bidirectional decision feedback equalization and...
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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�\�$�<ÈP�r�1�l¡u¡u¡6ÈP�_²<Ég�x²6³��±¼ �cµF¶ hasalength ²�Ê � ²<ÉË�x²<³��Ì� taps.Equation
(2.7)canberewrittenas,
� �o�X��� ÈP� Å � �>�o�l� Å �Í ÎtÏ Ð �sÑ ���� �g� �ÈÒ�\�9�j�k�o�l�+�9�Í ÎtÏ Ð � � Ä� �g���tÓ ÈP�\�Ô� Å �°�"»(�\�9�ÖÕ&�k�_�l� Å �+�9�Í ÎtÏ Ð
cursor res.pre-cursorISI modeledpost-cursorISI
� � � �9� Á � ¥��g� Ñ �9� Ä ���ÈP�����j�k�_�l�+�9�Í ÎtÏ Ð � � Á ���� �g� �
¬®�����r�T�o�Ã�+�9�Í ÎtÏ Ðresidualpost-cursorISI filterednoise
(2.8)
An effectivewayof equalizingthechannelwouldbefor theFFFto× shapethecombinedchannel-feedforwardfilter responseÇ to maximizetheenergy
of thecursor, i.e., ÈP� Å ��Ø�� in theunbiaseddesign
× result in a small residualISI, i.e., ÈP�����±ØÙ� for �ÛÚÜ�LÝ Å and Å ��²<¸Tݺ�)Ú²�ÉÞ�)²<³-�+¼× keepthenoisegain, © ¶ © assmallaspossible
andfor theFBF to× cancelexactly the modeledpost-cursorISI by matchingthe FBF tapsto that of
thecombinedresponse,i.e., »(�\�9�<��ÈP���´� Å �Æßu�´Ú��lÚ%²�¸ .This is essentiallywhat theMSE optimizedDFE does,aswill becomeapparentin the
next subsection.
23
2.2.1 Finite length MMSE-DFE
Thecoefficientsof theDFE canbedesignedto beeitherzero-forcing(perfectlycancel
the ISI) or to minimize the mean-squarederror (MSE). TheMMSE optimizedDFE is
popularandthefinite lengthdesignwasderivedby Al-Dhahir [1]. A simplifiedderiva-
tion of the finite lengthoptimal DFE coefficientswill be reviewed in this subsection.
TheMSE at theoutputof theDFErepresentedin Figure2.5 is givenby,
£ ¥à � E á Ó �k�_�l� Å �°� � �o�X�ÖÕ ¥râ ß (2.9)
whereã �o�X�<��k�o�±� Å �ä� � �_�`� is thesoftdecisionerrorof theequalizer. Thegoalof the
MMSE designof theDFEis to determinethefilter pair � © mmseß ¹ mmse� thatminimizes£ ¥àof equation(2.9). Wenow definethe ²<ʯ彲<³ channelconvolutionmatrix æ as
æ �
çèèèèèèèèèèèèèèèé
�&�\��� � ê�ê�ê �... �&�\�$� 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, � � � �� , 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,ÃMSEà 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_ «oغ٠S�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 «gغ٠S�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
4See[19] for Matlabsourcecode
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�-ÁÀ 2 3uv w ���A� �>�$� ��� y v y �v1ÃÄ N � j (3.12)
Ariyavisitakul also demonstratedthat the MSE of both the normal modeand time-
reversalmodeDFEsarethesamefor theinfinite lengthscenario.
Lemma 3.1 The MSE of the output of the normal modeDFE and the time-reversal
modeDFE are thesame, underthehigh SNRassumption.
Proof: To prove the above statement,it is sufficient to show that the energy of the
feedforwardfilter of boththenormalmodeandtime-reversalmodeDFEsarethesame,
i.e., �n� ź�.�k� " a�n� �"��k� " . By usingargumentssimilar to thosein equations(3.10)- (3.12),we
canshow that £AW�"H*\X�-fW �" *\XQ{4|�- ¨ » a �t4*\�(f-ft � */�(f-ÁÀ 2 3uv w ���A� �>�`� ��� y v y �v1ÃÄ j (3.13)
63
Applying theresultof equation(3.5) to equation(3.13),
£AW�"H*\X�-fW �" */XQ{�|!- ¨ » a �t4*\M�-ft � */M�-ÇÀ 2 3uv w ���A� �>�$� ��� 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 À 243uv w ���A� �>�$� ��� y v y �v1ÃÄ N � � (3.15)
As boththenormalmodeandtime-reversalmodestreamsyield thesameMSE,anequal
gain combiningscheme,i.e., bËa �" , is optimal for the diversity combiningblock of
theLC-BiDFE of Figure3.1. As theresidualISI componentsareabsentunderthehigh
SNRapproximationfor the feedforwardfilters in equations(3.6) and(3.7), theoverall
MSE of theLC-BiDFE canbewrittenas
MSEBiDFE ac¢¤£¥� ^ */_)-Yg�§O*/_�-.� " ¨ac¢oª¬�® ¯ÎÍ ° ��*\³+-�� ° "1*fg]³+-#Ϧ´Ð*/_¶g�³+-
" ¸¹ �(3.16)
TheMSE canbesimilarly evaluatedin thefrequency domainas
MSEBiDFE a q "rÒÑ W�Ó!Ô&*\X�-fW �Ó!Ô *\X {4| - Õ » (3.17)
where WYÓ!Ô&*\X�-�a W���*\X�-d��WY"1*/X {�| - �(3.18)
Theperformancegainprovidedby theLC-BiDFE over theDFE is givenby,pBiDFEpDFE
a £AW���*\X�-fW��� */X {�| - ¨ »Ñ W�Ó!Ô.*/X�-ºW �Ó!Ô *\X {4| -�Õ » � (3.19)
64
Lemma 3.2 Thegain in outputSNRof the infinite lengthLC-BiDFE over the infinite
lengthDFE, undera highSNRassumptionispBiDFEpDFE
a �%��Ö Í.× */M�->Ï } (3.20)
where Ö Í.× */M�->Ï denotesthereal part of × *\M�- , andthe~
-transformof theresponse× *!³+-is Ø */X�-�a t4*\M�-fX � 3t4*\�(f- ��F� 243uv w ���A� �>�$� ��� y
v */y �v giX {4| -*f�xg�y v *fg�X�- {4| - �F�� ��F� 2H3uv w ���A� �>�$� ��� y �v *f�xgzy v X {�| -*/y �v gzX {�| - �F�� � (3.21)
Proof: Fromequation(3.18),thedenominatorof equation(3.19)canberewrittenas£AW�Ó!Ô.*/X�-ºW �Ó!Ô *\X {4| - ¨ » a�£ Í W���*/X�-d� W�"1*\X {4| ->Ï Í W �� *\X {�| -�� W �" *\X�->Ï ¨ »aË£FW��J*/X�-ºW �� *\XQ{4|!-�� W�"H*\XQ{4|!-ºW �" *\X�-d��W���*\X�-ºW �" *\X�-d��W �� */XQ{�|�-fWY"1*/XQ{�|!- ¨ » j (3.22)
FromLemma3.1andequation(3.12),£AW���*\X�-fW �� *\X {4| ¨ » a�£FWY"H*\X {�| -ºW �" */X�- ¨ » a �t4*\M�-ºt � *\M�- À 2H3uv w ���A� �>�`� ��� y v y �v1ÃÄ N � j (3.23)
Now define,Ø */X�-�� W��� *\X {4| -fWY"1*/X {�| -£FW���*/X�-ºW �� *\X {4| ¨ »a t4*\M�-J*fg�X�- � 3t4*\�(f- ��F� 243uv w ���A� �>�$� ��� 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
1See[19] for Matlabsourcecode
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 £FźRH}©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
1See[19] for Matlabsourcecode
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
2See[19] for Matlabsourcecode
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-
3See[19] for Matlabsourcecode
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�à 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 äN 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.ߤŠ...
. . . ßß Û Á&ߤŠ. . .Û Á&Ì]ÎÏ Ç Å
... ß . . ....ß ... ð·ð·ð Û Á.ߤÅ
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. optÆGøeù�úMûýü�þÿ Ç� � Á�� äNÂ\Ï����XÅ�xÁ� Õ Ý� ¿�Å�xÁ� Õ Ý� ¿�Å_Á�� äNÂ0Ï����XÅÝ� �]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� and ë3 ,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)is 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�à 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.÷�Å6à 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�à 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
1See[19] for Matlabsourcecode
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 � -&ä¾ Y 7 Z Æ)( Ï\ß ãiM$M$h ßfÃ©ß ãmO � l$l -&þ Z 7 Y ÆR( ß ãih¤Ê K$j ÃMß ãilkO Ç � -&ä¾ Z 7 Z Æ)( ß ãnM M$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.ߤÅ6à 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�{ Å6à ã^ã^ã à 3y4 Á.Ì©Å.-�� (6.3)
is thevectorof observations,¾ 4�;�( Û Õ 7 4 Á.ߤÅ6à ã^ã_ã Ã Û Õ 7 4 Á.Ì Ï Ç Å�à ã^ã^ã Ã Û 5 7 4 Á&ߤÅ�à ã_ã^ã Ã Û 5 7 4 Á.Ì Ï Ç Å-�� (6.4)
is thestackedvectorof channelimpulseresponses,} 4~;�( õ 4 Á�{ Å6à 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 à � � 5�Ñ � � & � � Æ � Ç� � Õ Ã ã^ã_ã à Ç� � 5�Ñ � � ã (6.12)
By theCauchy-Schwartzinequality,� � Ã�� ��� ¿ à � � Ã�� � � � � Ã_� ��� (6.13)
where� à � 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 ¦�Çï+È » ¦�¶É¨«ÇïD¬«Ê µ ¦�¶·¯ Ë (6.18)¸ ¹º»½¼�¾�¿0À ¾º Á ¼ÃÂ Ä LS»zÆ µ ¦�Çï+È » ¦�¶É¨«ÇïD¬ ¹º»½¼�¾�¿0À ¾º Á ¼ÃÂTÌ Ä »zÆ µ ¦Í¶Å¯Î¨ Ä LS»zÆ µ ¦�ÇïNÏzÈ » ¦�¶Ð¨,ÇѯŬ,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 of��
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��
, 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 ¦�Çï+È0¦Í¶ò¨�Ç ¨��)¬®'¯ï¬,Ê µ ¦�¶·¯¸ ¿º Á ¼Ã ð��Ä ¾Æ µ ¦#ÇѯŬ��Ä Ù Æ µ ¦#Çï_ó�È�¦Í¶Ð¨,ÇïŬ,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 ¦#Çï:È » ¦�¶Ð¨,Çt¨��R¬®'¯Å¬«Ê µ ¦�¶·¯¸ ¿º Á ¼Ã ð��Ä ¾Æ µ ¦�Çï+È ¾ ¦�¶Ð¨�ÇïD¬��Ä Ù Æ µ È Ù ¦�¶Ð¨,Çï_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 ˳²�²�²£Ë 9 ¿ � ¾ betherow vectorsof theaugmentationmatrix%
. Eachof thisrow
vectors9 Á hasnon-zeroentries,namelyunity, at theÇ-th locationandthe
¦�ªõ¬�Ŭ Çï-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�ËЪonecanseethatD 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#ɦ�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�¨\�Ѳ^]`_ ý �a�Ѳ^�Ñcb ý �ù²ed`] ý ���Ѳ^�Ñcb ý ¨U�Ѳe]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
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
Normal Mode DFELC−BiDFE : HardLC−BiDFE : SoftBAD
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
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 [jiXkml6n`oqp�r�sgnWn`o�nutwv�sWn`oqx�p�sgnyn`o�nutwv�szl{n`oqp�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�Ѳ ý �ubf]a�ù² �]f�`���Ѳ^_`]`_ù��Ѳ��]`�`� ¨��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 n`oqp�r�sgnWn`o�nutwv�sWn`oqx�p�sgnWn`o�nutwv�s�l{n`oqp�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 n`oqsgncv�p�n`o�twpgn�nyn`oqr�p�r`t�n`o�twpgn�n�l{n`oqsgncv�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
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 3
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 4
Normal Mode DFETime−reversal Mode DFELC−BiDFE : (Soft)
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
1See[19] for Matlabsourcecode
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
2See[19] for Matlabsourcecode
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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