.5)

where£ ¥¦ is thevarianceof thesourcesymbols.TheMFB canbeattainedat thereceiver

underthefollowing scenarios,

1. Whenonly onesymbolis transmittedin thepacket. Thisresultsin nointersymbol

interferenceandthematchedfilter receiver is theoptimaldetector.

2. If all transmittedsymbolsin thepacket,exceptthesymbolof interest,areknown.

In this case,the ISI componentsaffecting thesymbolof interestat theoutputof

thematchedfilter receiver (seeequation(2.2))canbecanceledperfectly.

In reality, whentheabove two conditionsarenot satisfied,thematchedfilter bound

cannotbe attainedundermostcircumstances.Even the optimal equalizer, namelythe

MLSE, hasagapfrom theMFB in mostscenariosandthisgapcanbedeterminedbased

on thecoefficientsof thechannelimpulseresponse[76]. In fact, theMLSE attainsthe

MFB only whenthechannelimpulseresponseis limited to a lengthof 2 symboltaps.

In spiteof the fact that theMFB cannotbeattained,thegapfrom theMFB is a useful

metricto compareequalizerstructures.

2.2 DFE Review

Thedecisionfeedbackequalizerconsistsof a feedforwardfilter (FFF),a feedbackfilter

(FBF) anda decisiondevice. TheDFE block diagramis illustratedin Figure2.4. Let

the feedforward filter ¨ have an impulseresponse,© �«ª¬®�_�$�¯¬°�r�1�"¡u¡u¡±¬®�\²<³´��1�cµF¶ ,

where ²<³ is thenumberof tapsin theFFF. Similarly, thefeedbackfilter · is an ²�¸ -tap

filter with animpulseresponse,¹ �ºª»(�r��D»&�\¼��½¡u¡u¡¾»(�\²�¸r�cµF¶ . Thedecisiondevice, ¿ is

21

^

Q (.)

y(n)r(n)

s(n)B

F

Figure 2.4: Block diagramof aDFE

usuallya quantizerthatmakesharddecisionsÀ�k�_�X� on theoutput � �o�X� of theDFE. The

output � �_�X� of theDFEcanthenbemathematicallyexpressedas,

� �_�X�� ÁÂ�� g� �¬®��r��_�Ã�+�9�°� � Ä� �g�� »(�\�9� À�>�_�l�+�9�@¡ (2.6)

^

−δ

C F Q (.)

Bz

s(n) r(n) y(n) s(n)

w(n)

Figure 2.5: Block diagramof aDFE with idealfeedbackassumption

Assumption2.5 Thedecisionsfed back to the FBF are error-free, i.e., ideal decision

feedback.

Although,this is almostnever truein a practicalimplementation,it is akey assumption

madewhile designingDFEs. The “ideal feedback”assumptionsimplifiesthe mathe-

maticsinvolvedandmakesanalysistractable.However, theeffect on thesymbolerror

rateperformance,whenthis assumptionis violated,will alsobeinvestigated,albeitnu-

merically.

22

Let Å bethedetectiondelayof theDFE.Then,underassumption2.5,theequivalent

blockdiagramof theDFE is shown in Figure2.5andequation(2.6)canberewrittenas

� �_�`�� Á �� g� �¬®�\�9�r��_�l�"�� Ä� �g�� »(�\�9�j�k�_�l� Å �+�9�Æ¡ (2.7)

Let Ç representthecombinedchannel-feedforwardfilter impulseresponse,i.e, Ç � |�� © .Then,Ç ��ªÈP�\�$�<ÈP�r�1�l¡u¡u¡6ÈP�_²<Ég�x²6³��±¼ �cµF¶ hasalength ²�Ê � ²<ÉË�x²<³��Ì� taps.Equation

(2.7)canberewrittenas,

� �o�X�� ÈP� Å � �>�o�l� Å �Í ÎtÏ Ð �sÑ �� g� �ÈÒ�\�9�j�k�o�l�+�9�Í ÎtÏ Ð � � Ä� �g��tÓ ÈP�\�Ô� Å �°�"»(�\�9�ÖÕ&�k�_�l� Å �+�9�Í ÎtÏ Ð

cursor res.pre-cursorISI modeledpost-cursorISI

� � � �9� Á � ¥��g� Ñ �9� Ä ��ÈP��j�k�_�l�+�9�Í ÎtÏ Ð � � Á �� g� �

¬®��r�T�o�Ã�+�9�Í ÎtÏ Ðresidualpost-cursorISI filterednoise

(2.8)

An effectivewayof equalizingthechannelwouldbefor theFFFto× shapethecombinedchannel-feedforwardfilter responseÇ to maximizetheenergy

of thecursor, i.e., ÈP� Å ��Ø�� in theunbiaseddesign

× result in a small residualISI, i.e., ÈP��±ØÙ� for �ÛÚÜ�LÝ Å and Å ��²<¸TÝº�)Ú²�ÉÞ�)²<³-�+¼× keepthenoisegain, © ¶ © assmallaspossible

andfor theFBF to× cancelexactly the modeledpost-cursorISI by matchingthe FBF tapsto that of

thecombinedresponse,i.e., »(�\�9�<��ÈP��´� Å �Æßu�´Ú��lÚ%²�¸ .This is essentiallywhat theMSE optimizedDFE does,aswill becomeapparentin the

next subsection.

23

2.2.1 Finite length MMSE-DFE

Thecoefficientsof theDFE canbedesignedto beeitherzero-forcing(perfectlycancel

the ISI) or to minimize the mean-squarederror (MSE). TheMMSE optimizedDFE is

popularandthefinite lengthdesignwasderivedby Al-Dhahir [1]. A simplifiedderiva-

tion of the finite lengthoptimal DFE coefficientswill be reviewed in this subsection.

TheMSE at theoutputof theDFErepresentedin Figure2.5 is givenby,

£ ¥à � E á Ó �k�_�l� Å �°� � �o�X�ÖÕ ¥râ ß (2.9)

whereã �o�X�<��k�o�±� Å �ä� � �_�`� is thesoftdecisionerrorof theequalizer. Thegoalof the

MMSE designof theDFEis to determinethefilter pair � © mmseß ¹ mmse� thatminimizes£ ¥àof equation(2.9). Wenow definethe ²<Ê¯å½²<³ channelconvolutionmatrix æ as

æ �

çèèèèèèèèèèèèèèèé

�&�\�� ê�ê�ê �... �&�\�$� ê�ê�ê ...� �_²�Éë�s�1� ...

. . . �� &�\²�ÉÒ�ì�1� . . . � �_�$�... � . . .

...� ... ê�ê�êA�&�\²�É°�ì��

íHîîîîîîîîîîîîîîîï¡ (2.10)

WealsodefinethevectorsðÂñ , ÀðÆñ and òÌñ as,

ðÂñ � ª�>�o�X�<�k�_��ì�1�l¡u¡u¡B�k�_�l�"²�Êó�s�1�cµ ¶ÀðÂñ � ª�>�o�� Å �ì�1�l¡u¡u¡B�k�_�l� Å �+¼ ��>�o�l� Å �M²�¸r�cµ ¶ (2.11)òÌñ � ªô�T�_�`�®�T�_�2�s�1��¡t¡u¡<�T�_�l�õ²6³ó�s�1�cµ ¶ ¡

Fromequations(2.1)and(2.6),equation(2.9)now becomes,

£ ¥à � E á Ó �k�_�l� Å �°� ð ¶ ñ æ´© � Àð ¶ ñ ¹ � ò ¶ñ © Õ ¥jâ (2.12)

24

which canbeminimizedby settingthegradientof £`¥à with respectto © and ¹ aszero.

First, öø÷ £ ¥à � E á �I¼ ÀðÂñ ð ¶ñ æ´© �)¼ ÀðÂñ Àð ¶ ñ ¹ �"¼ ÀðÆñ&ò ¶ñ æÔ© �L¼ �>�o�l� Å � ÀðÂñ â ¡ (2.13)

Assumption2.6 Thesourcesymbolsare independentandidenticallydistributed(i.i.d)

with a variance£ ¥¦ .Fromassumptions2.3,2.5and2.6,equation(2.13)simplifiesto,öÌ÷ £ ¥à �E��¼ £ ¥¦uù æ´© �)¼ £ ¥¦ ¹ (2.14)

where ù ��ª�� Ä ú Ñ�û �&ÄËú �&Ä � �&Ä ú � Ä��>��üt� Ñ µ�¡ (2.15)

Hence,theMMSE FBFfilter tapsare

¹ mmse � ù æÔ© ¡ (2.16)

Similarly, for theFFFparameterswehaveöÌý £ ¥à �¼ £ ¥¦ æ ¶ æ´© �+¼ £ ¥¦ æ ¶ ù ¶ ¹ �+¼ £ ¥¦ æ ¶ÿþ Ñ �L¼ £ ¥§ © (2.17)

where þ Ñ �«ª�.¡t¡u¡±�p�Ã�E¡u¡u¡¯�&µ ¶ is an ²�Ê lengthunit vectorwith a unity at positionÅ �� , with ��Ú Å ÝA²�Ê . Substitutingtheresultof equation(2.16)into equation(2.17)

yields öÌý £ ¥à �¼ £ ¥¦ æ ¶ � û � ù ¶ ù � æ´© �M¼ £ ¥¦ æ ¶ÿþ Ñ �)¼ £ ¥§ © ¡ (2.18)

TheMMSE feedforwardequalizeris

© mmse �� æ ¶�� æ �� û � �� æ ¶ þ Ñ (2.19)

where �Ã� £ ¥§�� £ ¥¦ ß � �.� û � ù ¶ ù �Æ¡ (2.20)

25

AlDhahir [1] emphasizesthe importanceof an unbiasedDFE, in which the cursorÈP� Å � � � . Although the MSE of an unbiasedDFE designis larger than the MMSE-

DFE, it hasa lower probabilityof error. TheunbiasedMMSE-DFEcanbeobtainedby

minimizing the MSE of the DFE undera unit tap constraintfor the cursor. It canbe

shown that, the unbiasedMMSE-DFE tapsareobtainedfrom the biasedMMSE-DFE

by normalizingthetapcoefficientsby thecursorterm,i.e.,

© mmse� � © mmse� æ´© mmse� ¶ þ Ñ ß ¹ mmse� � ¹ mmse� æÔ© mmse� ¶ þ Ñ ¡ (2.21)

2.2.2 Infinite length MMSE-DFE

In subsection2.2.1theMMSE-DFEsolutionwasderivedunderafinite lengthconstraint

on thefeedforwardandfeedbackfilters of theDFE. If thelengthconstrainton theDFE

filters is relaxed,we obtainthe infinite lengthor the idealDFE solution. In [69] Mon-

senformulatedtheoptimuminfinite lengthDFE undertheMMSE criterion. Let æ ��$�denotethe -transformof thechannelimpulseresponse,namely

æ ��$� � � � �� g� ��&�� >� ¡ (2.22)

Let ¨ ��$� and · ��$� denotetheFFFandFBFin -transformrepresentation.Then,under

theidealfeedbackassumption2.5,theoptimumfilters ¨ ��$� and · ��$� aregivenby,

¨ ��$�<� æ } �� $� � � �_�$� (2.23)

and · ��B� � � ��$�� _�$� �s� (2.24)

where � ��$�B� æ ��$� æ } �� X��X¡ (2.25)

26

Theterms� � �� and

� � ��$� correspondto theminimumphase(all rootsinsidetheunit

circle)andthemaximumphase(all rootsoutsidetheunit circle)spectralfactorsof� ��$� ,

respectively. The term� � �\�� , in equations(2.23) and (2.24), refersto the constant

term in the polynomialexpansionof the maximumphasespectralfactor� � ��$� . The

feedforwardfilter canonly beimplementedasastable,anti-causalfilter with aninfinite

impulseresponse,while theFBF is strictly causal.Fromequation(2.23),it is clearthat

theFFFis composedof afilter matchedto thechannelimpulseresponseandawhitening

filter. Whenthenoisevarianceis very small, i.e., £ ¥§�� , thentheFFFapproachesan

all-passfilter that reflectsthe maximumphaserootsof the channelimpulseresponse

acrosstheunit circle into minimumphaseroots.

2.2.3 Gap fr om the MFB

In subsection2.1.1,it wasmentionedthat the MFB canbe attainedonly whenall the

interferingsymbolsareperfectlyknown at thereceiver. TheDFE,undertheidealfeed-

backassumption,hasperfectknowledgeof at leastthe interferingsymbolstransmitted

prior to the symbolbeingdetected.Hence,onecanexpectthe infinite lengthDFE to

performbetteror at leastaswell asan infinitely long linear equalizer. This is indeed

trueandhasbeenshown in [69, 70]. However, theDFE still hasa gapfrom theMFB

andthisgapwasquantifiedfor aninfinite-lengthDFEby Ariyavisitakul in [6]. Thiswill

bediscussedin greaterdetail in Chapter3. To demonstratethegapfrom theMFB, we

considerthefollowing example.

Example2.1 Considera real,2-tapchannelwithanimpulseresponseæ ��$�<� � �� ,with thesecondtap of magnitudelessthanunity, i.e., � � � Ý.� . Let theinput SNRof the

systembe high, such that � � £ ¥§�� £ ¥¦�� . Then, the infinite length MMSE-DFE

27

coefficientsaregivenby, ¨ ��$�<Ø.��ß · ��$�6Ø�� (2.26)

andtheoutputSNRof theDFE is

¢dfe

� £`¥¦£ ¥§ ¡ (2.27)

However, theMFB definedin equation(2.5) for this channelis

¢mfb

� £`¥¦ �r�B�� ¥ �£ ¥§ ¡ (2.28)

Theinfinite lengthMMSE-DFEhasan outputSNRthat is lower by a factor �c�� ¥ �fromtheMFB, for this channelexample.

2.2.4 Err or Propagation

As long asthe decisionsmadeby the decisiondevice arecorrect,the performanceof

theDFEis asdesired(thoughshortof theMFB). However, in thepresenceof noiseand

residualISI, decisionerrorsareinevitable.Thefirst noiseandresidualISI inducederror

is known asa primaryerror. As thedecisionerror is fed backthroughtheFBF, instead

of cancelingthepost-cursorISI components,it addsadditionalinterference.Thisresults

in an increasedprobabilityof errorduring thesubsequentsymboldecisions.Hence,a

primaryerrortypically inducesaburstof errors.Theburstterminateswhentheresidual

errorsaddup in theright way [52, 54] to flushout all thedecisionerrorsfrom theFBF

tappeddelayline. The averagedurationof the errorburst is a function of thechannel

andDFE coefficients,thesourceconstellationandthenoiselevel. Figure2.6 illustrates

theburstinessof theerrorpropagationphenomenonin a DFE. Thesourcesymbolsare

drawn from a binary phase-shift-keying (BPSK) constellation.The soft outputof the

DFE is plottedin Figure2.6alongwith a “*” to markthelocationsof theharddecision

28

errors.Thedegradationdueto errorpropagationgenerallyincreaseswith themagnitude

of theFBFcoefficientsandthelengthof theFBF.

6100 6150 6200 6250−4

−3

−2

−1

0

1

2

3

4

time index

Sof

t / H

ard

Err

ors

Figure 2.6: Errorpropagationin DFE

2.2.5 Numerical Example

The gap from the matchedfilter boundand the degradationdue to error propagation

for a DFE hasbeenillustratedin Figure2.7. Thesymbolerror rate(SER)curveshave

beenplottedasa function of theSNR in Figure2.7. The sourcesymbolsweredrawn

from aBPSKconstellationandtransmittedthroughachannelwith animpulseresponse| � ª��¡�¼�� ¡��1¼�� ¡�¼��(µ . The DFE was allocated4 tapsfor the feedforward

filter and3 tapsfor thefeedbackfilter. Theperformancecurvesfor anidealDFE anda

conventionalDFE areplottedalongwith thematchedfilter bound. For the idealDFE,

perfectdecisions,i.e., error-free decisions,are fed back to the FBF. To plot the SER

curve correspondingto the matchedfilter bound,assumption2.3 was invoked. The

29

symbolerrorratefor thematchedfilter boundis givenby,

SERmfb�! #" �£ §%$ ß (2.29)

where£ ¥§ is thenoisevarianceand 2�cêH� is theQ-functiondefinedas, �'&��6� �( ¼*),+�-. /1032 " �4� ¥¼ $65 ��¡ (2.30)

Theperformancedegradationof theconventionalDFE with respectto thematched

filter boundcan be decomposedinto two parts. The first, namelyerror propagation

gap is the performancedegradationof the conventionalDFE when comparedto the

idealDFE. For anSERof �u� � 7 , theerrorpropagationgapis about0.7dB. Thesecond

componentof theDFE performancedegradationis known asthegapfromthematched

filter boundandis givenby theperformancedifferenceof the idealDFE curve andthe

MFB. In this example,thegapfrom MFB is about1.5dB at anSERof �� 7 . Notethat

thegapfrom thematchedfilter boundincreaseswith increasingSNR.This is dueto the

factthatathighSNRscenarios,ISI is thedominantfactor.

2.3 DFE Enhancements

Therehave beena few attemptsin the literatureto addressthedrawbacksof theDFE,

namely the gap from the MFB and error propagation,individually. A few of these

attemptswerebriefly discussedin Chapter1. In thissection,someof theenhancements

proposedto theDFEstructureto mitigatetheseproblemsarediscussedin greaterdetail.

2.3.1 Non-causalDFE

As indicatedin subsection2.1.1,theMFB canbeattainedwhenall interferingsymbols

areperfectlyknown. In 1970,Proakis[75] proposedtheprincipleof non-causaldecision

30

6 7 8 9 10 11 12 1310

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

Normal mode DFEIdeal FeedbackMatched Filter Bound

Figure 2.7: Gapfrom thematchedfilter boundandidealDFE

feedbackequalization(NCDFE)to attaintheMFB. TheMMSE optimizedNCDFEtaps

werederived by GershoandLim [43]. The block diagramof an NCDFE is depicted

in Figure2.8. TheNCDFEconsistsof a preliminaryequalizerin parallelwith theDFE

structure.Thepreliminaryequalizeris usedto provideestimatesof the“future symbols”

to theNCDFE.Thepreliminaryequalizercanbeeitheralinearequalizeror evenaDFE.

The DFE componentof the NCDFE is precededby a delayelementandconsistsof a

feedforwardfilter ¨ anda feedbackfilter · . Typically, thedelayprecedingtheDFE is

chosento beat leastaslargeasthedetectiondelayof thepreliminaryequalizer. Then,

theinterferencedueto the“future symbols”,i.e., thepre-cursorISI, at theoutputof the

filter ¨ canbecanceledusingtheestimatesÀ� � �o�X� from thepreliminaryequalizer. The

decisionsÀ�k�_�X� of theNCDFEareusedto cancelthepost-cursorISI component.

GershoandLim have shown in [43] that for the MSE optimizedNCDFE, the for-

wardfilter ¨ is proportionalto thematchedfilter æ } �� andthefeedbackfilter · is a

31

DeviceDecision s^

s1^

Delay

r

F

B

y

‘‘past’’

Non−causal DFE

PreliminaryEqualizer

‘‘future’’

Figure 2.8: Structureof aNon-causalDFE

cascadeof thechannelandfeedforwardfilter impulseresponse,withoutthecentralcoef-

ficient. In thepresenceof decisionerrors,theNCDFEsuffersfrom anerrorpropagation

phenomenonthatresultsin performancedegradation.UnliketheDFE,theNCDFEcan-

celsboththepre-cursorandpost-cursorISI usingthesymboldecisions.Hence,theFBF

in anNCDFEis, in effect,twiceaslongasthatof aDFE.As thedegradationdueto error

propagationis moreseverewith a longerFBF, someof theperformanceimprovement

expecteddue to closing the gap from the MFB is lost to increasederror propagation

gap. In [27], Slock andde Carvalho have proposedthe useof soft decisionsandan

iterativeNCDFEapproachto furtherimprovetheperformanceof theNCDFE.In theit-

erativeapproach,themorereliablesymboldecisions,i.e., À�k�_�X� , of theNCDFEareused

asa tentative estimateof the “future symbols”to cancelthepre-cursorISI component

for anotherNCDFE.This operationis performediteratively, in anattemptto refinethe

symboldecisions.Although,thesesolutionsareusefulin decreasingtheeffect of error

propagation,they comeat thecostof increasedcomplexity.

In [20, 21], ChanandWornell haveproposedablock-iterativeDFE,whichattempts

to attaintheMFB. Theblock iterativeDFE hasa structurevery similar to theNCDFE,

but insteadof usingdecisionsfrom apreliminaryequalizer, theiterativeDFEusesdeci-

32

sionsfrom thepreviousiteration.Furthermore,thetapcoefficientsof theblock iterative

DFE arerecomputedduringeachiterationto incorporatetheeffect of the reliability of

thedecisionsthatareusedto cancelthepre-cursorandpost-cursorISI. Thiswork, how-

ever, assumesthatthechannelimpulseresponseis infinitely long, i.e., ²�É � 8 andthe

frequency responseof thechannelis drawn from a complex Gaussianrandomprocess.

Numericalresultsshow thatwhenthechanneldelayspreadsarelarge (over 100 taps),

the high SNR performanceof the block-iterative DFE, after a small numberof itera-

tions, tendstowardstheMFB. It is not clearwhethertheseresultsreadily translatefor

practicalchannels(say, for delayspreadstypical to wirelesschannels).

2.3.2 DecoderFeedback

In systemsemploying errorcontrolcodes,a popularideato addresstheerrorpropaga-

tion problemin DFEshasbeento usefeedbackfrom thedecoder. As thedecisionsatthe

outputof thechanneldecoderaremorereliablethanthequantizerdecisions,thedecoder

decisionscanbefedbackthroughtheFBFto mitigateerrorpropagation.However, there

is adelayassociatedwith thechanneldecoderandhenceAriyavisitakul [8, 7] proposed

the ideaof splitting theFBF of theDFE into two parallelfilters, oneof which usesthe

decoderdecisionsto cancelthepost-cursorISI componentscorrespondingto symbolsª�>�o��9¤� Å �Æß¾�k�_��:9¤� Å �s��Æß�¡u¡t¡�µ , where 9 is thedecoderdelay, while theother

cancelstheremainderof thepost-cursorISI usingthetentativeDFEdecisions.

An alternatemethodof decoderfeedbackproposedin [10, 62] is theiterative DFE-

decoderarchitecture,referredto as the Turbo DFE. In the Turbo DFE, the received

signalisequalizedusingaDFEandthenthetransmittedbitsaredecodedusingachannel

decoder. Themorereliabledecoderdecisionsarenow usedastheFBFinputsfor another

DFE thatoperateson a delayedversionof the input signal. Theequalization-decoding

33

operationcanbeperformediteratively to mitigateerrorpropagation.Numericalresults,

provided in [10], show that the error propagationgapcanbe decreasedby about75%

with asmallnumberof iterations.

2.3.3 DecisionDevice Optimization

Thereplacementof thedecisiondevice,shown asaquantizerin theDFEblockdiagram

of Figure 2.4, with a soft decisiondevice is an effective techniqueto mitigate error

propagationfor both codedanduncodedsystems.The useof a soft decisiondevice

hasbeenproposedin [8, 27, 10, 14]. In [10, 14], Balakrishnanet al. optimize the

soft decisiondevice in thepresenceof errorpropagationunderboththeminimumMSE

andthe minimum bit-error rate(BER) criteria. For instance,the soft decisiondevice

(see[14] for details)that minimizesMSE at the DFE output, in the presenceof error

propagation,for a BPSK sourceconstellationis a sigmoidfunction given by ¿ � � �x�tanh� � � £`¥à � , where£`¥à is thevarianceof thefilterednoiseandresidualISI.

2.4 Time-reversal

Considerthe operationof time-reversal applied to the received sequence. A time-

reversaloperationon the sequence�9�o�X� of equation(2.1) canbe mathematicallyex-

pressedas,

~��o�X� � ��r�ó�X�� g��<;�� = ~�&��9� ~�k�_�l�+�9�X� ~�x�o�X� (2.31)

where“ > ” impliestime-reversal.Now,

~� �\�9� � � �c��9�Öß ~�k�_�X� � �k�c�-�`�Æß and ~�x�_�X� � �x�r�ó�X� (2.32)

34

arethe time-reversedchannelimpulseresponse,sourcesequenceandnoisesequence,

respectively. In equation(2.31), the linearconvolution structureof thechannelis pre-

served. Hence,this is equivalent to transmittingthe time-reversedsourcesequence

througha channelimpulseresponse,whosecoefficientsarethe time-reversedversion

of | .Property 2.1 Thetime-reversedchannelhasroot locationsthat are reciprocal of the

rootsof the original channelimpulseresponse, i.e., the rootsare reflectedacrossthe

unit circle.

Proof: Recall the z-transformrepresentationof equation(2.22). We let ?A@ ßCBED ��ß¾¡u¡u¡Âß@²�ÉÒ�ì� representthezerosof theFIR channelæ ��$� . Then,

æ ��6� ? � � � ��F @ �� r�¾� ?A@ � �� (2.33)

where ? � � � �_�$� is thefirst tapcoefficient of thechannel.Let ~æ ��$� bethez-transform

of the time-reversedchannelimpulseresponse{| . Then ~æ ��$� � æ �� andcan be

expressedas,

~æ ��$�<� ? � �&�c��F @ �� c�¾� ?A@ �$�� ? � � � ��F @ �� c� ?A@ �$�G"X�¾� � ��?A@ $ ¡ (2.34)

Hence, � � ?A@ ßHBIDT� ��ß ¡u¡u¡Âßl²�ÉD� � arenow the zerosof the time-reversedchannel

impulseresponse. JEffectively, the time-reversaloperationconverts the minimum phaseroots of the

channelin to maximumphaserootsandvice-versa.

35

2.4.1 SelectiveTime-ReversalDFE

Property(2.1) wasexploitedby Ariyavisitakul in [5, 6] to improve theperformanceof

aDFEwith afinite lengthconstraint,in apacketbasedsystem.Ariyavisitakulproposed

the useof two parallelDFE structures,oneof which is a normalmodeDFE that op-

erateson thereceivedsignal. Thereceivedpacket of datais time-reversedandanother

DFEis usedto equalizethetime-reversedreceivedsequence.Ariyavisitakularguedthat

undera finite lengthconstraint,oneof thetwo DFE structureswould have a betterper-

formance.If thechannelimpulseresponseis known at thereceiver(seeassumption2.4)

thenthefinite lengthMMSE-DFEcoefficientscanbedeterminedandtheperformance

of thenormalmodeandtime-reversalmodeDFEsdetermined.Basedon theMSE per-

formancemetric,Ariyavisitakulchoosestheoutputof oneof theDFEstructuresfor the

durationof thepacket. It wasfurthershown that,asthenumberof DFE tapsincreases,

the differencein performancebetweenthe normalmodeandtime-reversalmodeDFE

structuresdiminishandtheadvantageof usingtheselectivetime-reversal structure for

the DFE disappears.The differencein performancesof the two DFE modescan be

illustratedby thefollowing example.

Example2.2 Considera real,2-tapchannelwith animpulseresponseæ ��6�I�<�C� �� ,with the first tap of magnitudelessthan unity, i.e., � � � Ý � . Let us constrain theFFF

andFBF tapsto beof lengthoneeach. ThentheunbiasedMMSE-DFEtapsare,

¨ ��6� �� ß · ��$�<� � �� (2.35)

andtheoutputSNRof thenormalmodeDFE is

¢dfe

� £`¥¦� ¥ £ ¥§ ¡ (2.36)

On theotherhand,for thetime-reversedchannelimpulseresponse~æ ��$� , theunbiased

36

MMSE-DFEtapsare ¨ tr��$�<�E� ß · tr

��B�� (2.37)

andtheoutputSNRof thetime-reversalmodeDFE is¢trdfe

� £ ¥¦£ ¥§ ¡ (2.38)

Since, � � � Ý�� , theoutputSNRof thetime-reversalmodeDFE is greaterthanthatof the

normal modeDFE and the selectivetime-reversal structure will choosetheoutputsof

thetime-reversalmodeDFE.

2.5 Bidir ectionalDFE

The ideaof time-reversalin a DFE canbe extendedfrom selective time-reversalto a

bidirectionalDFE(BiDFE). Theblockdiagramof aBiDFE is similar to thatof aselec-

tive time-reversalDFE,but with anadditionaldiversitycombiningblock thatcombines

theoutputsof thenormalmodeandtime-reversalmodeDFE.Thestructureof aBiDFE,

proposedin [11], is illustratedin Figure2.9. TheBiDFE processesthereceivedpacket

of datain two parallel streams.The received sequence��_�`� is equalizedin streamI

using a DFE with an inherentdetectiondelay Å � . The feedforward filter ©K has ²<³ �taps,while thefeedbackfilter ¹LK has²�¸ � taps.Thecombinedchannel-feedforwardfilter

impulseresponsefor thenormalmodeDFE,namelystreamI, is denotedby ÇLK � |6� ©K .A block time-reversaloperationis appliedto the receivedsequence�9�o�X� in stream

II. Theonly differencein theblocktime-reversaloperation,whencomparedto thetime-

reversaloperationdefinedin equation(2.32), is a time-shift operationthat manifests

asanadditionof a constantterm to the index of the time-reversedsequence.We now

define, ~��_�`� � �9� � �)²�Éë�M�`� B�� E��ß¾¡u¡t¡tß � �)²�Éë�s� (2.39)

37

TimeReversal

w

rs DecisionDevice

DeviceDecision

ReversalTime

b

b

DiversityCombining Device

Decisiony

2

1

II

I

C f1s1^

s^ 2

y2

y1

f2

s^

Figure2.9: Structureof aBidirectionalDFE

Fromequation(2.1),equation(2.39)canberewrittenasMNAO'PRQLSIT*UWV<XY Z\[A]G^ O�_AQa`�O'Pcb�dLegfhPif:_jQRb:klO�mnb�dLeofhPRQ (2.40)

Now, we definethe block time-reversedsourcesequence,channelresponseandnoise

sequenceas M`�O�P�QLpq`�O�mrfhPcb�stQvu wxPySzs�u|{}{~{}u�myu (2.41)M^ O�_jQ�p ^ O�dLeof�s�f:_AQ1u wy_cSI�ju�{~{~{�u%dLeof�s�u (2.42)Mk�O�P�Q�p!k�O�mnb:d�eofhPRQvu wiP,Sqs�u�{~{~{}u�m#b�dLeof�s�u (2.43)

respectively. With a few simplemanipulations,equation(2.40)canbesimplifiedtoMNjO�P�Q�S�T U V<XY Z\[A] M^ O�_AQ M`�O�Pif:_jQRb MklO'PRQ (2.44)

where“ � ” implies block time-reversal. Let �� with d��\� tapsand �g� with dL�� tapsbe

the feedforward andfeedbackfilter settingsof the time-reversalmodeDFE, of stream

II, thatequalizesthetime-reversedchannel�� . We furtherdefinethecombinedchannel-

feedforwardfilter impulseresponsefor thetime-reversalmodeDFEas �g� S �� with� � denotingthedetectiondelay.

38

2.5.1 Rationale

Whatis therationalein goingfromtheselectivetime-reversalDFEof subsection2.4.1to

theBiDFE structureproposedin Figure2.9?Theadvantagesof theBiDFE structureare

two-fold. Firstly, errorpropagationis a causalphenomenon.A primaryerror inducesa

burstof secondaryerrorsthatproceedin theforwarddirectionin time. Whenthesignal

is processedusinga time-reversalmodeDFE, the error propagationrunsin backward

time. Basedon the fact that the error burstsrun in oppositedirectionsin time for the

normalmodeandthetime-reversalmodeDFE,onecanexpectalow correlationbetween

theseerrorbursts.Perhaps,this featureof theBiDFE canbeexploitedto mitigateerror

propagation.

Recallfrom subsection2.1.1that theMFB canbeattainedwhenall the interfering

symbols,namelythepastandthefuture interferingsymbolsareperfectlyknown. This

is thereasonwhy anNCDFEattainstheMFB. Undertheidealfeedbackassumption2.5,

theproposedBiDFE structurehasperfectknowledgeof boththepastsymbols(dueto the

normal-modeDFE)andthefuturesymbols(dueto thetime-reversalmodeDFE).Hence,

onecanexpecttheBiDFE to havea lessergapfrom theMFB thanaconventionalDFE.

The known symbolinformationfor variousequalizerstructureshasbeencomparedin

Table2.1.

2.5.2 BAD: Bidir ectional Arbitrated DFE

ThebidirectionalarbitratedDFE(BAD) proposedby McGahey etal. in [64] is aspecial

caseof theBiDFE with anelaboratediversitycombiningblock. Thediversitycombin-

ing block is shown in Figure2.10andconsistsof a reconstructionandarbitrationstage.

In the reconstructionstage,thesymboldecisions �` X O�P�Q and �`~�tO�P�Q of thenormalmode

andtime-reversalmodeDFEsarefiltered throughan estimateof the channelimpulse

39

Table2.1: Known symbolinformationof equalizers

Equalizer Structure Past symbols Future symbols

LinearEqualizer Unknown Unknown

DFE Known Unknown

BiDFE Known Known

NCDFE Known Known

responseto reconstructthe received sequence.The reconstructedreceived sequences,

namely �N X O�PRQ and �N~�*O'PRQ arecomparedto the true received sequenceNjO�P�Q anda win-

doweddistancemetricis computed.For instance,�� X O�P�Q�SZ\[<�YZ\[ V �,� �N X O�Pif�_AQ�f�NjO�Pif:_jQ � � { (2.45)

Thewindoweddistances �� X O�PRQ and �� tO'PRQ arecomparedandtheoutputof theDFE

resultingin a smallerwindoweddistanceis selectedastheoutputof thearbiterfor that

symbolinstant,i.e., �`�O'PRQ�S �� ` X O�P�Q1u if �� X O�P�Q%¡ �� tO�P�Q�`~�*O�P�Q1u if �� X O�P�Q%¢ �� tO�P�Q { (2.46)

BAD wasproposedto exploit thedirectionalityof errorpropagationto improveperfor-

manceandnumericalresultsdemonstratingperformanceimprovementswereprovided

in [64].

2.5.3 Preliminaries

In thissubsection,themathematicalmodelthatleadsto thederivationof asimplelinear

diversitycombingschemeis provided. For theBiDFE, we make thefollowing simpli-

fying assumptions.

40

1^s

^s

1d^

2^d

1^s

s^ 2

s^ 2

C

C

r

^r 1

^r2

~

~

windoweddistance

windoweddistance

Arbiter

ArbitrationReconstruction

Figure 2.10: Structureof aBidirectionalArbitratedDFE

Assumption2.7 TheDFEs for the normal modeand time-reversal modestreamsare

finite lengthunbiasedMMSE-DFEs,i.e., £ X O � X Q�S £ �*O � �vQ�Szs .As the outputSNR,or equivalently thegapfrom theMFB, will be usedasthemetric

to evaluateequalizerstructures,it is meaningfulto assumetheDFEsto beunbiasedand

alsoto minimizetheMSE.

Assumption2.8 Thesourcesequence�O�P�Q hasunit variance, ¤ �¥ Szs .This assumptioncanbe madewithout lossof generalityasour interestis in the SNR

termandnot onany absolutevalues.

Let ¦ X O�PRQ be the soft output (also the input to the decisiondevice) of the normal

modeDFE and ¦ �§O�PRQ betheblock time-reversed(andsynchronized)soft outputof the

time-reversalmodeDFE.Thesequences¦ X O�P�Q and ¦ �tO�P�Q canbeexpressedas,¦ X O�P�Q�S!`�O�P�QRb:¨ X O'PRQ (2.47)¦ �tO�P�Q�S!`�O�P�QRb:¨��*O'PRQ (2.48)

41

where,undertheidealfeedbackassumption2.5¨ X O�P�Q�S T�©aª�V<XY« [A]c¬ X O'®QWklO'Pcb � X fh®QRb�¯ ª V<XY« [A] £ X O�®Q`�O�P°b � X fh®Qb T�©aª�±AT�U²V �Y« [ ¯ ª ±AT�³'ª�±´X £ X O�®Q`�O�P°b � X fh®Q (2.49)

¨��tO�P�Q�S T ©²µ V<XY« [A] ¬ �tO'®QWk�O�Pcb�dLeof�s�f � �gb:®Q¶b:¯ µ V<XY« [A] £ �*O�®Q` O'Pif � �·b:®Qb T ©²µ ±AT�U�V �Y« [ ¯ µ ±AT ³¸µ ±´X £ �tO'®Qa`�O�Pxf � ��b�®Q1{ (2.50)

Let ¤ �X and ¤ �� bethevariancesof ¨ X OW¹ºQ and ¨��*O²¹»Q , respectively. We use ¼ to denotethe

coefficientof correlationbetweenthesequences¨ X OW¹»Q and ¨��*OW¹»Q , i.e.,¼ S E ½ ¨ X ¨��¿¾¤ X ¤ � (2.51)

whereE[ ¹ ] is theexpectationoperation.

2.5.4 Diversity Combining

In subsection2.5.2,wediscussedanelaboratereconstructionbasedarbitrationtechnique

for thediversitycombiningblock. Although,this techniquecouldbequiteeffective in

exploiting the low correlationbetweentheerrorbursts,it is extremelyhardto analyze.

Furthermore,it is not clearwhat fractionof theperformancegainof theBAD is dueto

error propagationmitigation andhow muchis dueto a decreasedgapfrom the MFB.

Wenow consideramuchsimplifiedandlow complexity diversitycombingblockwhich

generatestheoutputs¦ O�P�Q asaweightedlinearcombinationof thesequences¦ X O'PRQ and¦ �§O�PRQ , namely ¦ O�P�QLS!À ¦ X O�P�QRb!O²s|f�À·Q ¦ �tO�P�Q (2.52)S�`�O�PRQRb:Àg¨ X O�P�QRb!OWs�f�À·QW¨��§O�PRQ

42

where À is theweightingfactor. Thearchitectureemploying this weightedlinearcom-

binationof theoutputsof theconstituentDFEswill bereferredto asa linear-combining

bidirectionalDFE (LC-BiDFE). Although, sucha simplified diversity combineris ill-

equippedto handletheburstynatureof errorpropagation,wecanobtainusefulintuition

on how well the gapfrom the MFB hasbeendecreased.TheMSE for thesoft output¦ O�PRQ is

MSE S E ½ � ¦ O'PRQgf:`�O�P�Q � � ¾S�À � ¤ �X bIOWs�f�À�Q � ¤ �� b�Á ¼ À%OWs�f�À�Q ¤ X ¤ �4{ (2.53)

Lemma 2.1 Theweightingfactor À , thatminimizestheMSEof equation(2.53)is given

by À opt S ¤ �� f ¼Â¤ X ¤ �¤ �X b ¤ �� f:Á ¼Â¤ X ¤ � { (2.54)

Proof: To minimizetheMSEof equation(2.53),setthefirst derivativeof theMSEwith

respectto À to zero.Now,ÃMSEÃ À S!Á�À ¤ �X fhÁAOWs�fhÀ·Q ¤ �� b�Á ¼ O²s�f�Á�À�Q ¤ X ¤ �§{ (2.55)

Settingthegradientof MSE to zero,i.e.,ÃMSEÃ À S!�ju (2.56)

we obtaintheoptimalweightingfactorof equation(2.54). When ¤ X S ¤ � , theoptimal

weightingfactoris ÀÄS X� andthis is known asequal-gaincombining. ÅFurthermore,it can be shown that the minimum MSE obtainedwith the optimal

valueof À is

MMSE S OWs�f ¼ � Q ¤ �X ¤ ��¤ �X b ¤ �� f:Á ¼Â¤ X ¤ � { (2.57)

43

The MMSE of equation(2.57) is lessthanthe MSE of both the normalmodeandthe

time-reversalmodeDFEs. Hence,the performanceof a LC-BiDFE is betterthanthe

performanceof aDFEwith selective time-reversalstructure.

2.5.5 Time-ReversalDiversity

In Lemma2.1, it wasdemonstratedthat theLC-BiDFE hasa betterMSE performance

than eachof the constituentDFEs. In this subsection,a symmetricchannelimpulse

responsewill be consideredand the LC-BiDFE will be shown to result in a smaller

valueof noisegainandresidualISI terms.For a symmetricchannel,thetime-reversed

channelimpulseresponse�� is equalto thechannelimpulseresponse� . Hence,thesame

DFE tapsettingscanbeusedfor thetwo streams,i.e., �WÆ S �� S � and ��Æ S �·� S � .

Further, the mean-squarederror (MSE) for the two streams,¤ X and ¤ � , areequaland

theoptimalweightingfactorof equation(2.54)is À�S X� . Thesoft outputs,¦ O�PRQ of the

diversitycombiningblockcanbeexpressedas,¦ O�P�QLS!`�O�P�QRb:¨�O�PRQ (2.58)

where�O�P�Q is thesumof thenoiseandtheresidualISI componentsand,¨�O'PRQ�S ¨ X O�P�QRb:¨��*O'PRQÁ { (2.59)

Wedefinethevector Ç , È O�_AQLS �� ju ��É _ ÉÊ� b�dL�£ O�_AQ1u otherwise

{ (2.60)

Equation(2.59)canberewrittenas,¨�O�PRQ�S sÁ T�Ë�V<XY Z\[A] È O�_AQvÌ�`�O�Plb � f:_AQRb�`�O�Pif � b�_AQvÍb sÁ T © V<XY Z\[A] ¬ O�_AQvÌtk�O�PHb � f�_AQRb:k�O�Pcb:d�eof�s�f � b�_jQvÍÎ{ (2.61)

44

The variance(or equivalently the MSE) of the overall systemerror of equation(2.61)

canbeexpressedasasumof thenoisegainandtheresidualISI terms.

Noisegainterm S ZÏ3Ð�ÑYZ\[AZ ÏAÒÔÓ%Õ ¬ O�_AQRb ¬ O�Á � b�s�f�dLegf:_AQÁ Ö � ¤ �× (2.62)

where_ «oØºÙ S�ÚcÛÝÜ�O��ju\Á � bÊs|f�dLÞ*Q and _ «·ß²à SIÚâátã�O�d��Gf�s�u\Á � bÊs|f�d�eQ (2.63)

and

ResidualISI term S ä ÏjÐ�ÑYZ\[ ä ÏjÒåÓæÕÈ O�_AQRb È O�Á � f�_jQÁ Ö � (2.64)

where ç «gØºÙ S�Ú°Û¸ÜRO��3uvÁ � bÊs�fhdLÞ�Q and

ç «gßWà SIÚ°á*ã�O�dLÞ|fès�uvÁ � Q%{ (2.65)

For eachof the individual streams,the noisegain term andthe residualISI termsare¤ �× O ��éê� Q and ÇêéëÇ , respectively. From equations(2.62) and (2.64), we observe that

theequal-gaindiversitycombiningschemehasdecreased1 thecontributionsof boththe

noisegainandtheresidualISI termson theeffectiveMSEwhencomparedto eitherthe

normal-modeor time-reversalmodeoperationof theDFE. This resultsin a decreased

gapfrom theMFB for theLC-BiDFE structure.

2.5.6 Simulation Results

In thissubsection,theMSEandSERperformanceimprovementsprovidedby theBiDFE

architecture(LC-BiDFE andBAD) over theconventionalDFE structurewill be inves-

tigatednumerically, with theaid of two samplechannelimpulseresponses.Additional

simulationexamplesareprovidedin AppendixA. Matlabscriptfiles for generatingthe

1ThiscanbeeasilyprovedusingtheCauchy-Schwartzinequality.

45

simulationplotsprovidedin this sectionareavailableat [19]. TheSERperformanceof

the LC-BiDFE will alsobe comparedwith BAD andthe NCDFE.We first considera

real,asymmetricchannelwith animpulseresponseof� X S ½ �j{ìs§í�îI�j{åïjs§ð��j{�Á�í�ï�f��j{ìs§í�î!�3{��ï�Á�f��j{å��ñÂðI�j{��3s§í�¾�{ (2.66)

Eachof the constituentDFEsin the BiDFE structurewererestrictedto 4 feedforward

tapsand3 feedbacktaps. As statedin assumption2.7, unbiasedMMSE-DFEswere

used.Thesourcesymbolswereselectedfrom a BPSKsourceconstellation.TheMSE

performancecurves for a normal modeDFE, time-reversalmodeDFE and the LC-

BiDFE arecomparedin Figure2.11. Correctdecisionfeedback(assumption2.5) was

invoked for computingthe MSE values. In Figure2.11, the normalmodeDFE hasa

marginally betterperformancethanthetime-reversalmodeDFE. Ariyavisitakul’s DFE

with selective time-reversalwill choosethe normalmodeDFE for this example. The

LC-BiDFE, on theotherhand,hasa significantlybetterperformancethaneitherof the

two DFEmodesandyieldsagainof about1 dB for anMSE of 0.1(i.e., -10 dB).

Thesymbolerror rate(SER)curvesfor thenormalmodeDFE, time-reversalmode

DFE and the LC-BiDFE are shown in Figure 2.12 for the sameasymmetricchannel

impulseresponse� Æ . Thesesimulationresultsdonot invokeassumption2.5,andhence,

incorporatethe effect of error propagation,i.e., decisionfeedbackwas assumed.In

Figure2.12,therearetwo performancecurvesshown for theLC-BiDFE, namelywith a

hard/softDFE.In asoftDFE,thehard-limiteror quantizerof ahardDFEis replacedby

a schemewith a soft decisiondevice [14], which for a BPSKsourceconstellationis a

tanhOW¹»Q nonlinearity. Ascanbeseenfromtheplot, theLC-BiDFEstructureincorporating

a soft decisiondevice hasa gainof about1.2 dB at an SERvalueof s§� V ò . Although,

thedesignof thediversitycombiningblockof theLC-BiDFE structuredoesnot takethe

errorpropagationinto account,theperformancegainobservedin thepresenceof ideal

46

6 7 8 9 10 11 12 13 14−14

−12

−10

−8

−6

−4

SNR in dB

MS

E i

n d

B

Normal Mode DFE Time−Reversal Mode LC−BiDFE

Figure 2.11: MSE performancefor an asymmetric channel with an impulse responseó X·ô�õ ö�÷ùøvú§û�ö�÷Ôü�øvý�ö�÷Ôþ§ú§ü�ÿxö�÷ùøvú§û�ö�÷ ötü§þ�ÿxö�÷ ö � ý�ö�÷ öÂøvú �feedbackalso translatesto the casewith decisionfeedback. Also, as expectedfrom

thediscussionin subsection2.3.3,theperformanceof a LC-BiDFE with soft decision

feedbackis betterthanthatof aLC-BiDFE with harddecisionfeedback.

In Figure2.13,theSERperformancecurvesof thenormalmodeDFE andtheLC-

BiDFE (with soft decisions)are comparedwith the BAD and MLSE. A window of

size � S�� , wasusedin BAD to computetheaveragedEuclideandistancemetric (see

equation2.45) in thearbitrationstage.TheMLSE wasimplementedusingtheViterbi

algorithm.Comparedto the1.2dB of improvementofferedby theLC-BiDFE (with soft

decisions),over thenormalmodeDFEatanSERof s§� V ò , theBAD offersanadditional

improvementof only 0.2 dB. Furthermore,thegapfrom theMLSE for theLC-BiDFE

is 0.5 dB. It shouldbenotedthat theMLSE is theoptimaldetectorfor the transmitted

symbols.

47

6 7 8 9 10 11 12 13 1410

−5

10−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

Normal Mode DFE Time−Reversal Mode DFE LC−BiDFE : (Hard) LC−BiDFE : (Soft)

Figure 2.12: SER performancefor an asymmetric channel with an impulse responseó X·ô�õ ö�÷ùøvú§û�ö�÷Ôü�øvý�ö�÷Ôþ§ú§ü�ÿxö�÷ùøvú§û�ö�÷ ötü§þ�ÿxö�÷ ö � ý�ö�÷ öÂøvú �In anotherexperimenta symmetricbaud-spacedchannelwith an impulseresponse

of � � S ½ �j{�Á�í�í��j{�ï3stÁ�ï��j{�Á�í�í��*¾ (2.67)

wasconsidered.A finite lengthconstraintof 4 feedforwardtapsand3 feedbacktapswas

imposedontheconstituentDFEsof theBiDFE structure.As � � is asymmetricchannel,

the DFE settings,andhencethe performance,arethe samefor both the normalmode

andthe time-reversalmodeDFEs. Figure2.14comparestheMSE performanceof the

DFE andLC-BiDFE with the MFB. Correctdecisionfeedback(assumption2.5) was

invokedfor computingtheMSE values.TheLC-BiDFE hasa MSE performancegain

of about1 dB over the conventionalDFE structureat an MSE of 0.1, but still suffers

from agapof 0.6dB from theMFB.

The SERperformanceof a conventionalDFE with andwithout ideal feedbackis

48

6 7 8 9 10 11 12 1310

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

Normal Mode DFELC−BiDFE : (Soft)BADMLSE

Figure 2.13: SER performancecomparisonwith MLSE for an asymmetricchannelwith an

impulseresponseó XLô�õ ö�÷ùøvú§û�ö�÷Ôü�øvý�ö�÷Ôþ§ú§ü�ÿxö�÷ùøvú§û�ö�÷ ötü§þ�ÿ®ö�÷ ö � ý:ö�÷ öÂøvú �comparedwith the performanceof a LC-BiDFE and an NCDFE, both with decision

feedback,in Figure2.15. Theerrorpropagationgapfor this exampleis nearly0.7 dB

andat an SERvalueof s§� V ò , the LC-BiDFE (with harddecisions)hasa performance

gain of 0.7 dB over a conventionalDFE. The NCDFE, on the otherhand,hasonly a

marginal performanceimprovementof 0.25dB over the conventionalDFE. This is in

tunewith theincreasederrorpropagationin anNCDFEthatwasdiscussedin subsection

2.3.1. For theNCDFE,a conventionalDFE with 4 FFFtapsand3 FBF tapswasused

asa preliminaryequalizer. Theoptimal feedforwardfilter for theNCDFEis a channel

matchedfilter. Hence,3 tapswereallocatedto theFFFof theNCDFEand4 tapsto the

FBF(to cancelall thepre-cursorandpost-cursorISI terms)of theNCDFE.

As thebidirectionalarbitratedDFE (BAD) hasa diversitycombiningschememore

suitedtowardsexploiting theburstinessof errorpropagation,in additionto theproperty

49

6 7 8 9 10 11 12 13 14−14

−12

−10

−8

−6

−4

SNR in dB

MS

E i

n d

B

Normal Mode DFELC−BiDFEMatched Filter Bound

Figure 2.14: MSE performancecurves for a symmetricchannelwith an impulse responseó � ôÊõ ö�÷Ôþ§ú§ú ö�÷Ôü�øvþ§ü�ö�÷Ôþ§ú§ú �of closingthegapfrom theMFB thatis inherentto theBiDFE structure,theSERperfor-

manceof a BAD is comparedwith theperformanceof a LC-BiDFE (with andwithout

soft decisions),thenormalmodeDFE, andtheMLSE in Figure2.16. As expectedthe

BAD outperformsthesoftLC-BiDFE by about0.3dB atanSERof s§� V ò , while thesoft

LC-BiDFE hasa performancegainof 0.9dB over thenormalmodeDFE performance.

The MLSE, on the otherhand,offers a performanceimprovementof 0.5 dB over the

LC-BiDFE (with soft decisions).Theperformanceimprovementof BAD comesat the

costof additionalcomplexity, dueto theelaboratediversitycombiningstructure.Figure

2.16 illustratesthat near-optimal performancecanbe attainedat a significantly lower

complexity by employing theBidirectionalDFEarchitecture(BAD or LC-BiDFE).

50

8 9 10 11 12 13 14

10−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

Normal mode DFENon−causal DFELC−BiDFE : (Hard)Ideal feedback

Figure 2.15: SER performancecurves for a symmetricchannelwith an impulse responseó � ôÊõ ö�÷Ôþ§ú§ú ö�÷Ôü�øvþ§ü�ö�÷Ôþ§ú§ú �2.5.7 Implementation Issues

In theprevioussubsection,theperformanceimprovementpotentialof theBiDFE (e.g.,

BAD, LC-BiDFE),overtheconventionalDFEandtheNCDFE,wasnumericallydemon-

strated.In thissubsection,theimplementationissuesor challengesposedby theBiDFE

are discussed.As the goal of advancedsignalprocessingat the receiver in a digital

communicationsystemis to improve theperformanceat very little additionalcost,we

discussthecomplexity of theBiDFE andcompareit with otherequalizerarchitectures.

Since,a BiDFE relieson non-causalprocessingof the receivedsignalthroughtheop-

erationof time-reversal,latency or theoverall detectiondelayof theBiDFE is of prime

concern.Theotherissuesthatarediscussedin this subsectionarecomplexity, robust-

nessto channelestimationerrorsandemployability of theBiDFE to non-packet based

streamingapplications.

51

6 7 8 9 10 11 12 1310

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

Normal mode DFELC−BiDFE : (Hard)LC−BiDFE : (Soft)BADMLSE

Figure 2.16: Performanceof a BidirectionalArbitratedDFE (BAD) for a symmetricchannel

with animpulseresponseó � ô�õ ö�÷Ôþ§ú§ú ö�÷Ôü�øvþ§ü�ö�÷Ôþ§ú§ú �Latency

Thetime-reversalmodeDFEcanprocessthereceivedsignalonly aftertheentirepacket

of datais received. So, the BiDFE will have an overall detectiondelaywhich is, at

least,as large as the size of the packet. Sucha detectiondelay may not be always

acceptable,especiallyin certainapplicationslikevoicetransmission,andis adrawback

of theBiDFE structure.Hence,thechoiceof employing a BiDFE hingescritically on

thelatency requirementsof thesystem.Ontheotherhand,in somestandardslikeGSM,

thetrainingsequenceusedfor channelestimationis locatedat thecenterof thepacket,

namelyasa mid-amble(seeFigure2.3). Hence,thedetectiondelayfor a GSM system

with aconventionalequalizerwill bealittle morethanhalf thepacketsize.Furthermore,

for GSM systems,theencodeddataburst is interleavedandsplit in to four datapackets

beforetransmission.Thesefour datapacketshave to beassembledat thereceiver prior

52

to channeldecoding.So,thedelayof theBiDFE, mayin fact,not bethelimiting factor

for sucha system. In addition,whena hybrid ARQ protocol is used,a packet error

resultsin a requestfor packet retransmissionandtheassociateddelaybeforethepacket

canbe detectedcorrectly. The performancegain offeredby the BiDFE, on the other

hand,will resultin a decreasedpacket error rate(PER)andhencereducetheneedfor

packet retransmission.

StreamingApplications

Although, the BiDFE structurehasbeendesignedfor packet basedtransmissionsys-

tems,it can be modified for streamingapplicationsthat lack a definite packetization

structure.For astreamingsystem,thereceivedsignalcanbebrokendown into overlap-

pingvirtual packetsof someuser-definedsize.Thevirtual packetscanthenbeprocessed

usingaBiDFE structureto improveperformanceat thecostof increaseddetectiondelay

for thesystem.Thereasonfor thepacketsto have somedegreeof overlapis to remove

theedgeeffectsthatwouldbecreateddueto thecreationof thesevirtual packets.

Complexity

The BiDFE structureof Figure 2.9 usestwo parallel DFE structuresand, all things

beingequal,hasa two-fold increasein complexity. For the performancecomparisons

providedin Figures2.11- 2.16,theBiDFEstructureusestwiceasmany tapsaseitherthe

normalmodeDFEor thetime-reversalmodeDFE.In anattemptto makethecomparison

fair, Figure 2.17 illustrates2 the MSE performanceof the LC-BiDFE structurewhen

comparedwith thenormalmodeDFE andthe time-reversalmodeDFE, bothof which

have thesametotal complexity astheLC-BiDFE. For instance,whena total numberof

2See[19] for Matlabsourcecode

53

14 tapsareused,eachof theconstituentDFEsin theLC-BiDFE structureareallocated

only 7 taps,while eachof the normalmodeandtime-reversalmodeDFEs, to which

the LC-BiDFE performanceis compared,are allocated14 taps. In other words, the

performanceof a “double-length”DFE is comparedwith the performanceof a LC-

BiDFE structurethathastwo “single-length”constituentDFEs.TheMSEplot is for the

asymmetricchannelwith animpulseresponse� Æ (seeequation(2.66))at anSNRof 15

dB. FromFigure2.17it is evident that, for a reasonablechoiceon the total numberof

DFEtaps,theperformanceimprovementprovidedby theLC-BiDFE structureis similar

to thatshown in Figure2.11.However, whenthetotalnumberof tapsin theLC-BiDFE

is very small, the performanceof eachof the “single length” constituentDFEsin the

BiDFE architectureis badandthis resultsin apooroverall performance.

2 4 6 8 10 12 14 16 18−15

−14

−13

−12

−11

−10

−9

−8

−7

Number of Taps

MS

E i

n d

B

Normal Mode DFETime−Reversal ModeLC−BiDFE

Figure2.17: Comparative MSEperformancewith sametotalnumberof tapsfor aBiDFE anda

DFE

Thecomplexity of alinearequalizer, aconventionalDFE,anLC-BiDFE,anNCDFE,

BAD andMLSE arecomparedin Table2.2. The numberof multiplicationsandaddi-

54

tions requiredfor the computationof eachsymbol is tabulated. No attempthasbeen

madeto simplify the architectureof any of the schemes.For instance,whena BPSK

sourceis equalized,theconvolutionoperationof theFBF, which is typically amultiply-

and-addoperationcan be replacedwith a simple add/subtractoperation3. However,

thesesimplificationsto theequalizerarchitectureareapplicationspecificandhenceig-

noredhere. For anNCDFE, �� is thenumberof tapsin thepreliminaryequalizer, for

theBAD, �� is thesizeof thewindow insidewhich theEuclideandistancemetric

is computed,andfor theMLSE, � �� is thenumberof elementsin thesourcealphabet.

Table2.2: Complexity of equalizerstructures

Equalizer Structure Number of Mult/Add Operations

LinearEqualizer ��DFE ��LC-BiDFE �� !�� #"�� $"NCDFE ��%� ��BAD ��&�� !�� #"'��$"'� ��()� +*! ��,��.-MLSE */�()�0�1-1� �� 243

Channel Estimation

Although, we assumethat the channelimpulseresponseis perfectlyknown at the re-

ceiver (assumption2.4), in practicethechannelis estimatedwith theaid of thetraining

sequence(recall Figure2.3) andat timesalsothedata. Theestimatedchannelis typi-

cally imperfectandtheseverity of theestimationerroris afunctionof thenoisevariance

3Thiswouldnot bepossibleif asoftDFE [14] is employed

55

andthe lengthof the training sequence.The channelestimationerror canbe incorpo-

ratedin to theequivalentnoiseandit resultsin performancedegradation.However, the

degradationcanbe expectedto have a comparableeffect on performanceof the nor-

mal modeDFE, thetime-reversalmodeDFEandtheLC-BiDFE. As theBAD relieson

channelestimatesfor the reconstructionbasedarbitrationstage,it may experiencean

additionalperformancedegradation.

6 7 8 9 10 11 12 13 1410

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

Normal Mode DFETime−Reversal Mode DFELC−BiDFE : (Hard)LC−BiDFE : (Soft)BAD

Figure 2.18: Effectof channelestimationon SERperformancefor anasymmetricchannelwith

animpulseresponseof 5 �'687 9�:<;>=.?�9�:A@�;>B�9�:AC.=.@�DE9�:<;>=.?�9�:[email protected]�DE9�:F9&GHB 9�:F9�;>=JITherobustnessof theBiDFE wastestedwith asimulationexample.Theasymmetric

channelimpulseresponseKL� , of equation(2.66),wasconsidered.Assumption2.4 was

relaxedandthechannelimpulseresponsewasestimatedat the receiver usinga Least-

squareschannelestimator[48]. A GSM trainingsequence[33], with a training length

of 26 symbols,wasusedto aid thechannelestimation.TheSERperformancecurves4


56

for a normalmodeDFE, a time-reversalmodeDFE, an LC-BiDFE (with andwithout

softdecisions)andBAD areplottedin Figure2.18.Thenormalmodeandtime-reversal

modeDFEs were parameterizedwith 4 FFF tapsand3 FBF taps. The performance

degradationexhibitedby eachequalizerstructurein Figure2.18,dueto channelestima-

tion errorsat an SERof �.MONQP , whencomparedto theSERcurvesof Figures2.12and

2.13is tabulatedin Table2.3.TheLC-BiDFE (with soft decisions)andBAD exhibit an

additionalperformancedegradationof about0.25dB - 0.3dB at anSERof �.M NQP , when

comparedwith the normalmodeDFE. However, even with channelestimationerrors,

theperformanceimprovementof theLC-BiDFE (with soft decisions)is still about0.9

dB.

Table2.3: Performancedegradationdueto channelestimation

Equalizer Structure PerformanceDegradation

DFE 0.8dB

LC-BiDFE (Hard) 0.9dB

LC-BiDFE (Soft) 1.06dB

BAD 1.12dB

2.6 Summary

A BiDFE structurethatemploys time-reversedprocessingof thereceivedsignalin con-

junction with a normalmodeDFE hasbeenproposedin this Chapter. The proposed

BiDFE structurehas the twin advantageof simultaneouslydecreasingthe gap from

the MFB andmitigating error propagation(for instanceBAD). The performancegain

57

providedby theLC-BiDFE structurehasbeendemonstrated,bothanalyticallyandnu-

merically. The performanceof the BiDFE structures,LC-BIDFE andBAD, hasbeen

numericallycomparedwith DFEenhancementsof similarcomplexity, like theNCDFE.

Theperformancegapfrom theMLSE for thevariousBiDFE structuresis tabulatedin

Table2.4 for the asymmetricchannelimpulseresponseKLR andan SERvalueof �&MQNQP .Implementationissues,namelylatency, complexity androbustnessto channelestima-

tion errors,associatedwith the practicalrealizationof the BiDFE structurehave been

discussedin detail.

Table2.4: Performancecomparisonof equalizeretructuresfor 5 REqualizer Structure PeformanceGap fr om MLSE

DFE 1.66dB

LC-BiDFE (Hard) 1.07dB

LC-BiDFE (Soft) 0.49dB

BAD 0.27dB

In thenext Chapter, theperformancelimits of theLC-BiDFE will beinvestigatedby

consideringaninfinite lengthdesignandtheMSEperformanceof theinfinite lengthLC-

BiDFE will beevaluated.Givenapacketbasedsystemassumptionandtheemployment

of time-reversal,suchaninfinite lengthLC-BiDFE designseemscounter-intuitivewith

its infinite latency. However, our interestis in characterizingtheperformancelimits of

theLC-BiDFE, by relaxingthefinite lengthconstraint.Hence,by lettingthepacketsize

tendto infinity asymptotically, wecanobtainsuchanintuition.

Chapter 3

Infinite Length BiDFE

In this Chapter, an infinite lengthLC-BiDFE will be consideredandtheboundon the

performancegainof this structureover theconventionalDFE structurewill be investi-

gated. The performanceof a finite lengthLC-BiDFE will asymptoticallyconverge to

thatof theinfinite lengthLC-BiDFE, whenthetotal numberof tapsusedby eachof the

constituentDFEsis increased.For apacketbasedcommunicationsystemtheideaof us-

ing aninfinite lengthLC-BiDFE mayseemcounter-intuitiveastheBiDFE architecture

employs time-reversalof thereceivedsignal.However, our interestis in characterizing

the performancelimits of the LC-BiDFE by relaxing the finite lengthconstraint,and

thereby, obtainusefulperformancebounds. To employ an infinite lengthBiDFE the

packet sizealsoneedsto beinfinitely long; soweassumethat SUT V .

This chapteris organizedasfollows. Section3.1describesthenotationusedin this

chapter. Section3.2 quantifiesthe advantageof theLC-BiDFE, over the conventional

DFE,by computingthedecreasein thegapfrom theMFB at high SNRscenarios.This

motivatesthetapoptimizationproblemfor theinfinite lengthLC-BiDFE. In Section3.3,

the infinite lengthLC-BiDFE coefficientsareoptimizedto minimize theoverall MSE,

ratherthantheMSE at theoutputof eachthe individual DFE streams,andSection3.4

58

59

summarizestheresultsin this chapter.

3.1 SystemModel

CombiningDecisionDevice

y

TimeReversal

TimeReversal

DecisionDevice

s2^

DecisionDevice

s1^

y1

y2

s^Linear

F1(z)

F2(z)

2B (z)

(z)1B

II

Ir

Figure 3.1: Structureof anLC-BiDFE.

The structureof an LC-BiDFE is depictedin Figure3.1. Now, W��*/X�- and WY"1*/X�-representtheZ-transformof theFFF tapsof thenormalmodeandtime-reversalmode

DFEs,respectively, and Z[�J*\X�- and Z]".*\X�- arethe Z-transformresponsesof the corre-

spondingFBF taps. As statedin equation(2.52), thesoft outputsof the normalmode

and the time-reversalmodeDFEs arecombinedusinga memorylessweightedlinear

combiner, i.e., ^ *`_�-�acb ^ �>*/_�-d�e*f�hgib'- ^ "1*`_�->j (3.1)

Also recalltheMSEoptimizedcombinercoefficient b of equation(2.54).In thischapter

we invoke assumptions2.1-2.6and2.8 of Chapter2.5. For simplicity of notation,we

alsomake thefollowing additionalassumption.

60

Assumption3.1 Thechannelimpulseresponsehasa unit norm, i.e., �k�lKm�n� " ao� , and

doesnothavea spectral null, i.e., no rootson theunit circle.

Theassumptionon theenergy of thechannelcoefficientscanbemadewithout lossof

generality, asit would not affect theoutputSNRof theequalizer, but only theabsolute

energy. The presenceof spectralnulls is undesirablefor the designof baud-spaced

equalizers.Hence,we make theassumptionon the absenceof rootson theunit circle

andthis precludesonly asmallclassof channelsfrom thefollowing analysis.

3.2 Performanceof an Infinite Length LC-BiDFE

In this section,thegapfrom theMFB for theLC-BiDFE will bederivedasa function

of thechannelimpulseresponsefor ahigh SNRscenario.Recallthedefinitionof MFB

from equation(2.5),namely pmfb a �k�lKm�k� "q)"r j (3.2)

The infinite lengthMMSE-DFEtapcoefficientsderived in [70] werereviewed in sub-

section2.2.2. The normalmodeandtime-reversalmodeDFE tapsareassumedto be

parameterizedwith theMMSE-DFEcoefficients. Recallthe factorizationof thechan-

nel impulseresponsefrom equation(2.33),s *\X�-�aet�*/M�- 2 3 N �u v w � *f�xgzy v XQ{�|!-�} (3.3)

61

where y v aretheroot locationsof thechannelimpulseresponsepolynomial. Then,the~-transformof thetime-reversedchannelimpulseresponse�s */X�- is,�s */X�-�� s */XQ{�|�-%}act�*/M�- 2H3u v w � *f�xgzy v X�-h}act�*/�(f-�*fg�X�-f� 3 243u v w � � �xg X {4|y v�� (3.4)

In equation(3.4), the secondequality follows from the identity on the productof the

rootsof apolynomial, t�*/�(f-t4*\M�- a�*fg��1- 2H3 2H3u v w � y v � (3.5)

Underthe high SNR assumption,i.e., ��T M , the feedforward filter (FFF) for the

normalmodeandthetime-reversaloperationsaregivenby (recallequations(2.23)and

(2.25)) W��J*/X�-�� t4*\M�- 2 3uv w ��A� �>�$� �� *`y��v g�X {�| -y �v *f�hgzy v X {�| - } (3.6)

and WY"H*/X�-�� *fg�X�- { � 3t�*/�(f- 2 3uv w ��A� �>�$� �� yv *f�xg�y��v X {4| -*/y v giX {4| - �

(3.7)

Thefeedbackfilters Z��*/X�- andZ]"H*\X�- arechosensuchthatthepost-cursorISI is perfectly

canceled.Mathematically,

Z��*/X�-�a ��F� 2H3uv w ��A� �>�/� �� *f�xgzy v X {4| -O�F�� F� 2H3uv w ��A� �>�$� �� *f�xg�*/y �v X�- {�| -��F�� g��} (3.8)

and Z]"H*/X�-�a �� 2 3uv w ��A� �>�/� �� *f�xgzy �v X {�| - �F�� 2 3uv w ��A� �>�$� �� * �¡g8*/y v X�- {�| - �F�� g��j (3.9)

In [6], AriyavisitakulderivedtheMSEandtheoutputSNRof aninfinite lengthDFE,

undertheideal feedbackassumption.Theseresultswill bebriefly discussedhere.The

62

MSE of theinfinite lengthDFE for ahighSNRscenariois

MSEDFE ac¢¤£¥� ^ �J*/_)-¦g�§O*/_�-.� "©¨ac¢«ª¬4®O¯±° �²*!³+- ´µ*/_¶g�³·-4

"f¸¹ �(3.10)

Since,the energy of an impulseresponsein time domainis the sameas the constant

termin the~

-transformpolynomialexpansionof its auto-correlationfunction[72], the

MSE canbeevaluatedin thefrequency domainas

MSEDFE a q "r £FW��*/X�-ºW �� *\X {4| - ¨k» (3.11)

wherethenotation £½¼¾*/X�- ¨ » representstheconstanttermin thepolynomialexpansionof¼¾*\X�- , namely ¿Y*\M�- . With a few simplemanipulationsit canbeshown that

MSEDFE a q "rt�*/M�-ºt � *\M�-ÁÀÂ 2 3uv w ��A� �>�$� �� y v y �v1ÃÄ N � j (3.12)

Ariyavisitakul also demonstratedthat the MSE of both the normal modeand time-

reversalmodeDFEsarethesamefor theinfinite lengthscenario.

Lemma 3.1 The MSE of the output of the normal modeDFE and the time-reversal

modeDFE are thesame, underthehigh SNRassumption.

Proof: To prove the above statement,it is sufficient to show that the energy of the

feedforwardfilter of boththenormalmodeandtime-reversalmodeDFEsarethesame,

i.e., �n� Åº�.�k� " aÆ�n� Å�"��k� " . By usingargumentssimilar to thosein equations(3.10)- (3.12),we

canshow that £AW�"H*\X�-fW �" *\XQ{4|�- ¨ » a �t4*\�(f-ft � */�(f-ÁÀÂ 2 3uv w ��A� �>�`� �� y v y �v1ÃÄ j (3.13)

63

Applying theresultof equation(3.5) to equation(3.13),

£AW�"H*\X�-fW �" */XQ{�|!- ¨ » a �t4*\M�-ft � */M�-ÇÀÂ 2 3uv w ��A� �>�$� �� y v y �v1ÃÄÉÈ 2 3u v w � y v y �v1Ê N � (3.14)

aË£FW��J*/X�-ºW �� *\XQ{4|!- ¨k» j ÌThe MMSE-DFE suffers a performancelosswhencomparedto the MFB andthe

outputSNRof theMMSE-DFEispDFE a t�*/M�-ºt²�.*/M�-q "r ÀÂ 243uv w ��A� �>�$� �� y v y �v1ÃÄ N � � (3.15)

As boththenormalmodeandtime-reversalmodestreamsyield thesameMSE,anequal

gain combiningscheme,i.e., bËa �" , is optimal for the diversity combiningblock of

theLC-BiDFE of Figure3.1. As theresidualISI componentsareabsentunderthehigh

SNRapproximationfor the feedforwardfilters in equations(3.6) and(3.7), theoverall

MSE of theLC-BiDFE canbewrittenas

MSEBiDFE ac¢¤£¥� ^ */_)-Yg�§O*/_�-.� " ¨ac¢oª¬�® ¯ÎÍ ° ��*\³+-�� ° "1*fg]³+-#Ï¦´Ð*/_¶g�³+-

" ¸¹ �(3.16)

TheMSE canbesimilarly evaluatedin thefrequency domainas

MSEBiDFE a q "rÒÑ W�Ó!Ô&*\X�-fW �Ó!Ô *\X {4| - Õ » (3.17)

where WYÓ!Ô&*\X�-�a W��*\X�-d��WY"1*/X {�| - �(3.18)

Theperformancegainprovidedby theLC-BiDFE over theDFE is givenby,pBiDFEpDFE

a £AW��*\X�-fW�� */X {�| - ¨ »Ñ W�Ó!Ô.*/X�-ºW �Ó!Ô *\X {4| -�Õ » � (3.19)

64

Lemma 3.2 Thegain in outputSNRof the infinite lengthLC-BiDFE over the infinite

lengthDFE, undera highSNRassumptionispBiDFEpDFE

a �%��Ö Í.× */M�->Ï } (3.20)

where Ö Í.× */M�->Ï denotesthereal part of × *\M�- , andthe~

-transformof theresponse× *!³+-is Ø */X�-�a t4*\M�-fX � 3t4*\�(f- ��F� 243uv w ��A� �>�$� �� y

v */y �v giX {4| -*f�xg�y v *fg�X�- {4| - �F�� F� 2H3uv w ��A� �>�$� �� y �v *f�xgzy v X {�| -*/y �v gzX {�| - �F�� (3.21)

Proof: Fromequation(3.18),thedenominatorof equation(3.19)canberewrittenas£AW�Ó!Ô.*/X�-ºW �Ó!Ô *\X {4| - ¨ » a�£ Í W��*/X�-d� W�"1*\X {4| ->Ï Í W �� *\X {�| -�� W �" *\X�->Ï ¨ »aË£FW��J*/X�-ºW �� *\XQ{4|!-�� W�"H*\XQ{4|!-ºW �" *\X�-d��W��*\X�-ºW �" *\X�-d��W �� */XQ{�|�-fWY"1*/XQ{�|!- ¨ » j (3.22)

FromLemma3.1andequation(3.12),£AW��*\X�-fW �� *\X {4| ¨ » a�£FWY"H*\X {�| -ºW �" */X�- ¨ » a �t4*\M�-ºt � *\M�- ÀÂ 2H3uv w ��A� �>�`� �� y v y �v1ÃÄ N � j (3.23)

Now define,Ø */X�-�� W�� *\X {4| -fWY"1*/X {�| -£FW��*/X�-ºW �� *\X {4| ¨ »a t4*\M�-J*fg�X�- � 3t4*\�(f- ��F� 243uv w ��A� �>�$� �� yv */y��v gzX {�| -* �¡gzy v X {�| - �F�� F� 243uv w ��A� �>�`� �� y��

v *f�xgzy v X {�| -*/y �v giX {�| - �F�� (3.24)

Let × */M�- denotetheconstanttermin thediscretetimedomainexpansionof

Ø */X�- . Then,£AW �� */X {�| -fWY"H*\X {�| - ¨ » a × *\M�-J£AW��²*\X�-ºW �� *\X {4| - ¨ » j (3.25)

Sincetime-reversalof a discretetime sequencedoesnot affect theentrywith theindex

zero, £FW��J*/X�-ºW �" *\X�- ¨ » aË£FW �� *\X {4| -ºWY"H*/X {�| - ¨ �» }a × � */M�-�£FW��*/X�-ºW �� */X {�| - ¨ » j (3.26)

65

Hence,wecanconcludethat£AW�Ó!Ô&*\X�-fW �Ó!Ô *\X {4| - ¨ » a�£AW��*\X�-fW �� */X {�| - ¨ » £A x�� Ö Í.× *\M�-#Ï ¨ j (3.27)Ì3.2.1 Numerical Results

0 0.2 0.4 0.6 0.8 1−1

0

1

2

3

4

5

6

7

8

Root Location β

Noi

se G

ain

(in

dB

)

Conventional DFELC−BiDFEMatched Filter Bound

Figure 3.2: Performancegapfrom thematchedfilter boundfor a symmetric3-tapchannelwith

root locationsat Ù#ÚdÛ ��+Ü .To illustrate the performanceimprovementthat canbe obtainedby the useof an

LC- BiDFE, we considertwo testcases.Figure3.2 illustrates1 thegapfrom theMFB,

for theconventionalDFE andtheLC-BiDFE, for a 3-tapreal symmetricchannelwith

root locationsat y and �� . Theseresultscorrespondto thehighSNRapproximationthat

hasbeenusedin this section. It is clear that as the channelbecomessevere, i.e., as


66

theroot locationsmove closerto theunit circle, theperformancegainprovidedby the

LC-BiDFE over theconventionalDFE is morethan3 dB.

0 0.2 0.4 0.6 0.8 1−1

0

1

2

3

4

5

6

Root Location β

Noi

se G

ain

(in

dB

)

Conventional DFELC−BiDFEMatched Filter Bound

Figure 3.3: Performancegapfrom the matchedfilter boundfor an asymmetric3-tapchannel

with root locationsat Ý$ÚdÛ �"HÞ .Figure3.3 illustrates2 similar performanceimprovementsfor theLC-BiDFE, when

appliedto a 3-taprealasymmetricchannelwith root locationsat y and �" . Althoughthe

LC-BiDFE performsbetterthan the MMSE-DFE, it still suffers from a small perfor-

mancepenaltywhencomparedto theMFB. Onepossiblereasonis thefactthateachof

theconstituentDFEsis chosento minimizetheMSE at its respectiveoutput.However,

for theLC-BiDFE, themetricof interestis theoverall MSE. Perhaps,if we attemptto

optimizethe tapsof theLC-BiDFE to minimize theoverall MSE, it might bepossible

to attaintheMFB.2See[19] for Matlabsourcecode

67

3.3 LC-BiDFE Tap Optimization

In the analysisof Section3.2 and in the examplescorrespondingto Figures3.2 and

3.3,theDFEcoefficientsof eachstreamof theLC-BiDFE areoptimizedindependently;

anMMSE-DFEsettingis chosen.In this section,we formulatethe infinite lengthLC-

BiDFE tapoptimizationproblemto minimize theoverall MSE of theLC-BiDFE. The

resultingMSE would provide a boundon thepotentialperformanceimprovementsthat

can be achieved with this structure. It shouldbe notedthat the finite length results,

whicharemoreusefulin practice,asymptoticallyconvergeto theinfinite lengthresults.

A generalizedformulationof the optimizationprobleminvolvesdeterminingthe LC-

BiDFE tap coefficients, namely £FÅºRH}©ß�R ¨ }i£lÅ�à�}áß¦à ¨ , and the coefficient of the diversity

combiner b , that minimizesthe overall MSE of the LC-BiDFE. Suchan optimization

problemis mathematicallyintractable. Hence,the LC-BiDFE structureof Figure3.1

is simplifiedby introducinga front-endfilter matchedto thechannelimpulseresponse,

namelys �1*\X {�| - , asshown in Figure3.4.

CombiningDecisionDevice

y(z )

TimeReversal

TimeReversal

DecisionDevice

s2^

DecisionDevice

s1^

y1

y2

s^Linear

B (z)*

(z)*F

r

II

I

C* −1 F(z)

B(z)

Figure 3.4: LC-BiDFE structurefor tapoptimizationin aninfinite lengthscenario

Theadvantageof thefront-endmatchedfiltering aretwo fold. Firstly, thecombined

68

channelandmatchedfilter impulseresponseâ¾ã/ä�å is conjugatesymmetric.â¾ã\ä�å�æ�çèã/ä�åºç�é1ã\äQê4ë�å%ì (3.28)

Secondly, astheadditivenoisesequenceíµã/î)å is assumedto bewhite,thepowerspectral

densityof thenoiseattheoutputof thefront-endmatchedfilter is â¾ã/ä�å . Let ïFðèã/ä�å²ñ#ò¤ã\ä�å�ódenotetheDFE tapcoefficientsof thenormalmodeDFE.As thetime-reversalof â¾ã/ä�å ,dueto its conjugatesymmetryproperty, is â é ã\ä�å , a suitablechoicefor the tap coeffi-

cientsof the time-reversalmodeDFE is ïAð é ã\ä�å²ñ#ò é ã/ä�å ó . It is straightforward to show

that for this modifiedstructureandthe choiceof DFE filter coefficients,both the nor-

mal modeandtime-reversalmodestreamswill have the samevalueof MSE. Hence,

an equal-gaincombiningscheme,with ôõæ�ö1÷�ø , is optimal. The MSE minimization

problemcannow becastin theformã/ð opt ñáò opt å�æcù�úºû�üþýkÿ�� E�� ã/î)å��Oã/î)å � �� ì (3.29)

Whenoptimizing theLC-BiDFE coefficients,a meaningfulconstraintto imposeis

for the FBF of eachconstituentDFE to remove all the post-cursorISI in that stream.

Otherwise,it is possibleto indefinitelyincreasetheMSEin eachstreamwithout affect-

ing theoverall MSE of theLC-BiDFE. Since,in reality the ideal feedbackassumption

hasto berelaxedto allow for decisionerrors,this is critical. Thefollowing lemmawill

beusedto optimizethetapsof theinfinite lengthLC-BiDFE.

Lemma 3.3 If any pair of polynomials ��¤ã/ä�å²ñ#çþã\ä�å�� aresuchthat ï��èã\ä�å ó��æ ö andïFçèã/ä�åºç é ã\ä ê4ë å ó��æõö , then � �¤ã/ä�å�� é ã\ä ê4ë åçèã/ä�åºç é ã\ä ê�ë å�� ö (3.30)

andequalityis attainedif andonly if �þã\ä�å�æeçþã\ä�åfç é ã/ä ê�ë å .

69

Proof: The term ï��¤ã\ä�å�ó�� canbe evaluatedusing the well-known identity basedon a

convolution integralaroundtheunit circle, i.e.,ï��èã/ä�å ó��hæ öø��! #"$ " �¤ã&%('*)#å*+-,µì (3.31)

Applying Cauchy-Schwartz inequalityto thetwo continuousfunctions, �èã.% '�) åº÷�çèã.% '�) åand ç é ã&% '*) å , we have� öø��! "$ " �þã&% '*) åçþã&% '*) å çþã&% '*) å�+�, � �0/ � öø��! "$ " �þã&% '*) å�� é ã&% '�) åçþã&% '�) åºç é ã&% '*) å +�, �1 � öø��! #"$ " çèã.% '�) åºç�é1ã&% '�) å�+�, � ñ (3.32)

whereequalityis attainedif andonly if�þã&% '�) åçþã&% '�) å32 ç�é1ã.% '�) å (3.33)

or equivalently, �èã/ä�åçèã/ä�å32 ç�é.ã/äQê�ë!å�4 (3.34)

Since ï5�èã\ä�å�ó��æõö and ïFçþã\ä�åfç é ã/ä ê�ë å�ó��%æ�ö , equation(3.32)reducesto� öø��! "$ " �èã&% '�) å�� é ã&% '�) åçþã&% '�) åºç é ã&% '*) å +�, � � ö (3.35)

with equalityif andonly if �¤ã/ä�å�æcçèã/ä�åºç é ã\ä ê�ë å64 7Theorem 3.1 Theunbiasedinfinite lengthMMSE-BiDFE,undertheideal feedback as-

sumption,attainstheMFB.

Proof: ConsidertheLC-BiDFE receiver structureillustratedin Figure3.4. Underthe

constraintthattheFBFperfectlyremovesthepost-cursorISI, wehaveðèã/ä�å©â¾ã/ä�å�æ98µã/ä�å;:�ö<: ò¤ã\ä�å (3.36)

70

where 8µã/ä�å is a purelyanti-causalresponsethat representstheresidualpre-cursorISI,ò¶ã/ä�å is thepurelycausalFBF. TheMSE of theLC-BiDFE is thengivenby

MSE æ ö= � �>8Ðã\ä�å;:?8 é1ã/äQê�ë!å(� � �:A@ �B= � â¾ã\ä�åC�Hðþã\ä�å;:�ð é ã/ä ê�ë å(� � � ì (3.37)

Since 8Ðã\ä�å is purelyanti-causal,thefirst termcanbesimplifiedtoö= � �>8µã/ä�å;:D8]é1ã\äOê�ë!å�� æ öø ïE8µã/ä�å(8]éHã\äOê�ë�å ó � (3.38)

andattainsa minimumvalueof zero,if andonly if 8Ðã/ä�å%æGF . By usingequation(3.36)

andtheconjugatesymmetrypropertyof â¾ã\ä�å , the secondterm of equation(3.37)can

berewrittenas@ �B= � â¾ã\ä�åC�Hðèã/ä�åH:�ð�é.ã\äQê4ë!å�� æ@ �B= I �4øJ: ò¤ã\ä�åH:�ò é ã\ä ê4ë åH:D8Ðã\ä�å:?8 é ã/ä ê�ë å(� �â¾ã/ä�å K � ì (3.39)

Let usdefine �¤ã/ä�åML�öN: ò¶ã\ä�åO:�ò é ã\ä ê�ë åH:D8Ðã/ä�å;:D8 é ã\ä ê4ë åø ì (3.40)

Then,equation(3.39)simplifiesto@ �B= � â¾ã/ä�å��Hðþã\ä�å;:�ð�é.ã/äQê�ë�å(� �� æ @ �B � ��þã\ä�å(� �â¾ã/ä�åP� � ì (3.41)

From Lemma3.3, the secondterm in equation(3.37) is minimizedwhen �èã\ä�åµæâ¾ã\ä�å . Hence,theMSE of equation(3.37)attainsaminimumvaluewhen 8Ðã\ä�å�æQF andö<: ò¶ã\ä�å;:�ò é ã\ä ê4ë åø æcâ¾ã\ä�åxì (3.42)

As thefeedbackfilter ò¶ã/ä�å is purelycausal,theoptimaltapcoefficientsaregivenby,R opt ã&S+å�æ ø>TYãUS+å%ñ V!S æ�ö�ñ#øLñ&ì.ì.ìmñXWMY (3.43)

71

andtheoptimumFFFis ð opt ã\ä�å�æ öN:�ò opt ã/ä�åâ¾ã\ä�å ì (3.44)

Fromequations(3.30),(3.42)and(3.44),theminimumMSE attainedby this choiceof

BiDFE coefficientsis

MSEMMSE-BiDFE æ @ �B (3.45)

andhencethemaximumoutputSNRisZMMSE-BiDFE æ ö@ �B ñ (3.46)

whichissameastheMFB. In otherwords,theMSEoptimizedinfinite lengthLC-BiDFE

attainstheMFB. 73.3.1 Relation to MSE optimized NCDFE

In thediscussiononNCDFEin Chapter2,subsection2.3.1,it wasstatedthattheoptimal

feedforwardNCDFEtapsareproportionalto thematchedfilter responseof thechannel.

Underthe unit norm assumptionon the channel,the constantof proportionalityturns

out to beunity. Thefeedbackfilter of theNCDFEis thengivenby,R ncdfeãUS·å�æGT¦ãUS·å%ñ V[SþæA�\W]Y>ñ^4^4^4Jñ^��ö�ñ&ö�ñ^4_4^4JñXWMY (3.47)

Comparingequations(3.43) and (3.47), we seethat althoughthe length of the feed-

backfilter in theMMSE-BiDFE is only half aslong astheFBF of theNCDFE,thetap

coefficientsaretwice in magnitudewhencomparedto thoseof theNCDFEFBF taps.

3.3.2 Uniquenessof MMSE-BiDFE

In the earlier part of the section,the MMSE-BiDFE tap coefficients were optimized

by restrictingthe structureof the receiver architecture.The resultingMMSE-BiDFE

72

wasshown to attaintheMFB, but aretheLC-BiDFE filter settingsthatattaintheMFB

unique?In otherwords,arethereotherfilter settingsthatwould let theBiDFE attainthe

MFB. In this subsection,we show that the MSE optimizationof the LC-BiDFE is not

uniqueby consideringa simple2 tapexample.

Example3.1 Considera real2-tapchannelimpulseresponseçþã\ä�å�æG`M:ba ö0�c` � ä ê�ë ,where

� ` �-d ö . ThentheMMSE-BiDFEfilter settingsthatattaintheMFB canbewritten

fromequations(3.43)and(3.44)asò opt ã\ä�å�æcøe` a ö0��` � äOê�ë (3.48)

and ð opt ã\ä�å�æ ö<:�øe`fa öJ�g` � ä ê4ë` a öh��` � ä ê�ë :�ö<:#` a ö0��` � ä ê4ë ì (3.49)

On theotherhand,considerthefollowing filter settings.ðMiJã/ä�å�æ ö` ñ%ò3iJã\ä�å�æ a öJ�g` �` äQê�ë (3.50)

and ð � ã/ä�å�æ ä ê�ëa öJ��` � ñ�ò � ã/ä�å�æ `a öJ�g` � äQê4ë*4 (3.51)

For thesefilter settings,theeffectivenoisetermsat theoutputof thenormalmodeand

time-reversalmodeDFEsare,j i²ã/î�å�æ íµã/î)å` ñ j � ã/î)å�æ íµã`îk:�ö1åa öJ�g` � (3.52)

and @ �i æl@ �B` � ñ @ �i æ @ �BöJ��` � ñ mµænFf4 (3.53)

Hence, fromequations(2.54)and(2.57),theoptimalweightingfactor ô and theMSE

aregivenby ô æG` � ñ MSE æ @ �B 4 (3.54)

Hence, thefilter settingsof equation(3.50)and(3.51)alsoattain theMFB.

73

The 2-tapchannelexample3.1 demonstratesthat thesettingsof the infinite length

LC-BiDFE, that attain the MFB, are not unique. In this chapter, the LC-BiDFE tap

coefficients have beenderived underthe ideal feedbackassumption.When the ideal

feedbackassumptionis relaxed, the additionalobstacleof error propagationhasto be

overcome.Theperformancelossdueto errorpropagationincreaseswith themagnitude

of the feedbackfilter tap coefficients. If all the LC-BiDFE tap settingsthat attainthe

MFB areknown,perhaps,alowerperformancedegradationcanbeachievedbychoosing

thetapsettingwith theleastenergy in its feedbackfilter taps.

3.4 Summary

In this chaptertheperformanceof aninfinite lengthLC-BiDFE with MMSE-DFEtaps

coefficientshasbeenquantified. Although, the LC-BiDFE offers a significantperfor-

mancegainfrom theconventionalDFEstructure,it still hasapenaltyfrom theMFB. In

anattemptto decreasethis gapfurtherwe formulatedandsolvedtheMSEoptimization

of the LC-BiDFE taps. The MMSE-BiDFE hasbeenshown to attainthe MFB, under

the ideal feedbackassumption.Since,in practice,only finite lengthfilter realizations

arepossible,theoptimizationof thefinite lengthLC-BiDFE tapswill beconsideredin

thenext chapter.

Chapter 4

Finite Length BiDFE

In chapter3, the optimizationof the infinite lengthLC-BiDFE to minimize the MSE

wasconsidered.In this chapter, the tap optimizationproblemwill be extendedto an

LC-BiDFE with a finite lengthconstraint. In Section4.1, a specialclassof channels,

with asymmetricchannelimpulseresponse,isconsideredandtheoptimalunbiasedDFE

tapsettingthatminimizestheMSEat theoutputof thediversitycombiningblockof the

LC-BiDFE is derived.Theeffectivenessof theLC-BiDFE tapoptimizationis evaluated

by numericalsimulations.Although, the ideal feedbackassumptionis invokedfor the

tap optimizationproblem, the effect of decisionfeedbackon the tap optimizedLC-

BiDFE is alsostudiedvia simulations.A drawbackof theLC-BiDFE tapoptimization,

underdecisionfeedback,is notedand two possiblesolutionsare discussed.First, a

modificationof the MSE cost function to include an additionalterm, proportionalto

the energy in the FBF, is proposedin Section4.2. Secondly, an iterative LC-BiDFE

approachis proposedin Section4.3. TheLC-BiDFE tapoptimizationis extendedto a

generalclassof channelswith anasymmetricchannelimpulseresponsein Section4.4,

andasummaryof resultsis providedin Section4.5.

74

75

4.1 MMSE-BiDFE for a Symmetric Channel

In Chapter2, eachof thenormalmodeandtime-reversalmodeDFEswereparameter-

ized with the MMSE-DFE tap coefficients. However, thesetap settingminimize the

MSE at theoutputof eachof theindividualstreamsratherthantheoverallMSE.In this

section,thefilter tapsof thefinite lengthLC-BiDFE will beoptimizedto minimizethe

overallMSE of thesystem,namelyE ï j � ó . In this section,weonly considerchannelim-

pulseresponsesthataresymmetric.Thetapoptimizationfor thecaseof anasymmetric

channelwill beaddressedin Section4.4.

For a symmetricchannelo , theblock time-reversedimpulseresponse,namely po is

thesameas o . So,thesamesetof filter coefficientscanbeusedfor boththenormalmode

andthetime-reversalmodeDFE streams.Furthermore,this resultsin thepropertythat

both the normalmodeand the time-reversalmodeDFE streamshave the sameMSE

performance.This leadsto an equal-gaincombiningscheme,i.e., ô,æ ö1÷�ø , for the

diversity combiningblock of Figure3.1. Let q denotethe detectiondelayof eachof

theDFE streams.For theconvenienceof the reader, the W]r 1 WMs channelconvolution

matrixof equation(2.10)is restatedhere.

ç ætuuuuuuuuuuuuuuuv

w ã&F�å F ì.ì.ì F... w ã.F�å ì.ì.ì ...w ã.WMYO�8ö1å ... ì.ì.ì FF w ã&W]Y��ö1å ì.ì.ì w ã&F�å... F ì.ì.ì ...F ... ì.ì.ì w ã.W]Yx� ö1å

y{zzzzzzzzzzzzzzz|(4.1)

Now, thecombinedchannelfeedforward impulseresponseis givenby }�æÆç3~ . From

equation(2.16),to minimizetheMSEundertheidealfeedbackassumption,theoptimal

FBF tapsof the DFE shouldexactly cancelthe W]� tapsof } that follow the cursor ��

76

(recallequation(2.16)).Recallthe W]r 1 W]r matrix � definedin equation(2.20).Define

thevector � as �ÁæG� ç3~J�g�� (4.2)

where�� is acolumnvectorof length W]r and,��æËï�F-i��e�xö�F�i��e�>� $ � $ i ó�� (4.3)

Recall the expressionfor the residualISI term of the noisesequencefrom equation

(2.64),namely

ResidualISI term æQ�-�� ã&S+å�: � ã\øeq��S·åø � � 4 (4.4)

TheresidualISI componentcanbeexpressedin vectorform as

ResidualISI term æ � � ãU�OiH:�� å � ãU�OiH:�� å��= (4.5)

where �Oi and � � are ã!øeW]r�� ö1å 1 WMr matricesthatperformshifting andtime-reversal

operationson thevector � , prior to averaging.Thematrices�Oi and � � areconstructed

as, �Oi�æ tuuuuv F-� � � $ i $ ��&�e� �� e� �Fe�(�e� �y zzzz|�� æ tuuuuv Fe�(�e� �p� � � �e� �F-� � � $ i $ ��&�e� �

y zzzz| (4.6)

where F��(�e�>� refersto a q 1 W]r dimensionalzeromatrix,�

refersto an identity matrix

and p� is a time-reversalmatrix, i.e.,amatrixwith unit anti-diagonalentriesandzerofor

all otherentries. p� �>��e�>�Oã��Òñáî�å�æ � ¡ ¢ ö if �£:�îÁæQW]rh:0öF otherwise(4.7)

Recalltheexpressionfor thenoisegaintermfrom equation(2.62),

Noisegainterm æ �¤�P��¥ ãUS+åH: ¥ ã!øeqN:�öJ��W]Y��S·åø � � @ �B 4 (4.8)

77

Thiscanbeexpressedin vectorform as,

Noisegainterm æ ~ � ã&¦Mi�:�¦ � å � ã.¦Mi�:#¦ � å*~ @ �B= (4.9)

where¦Mi and ¦ � are ã&W]r§:¨W©sH� ö1å 1 WMs matricesthatperformshiftingandtime-reversal

operationson thevector ~ , prior to averaging.Thematrices¦Mi and ¦ � areconstructed

asshown below. If øeqh:�öJ�gW]r � F then,

¦]i�æ tuv � �eª��e�>ªF � �>� $ i&�&�e�>ª y z| � ¦ � æ tuuuuv F � � ��«Ci $ �e�¬�&�e�>ªp� �eª��e�>ªF � � � � $ � � $ � �&�e� ªy{zzzz| (4.10)

andif øeqN:�öJ��W]r d F then,

¦Mi�æ tuuuuv F-� �>� $ � � $ i&�&�e�>ª� �eª��e�>ªF � �(�e�eªy zzzz| � ¦ � æ tuv p� �eª��e�>ªF-� �>� $ i&�&�e�>ª y z| 4 (4.11)

Theoptimalfeedforwardfilter settingthatminimizesoverallMSE is givenby,~ opt æcù�úºû�üþýkÿ ö= � ã&� ç�~J�g��#å��¦ãU�Oi�:�� å��¦ãU�Oi�:�� åJãU��ç�~0�c��#å:�~¬�'ã&¦Mi�:�¦ � å��¦ã&¦MiH:#¦ � å*~ @ �BC® 4 (4.12)

Thesolutionto theoptimizationproblemof equation(4.12)is,~ opt æ�ï ãU� ç�å¯�¦ã&�OiH:#� � å¯�¦ã&�OiH:#� � å��çD: @ �B ã&¦]i;:�¦ � å��¦ã&¦Mi�:�¦ � å ó $ iìQã&��ç�å � ã&�Oi;:#� � å � ã&�Oi;:#� � å*�� (4.13)

The above MMSE filter settingfor the BiDFE will be biasedandcanbe unbiasedby

scaling~ opt by thereciprocalof thecursor.~ bidfeopt æ ~ opt�dã&q4å (4.14)

78

andtheoptimumMMSE-BiDFE feedbackfilter is,° bidfeopt æ²±cç3~ bidfe

opt (4.15)

where ± is the W]� 1 W]r matrixdefinedin equation(2.15).

4.1.1 Numerical Results

To testtheeffectivenessof theMMSE-BiDFE,in reducingthegapfrom theMFB, areal

symmetricchannelimpulseresponsewith animpulseresponseofo¤³�æËï�Ff4 øe´>´¤µnFf4E¶Lö1øe¶²Ff4 øe´>´>µHó (4.16)

wasconsidered.An FFFwith 4 tapsandanFBF with 3 tapswereconsideredfor each

of the DFE streams.Recall from Figure2.14, that the LC-BiDFE with MMSE-DFE

tapssuffersa lossof 0.6dB from theMFB. Figure4.1compares1 theMSEperformance

of a conventionalDFE, an LC-BiDFE with MMSE-DFE taps,andan MSE optimized

LC-BiDFE with theMFB. In thefigure,thelegend“suboptimal”refersto thecasewhen

MMSE-DFE tap settingsare used,while “optimal” refersto the casewhen MMSE-

BiDFE tap settingsareused. From Figure4.1, it canbe seenthat the MMSE-BiDFE

almostclosesthegapfrom theMFB, undertheidealfeedbackassumption.

In thepresenceof decisionerrors,onewondersif theLC-BiDFE with theMMSE-

BiDFE tapsettingscanbeexpectedto offer similar performanceimprovementover the

LC-BiDFE with the MMSE-DFE filter settings? The answerto the above question,

unfortunately, is a “no”. The SERperformancecurvesfor thesamechannelexample,

assumingaBPSKsourceconstellation,areshown in Figure4.2. It canbeseenin Figure

4.2, that theMMSE-BiDFE, insteadof improving theperformance,resultsin a perfor-

manceworsethanthat of the “suboptimal” BiDFE. For instance,at an SERvalueof


79

6 7 8 9 10 11 12 13 14−14

−12

−10

−8

−6

−4

SNR in dB

MS

E i

n d

B

Normal Mode DFELC−BiDFE : SuboptimalLC−BiDFE : OptimalMatched Filter Bound

Figure 4.1: MSEperformanceof afinite lengthMMSE-BiDFE

ö·F $-¸ , theMMSE-BiDFE (with soft feedback)hasa performancelossof about0.2 dB.

In the presenceof decisionfeedback,the SERcurve is influencednot only by the de-

creasedgapfrom MFB, but alsoby theerrorpropagationeffect. For anMMSE-BiDFE,

theerrorpropagationis largerbecauseof thefollowing two reasons:¹ For theMMSE-BiDFE,althoughtheoverallMSEis lower, theMSEat theoutput

of thenormalmodeandtime-reversalmodeDFEstreamsarehigher, sinceneither

is individually optimized,andthis resultsin increasederrorpropagation.¹ The FBF tap settingsaredifferent for the MMSE-DFE andthe MMSE-BiDFE.

Simulationssuggestthat theFBF tapweightsfor theMMSE-BiDFE areusually

larger in magnitudethantheMMSE-DFEtapsettings.As theerrorpropagation

gapincreaseswith an increasein themagnitudeof theFBF taps,this hasanad-

verseeffecton theSERperformanceof eachof theconstituentDFE streams,and

hencetheperformanceof theLC-BiDFE.

80

6 7 8 9 10 11 12 13 1410

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

DFE : OptimalDFE : SuboptimalLC−BiDFE : OptimalLC−BiDFE : Suboptimal

Figure 4.2: SERperformanceof afinite lengthMMSE-BiDFE

Theeffect of increasederrorpropagationcanbeseenin Figure4.2,wheretheSER

performanceof thenormalmodeDFEfor theMMSE-BiDFEfilter setting(labeled“Op-

timal” in thefigure) is worsethanthenormalmodeDFE performancefor theMMSE-

DFEfilter setting(labeled“Suboptimal”in thefigure)by about0.8dB for anSERvalue

of ö·F $-¸ . In effect,althoughtheMMSE-BiDFEtapsettingsoffersagainof about0.6dB

in bridgingthegapfrom theMFB, theerrorpropagationgapincreasesby about0.8dB.

4.2 LC-BiDFE tap optimization with modified cost

In this section,we will modify thecostto beoptimizedin sucha way that theoptimal

LC-BiDFE tapsminimizestheeffect of errorpropagationwhile minimizing theoverall

systemMSE.Thereexist a few techniquesin theliteraturethatoptimizetheDFEtapsto

minimizetheeffectof errorpropagation.In [44, 45], Ghoshusesanapproximatemodel

81

for error propagationto optimizethe DFE tapsby iteratively solving a setof coupled

non-linearequations.However, this techniquehasahighcomputationalcomplexity and

cannotbe readily appliedto this problem. In [56], Kosutet al. modify the MSE cost

functionto includea normconstrainton theFBF tapsandattemptto achievea reduced

errorprobability. Suchanapproachwill beattemptedhere.

As the error propagationgapincreaseswith an increasein the weightsof the FBF

tapcoefficients,we includethenormof theFBFastheadditionaltermin theMSE cost

functionof equation(4.12).Themodifiedcostwith theadditionaltermis givenby,~ opt æ�ù�ú©û�ü ýnÿ ö= � ãU��ç3~0�g�-�áå¯�Yã&�Oi;:#� � å¯�¦ã&�Oi;:#� � å�ã&� ç3~0�g��áå:�~¬�'ã&¦]iH:�¦ � å��¦ã&¦Mi�:�¦ � å�~ @ �B :�º�ãU±cç3~¬�må�ã&±cç3~1å ® (4.17)

where º is a user-definedweightingfactor. When º æ9F , we obtaintheregularMMSE-

BiDFE coefficientsof equation(4.13), while otherchoicesof º result in BiDFE taps

with different MSE valuesand error propagationgap. As the weighting factor º is

increased,settingswith largevalueof FBFtapmagnitudesincurahigherpenalty. How-

ever, the overall systemMSE correspondingto the optimal tapsof the modifiedcost

will behigher. Thesolutionto theoptimizationof themodifiedcostfunctionof equa-

tion (4.17)is,~ opt æ � ã&��ç�å¯�¦ã&�Oi;:#� � å¯�¦ã&�OiH:#� � å��çD: @ �B ã&¦Mi�:�¦ � å��¦ã&¦MiH:#¦ � å:9º ãU±eç å��¦ãU±cç�å $ i ì�ãU� ç�å¯�¦ã&�OiH:#� � å¯�¦ã&�OiH:#� � å*�� (4.18)

As in section4.1,theDFE tapscanbeunbiasedasfollows.~ bidfemopt æ ~ opt�dã&q4å (4.19)

andthe“modified” MSE optimizedLC-BiDFE feedbackfilter is,° bidfemopt æ²±cç3~ bidfe

mopt (4.20)

82

where ± is the W]� 1 W]r matrixdefinedin equation(2.15).

6 7 8 9 10 11 12 1310

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

DFE : Mod. Opt.DFE : SuboptimalLC−BiDFE : Mod. Opt.LC−BiDFE : Suboptimal

Figure4.3: SERperformanceof afinite length“modified” MMSE-BiDFE

Table4.1: » vs. SNRfor theLC-BiDFE tapoptimizationwith themodifiedcostfunctionfor ¼ �SNR 6 7 8 9 10 11 12 13 14º 0.27 0.23 0.19 0.16 0.13 0.11 0.09 0.07 0.055

Theeffectivenessof includingthenormtermin theMSEcostfunctioncanbeevalu-

atedby consideringtheSERperformancecurvesfor thesymmetricchannelo � of equa-

tion (4.15).Figure4.3compares2 theperformanceof a BiDFE (with soft feedback)for

theMMSE-DFEtapsettingandthe“modified” optimal tapsettingof equations(4.19)

and(4.20). The º valueschosenfor this simulationaretabulatedin Table4.1. Unlike,

Figure4.2,we observe a performanceimprovement,althoughmarginal, for the“modi-

fied” optimalBiDFE tapsettings.Also noticethatthenormalmodeDFEperformanceis


83

only marginally worsewhencomparedto theMMSE-DFE(labeled“DFE: Suboptimal”

in thefigure)tapsettings.Theinclusionof thenormtermensuresthattheFBFtapcoef-

ficientsarecomparablein magnitudeto thoseof theMMSE-DFEtapweights.Thishas

beenillustratedby plottingtheFBFtapweightsfor theMMSE-DFE,theMMSE-BiDFE

andtheBiDFE optimizedwith the“modified” costin Figure4.4. Thisplot corresponds

to an SNR valueof 10 dB and º æ�Ff4 ø . The FBF tap weightsof the MMSE-BiDFE

areabout60 % aslargeastheMMSE-DFEtapsandhenceresultin an increasederror

propagationgap.

0

0.2

0.4

0.6

0.8

1

1.2

Fee

dbac

k F

ilter

Tap

Wei

ghts

MMSE−DFEMMSE−BiDFELC−BiDFE : Mod. Opt.

Tap 1 Tap 2 Tap 3

Figure 4.4: Comparisonof feedbackfilter tapweights

However, onedrawbackof this techniqueis thechoiceof theuser-definedparameter½. The

½valuesusedfor the simulationexampleof Figure4.3 aretabulatedin Table

4.1. In theaboveexample,thevalueof½

waschosenbasedon simulationsto maximize

the performancegain of the BiDFE. Hence,the difficulty in choosing½

rendersthis

approachimpractical.

84

4.3 LC-BiDFE with Iteration

Theeffectof increasederrorpropagationwhenusingtheMMSE-BiDFEtapcoefficients

wasillustratedin Figure4.2. In this sectionaniterative solutionis proposedto address

this issue.Theideais to equalizethereceivedsignalusinganLC-BiDFE with MMSE-

DFE tap coefficients to provide an initial estimateof the transmittedsymbols. In the

seconditeration,thereceivedsignalis re-equalizedusinganMMSE-BiDFE.Duringthis

iteration,theestimatesof thefirst iterationareusedto cancelthepost-cursorISI in both

the normalmodeandtime-reversalmodeDFEsandthe outputsof the two modesare

combinedasbefore.TheSERperformancecurve of theproposediterative LC-BiDFE

schemeis illustrated3 in Figure4.5for thechannel¾>¿ . Theperformanceimprovementis

similar to thatof Figure4.3,andthis techniquedoesnot have practicaldifficulties like

choosing½

encounteredin Section4.2. However, thisperformanceimprovementcomes

at the cost of a two-fold increasein complexity. An additionalsimulationexample,

demonstratingtheeffectivenessof this approachin offeringa marginal improvementin

SERperformance,is providedin AppendixA.

4.4 MMSE-BiDFE for an Asymmetric Channel

The MSE optimizationof the LC-BiDFE for an asymmetricchannelis moredifficult

asit involvesthejoint optimizationof boththeDFE filter tapsandtheweightingfactorÀ . Hence,we proposea finite lengthequivalentof thestructureproposedin Figure3.4

of section3.3. In this structure,the receivedsignalis processedwith a front-endfilter

matchedto the channelimpulseresponse.Hence,the effective responseseenby the

normalmodeandtime-reversalmodeDFEswill besymmetric.Furthermore,theauto-


85

6 7 8 9 10 11 12 13 1410

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

DFE : SuboptimalLC−BiDFE : SuboptimalLC−BiDFE : Iterated

Figure4.5: SERperformanceof aniterative finite lengthLC-BiDFE

correlationof thenoisefilteredthroughthechannelimpulseresponseis alsosymmetric.

Hence,thesametapcoefficients Á.Â>ÃXÄxÅ canbeusedfor boththenormalmodeandtime-

reversalmodeDFE streams.An equalgaincombiningscheme,ÀbÆÈÇ�ÉeÊ , is optimalfor

thediversitycombiningblock.

Let Ë be theauto-correlationvectorof thechannelimpulseresponsewith a length

of Ì]Í ÆGÊ Ì]ÎÏ Ç ,ËxÁ.Ð.Å ÆÒÑ�Ó�ÔÖÕ× Ø(Ù�Ú\Û ÁUÜfÅ Û ÁUÜ3Ý#Ì]ÎOÏ Ç Ï�Ð.Å�Ã Þ[ßáàÒÐxâ Ê Ì]ÎÏ Ç>ã (4.21)

Let ä representthe Ì]åçægÌMè channelconvolution matrix, where Ì]å Æ Ì]Í3ÝQÌ©è�Ï Ç .Let é Æ äNÂ denotethe combinedchannel,front-endmatchedfilter and feedforward

filter impulseresponse.Let usnow definethe Á.Ì]Î�Ý�Ì©è0Ï Ç Åêæ�ÌMè channelconvolution

86

matricesë (seeequation(4.1))and ìë , with

ìë ÆíîîîîîîîîîîîîîîîïÛ Á&Ì]ÎxÏ Ç Å ß ð·ð·ð ß

...Û Á&Ì]ÎOÏ Ç Åñð·ð·ð ...Û Á.ß¤Å ...

. . . ßß Û Á&ß¤Å . . .Û Á&Ì]ÎÏ Ç Å

... ß . . ....ß ... ð·ð·ð Û Á.ß¤Å

ò{óóóóóóóóóóóóóóóôã (4.22)

The residualISI termwill have thesameform asthatof equation(4.5), exceptfor the

useof theconvolution matrix ä , insteadof ë . Thenoiseterm,on theotherhand,will

bea little differentdueto thefront-endmatchedfilter throughwhich theadditivewhite

noise õöÁ.÷�Å is filtered. Incorporatingthefront-endmatchedfilter, thefeedforwardfilter

optimizationproblemto minimizetheMSE canbecastin thefollowing form.Â optÆGøeù�úMûýü�þÿ Ç� � Á�� äNÂ\Ï��XÅ�xÁ� Õ Ý� ¿�Å�xÁ� Õ Ý� ¿�Å_Á�� äNÂ0Ï��XÅÝ�Â �]Á�� Õ ìë?Ý��¿Xë Å��xÁ�� Õ ìëÒÝ��¿Xë Å�Â�� ¿�� (4.23)

wherethe shifting and time-reversalmatrices Õ Ã� ¿ are the sameas thosedefinedin

equation(4.6). The matrices � Õ and �¿ are Á&Ì]å Ý Ì]Î<Ý ÌMèáÏ Ê Åöæ9Á.Ì]ÎNÝ ÌMèkÏ Ç Åmatricesthatperformshifting andtime-reversaloperationson thevectors ìë�Â and ë3Â ,respectively. If Ê�� Ý Ç ÏgÌ]å��Òß then,

� Õ Æ íîï�� Ñ Ó� Ñ �¬ÔÖÕ�!#" � Ñ Ó#� Ñ$�¬ÔÖÕ�!ß � Ñ %_ÔÖÕ�!#" � Ñ�Ó � Ñ$�6ÔÖÕ�!ò{óô & �¿ Æ íîîîîï ß � ¿� � Õ¯Ô-Ñ$% !#" � Ñ Ó�� Ñ$�6ÔÖÕ�!ì� � Ñ�Ó � Ñ �¬ÔÖÕ�!#" � Ñ�Ó � Ñ$�¬ÔÖÕ�!ß � ¿ Ñ %_Ô ¿� Ô ¿ !�" � Ñ�Ó � Ñ$�¬ÔÖÕ�!

ò{óóóóô (4.24)

andif Ê � Ý Ç Ï�Ì]å â?ß then,

� Õ Æíîîîîï ß � Ñ % Ô ¿� ÔÖÕ�!�" � Ñ�Ó � Ñ � ÔÖÕ�!�'� Ñ�Ó � Ñ � ÔÖÕ�!#" � Ñ�Ó � Ñ � ÔÖÕ�!ß�¿� " � Ñ�Ó � Ñ � ÔÖÕ�!

ò{óóóóô & �¿ Æ íîï ì�'� Ñ�Ó � Ñ � ÔÖÕ�!#" � Ñ�Ó � Ñ � ÔÖÕ�!ß � Ñ % ÔÖÕ�!#" � Ñ�Ó � Ñ � ÔÖÕ�!ò{óô (4.25)

87

TheoptimalFFFtapsettingof equation(4.23)isÂ optÆ)( Á#� ähÅ�xÁ# Õ Ý* ¿(Å�xÁ# Õ Ý� ¿6Å+� ä Ý,� ¿� Á�� Õ ìëDÝ��O¿�ë Å��xÁ�� Õ ìëDÝ��O¿(ë Å.- ÔÖÕð-Á#� ähÅ�xÁ# Õ Ý* ¿(Å�xÁ# Õ Ý* ¿(Å/�0� ã (4.26)

As before,thesetapscanbeunbiasedby multiplying with the reciprocalof thecursor

term. A drawbackof the approachoutlinedhereis the fact that a front-endmatched

filter is necessaryin additionto thenormalmodeandtime-reversalmodeDFEs.Hence,

this increasesthetotal complexity of thereceiver.

4.5 Summary

In thischapter, theoptimalfinite lengthMMSE-BiDFEtapswerederivedundertheideal

feedbackassumptionfor both symmetricandasymmetricchannelimpulseresponses.

Although, the MMSE-BiDFE tapsdecreasethe gap from the MFB, when compared

to the LC-BiDFE with MMSE-DFE tap settings,thesegainsdo not translateto the

SERperformancecurvesin thepresenceof decisionfeedback.The useof a modified

costfunction, incorporatingan additionterm basedon the norm of the FBF taps,was

proposedto ensurethat performancegains,even if marginal, canalsobe obtainedin

thepresenceof decisionfeedback.However, thedifficulty in appropriatelychoosingthe

weightingfactor,½, for the norm term makesthis approachimpractical. An alternate

methodincorporatinganiterative LC-BiDFE structurehasalsobeenproposed,andthe

performanceimprovementsaresimilarto thoseobtainedwith themodifiedcostfunction.

In thenext Chapter, weconsidertheextensionof theBiDFE approachfor amulti-input-

multi-output(MIMO) channelequalizationproblem.

Chapter 5

BiDFE for MIMO Channel

Equalization

In theearlierchapters,equalizationof a single-inputsingle-output(SISO)systemwas

considered.In thischapter, thefocuswill beonmultiple-inputmultiple-output(MIMO)

systems. Channelswith MIMO characteristicsoccur frequently in moderncommu-

nication systems.A MIMO systemis typically characterizedby the transmissionof

multiple-inputsignalsthroughalinear, dispersive,noisychannelandresultsin multiple-

output signalsat the receiver. The received signalsare composedof a sum of sev-

eraltransmittedsignalscorruptedby ISI, co-channelinterference(alsoknown asmulti-

accessinterference,i.e.,MAI) andnoise.Examplesof MIMO channelsincludeTDMA

digital cellularsystemswith multipletransmitterandreceiverantennas,wide-bandasyn-

chronousCDMA systems,duallypolarizedradiochannelsandmagneticrecordingchan-

nels.Evenin asingleusercommunicationsystem,thereexist scenarioswhereaMIMO

modelingof the systemprovesto be useful. Onesuchexampleis a pulseamplitude

modulated(PAM) cyclostationarysequencein thepresenceof ISI [30].

Recentantennatechnologyadvanceshavemadeit possibleto supportmultipletrans-

88

89

mit andreceive antennasat theterminal[38, 90]. Particularlyfor largesizedatatermi-

nalssuchaslaptops,it is possibleto have up to four integratedantennaswith sufficient

spacingso that the correlationof the transmitted/received signalsacrossthe antennas

is small. Phasedarrayantennasandwidely-spaceddiversityantennasaretwo waysto

usemultipleantennasto provide improvedspectralefficiency. In thefirst caseanarrow

beamdirectedtowardsthe terminal is formedby transmittingthe samesignal,appro-

priatelyweightedin amplitudeandphase,from eachantennaelement,while in thelater

casedifferentsignalsaretransmittedfrom thedifferentantennasin orderto takeadvan-

tageof scatteringthroughspace-timecoding. Space-Time Coding(STC) canbe used

in differentways: someusethe additionalantennaelementsto provide diversity gain

(e.g. [84]), while other techniques,suchasBLAST (Bell LabsLayeredSpace-Time)

[39], achieve higherdataratesby transmittingindependentdatastreamsthrougheach

transmitantennaelement.

TheBLAST schemewasinitially proposedin [37] to increasethespectralefficiency

in aflat fadingwirelessenvironment.If thenumberof receiveantennasis greaterthanor

equalto thenumberof transmitantennas,it is possibleto separatethesignalsfrom the

differenttransmitantennas.Hence,onecanpotentiallysendindependentinformation

from thedifferenttransmitantennasandtherebyincreasethedatarateof the link. The

extensionof suchaschemeto afrequency- selectivefadingenvironmentis illustratedin

Figure5.1.

5.1 SystemModel

A MIMO channelmodelwith 1 inputsand � outputsis considered.An independent

datastreamis assumedto betransmittedthrougheachinputof theMIMO channel.The

90

. . .

. . .

. . .

Stream 1

Stream 2

Stream M

M - Tx Antennas P - Rx Antennas

M,p

2,p

1,p

c

c

c

Figure5.1: BLAST schemefor amulti-elementantennasystem

receivedsampledsignalvectorat the 2 -th outputof thechannelduringthe ÷ -th symbol

periodcanbeexpressedby thediscrete-timemodel,3+4 Á.÷�Å Æ 5×6 Ù Õ Ñ Ó ÔÖÕ× Ø(Ù�Ú\Û 687 4 ÁUÜ�Å:9 6 Á�÷¨Ï�Ü�ÅHÝ�õ 4 Á�÷HÅ (5.1)

where ¾ 687 4<; ( Û 687 4 Á&ß>Å6Ã ã^ã^ã Ã Û 687 4 Á&Ì]ÎhÏ Ç Å- � is the impulseresponseof the channel

betweenthe = -th input andthe 2 -th output, 9 6 Á.÷HÅ is the transmittedsymbolfrom the= -th input and õ 4 Á�÷HÅ is the additive noiseat the 2 -th output. Eachof the channel

impulseresponses¾ 687 4 is assumedto be time-invariant(assumption2.1), FIR with Ì]Îtaps(assumption2.2)andknown at thereceiver (assumption2.4). Thenoisesequencesõ 4 Á�÷HÅ areassumedto bewhite,uncorrelatedwith eachotherandthesourcesequences,

andof variance� ¿� (assumption2.3).

91

5.1.1 Multichannel Matched Filter Bound

Unlike theSISOsystem,eachtransmittedinformationstreamencountersasingle-input

multiple-output(SIMO) channel.Hence,theMFB for eachstreamhasto beappropri-

atelymodifiedto reflectthis. If themultiple-outputchannelfor eachstreamis known,

thena spatio-temporalmatchedfiltering operationcanbe performedat the receiver to

obtaina single-channelequivalent. Sucha multichannelMFB wasproposedby Slock

anddeCarvalhoin [82]. For user> , theMFB is givenby,?A@mfb

Æ � ¿BDCFE4 Ù Õ C Ñ Ó ÔÖÕØXÙ�ÚHG Û @ 7 4 ÁUÜ�Å G ¿� ¿� ã (5.2)

As in theSISOcase,theMFB canbeattainedin theMIMO scenariounderthefollowing

conditions,

1. Whenonly onesymbol,pertainingto the stream> , is transmittedwith no sym-

bols transmittedfor all otherstreams.This resultsin no ISI andno CCI andthe

matchedfilter receiver is theoptimaldetectorfor stream> .2. If all transmittedsymbolsof all streamsin thepacket,exceptthesymbolof interest

of stream> , areknown. In this case,the ISI andCCI componentsaffecting the

symbol of interestat the output of the matchedfilter receiver can be canceled

perfectly.

5.2 MIMO Equalization

As in thecaseof a SISOsystem,theMIMO channelcanbeequalizedusingsequence

estimationalgorithms,suchas the MLSE [34]. Although sequenceestimatorshave

superiorperformancewhen comparedto symbol-by-symbolestimators,they have a

high computationalcomplexity. TheMIMO MLSE schemerequires

G IJG 5�Ñ Ó statesfor

92

the detectionof a sourceselectedfrom an alphabetset

Itransmittedusing 1 anten-

nasthrougha channelwith a delayspreadof Ì]Î symbols. For instance,in the third-

generationwirelessTDMA proposal[40] 8-PSKmodulationis used. A typical urban

EDGE channel(including the transmitpulseshape)hasa delay spreadof at least4

symbols. Even with two transmitantennas,the numberof MLSE statesrequiredisK L Æ ÇNM$O$O$O>Ê Ç�M .Thelow complexity symbol-by-symboldetectorthatis consideredhereis theMIMO

DFE [30, 22, 2]. In [30], Duel HallenderivedtheMMSE-DFEfor theMIMO channel

equalizationproblem,while in [2], theMIMO extensionto theMMSE-DFEwasderived

undera finite lengthconstraint.TheMIMO DFE employs a feedforwardfilter P anda

purely causalfeedbackfilter Q . The block diagramof a MIMO DFE is illustratedin

Figure5.2. Typically, the MIMO feedforward filter suppressesthe pre-cursorISI and

CCI (co-channelinterference),while theMIMO feedbackfilter cancelsthepost-cursor

ISI andthecursor/post-cursorCCI usingthesymboldecisions.As in thecaseof aSISO

DFE, the MIMO DFE suffers from error propagationanda gap from the MFB. One

aspectthatmakestheMIMO DFE differentfrom theSISODFE is themany variations

thatarepossiblein the realizationof theMIMO DFE. Thesevariationsarisefrom the

differentwaysin which thetaskof cancelingthepost-cursorCCI andthecursorCCI of

thepreviouslydetectedstreamsis partitionedbetweentheFFFandtheFBF.

Scenario1

In this case,theFBF is usedonly to cancelthe post-cursorISI of the user(stream)of

interest.TheCCI from theinterferingusersareto besuppressedby theFFF. Thisstruc-

tureis naturalwhenonly oneof thestreamsis to bedetectedatthereceiver, for example,

detectionat themobilein downlink transmissionfor awidebandCDMA system.

93

FeedbackFilter

B (D)

. . .

. . .

. . .

. . .

. . .

Q

Q

ForwardFilter

F (D)r (n)N

r (n)1

s (n)N

^

1s (n)^y (n)1

y (n)N

Figure 5.2: Structureof aMIMO DFE

Scenario2

This assumesthat all the usersare equalizedat the receiver and so the FBF hasac-

cessto thepost-cursorCCI from all interferingusersin additionto thepost-cursorISI.

Hence,the post-cursorCCI componentsfrom all detectedstreamsarecanceledusing

theFBF. TheFFFis only usedto suppressthepre-cursorISI, pre-cursorandcursorCCI

components.

Scenario3

In scenario2, if we assumethat theusersaredetectedin a certainorder, thentheFBF

canbepotentiallymodifiedto cancelnot only thepost-cursorCCI, but alsothecursor

CCI of theinterferingusersthathavebeendetectedprior to theuserof interest.In such

a scheme,while cancelingthecontribution of thecursor/post-cursorISI of thedetected

users,thecorrespondingfeedbackfilter sectionwill have anadditionaltapwhencom-

paredto the feedbackfilter sectionsof the usersthat have not beendetectedyet. The

MMSE, underthe ideal feedbackassumption,improvesprogressively for eachof the

94

structuresdescribedin scenarios1-3 (see[2] for proof). Intuitively, whencomparedto

scenario1, asscenario3 assumesthat moreof the interferingsymbols(both ISI and

CCI) areknown, it hasa lesserpenaltyfrom theMFB anda betterperformance.How-

ever, theseverity of errorpropagation,i.e., whenthedecision-error-freeassumptionis

violated,alsoprogressively increasesfrom scenario1 to scenario3.

5.3 BiDFE Extensionto MIMO DFE

In thissectiontheextensionof theideaof abidirectionalDFEto MIMO channelequal-

ization is considered.The MIMO DFE structureof scenario3 is consideredfor this

extension. The reasonbeing that the MIMO DFE structureof scenario3 affords the

possibilityof notonly time-reversalbut alsouserre-ordering.In otherwords,onecould

usetwo MIMO DFE structures,eachwith a differentorderingfor thedetectionof the

usersandcombinethe two MIMO DFE outputsto improve performance.The ideaof

employing userre-orderingto improveperformancewasproposedby BarriacandMad-

how in [15] for a successive interferencecancellationbasedmultiuserdetectorfor a

CDMA system.The MIMO systemmodelconsideredin [15] wasassumedto be free

from time dispersion,namelyflat fading channels.Here, the useof time-reversalin

conjunctionwith userre-orderingis proposedfor channelswith timedispersion.

ConsideraMIMO BiDFE structuresimilar to theSISOBiDFE illustratedin Figure

2.9.Thevectorof receivedsignalsareprocessedusinganormalmodeMIMO DFE.Let

ord Õ ÆR( 9 Õ Á�÷HÅ�ÃS9^¿�Á�÷HÅ�Ã ã^ã_ã ÃT9 5 Á.÷�Å.- bethevectorthatdenotestheorderof detectionfor

the normalmodeMIMO DFE. The received signalsarenow block time-reversedand

equalizedusinga time-reversalmodeMIMO DFE. Further, the orderof detectionfor

time-reversalmodeMIMO DFE, ord ¿ is the reversedversionof ord Õ . Theoutputsof

95

thetwo modesarecombinedusingaweightedlinearcombination,namelyUV Á�÷HÅ ÆXW UV'Y Á.÷�ÅOÝQÁ � Ï W Å UV�Z Á.÷�Å (5.3)

whereW is an 1�æ[1 diagonalweightingmatrix,UV�Y Á�÷HÅ is thevectorof symbolestimates

from the normal modeMIMO DFE for the ÷ -th time instant,UV�Z Á.÷�Å is the vector of

symbolestimatesfrom thetime-reversalmodeMIMO DFEandUV Á.÷�Å is theoutputof the

linearcombiningblock. Let À Õ Ã À ¿·Ã ã^ã_ã Ã À 5 bethediagonalentriesof thematrix W .

Then,theoptimalchoicefor À @ is givenby,À opt@ Æ � ¿@ 7 ¿ Ï�\ @ � @ 7 Õ � @ 7 ¿� ¿@ 7 Õ Ý,� ¿@ 7 ¿ Ï Ê \ @ � @ 7 Õ � @ 7 ¿ (5.4)

where� ¿@ 7 Õ , � ¿@ 7 ¿ and \ @ aredefinedasin Section2.5,but for user > .The rationalefor theproposedstructureis asfollows. While detectingthe symbol9 @ Á�÷HÅ , the normal modeMIMO DFE assumesthat the interfering co-channelsignals] 9 Õ Á.÷�Å6ÃS9^¿�Á.÷�Å6Ã ã^ã^ã ÃT9 @ ÔÖÕ Á�÷HÅ_^ transmittedat thesametimeinstant,÷ areknown in addi-

tion to the“past” interferingsymbols(bothISI andCCI). Thetime-reversalmodeDFE,

on theotherhand,assumesknowledgeof] 9 5 Á.÷�Å6Ã`9 5 ÔÖÕ Á�÷HÅ�Ã ã^ã_ã ÃS9 @ � Õ Á.÷�Å�^ in addition

to the“future” interferingsymbolswhile detecting9 @ Á.÷�Å . In effect,all interferingsym-

bolsareassumedto beknown andthiswill resultin aperformanceimprovement.Recall

that,theknowledgeof all theinterferingsymbolsis anecessaryconditionto achievethe

MFB.

5.4 Numerical Results

In this subsection,theMSE andSERperformanceimprovementsprovidedby thepro-

posedMIMO LC-BiDFE structureover theconventionalDFE structurewill bedemon-

stratednumerically1. Additional simulationexamplesareprovidedin AppendixA. We


96

3 4 5 6 7 8 9 10 11 12

−14

−12

−10

−8

−6

−4

SNR in dB

MS

E i

n d

B

Normal Mode DFETime−reversal Mode DFELC−BiDFE

Figure 5.3: MSE performanceof user1 for theMIMO testchannelC with acb�d egf�b�dconsidera synthetic Ê æ Ê MIMO channelwith Ì]Î Æ Ê taps. The MIMO impulse

responseof thechannelC is givenby,¾ Y 7 Y ÆR( ß ãih¤Ê Kkj ÃJÏ�ß ãilkO Ç � -&Ã¤¾ Y 7 Z Æ)( Ï\ß ãiM$M$h ßfÃ©ß ãmO � l$l -&Ã¾ Z 7 Y ÆR( ß ãih¤Ê K$j ÃMß ãilkO Ç � -&Ã¤¾ Z 7 Z Æ)( ß ãnM M$h ßfÃMß ãnO � l l - ã (5.5)

The lengthof theFFF andFBF for the normalmodeMIMO DFE were, Ì©è Õ Æ �andÌpo Õ ÆAÇ , respectively. For thetime-reversalmodeMIMO DFE,achoiceof ÌMè�¿ Æ KandÌpo ¿ ÆAÇ wasmadefor thefilter lengths.UnbiasedMMSE coefficientswereusedfor the

MIMO DFEsfor boththenormalmodeandtime-reversalmodeoperation.Thesource

symbolswereselectedfrom a BPSK sourceconstellation.The orderof detectionfor

normalmodeDFE was,ord Õ Æq( 9 Õ Á�÷HÅ�Ãr9^¿�Á.÷�Å.- while for thetime-reversalmodeDFE,

ord ¿ Æs( 9^¿�Á.÷�Å6Ãt9 Õ Á.÷HÅ- . The MSE performancecurvesfor a normalmodeDFE, time-

reversalmodeDFE andtheLC-BiDFE arecomparedin Figures5.3and5.4 for thetwo

users9 Õ Á.÷HÅ and 9·¿�Á�÷HÅ , respectively. Ideal feedbackwasinvoked to computethe MSE

97

3 4 5 6 7 8 9 10 11 12

−14

−12

−10

−8

−6

−4

SNR in dB

MS

E i

n d

B

Normal Mode DFETime−reversal Mode DFELC−BiDFE

Figure 5.4: MSE performanceof user2 for theMIMO testchannelC with acb�d egf�b�dvalues. In Figure5.3, the time-reversalmodeDFE hasa lower MSE whencompared

to thenormalmodeDFE. This is dueto the fact that, in the time-reversalmodeDFE,

user1 is detectedafter user2 andhasadditionalknowledgeof the interferingcursor

symbolof user2. TheLC-BiDFE, however, hasanadditionalgainof about0.5dB over

thetime-reversalmodeDFE.Similarly, for user2, theBiDFE hassuperiorperformance

whencomparedto eitherthe normalmodeor the time-reversalmodeDFEsandhasa

gainof at least0.5dB overthebestperformingconstituentDFE(in thiscase,thenormal

modeDFE).

Thesymbolerror rate(SER)curvesfor thenormalmodeDFE, time-reversalmode

DFE andtheLC-BiDFE areshown in Figures5.5 and5.6 for the two users9 Õ Á.÷HÅ and9^¿�Á.÷�Å , respectively, andthetestchannelC of equation(5.5).Thesesimulationresultsdo

not invoke the assumption(2.5) andhenceincorporatethe effect of error propagation,

i.e., decisionfeedbackwasassumed.For user1, the soft MIMO LC-BiDFE offers a

98

3 4 5 6 7 8 9 10 11 1210

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

Normal Mode DFETime−reversal Mode DFELC−BiDFE : (Hard)LC−BiDFE : (Soft)

Figure5.5: SERperformancecurvesof user1 for theMIMO testchannelCwith aub�d evfwb�dperformanceimprovementof nearly1 dB, at an SERof Ç ß Ô�x , over the time-reversal

modeMIMO DFE, while for user2 the gain is about0.6 dB for the soft MIMO LC-

BiDFE overthenormalmodeMIMO DFE.Theseresultsdemonstratetheviability of the

LC-BiDFE in providing performanceimprovementsin theMIMO channelequalization

problem.

5.5 Summary

In thischapter, theideaof usingtime-reversalto improvetheperformanceof aDFEhas

beenextendedto MIMO channelequalization.In addition,the MIMO DFE structure

consideredin this chapteraffordsthepossibilityof improvedperformanceby rearrang-

ing theorderof detectionof thevariouscochannelusers.Numericalresultsfor asample

channeldemonstratingachievableperformanceimprovementshasbeenprovided.It has

99

3 4 5 6 7 8 9 10 11 1210

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

Normal Mode DFETime−reversal Mode DFELC−BiDFE : (Hard)LC−BiDFE : (Soft)

Figure5.6: SERperformancecurvesof user2 for theMIMO testchannelCwith aub�d evfwb�dbeenfurtherassumedthattheMIMO channelimpulseresponseis perfectlyknown atthe

receiver. In practicethis conditionis often violatedandthe channelimpulseresponse

needsto be estimated.Most packet basedcommunicationsystemsprovide a training

segmentin eachpacket to facilitatechannelestimation.During this phase,a sequence

known to thereceiver is transmitted.Thedesignof sucha trainingsequencefor MIMO

channelestimationwill beconsideredin thenext chapter.

Chapter 6

Training SequenceDesignfor MIMO

ChannelEstimation

In Chapter5, the extensionof a BiDFE to a multi-input multi-output (MIMO) chan-

nel equalizationproblemwasconsidered.Knowledgeof thechannelimpulseresponse

from eachtransmitantennato any receive antennawasassumedto be availableat the

receiver. However, in practicethesechannelimpulseresponseshave to be estimated

at the receiver. Training-basedestimation,semi-blindestimationandblind estimation

arethreetypesof estimatorsthatcanbepotentiallyusedto estimatetheMIMO channel

impulseresponseat the receiver. Training-basedestimatorsassumethe presenceof a

trainingsequenceandrely solelyon theknowledgeof thesesymbols.Blind estimation

techniques,on the otherhand,rely on qualitative informationon the transmittedsig-

nals. Semi-blindtechniquesexploit the knowledgeof the training symbolsaswell as

thequalitative informationaboutthedatasymbols.

A packet basedsystem,for exampleGSM, typically containsa training sequence

andhencetraining-basedestimatorsaresuitable.Although,semi-blindchannelestima-

torsmayoffer a betterperformanceoverpurelytraining-basedchannelestimators[26],

100

101

purely training-basedchannelestimatorsarecommonin practice. The quality of the

training-basedchannelestimatedependson theparticularchoiceof trainingsequence.

It hasbeenknown that trainingsequenceswith impulse-like auto-correlationfunctions

aresuitablefor thesingleantennachannelestimationproblemandthesearchfor such

sequenceshasreceiveda greatdealof attentionin thepast[68, 18]. Theoptimality cri-

terion for training sequencesfor least-squareschannelestimationhasbeenconsidered

in [24] andsomeoptimal trainingsequenceshave beenprovided,but from thecontext

of asingleantennasystem.On theotherhand,near-optimaltrainingsequences,derived

usingdiscreteFouriertransformtechniques,aretabulatedin [86].

The designof optimal training sequencesfor least-squareschannelestimationin

multiple antennasystemswasconsideredandappropriateoptimality criterionwerede-

rived by Balakrishnanet al. in [13, 89]. Theseresultsare presentedin this chapter.

In section6.1, the systemmodel is describedandthe least-squareschannelestimator

is reviewed. The optimality criterion for training sequencedesignis derived and the

designtradeoffs in thechoiceof trainingsequencelengtharediscussedin section6.3.

Section6.4 describesa few heuristicmethodsfor the searchfor near-optimal training

sequences.A few near-optimalbinarysequencesobtainedusingthesesearchmethods,

for employmentin anEDGE(EnhancedDatafor GSMEvolution)system,aretabulated

in AppendixB. Thespecialcaseof designingtrainingsequencesfor a delay-diversity

schemeis discussedin section6.5andsection6.6concludes.

6.1 SystemModel

A systemwith 1 transmitterantennasand � receiver antennasis considered.Without

lossof generality, eachtransmitantennacanbeassumedto transmitdifferentinforma-

102

tion symbols.Thereceivedsampledsignalvectorat the 2 -th receiveantennaduringthe÷ -th symbolperiodcanbeexpressedby thediscrete-timemodel,3y4 Á�÷HÅ Æ 5×6 Ù Õ Ñ-ÔÖÕ× Ø(Ù�Ú Û 6z7 4 Á&Ü�Å+9 6 Á.÷¨Ï�Ü�ÅHÝ�õ 4 Á.÷�Å (6.1)

where ¾ 6z7 4�; ( Û 6z7 4 Á.ß¤Å6Ã ã^ã_ã Ã Û 687 4 Á.ÌQÏ Ç Å- � is the impulseresponseof the channel

betweenthe = -th transmitantennaandthe 2 -th receive antenna,9 6 Á.÷�Å is thetransmit-

tedsymbolfrom the = -th transmitantennaand õ 4 Á.÷HÅ is theadditive noiseat the 2 -th

receive antenna.Eachof the channelimpulseresponsesis assumedto be of length Ìtaps.During thetrainingphase,differenttrainingsequencesaretransmittedfrom each

of the transmitantennas.The training sequencesareassumedto be { symbolslong,

andthechannelimpulseresponseis estimatedat the receiver basedon theknowledge

of the training symbols. Here,the channelimpulseresponses¾ 687 4 refer to the digital

equivalentof theconvolutionof thetransmitpulseshapingfilter responsewith thephys-

ical multipathchannel.Furthermore,thechannelis assumedto betime-invariantfor the

durationof thepacket.

6.2 MIMO ChannelEstimation

Transmissionof space-timecodedsignalsrequirescoherentdemodulationat the re-

ceiver. Hence,thereceiver mustbecapableof estimatingthechannelfrom eachtrans-

mit antenna.CurrentEDGE standardsallow the transmissionof a training sequence

of length26 symbolsin eachburst for a singletransmitantennasystem,which allows

thereceiver to estimatethechannelfor equalization.For estimatingmultiple channels

from themultiple transmitantennas,weproposetheuseof differenttrainingsequences,

onefor eachtransmitantenna,transmittedsimultaneouslyduringeachburst. Thesese-

quenceshaveto bedesignedwith goodauto-correlationandcross-correlationproperties

103

to enableaccuratechannelestimationat thereceiver.

Thevectorof observationsat the 2 -th receiveantenna,duringthetrainingphasecan

bewritten in matrix form as, | 4\Æ ]¾ 4 Ý�} 4 (6.2)

where | 4~;�(�3+4 Á�{ Å6Ã ã^ã^ã Ã 3y4 Á.Ì©Å.-�� (6.3)

is thevectorof observations,¾ 4�;�( Û Õ 7 4 Á.ß¤Å6Ã ã^ã_ã Ã Û Õ 7 4 Á.Ì Ï Ç Å�Ã ã^ã^ã Ã Û 5 7 4 Á&ß¤Å�Ã ã_ã^ã Ã Û 5 7 4 Á.Ì Ï Ç Å-�� (6.4)

is thestackedvectorof channelimpulseresponses,} 4~;�( õ 4 Á�{ Å6Ã ã^ã^ã ÃMõ 4 Á&Ì©Å.-�� (6.5)

is thenoisevectorand is an Á�{ Ï#Ì Ý Ç Åhæ�1GÌ block-Toeplitzmatrix consistingof

thetrainingsymbols,

; íîîîîï 9 Õ Á�{ Å ð·ð·ð�9 Õ Á�{ Ï�Ì Ý Ç Åñð·ð·ðH9 5 Á�{ Å ð·ð·ð�9 5 Á�{ ÏgÌ Ý Ç Å...

. . .... ð·ð·ð ...

. . ....9 Õ Á.Ì<Å ð·ð·ð 9 Õ Á Ç Å ð·ð·ð�9 5 Á&Ì©Å ð·ð·ð 9 5 Á Ç Å

ò{óóóóô ã (6.6)

Thestackedimpulseresponsevector ¾ 4 is estimatedat thereceiver for eachof the �receiverantennas.A least-squares(LS) channelestimatoris consideredfor thispurpose.

An LS channelestimatorminimizesthesquarederrorbetweenthereceivedsignalvector

andthechannelestimate-basedreconstructedsignal.TheLS estimateis givenby,�¾ LS4 ;²øeù�ú©ûkü�þ� � | 4 Ï,]¾ � ¿Æ�� p�T8� ÔÖÕ p� | 4¤ã (6.7)

104

In equation(6.7) theauto-correlationmatrix � is assumedto be invertible. The

LS estimatecanbeexpressedin termsof thechannelimpulseresponseas,�¾ LS4 Æ ¾ 4 Ý � � p� ÔÖÕ � } 4¤ã (6.8)

If the additive noise is zero-meanand uncorrelatedto the training sequence,the LS

channelestimateis unbiased.TheLS channelestimationerroris,

E � � ¾ 4 Ï �¾ LS4 � ¿+� Æ tr � � � p� ÔÖÕ � E ��} 4 } �4 � � � 8� ÔÖÕN� ã (6.9)

Under the assumptionthat the noiseprocessis white with a varianceof � ¿� , the LS

estimationerrorsimplifiesto

E � � ¾ 4 Ï �¾ LS4 � ¿ � Æ tr � � ¿� � � � ÔÖÕN� ã (6.10)

6.3 Training SequenceDesign

Fromequation(6.10),it is evidentthattheLS estimationerrordependsonthechoiceof

trainingsequence.Hence,thetrainingsequence canbeoptimizedto minimizetheLS

estimationerror.

Theorem 6.1 TheminimumLSerror is obtainedif andonly if � Æ Á�{ Ï�Ì Ý Ç Å/� ¿� � 5�Ñ Ã (6.11)

where � ¿� is themaximumpermissiblevariancefor thetrainingsymbols.

Proof: Theargumentsoutlinedherearesimilar to thoseprovidedin [85]. Thetraining

sequenceis optimizedto minimize theLS channelestimationerrorof equation(6.10).

This is equivalentto minimizingtr � � � � ÔÖÕ � with respectto thetrainingsequence .

Let � Õ ÃT�Ö¿·Ã ã_ã^ã ÃT� 5�Ñ betheeigenvaluesof theauto-correlationmatrix � . Sincethe

105

Hermitianmatrix � is assumedto be invertible,all theeigenvaluesaregreaterthan

zero.Let usnow definethevectors� Æ��.� � Õ Ã ã_ã^ã Ã � � 5�Ñ � � & � � Æ � Ç� � Õ Ã ã^ã_ã Ã Ç� � 5�Ñ � � ã (6.12)

By theCauchy-Schwartzinequality,� � Ã�� ¿ à � � Ã�� Ã_� �� (6.13)

where� Ã � denotesavectordotproduct.Hence,Á#1GÌ©Å ¿ à¡ 5�Ñ× ØXÙ Õ �

Ø£¢ 5�Ñ× ØXÙ Õ Ç� Ø¢ ã (6.14)

Since,thesumof eigenvaluesis thetraceof amatrix,theaboveinequalitycanberewrit-

tenas,

tr � � � � ÔÖÕN� � Á#1GÌ©Å ¿tr] � S^ ã (6.15)

Equality occursin equation(6.15) only when � Æ À � for someconstantÀ , i.e, � mustbe proportionalto the identity matrix. Furthermore, the tr] � S^ canbe

maximizedby choosingthe maximumenergy symbolsfrom the sourceconstellation,

namelythe farthestpointsfrom the origin. Let � ¿� be the varianceof thesemaximum

energy sourcesymbolschosenfor training.Then,theminimumvalueof theestimation

erroris ûkü�þ E � � ¾ 4 Ï �¾ LS4 � ¿ � Æ 1GÌp� ¿�Á�{ Ï�Ì�Ý Ç Å/� ¿� ã (6.16)

Theabove resultis analogousto thesingletransmitantennascenarioandis equiva-

lent to choosingthetrainingsequencesto betemporallywhiteandspatiallyuncorrelated

(i.e., acrosstransmitantennas).The LS channelestimatorturnsout to be the best(in

termsof having theminimum mean-squarederror)amongall unbiasedestimatorsand

it is the mostefficient in the sensethat it achievesthe Cramer-Raolower bound[51].

106

Hence,the useof an LS estimatoris consideredto derive the optimality criterion for

trainingsequencedesign.Although,theoptimality conditionfor thetrainingsequences

have beenderived assumingthat the channelis time-invariantduring the durationof

theslot, they would still be valid if thechannelis slowly time varying. If thechannel

timevariationsarerapid,channeltrackingis essentialandpurelytrainingbasedchannel

estimatorsareno longersuitable.

6.3.1 Training SequenceLength Design

A critical parameterin training sequencedesignis the length of the sequence.The

training sequenceneedsto be long enoughfor the channelto be identified. A longer

training sequencehasthe addedadvantageof reducingthe channelestimationerror.

However, anincreasein trainingsequencelengthresultsin adecreasein theusefuldata

rateof the transmission.The designtradeoffs associatedwith the choiceof training

sequencelengtharediscussedin this subsection.

Identifiability

For thechannelimpulseresponseto beidentifiable,theauto-correlationmatrix ¤p¥T¤ of

equation(6.7) hasto be invertible. Hence,thetrainingsequencematrix ¤ hasto beof

full columnrank.Thenecessaryconditionfor ¤ to befull columnrankis,¦�§©¨«ªJ¬®N¯`°®±Xª³²(6.17)

Lossdue to Channel Estimation

Any errorresultingfrom channelestimationcanbeincorporatedinto thenoiseprocess

and can be quantifiedas a loss in effective SNR. During datatransmissionwe have

107

(recallequation(6.1)),´yµ ¦�¶·¯8¸ ¹º»½¼�¾�¿0À ¾º Á ¼ÃÂÅÄ »zÆ µ ¦�ÇÃ¯+È » ¦�¶É¨«ÇÃ¯D¬«Ê µ ¦�¶·¯ Ë (6.18)¸ ¹º»½¼�¾�¿0À ¾º Á ¼ÃÂ Ä LS»zÆ µ ¦�ÇÃ¯+È » ¦�¶É¨«ÇÃ¯D¬ ¹º»½¼�¾�¿0À ¾º Á ¼ÃÂTÌ Ä »zÆ µ ¦Í¶Å¯Î¨ Ä LS»zÆ µ ¦�ÇÃ¯NÏzÈ » ¦�¶Ð¨,ÇÑ¯Å¬,Ê µ ¦Í¶Å¯Ò ÓÕÔ ÖÊ~×µ ¦�¶·¯wheretheterm

Ê ×µ ¦�¶Å¯ denotestheequivalentnoise.However, thestatisticsof thenoise

processis no longerGaussian.However, simulations(see[89]) show that it is possible

to treat the noiseterm asGaussianwith an equivalentnoisevariance. Let us assume

that theequivalentnoiseÊ ×µ ¦Í¶Å¯ is uncorrelatedwith thesourcesymbolsandthesource

symbolsto be i.i.d with a varianceØ·ÙÚ . The equivalentnoisevariance(or the MSE) is

thengivenby,

MSE¸

E Û�Ü Ê~×µ ¦Í¶Å¯ Ü Ù+ÝÞ¸E ß à µ ¨Fáà LSµ ß Ù Ø ÙÚ ¬ Ø Ùâ (6.19)ã¸ Ø Ùâåä T¬ ±Fª Ø·ÙÚ¦�§æ¨�ªJ¬®'¯ Ø Ùç½è ²

In equation(6.19),Þ¸

is obtainedby usingthei.i.d assumptiononthesourcesymbolsand

thevectorrepresentationof thestackedchannelimpulseresponse,while stepã¸

results

from equation(6.16).If thesourcesymbolswereto bechosenfrom aconstantmodulus

type constellation(for example8-PSK),then the sourcevariancewill be the sameas

thevarianceof thetrainingsymbolsandequation(6.19)canbefurthersimplified. The

increasein MSE due to the channelestimationerror can be interpretedas a loss in

effectiveSNRat thereceiver.

108

Lossin Throughput

The throughputof a systemis the productof the datarateandthe probability of suc-

cessfultransmissionof a packet. Clearly, from equation(6.19),the longerthe training

sequence,thelesserthelossin effectiveSNR.A smallerchannelestimationerrorwould

resultin a decreasein packet error rate. However, an increasein thetrainingsequence

lengthreducesthe numberof informationbits that canbe transmittedin a packet and

hencethe datarate. An EDGE packet consistsof 26 training symbolsand118 data

symbols.Hence,any additionaltrainingsymbolscomeat thecostof thedatasymbols.

It is clearfrom the above discussionthat the throughputdependson the lengthof the

training sequence.Hence,a goodcriterion for the designof training lengthwould be

to maximizetheachievablethroughput.Theachievablethroughputfor a givenchoice

of traininglengthcanbecalculatedif theprobabilitydistribution function(PDF)of the

SINR, theachievabledatarateandthecorrespondingblock error rateareknown. The

averagethroughputis thencalculatedas,

AverageThroughputêé�ëë ì ¦îíï¯�ðk³¨�ñ × ¦ÍívË_§òË_ªz¯�ógôõ¦�§ö¯:÷$í

(6.20)

whereì ¦îí·¯ is thePDFof theSINR,ñ × ¦ÍívË_§ö¯

andôõ¦�§ö¯

aretheblockerrorratesandthe

datarates,respectively, asafunctionof theSINR,thetraininglength§

andthechannel

impulseresponselengthparameterª

.

6.4 Search for goodTraining Sequences

Oncethetrainingsequencelengthhasbeendecided,it becomesessentialto searchfor

training sequenceswith goodproperties.Equation(6.11)specifiesthe optimality cri-

terion for thesearchof suchsequences.Thesequenceshave to beof simplealphabets

in orderto guaranteelow complexity realization.Two typesof sequencesarecommon:

109

aperiodicandperiodicsequences.While aperiodicsequencesexist for many lengths,pe-

riodic onesaremuchharderto find. However, dueto anexpansiontheorem[79], short

periodicsequencescanbeconcatenatedto very largesequencespreservingtheirorthog-

onal properties.A list of known periodicsequencesof simplealphabetsaretabulated

in [13]. Originally appliedto singleantennasystems,the periodicsequencescanbe

usedfor multipleantennasystems.Constructionof multipleantennatrainingsequences

from theperiodicsequenceis describedlater in subsection6.4.3. TheQPSKsequence

of lengthªw¸�Nø

wasproposedto extendexisting OFDM systemsto four transmitand

receiveantennas[71].

It is possiblethatoptimumtrainingsequencesmaynot exist for a particularchoice

of training lengthandchanneldelayspread.In thatcase,trainingsequenceswith near

optimalpropertiescanbesearchedfor. A few heuristicmethodsfor thesearchof such

sequencesarediscussedin this section.A few near-optimalbinary trainingsequences,

suitablefor a multi-antennaEDGE system,are listed in AppendixB. The searchfor

thesesequencesarebasedon themethodsdescribedin thissection.

6.4.1 Full search

Near-optimal sequencescanbe obtainedby searchingover all possiblesequencesand

choosingthosewhich have the minimum valueof trðù¦ ¤p¥T¤ ¯ À ¾ ó . However, the search

hasto bedoneover Ü úJÜ ¹üû sequences,where Ü úJÜ is thenumberof pointsin thesource

constellation.Thissearchis computationallyprohibitive. Hence,heuristicmethodsthat

searchovera reducedsetof sequencesareof specialinterest.

110

6.4.2 RandomSearch

From equation(6.11) it is clear that near-optimal sequencesshouldhave good auto-

correlationandcross-correlationproperties,i.e., smallnon-peakauto-correlationterms

over a window of sizeª�¨�

on either side of the peak location and small cross-

correlationtermsfor a window of length ý ª«¨X. To begin with, sequenceswith good

auto-correlationpropertiescanbe determinedby searchingover all the Ü úJÜ û possible

sequences.The numberof suchsequencescanbe expectedto be muchsmallerthanÜ úJÜ ¹üû . This is followedby asearchfor±

sequenceswith goodcross-correlationprop-

ertiesfrom this reducedsetof sequences.

6.4.3 Cyclic Shift Search

Considerthesequenceþ ¾ ¸©ÿ È0¦/'¯�Ë ²Õ²�²ÕË~È0¦�§ � ¯��of length

§ �, where

§ � ¸H§�¨�ª�¬F.

The sequencesð þ Ù Ë,²�²�²£Ë þ ¹ ó

are now constructedby cyclic-shifts of the sequenceþ ¾ . For example,the sequenceþ Á � ¾ ¸�ÿiÈ0¦�Ç��ü¬ '¯�Ë ²�²�²£Ë È0¦�§ � ¯ Ë È0¦/'¯�Ë ²�²�²ÕËõÈ0¦#Ç��¯��is

obtainedby a cyclic-shift ofÇ��

of the sequenceþ ¾ , where�ö¸ � û�¹

. New sequencesð�� ¾ Ë[²�²�²£Ë � ¹ óareconstructedby addingacyclic-prefixof length

ªt¨òto thesequencesð þ ¾ Ë��ÅË þ ¹ ó

. For example,� ¾ ¸�ÿ È�¦�§ � ¨*ª ¬ ý ¯�Ë ²Õ²�²ÕË È0¦�§ � ¯ Ë~È0¦/'¯�Ë ²Õ²�²ÕË È0¦�§ � ¯��

is

onesuchsequencederivedfrom theoriginal sequence.Notethatthenew sequences� Á

areof length§

.

If the sequenceþ ¾ hasa cyclic auto-correlationfunction with zerooff-peak terms

andif�ò°�ª

, thenit is easyto seethatequation(6.11)will besatisfiedfor thechoice

of trainingsequences� ¾ Ër²�²Õ²ÕË�� ¹ . However, whensearchingfor near-optimal training

sequencestherestrictionof zerooff-peaktermsfor thecyclic auto-correlationfunction

canbe relaxed andsmall off-peakcyclic auto-correlationtermscanbe allowed. This

restrictsthesearchspaceto asize Ü úJÜ û À�¿ � ¾ .

111

6.5 Training Sequencefor Delay-diversity Scheme

A delay-diversityschemeis a simplespace-timecodingschemethat is usedto achieve

diversitywhenmultiple transmitantennasareavailable.Weconsidera two transmitan-

tennasystememploying delaydiversity. In this scheme,thesameinformationsymbols

aretransmittedfrom two antennaswith a singlesymboldelayon the secondantenna.

Thus,theinformationsymbolsarereceivedtwice with differentpathgainsresultingin

diversitygain. Thedelaydiversitytechniquecanalsobeviewedasa trellis space-time

code[84] andhastheadvantagethatanoptimizedequalizeris sufficient to decodethe

delay-diversityspace-timecode.

For the delay-diversity scheme,the received signalat the � -th receive antennafor

the¶

-th symbolperiodis,´yµ ¦�¶·¯p¸ Ùº»Î¼�¾�¿�À ¾º Á ¼ÃÂ Ä »zÆ µ ¦�ÇÃ¯+È0¦Í¶ò¨�Ç ¨��)¬®'¯ï¬,Ê µ ¦�¶·¯¸ ¿º Á ¼ÃÂ ð��Ä ¾Æ µ ¦#ÇÑ¯Å¬��Ä Ù Æ µ ¦#ÇÃ¯_ó�È�¦Í¶Ð¨,ÇÃ¯Å¬,Ê µ ¦�¶Å¯ (6.21)

where�à ¾Æ µ and �à Ù Æ µ areaugmentedchannelimpulseresponsevectorsof lengthª ¬J

taps

suchthat �à ¾Æ µ ¸�ÿ Ä ¾Æ µ ¦��k¯ Ë ²Õ²�²ÕË Ä ¾Æ µ ¦�ª�¨ê'¯�Ë��and �à Ù Æ µ ¸ ÿ��ÑË Ä Ù Æ µ ¦ �$¯ Ë ²�²�²£Ë Ä Ù Æ µ ¦�ª,¨'¯!�"�

. Theequivalentchannelimpulseresponseà eqµ is asumof theseaugmentedchannel

impulseresponsevectorsandit is sufficient to estimateà eqµ at thereceiver.

During the training phase,the transmitter, however, hasthe option of transmitting

differenttrainingsequencesfrom eachof the two transmitterantennas.The individual

channelimpulseresponsescanthenbeestimatedandsummedup with theappropriate

delay to obtain the equivalentchannelimpulseresponse.A secondpossibility would

be to transmitthe sametraining sequencefrom both the transmitterantennas,oneof

themwith a singlesymboldelay, andestimatetheequivalentchannelresponsedirectly.

Intuitively, theideaof usingthesametrainingsequenceis appealingandit will beshown

112

thatthis, in fact,is abetterchoice.

First, we considertheuseof differenttrainingsequencesfrom thetwo transmitan-

tennas.For the framingstructureto be preserved in a delaydiversityscheme,it is es-

sentialthatthetrainingsequencebedelayedin a fashionsimilar to thedata.Hence,the

trainingsequencefrom thesecondantennais transmittedwith a symboldelay. There-

ceivedsequenceat the � -th receiveantenna,duringthetrainingphase,canbeexpressed

as, ´yµ ¦Í¶Å¯p¸ Ùº»Î¼�¾ ¿�À ¾º Á ¼ÃÂÅÄ »zÆ µ ¦#ÇÃ¯:È » ¦�¶Ð¨,Çt¨��R¬®'¯Å¬«Ê µ ¦�¶·¯¸ ¿º Á ¼ÃÂ ð��Ä ¾Æ µ ¦�ÇÃ¯+È ¾ ¦�¶Ð¨�ÇÃ¯D¬��Ä Ù Æ µ È Ù ¦�¶Ð¨,ÇÃ¯_ó ¬«Ê µ ¦�¶·¯ (6.22)

and the stacked channelimpulseresponsevector �à µ can be estimatedas in equation

(6.7). Theonly differenceis that �à µ has ý ¦�ªJ¬XN¯tapsand ¤ is an

¦�§æ¨«ªT¯$# ý ¦�ª ¬®N¯block-Toeplitzmatrix. Theequivalentchannelimpulseresponseà eqµ canbeexpressedin

termsof�à µ as, à eqµ ¸&% �à µ Ë (6.23)

wherethe¦�ªö¬®'¯'# ý ¦�ªJ¬ '¯

matrix%

is constructedas,%X¸ ()* + ¿ � ¿ , Ù � ¾ , ¿� ¾ , ¿ � ¾ , Ù + ¿-/.0 ²

(6.24)

TheLS errorfor theequivalentchannelimpulseresponseis,

E Ûß à eqµ ¨Fáà eqµ ß ÙyÝ ¸ tr 1 Ø Ùâ %&2 ¤ ¥ ¤43 À ¾ % �65 ²(6.25)

The training sequence¤ hasto be optimizedto minimize the LS channelestimation

errorof equation(6.25).

113

Theorem 6.2 The leastpossibleLS channelestimationerror, whendifferent training

sequencesareusedfromthetwotransmitantennasin a delay-diversityscheme, is given

by

E Û ß à ×87µ ¨Fáà × 7µ ß Ù Ý ° Ø Ùâ ¦�ªö¬®'¯Ø Ùç ¦�§æ¨�ªz¯ ² (6.26)

Proof: TheLS estimationerrorof theequivalentchannelimpulseresponseis expressed

in equation(6.25).Thetrainingsequence¤ is to beoptimizedto minimizethis estima-

tion error. Letô ¸ ¤ ¥ ¤ denotetheauto-correlationmatrix of thetrainingsequence¤ .

Thematrixô

is Hermitianandpositivedefinite. We assumethat themaximumenergy

pointsof thesourceconstellationareusedasthetrainingsymbols.Hencethediagonal

entriesofô

areequalto Ø Ùç ¦�§æ¨«ªz¯.

Let 9 ¾ Ë³²�²�²£Ë 9 ¿ � ¾ betherow vectorsof theaugmentationmatrix%

. Eachof thisrow

vectors9 Á hasnon-zeroentries,namelyunity, at theÇ-th locationandthe

¦�ªõ¬�Å¬ ÇÃ¯-th

location,exceptfor 9 ¾ and 9 ¿ � ¾ . Thevectors9 ¾ and 9 ¿ � ¾ areunit vectorswith anentry

of oneat thefirst andthe ý ¦�ª�¬F'¯-th locations,respectively. TheLS errorof equation

(6.25)canbeexpressedasa functionof therow vectors9 Á as,

E Ûß à eqµ ¨Fáà eqµ ß Ù+Ý ¸ Ø Ùâ ¿ � ¾º Á ¼�¾;: ô À ¾ 9 Á Ë 9 Á�< ² (6.27)

Kantorovich inequality: Ifô =?>A@ , @

is a positive definite Hermitian matrix andB =C> @is avector, then D B Ë BFE ÙHG D ô B Ë BFE D ô À ¾ B Ë BFE ² (6.28)

Kantorovich inequality [63] canbe appliedto equation(6.27) anda lower boundfor

the LS estimationerror canbe obtained. For aÇ

valueof ý Ëö²�² ²�ËÐªonecanseethatD 9 Á Ë 9 Á E ¸ ý andD ô 9 Á Ë 9 Á E ¸ ´ Á Æ Á ¬ ´ Á Æ Á � ¿ � ¾ ¬ ´ Á � ¿ � ¾Æ

Á ¬ ´ Á � ¿ � ¾ÆÁ � ¿ � ¾ Ë (6.29)

114

where ´JI Æ K are the entriesof the auto-correlationmatrixô

. Auto-correlationmatrices

have the propertythat the off-diagonalentriesareno larger thanthe diagonalentries.

Hence, D ô 9 Á Ë 9 Á E GML Ø Ùç ¦�§©¨«ªz¯ ²(6.30)

Fromequations(6.28)and(6.30),it is clearthat for theparticularchoiceofÇ

between

2 toª

, D ô À ¾ 9 Á Ë 9 Á E ° Ø Ùç ¦�§æ¨�ªz¯ ² (6.31)

Equation(6.31) is alsosatisfiedwhenÇ«¸�

andÇ,¸ ª�¬

. Henceequation(6.27)

reducesto,

E Û ß à eqµ ¨Fáà eqµ ß Ù Ý ° Ø Ùâ ¦�ªö¬®'¯Ø Ùç ¦�§æ¨�ªz¯ ² (6.32)NNow, considertheuseof thesametrainingsequencefrom both thetransmitanten-

nas,with the appropriatesymboldelay. From equation(6.21), the LS estimateof the

equivalentchannelimpulseresponseà ×87µ canbedeterminedlike thatof anSISOsystem.

Since,thesametrainingsequenceis usedfrom boththeantennas,theToeplitzmatrix ¤hasasize

¦�§�¨Éªz¯O#É¦�ªt¬ '¯. Basedonanalysissimilar to Theorem6.1,it canbeshown

thattheminimumpossibleLS estimationerroris,PRQTS E Û ß à × 7µ ¨Fáà ×87µ ß Ù Ý ¸ Ø Ùâ ¦�ª ¬®N¯Ø Ùç ¦�§©¨«ªz¯ Ë (6.33)

andis obtainedif andonly if ¤ ¥ ¤ ¸R¦�§æ¨«ªT¯ Ø Ùç + ¿ � ¾ ² (6.34)

Comparingequation(6.33)with the lower boundobtainedin equation(6.26), it is

clearthat thechoiceof identicaltrainingsequencesfor thetwo transmitantennas,pro-

videdequation(6.34)is satisfied,is indeedoptimal.Theaboveresultis not restrictedto

115

thecaseof a singlesymboldelayin thedelay-diversityscheme.Theoptimality of the

choiceof identicaltrainingsequencesis alsovalid for a delay-diversityschemewith a

symboldelay�. In suchascenario,the

%matrixwill beof size

¦�ª ¬U� ¯V# ý ¦�ª ¬U� ¯, Ä eqµ

will have ý ¦�ª ¬M� ¯tapsand ¤ will bean

¦�§ ¨wª�¨W�~¬ê'¯�# ý ¦�ª ¬X��¯block-Toeplitz

matrix. Usingsimilarargumentsasbeforeit canbeshown that

E Û ß à × 7µ ¨Fáà ×87µ ß Ù Ý ° Ø Ùâ ¦�ªJ¬W��¯Ø Ùç ¦�§æ¨�ª�¨U�`¬ '¯ Ë (6.35)

andthat the lower boundcanbe achieved if identical training sequences(satisfyinga

conditionanalogousto equation(6.34))areused.

6.6 Summary

In this chapter, the channelimpulseresponsefor multiple antennasystemswasdeter-

minedusingleastsquareschannelestimation.It wasshown that thelossin throughput

due to channelestimationcanbe minimizedby appropriatechoiceof training length

andtrainingsequences.Theseoptimalsequencesoughtto satisfythepropertyof both

temporalandspatialwhiteness,i.e., have a low auto-correlationaswell aslow cross-

correlation. After deriving an optimality criterion for training sequencedesignfor

MIMO systems,a few heuristicmethodsfor thesearchof near-optimalsequenceswere

proposed.It wasfurthershown thatthechoiceof identicaltrainingsequences,transmit-

tedwith theappropriatedelays,is optimalfor a two transmitantennasystememploying

thedelay-diversityscheme.Whenidenticaltrainingsequencesaretransmittedfrom the

two antennas,the estimationof the equivalentchannelimpulseresponseof equation

(6.23), at the receiver, is no different from that of estimatingthe channelimpulsere-

sponsefor a1x1antennaconfiguration.This impliesthatthereceiverarchitectureneeds

no modificationfor a 2x1 antennaconfiguration,employing a delaydiversityscheme,

116

except the capability to estimateand equalizea longer effective channelimpulsere-

sponsewhencomparedto the1x1antennaconfiguration.

Chapter 7

Conclusions

“Onealwayshastimeenough,if onewill applyit well.”

- Goethe

Equalizationapproacheshave traditionally relied on causalprocessingof the signalat

the receiver. The increasein popularityof packet basedtransmissionsystemslike the

GSM or EDGE offers thepossibility of block processingof the received signal. With

block processingcomesthe freedomto processthe signal in eithera causalor a non-

causalfashion. The advantagesof employing time-reversalhasbeenpresentedin this

dissertationfrom the context of a decisionfeedbackequalizer. A summaryof the re-

sultsin this dissertationarepresentedin Section7.1 andfuture researchdirectionsare

discussedbriefly in Section7.2.

7.1 Summary of Results

Firstly, the performanceand limitations of a DFE as an equalizerstructurewere re-

viewed. A novel bidirectionalDFE architecturethat employs time-reversalof the re-

ceivedblock of datawasproposed.A BiDFE consistsof two parallelDFE structures,

117

118

one to equalizethe received signaland the other the time-reversedversionof the re-

ceivedsignal.Thecausalnatureof errorpropagationcausestheerrorburststo proceed

in oppositedirectionsin thetwo parallelDFEsandresultsin a low correlationbetween

the error bursts. Error propagationcanbe mitigatedby combiningthe outputsof the

two parallelDFEs. In addition,underthe ideal feedbackassumption,the BiDFE, ef-

fectively, assumesknowledgeof boththepastandfutureinterferingsymbolsandhence

decreasesthegapfrom theMFB. TheBiDFE architecture(BAD andLC-BiDFE) was

shown to provideasignificantperformanceimprovementoveraconventionalDFEwith

little additionalcomplexity. Onedisadvantageof theBiDFE is theincreasein latency or

theoveralldetectiondelay.

In anattemptto gaininsighton theperformancelimitation of theLC-BiDFE, thefi-

nite lengthconstraintwasrelaxedandtheasymptotic(asthenoisevarianceapproaches

zero)MSEperformancewasevaluated,undertheidealfeedbackassumption.Although,

the infinite lengthLC-BiDFE offers performanceimprovementover the conventional

DFE, thegapfrom theMFB wasnon-zero.This led to theformulationof thetapopti-

mizationproblem,wherethecoefficientsof theLC-BiDFE wereoptimizedto minimize

theoverallMSE insteadof theMSE at theoutputof eachof theconstituentDFEs.The

MMSE-BiDFE coefficientsweredeterminedandwereshown to attaintheMFB. It was

further shown by a numericalexamplethat the LC-BiDFE tap settingthat attainsthe

MFB is notunique.

The tapsof the LC-BiDFE wereoptimizedundera finite lengthconstraintfor the

specialcaseof a symmetricchannel. The effectivenessof the tap optimizationin an

LC-BiDFE wasevaluatednumerically. It wasshown that,althoughtheMMSE-BiDFE

offeredanimprovedperformanceundertheidealfeedbackassumption,therewasnoim-

provementin SERwhendecisionfeedbackwasused.TheMSEcostfunctionwasmod-

119

ified by incorporatingthenormof thefeedbackfilter tapcoefficientsandtheLC-BiDFE

tapsthatminimizesthemodifiedcostfunctionwasshown to offer anSERimprovement,

albeitmarginal. An alternatesolution,basedontheuseof aniterativeBiDFE approach,

wasdemonstratedto be reasonablysuccessfulin offering an SERimprovementfrom

tapoptimization,whenthe ideal feedbackassumptionwasviolated. Thetapoptimiza-

tion wasalsoextendedto theasymmetricchannelscenario,by constrainingthereceiver

structureto havea front-endfilter, matchedto thechannelimpulseresponse.

TheBiDFE wasextendedto multiple-inputmultiple-output(MIMO) channelequal-

ization. Theeffectivenessof a MIMO LC-BiDFE wasdemonstratedwith simulations.

Oneaspectin which the MIMO BiDFE differs from the SISOBiDFE is in the useof

differentorderof detectionfor theusersin thetwo parallelMIMO DFEstreams.

In training basedMIMO channelestimationthe choiceof training sequenceand

lengthaffectsperformance.Trainingsequencesthatsatisfythepropertyof beingwhite

acrossbothtimeandspace(i.e.,antennas)wereshown to beoptimal.For adelaydiver-

sity scheme,transmittingthesameoptimal(in thesingleusersense)trainingsequence

from bothtransmitantennaswasshown to beoptimalfor LS channelestimation.Hence,

thereceiverarchitecturerequiresnochangefrom thesingleantennacase,whenmultiple

transmitantennasaredeployedanda delaydiversityschemeis used.

7.2 Futur eDir ections

In Chapter3, an MMSE-BiDFE tap settingthat attainsthe MFB wasderived. It was

shown, with theaid of a 2-tapchannelexample,that theLC-BiDFE filter settingsthat

attaintheMFB arenotunique.It will beinterestingto know, if thelackof uniquenessof

MFB attainingLC-BiDFEfilter settingsextendto all genericchannelimpulseresponses.

120

Furthermore,knowledgeof all suchsolutionswould help in choosingthe “best” filter

setting,whentheidealfeedbackassumptionis relaxed.

In Chapter4, the tap optimizationof the LC-BiDFE resultedin an MSE perfor-

manceimprovement,undertheideal feedbackassumption.However, this performance

improvementdid not translateinto a decreasein theSER,whenthe ideal feedbackas-

sumptionwasrelaxed. Two solutions,onebasedon the optimizationof the modified

cost function and the other using an iterative LC-BiDFE approach,proposedto mit-

igate this problemofferedonly a marginal improvement. Furthermore,this marginal

improvementcould be attainedonly if the userdefinedparameterY (for the modified

costfunction)is chosenappropriately. In thecaseof theiterativeLC-BiDFE approach,

theperformanceimprovementcomesat thecostof increasedcomputationalcomplexity.

The searchfor a low-complexity solutionthat canresult in the realizationof the per-

formanceimprovementdueto tap optimization,in the presenceof decisionfeedback,

needsfurtherinvestigation.

In Chapter5, theBiDFE architecturehasbeenextendedto theMIMO channelequal-

ization problem. Therearea numberof openproblemsin this area,including perfor-

manceanalysisof theMIMO LC-BiDFE andtapoptimizationof theMIMO LC-BiDFE.

Furthermore,thereis no known extensionof thebidirectionalarbitrateddecisionfeed-

backequalizer(BAD) schemeto theequalizationof aMIMO channelwith memory.

Appendix A

Additional Simulation Results

In this appendix,additionalsimulationexamplesdemonstratingthe performanceim-

provementcapabilitiesof the BiDFE arepresented.Matlab script files for generating

thesimulationplotsprovidedin thisappendixareavailableat [19]. Frequency-selective

fadingchannelmodelsfor a typical urbanenvironmentareconsideredin SectionA.1

to supplementthe simulationexamplesof Subsection2.5.6. SectionA.2 considersa

syntheticchannelimpulseresponsewith severeISI, namelyrootscloseto theunit cir-

cle, and comparesthe performanceof the LC-BiDFE with that of BAD and MLSE.

SectionA.3 providesan additionalsimulationexampleto demonstratethe efficacy of

the iterative LC-BiDFE approachproposedin Section4.3. Additional fadingchannel

simulationsto testtheefficacy of theMIMO LC-BiDFE areprovidedin SectionA.4.

A.1 BroadbandWir elessChannels

In this section,the equalizationof quasi-staticfrequency-selective fadingchannelim-

pulseresponsesis considered.Eachrealizationof thechannelimpulseresponse(corre-

spondingto the channelencounteredby eachpacket of data)is modeledbasedon the

121

122

0 1 2 3 4 5 6 7 8 9 10−30

−25

−20

−15

−10

−5

0

5

Delay ( µs )

Pow

er (

dB

)

FigureA.1: Powerprofile of fadingchannelfor a typical urbanenvironment

multipathdelayprofile specifiedby GSM [32] (with slight modifications)for theurban

environment.Thepower profile for theurbanenvironmentis shown in FigureA.1. All

the pathsin the above profile wereassumedto be independentlyRayleighfading. A

square-rootraisedcosine(SRRC)pulseshapingfilter with a 12.5%roll-off factorwas

usedat the transmitter. A symbol rate of 1 Mbaud (the symbol period is 1 Z s) was

assumedandthesourcewasdrawn from a BPSKconstellation.A total of 16 feedfor-

ward tapsand8 feedbacktapswereallocatedto the DFE. Perfectchannelknowledge

(assumption2.4)wasassumedat thereceiver.

Simulationswereperformedfor anensembleof multipathchannels,randomlygen-

eratedusingtheurbanenvironmentpowerprofile. In FigureA.2, theSERperformance

(averagedover the variouschannelrealizations)of an LC-BiDFE (with hard/softde-

cisions)is comparedwith the performanceof a normalmodeDFE andBAD. For the

fadingchannelsimulations,BAD offers a performanceimprovementof 1.1 dB, at an

123

5 6 7 8 9 10 11 1210

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

Normal mode DFELC−BiDFE : (Hard)LC−BiDFE : (Soft)BAD

Figure A.2: SERperformancecomparisonfor a fadingchannelenvironment

SERof�� À [ , over the normalmodeDFE, while the LC-BiDFE (with soft decisions)

offers about0.8 dB of improvement. Although, BAD offers betterimprovementthan

LC-BiDFE, theimprovementis smallandcomesat thecostof increasedreceiver com-

plexity. The SERperformancecurvesfor four randomlygenerated(sample)channels

with theurbanpowerprofile is illustratedin FigureA.3.

A.2 BiDFE Performancefor a Channelwith DeepNulls

In thissectionweconsiderasyntheticchannelimpulseresponsewith adeepnull. Such

achannelwill haverootscloseto theunit circle. Theroot locationsof thechannel,à [ ¸Rÿ�¨\�Ñ²^]`_ ý �a�Ñ²^�Ñcb ý �ù²ed`] ý ��Ñ²^�Ñcb ý ¨U�Ñ²e]f_ ý �� (A.1)

areillustratedin FigureA.4. TheSERperformanceof anormalmodeDFEis compared

with thatof an LC-BiDFE, BAD andMLSE in FigureA.5. The sourceis assumedto

124

5 6 7 8 9 10 11 12 1310

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

Normal Mode DFELC−BiDFE : HardLC−BiDFE : SoftBAD

5 6 7 8 9 10 11 1210

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate


5 6 7 8 9 10 11 1210

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate


5 6 7 8 9 10 11 12 1310

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate


Figure A.3: SER performancecomparisonfor a few samplechannelswith the urbanpower

profile

be drawn from a BPSK source.A DFE with 8 feedforward filter tapsand4 feedback

filter tapswasused.TheLC-BiDFE offersonly a smallgainof about0.4 dB from the

normalmodeDFE curve at an SERvalueofg� À [ , while BAD offers a gainof nearly

1.1 dB. Whenthe channelhasdeepnulls, the magnitudesof the FBF tapstendsto be

largeandtheerrorpropagationphenomenonis moresevere.As thereconstruction-and-

arbitrationschemeof BAD is betterdesignedtowardsexploiting the low correlationin

the error burstsbetweenthe normalmodeDFE andtime-reversalmodeDFE streams,

it offers a significantlybetterperformance.However, the gapfrom the MLSE for the

BAD algorithmis about0.8dB.

125

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Real Axis

Imag

inar

y A

xis

FigureA.4: Channelzerosfor h [jiXkml6nòqp�r�sgnWnò�nutwv�sWnòqx�p�sgnynò�nutwv�szl{nòqp�r�sgn}|A.3 Simulation Example for LC-BiDFE with Iteration

In Section4.3, an iterative LC-BiDFE approachwasproposedto ensurethat the per-

formanceimprovementobtainedwith tapoptimizationis attained,at leastpartly, when

the ideal feedbackassumptionis relaxed. In this section,an additionalsimulationex-

ampleto supplementthe numericalresult in Section4.3 is provided. We considerthe

symmetricchannelimpulseresponse,àf~ ¸Rÿ�¨H�Ñ² ý �ubf]a�ù² �]f�`��Ñ²^_`]`_ù��Ñ²��]`�`� ¨��ù² ý �`bf]��²(A.2)

Recallthat,in theiterativeLC-BiDFE architecture,proposedin Section4.3,thereceived

signalis first equalizedby usinganLC-BiDFE with MMSE-DFEsettings.This is fol-

lowed up by equalizingthe received signalusingan LC-BiDFE with MMSE-BiDFE

tap settings. The SERperformanceof the iteratedLC-BiDFE is illustratedin Figure

A.6. For anSERvalueof�� À [ , the iteratedLC-BiDFE (with soft decisions)offersan

additionalperformanceimprovementof nearly0.3dB over thesoft LC-BiDFE with the

“Suboptimal”tapsettings.Thelabel“LC-BiDFE : Suboptimal”refersto theuseof the

126

6 8 10 12 14 1610

−4

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

Normal Mode DFELC−BiDFE : (Soft)BADMLSE

Figure A.5: SER performancecomparisonfor a deepnull channelwith impulse responseh [ iMkml nòqp�r�sgnWnò�nutwv�sWnòqx�p�sgnWnò�nutwv�s�l{nòqp�r�sgn}|MMSE-DFEtapsettingsfor theLC-BiDFE. As notedbefore,the iterative LC-BiDFE

approachhasa two-fold complexity whencomparedto theLC-BiDFE.

A.4 Fading ChannelSimulation for MIMO LC-BiDFE

In this section,weconsidertheequalizationof aquasi-staticfrequency-selective fading

MIMO channel.The channelimpulseresponsebetweeneachinput andoutputof the

systemis assumedto be independent.Simulationswere performedfor an ensemble

of L # L MIMO multipathchannels,randomlygeneratedusingtheurbanenvironment

power profile illustratedin FigureA.1. Eachof the channelimpulseresponseswere

normalizedto possessa unit norm. The sourcesymbolswereselectedfrom a BPSK

sourceconstellationandperfectchannelknowledgewasassumedat thereceiver.

UnbiasedMMSE-DFE coefficientswithª6�J¸sNø

andª ã ¸¡g�

wereusedfor the

127

6 7 8 9 10 11 12 13

10−3

10−2

10−1

SNR in dB

Sym

bol

Err

or R

ate

DFE : SuboptimalLC−BiDFE : SuboptimalLC−BiDFE : Iterated

Figure A.6: SERperformanceof aniteratedLC-BiDFE for thechannelwith impulseresponseh ~ iMkml nòqsgncv�p�nò�twpgn�nynòqr�p�r`t�nò�twpgn�n�l{nòqsgncv�pJ|MIMO DFEsfor both the normalmodeandtime-reversalmodeoperation.The order

of detectionfor normalmodeDFE was,ord ¾ ¸ ÿiÈ ¾ ¦Í¶Å¯ Ë È Ù ¦Í¶Å¯ Ë È [ ¦Í¶Å¯ Ë È ~ ¦Í¶Å¯!� , while

for the time-reversalmodeDFE, ord Ù ¸�ÿiÈ ~ ¦Í¶Å¯�Ë�È [ ¦Í¶Å¯�Ë�È Ù ¦Í¶Å¯�Ë�È ¾ ¦Í¶Å¯!� . TheSERper-

formancecurves(averagedover all the channelrealizations)for a normalmodeDFE,

time-reversalmodeDFE andthe LC-BiDFE arecomparedin FigureA.7 for the four

usersÈ ¾ ¦�¶Å¯ , È Ù ¦Í¶Å¯ , È [ ¦�¶·¯ and

È ~ ¦Í¶Å¯ , respectively. Thesesimulationresultsdonot invoke

theassumption2.5,andhenceincorporatetheeffect of errorpropagation,i.e., decision

feedbackwasassumed.The MIMO LC-BiDFE (with soft decisions)offers a perfor-

manceimprovementof about1.5 dB, at an SERof�� À [ , over the time-reversalmode

MIMO DFE for all theusers.Theseresultsdemonstratetheviability of theLC-BiDFE

in providing performanceimprovementsfor theMIMO channelequalizationproblem.

128

0 2 4 6 8 10 1210

−4

10−3

10−2

10−1

100

SNR in dB

Sym

bol

Err

or R

ate

User 1

Normal Mode DFETime−reversal Mode DFELC−BiDFE : (Soft)

0 2 4 6 8 10 1210

−4

10−3

10−2

10−1

100

SNR in dB

Sym

bol

Err

or R

ate

User 2


0 2 4 6 8 10 1210

−4

10−3

10−2

10−1

100

SNR in dB

Sym

bol

Err

or R

ate

User 3


0 2 4 6 8 10 1210

−4

10−3

10−2

10−1

100

SNR in dB

Sym

bol

Err

or R

ate

User 4


FigureA.7: SERperformancecomparisonof users1-4for aMIMO fadingchannelenvironment

with � i��u�;�Wi��

Appendix B

Training Sequencesfor EDGE

In chapter6, the criterion of optimality for training sequencesanda few searchtech-

niquesfor optimal and near-optimal training sequenceswere discussed.In [89], the

authorsproposetheemploymentof multiple antennasto an EDGE systemto increase

systemthroughput.To enablechannelestimationin suchaMIMO system,near-optimal

trainingsequencessuitablefor anEDGEsystemaredeterminedandaretabulatedin this

appendix.

First,we consideranEDGEsystemwith 2 transmitantennas.Basedon thevarious

designmetricsdescribedin Section6.3, a training length of N = 26 was chosenfor

the2 transmitantennacase.Fromequation(6.17),for this choiceof trainingsequence

length,themaximumidentifiablenumberof thechannelimpulseresponsetapsis 9. A

typical urbanchannelimpulseresponse,with pedestrianmobility (about3 Km/h), has

mostof theenergy concentratedin either4 or 5 taps,andwemakeaconservativechoice

ofªæ¸��

for the channelimpulseresponseto be estimated. It shouldbe notedthatª, whenusedin thedesignof trainingsequences,is merelya designparameterandas

longastheactuallengthof thechannelimpulseresponseto beestimatedis lessthanthe

designparameterª

, thedesignedtrainingsequenceswill benear-optimal. Thetraining

129

130

TableB.1: Trainingsequencesfor anEDGEsystemwith � i�sAntenna 1 Antenna 2 Penalty Over Ideal Training

0FB5D8F 293BE29 0.1599dB

0391483 251F725 0.1599dB

3785377 0BB9F4B 0.1599dB

3BB287B 0B4188B 0.1599dB

1D2F9DD 21135E1 0.1599dB

11182D1 21EB221 0.1599dB

2F0A6EF 1773E97 0.1599dB

3DD943D 05A0C45 0.1599dB

symbolsweredeterminedbasedon therandomsearchmethod,describedin [13] andin

subsection6.4.2,andwererestrictedto be from a BPSKconstellation.A few pairsof

thesenear-optimal training sequencesareshown in hexadecimalformat in TableB.1.

The most-significant-bit(MSB) of the hexadecimalrepresentationcorrespondsto the

first symbolof thetrainingsequence.Thebit 1 correspondsto thesymbol“+1” andthe

bit 0 to the symbol “-1”. The penaltyincurred,in termsof the loss in effective SNR

dueto channelestimation,by thesub-optimaltrainingsequenceswhencomparedto the

ideal trainingsequencesis alsotabulatedin TableB.1 andis assmallas0.16dB. This

penaltyis computedastheadditionallossin effective SNRcausedby thechoiceof the

sub-optimaltrainingsequencesover theidealtrainingsequencesandis givenby (recall

equations(6.10)and(6.19)),

Penaltyover idealtraining¸ ��4�T�`�� T¬ tr

ðù¦�� ¥ �D¯ À ¾ óT¬ ¹ ¿� û À�¿ � ¾ � � ²(B.1)

131

FigureB.1 illustrates1 the loss incurredby the sub-optimaltraining sequences,when

estimatingchannelimpulseresponsesof varying lengths,over the ideal training se-

quences.Thepenaltyover idealtrainingbecomessignificant(about0.8dB) only when

the channelimpulseresponsehasa lengthof 8, which is beyond the designchoiceofª�¸a�.

1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

3.5

4

Channel length

Loss

in S

NR

(dB

)

Ideal training Suboptimal training

Figure B.1: Lossdueto channelestimation,��U�Now weconsidertheuseof four antennasatthetransmitter, namely� �� . For this

choiceof a transmitantennasit wasshown in [89] thata trainingsequencewith length� ��`� is suitablein thesensethat it nearlymaximizestheaveragethroughputof the

systemwhile affording identifiability of thechannelimpulseresponse.For this choice

of a training sequencelength, from equation(6.17) we note that a channelimpulse

responsewith a maximumof 7 tapscanbeidentified.For thetrainingsequencedesign,

a delayspreadof ��¢¡ wasassumed.This choicewasmotivatedby thefactthat,for a


132

TableB.2: Trainingsequencesfor anEDGEsystemwith �£��¤Antenna 1 Antenna 2 Antenna 3 Antenna 4 Penalty Over

Ideal Training

0A7076510 7076510A7 76510A707 510A70765 0.0738dB

2F9291822 9291822F9 91822F929 822F92918 0.0738dB

517A46305 7A4630517 4630517A4 30517A463 0.0738dB

C2D45980C D45980C2D 5980C2D45 80C2D4598 0.1433dB

2D8B8E402 8B8E402D8 8E402D8B8 402D8B8E4 0.1349dB

B6E05238B E05238B6E 5238B6E05 38B6E0523 0.1166dB

59B80A8E5 B80A8E59B 0A8E59B80 8E59B80A8 0.1191dB

CC876AEBC 876AEBCC8 6AEBCC876 EBCC876AE 0.1110dB

TU3 (typical urbanenvironmentwith a mobility of 3 Km/h) channelimpulseresponse

mostof the energy is concentratedin either4 or 5 taps. As in the 2 transmitantenna

case,thetrainingsymbolswererestrictedto befrom a BPSKconstellation.A few sets

of sub-optimaltraining sequencesare tabulatedin hexadecimalformat in Table B.2.

Further, thepenaltyincurreddueto thechoiceof thesesequencesovertheidealtraining

sequencesis alsotabulatedin TableB.2. Thesetrainingsequenceswereobtainedusing

the cyclic shift searchmethoddescribedin subsection6.4. For the designchoiceof ��¥¡ , thelossincurredin effectiveSNRdueto channelestimationby thesub-optimal

training sequencesover the ideal training sequencesis at most 0.14 dB. Figure B.2

illustrates2 the loss incurredby the sub-optimaltraining sequences,when estimating

channelimpulseresponsesof varying lengths,over the ideal training sequences.The


133

incurredpenaltyis in theorderof 1 dB, only whenthechannelimpulseresponsehas7

taps,which is higherthanthedesignchoiceof U��¡ .

0 2 4 6 80

1

2

3

4

5

Channel Length

Loss

in S

NR

(dB

)

Ideal training Suboptimal training

Figure B.2: Lossdueto channelestimation,��¦¤

Appendix C

Birding the CayugaLakeBasin

TheCayugabasinis looselydemarcatedby thebordersof thewatershedthatflow into

theCayugalake. The Cayugabasinbio-region playshostto a numberof bird species

- breeders,migrantsandwinter irruptives. Numerousbirding hot spotsin the Cayuga

basinprovideampleopportunityto abirderto seeandappreciatetheover300speciesof

birdsthathavebeensightedat onetimeor otherin thepast.A listing of thebirding hot

spotsin theCayugabasincanbe found in [66]. TheCayugabird club website[23] is

a usefulresourceto learnaboutlocal birdsandbirding localities.The244bird species

sightedby theauthorin theCayugalake basin,betweenJuly 1999andMay 2002,are

listedin this appendix.

TableC.1: List of bird speciesseenin theCayugalake basin

Commonloon Red-throatedloon Pied-billedgrebe

Hornedgrebe Earedgrebe Double-crestedcormorant

Leastbittern Greatblueheron Greategret

Greenheron Black-crown. night heron Woodstork

134

135

TableC.1(continued)

Turkey Vulture Snow goose Canadagoose

Brant MuteSwan Tundraswan

Woodduck Gadwall Americanwigeon

Americanblackduck Mallard Blue-wingedteal

Northernshoveler Northernpintail Green-wingedteal

Canvasback Redhead Ring-neckedduck

Greaterscaup Lesserscaup Surf scoter

Black scoter Long-tailedduck Bufflehead

Commongoldeneye Hoodedmerganser Commonmerganser

Red-breastedmerganser Ruddyduck Osprey

Baldeagle Northernharrier Sharp-shinnedhawk

Cooper’s hawk Broad-wingedhawk Red-tailedhawk

Rough-leggedhawk Goldeneagle Americankestrel

Merlin Peregrinefalcon Ring-neckedpheasant

Ruffedgrouse Wild turkey Virginia rail

Sora Commonmoorhen Americancoot

Sandhillcrane Black-belliedplover Americangoldenplover

Semipalmatedplover Pipingplover Killdeer

Greateryellowlegs Lesseryellowlegs Solitarysandpiper

Spottedsandpiper Uplandsandpiper Hudsoniangodwit

Marbledgodwit Ruddyturnstone Semipalmatedsandpiper

Leastsandpiper Pectoralsandpiper Purplesandpiper

Dunlin Stilt sandpiper Buff-breastersandpiper

136

TableC.1(continued)

Short-billeddowitcher Long-billeddowitcher Commonsnipe

Americanwoodcock Wilson’sphalarope Red-neckedphalarope

Bonaparte’s gull Ring-billedgull GreaterBlack-backedgull

Icelandgull Herringgull Lesserblack-backedgull

Caspiantern Commontern Forster’s tern

Black tern Long-billedmurrelet Rockdove

Mourningdove Yellow-billed cuckoo Easternscreechowl

Great-hornedowl Snowy owl Long-earedowl

Short-earedowl NorthernSaw-whetowl Commonnighthawk

Whip-poor-will Chimney swift Ruby-thr. hummingbird

Beltedkingfisher Red-headedwoodpecker Red-belliedwoodpecker

Yellow-belliedsapsucker Downy woodpecker Hairy woodpecker

Northernflicker Pileatedwoodpecker Olive-sidedflycatcher

Easternwood-peewee Yellow-belliedflycatcher Acadianflycatcher

Alder flycatcher Willow flycatcher Leastflycatcher

Easternphoebe Great-crestedflycatcher Westernkingbird

Easternkingbird Northernshrike Yellow-throatedvireo

Blue-headedvireo Warblingvireo Philadelphiavireo

Red-eyedvireo Blue jay Americancrow

FishCrow Commonraven Hornedlark

Purplemartin Treeswallow N. rough-wingedswallow

Bankswallow Clif f swallow Barnswallow

Black-cappedchickadee Tuftedtitmouse Red-breastednuthatch

137

TableC.1(continued)

White-breastednuthatch Brown creeper Carolinawren

Housewren Winter wren Marshwren

Golden-crownedkinglet Ruby-crownedkinglet Blue-graygnatcatcher

Easternbluebird Veery Gray-cheekedthrush

Swainson’s thrush Hermit thrush Woodthrush

Americanrobin Graycatbird Northernmockingbird

Brown thrasher Europeanstarling Americanpipit

Bohemianwaxwing Cedarwaxwing Blue-wingedwarbler

Golden-wingedwarbler Tennesseewarbler Orange-crownedwarbler

Nashvillewarbler Northernparula Yellow warbler

Chestnut-sidedwarbler Magnoliawarbler Cape-Maywarbler

Black-thr. bluewarbler Yellow-rumpedwarbler Black-thr. greenwarbler

Blackburnianwarbler Pinewarbler Prairiewarbler

Palm warbler Bay-breastedwarbler Blackpollwarbler

Ceruleanwarbler Black-and-whitewarbler Americanredstart

Worm-eatingwarbler Ovenbird Northernwaterthrush

Louisianawaterthrush Mourningwarbler Commonyellowthroat

Hoodedwarbler Wilson’swarbler Canadawarbler

Yellow-breastedchat Scarlettanager EasternTowhee

Americantreesparrow Chippingsparrow Fieldsparrow

Vespersparrow Savannahsparrow Grasshoppersparrow

Henslow’ssparrow Fox sparrow Songsparrow

Lincoln’ssparrow Swampsparrow White-throatedsparrow

138

TableC.1(continued)

White-crownedsparrow Dark-eyedjunco Laplandlongspur

Snow bunting Northerncardinal Rose-breastedgrosbeak

Indigobunting Bobolink Red-wingedblackbird

Easternmeadowlark Westernmeadowlark Rustyblackbird

Brown-headedcowbird Commongrackle Orchardoriole

Baltimoreoriole Pinegrosbeak Purplefinch

Housefinch White-wingedcrossbill Commonredpoll

Pinesiskin Americangoldfinch Eveninggrosbeak

Housesparrow

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