<%BANNER%>

A New Class of Sparse Channel Estimation Methods Based on Support Vector Machines

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Iwouldliketosincerelythankmyadvisor,Dr.JoseC.Principe,andco-advisor,Dr.LiuqingYang,fortheirsupport,encouragementandpatienceinguidingtheresearch.IwouldalsoliketothankDr.JohnHarrisandDr.DouglasCenzer,forbeingonmycommitteeandfortheirhelpfuladvice.IgratefullyacknowledgeDr.IgnacioSantamaraandJavierVaforprovidingnumeroussuggestionsandforhelpfulcommentsonseveralofmypapersthatarepartofthisdissertation.IalsoappreciatethehelpfuldiscussionsfrompeopleintheComputationalNeuroengineeringLab.Finally,Iwishtothankmywife,HyewonPark,forhersupport,aswellasmydearprincesses,EujinandJiwon,forprovidingbigmotivationstonishmystudy. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 11 1.1Background ................................... 12 1.2ContributionofthisThesis ........................... 14 1.3OrganizationoftheDissertation ........................ 15 2CONVENTIONALSPARSESOLUTION ..................... 17 2.1ObservationModel ............................... 17 2.2MatchingPursuit(MP)BasedMethod .................... 18 2.3SPARSE-LMS .................................. 19 2.3.1ConvolutionInequalityForRenyi'sEntropy ............. 20 2.4SimulationResults ............................... 22 3SUPPORTVECTORMACHINEBASEDMETHOD ............... 25 3.1Introduction ................................... 25 3.2SupportVectorClassication ......................... 26 3.2.1LinearlySeparablePatterns ...................... 26 3.2.2NonseparablePatterns ......................... 29 3.2.3TheKarush-Kuhn-TuckerConditions ................. 30 3.3SupportVectorRegression ........................... 32 3.4SparseChannelEstimationusingSVM .................... 35 3.5ExtensiontoComplexChannel ........................ 36 3.6SVMParameterSelection ........................... 37 3.7SimulationResults ............................... 48 4BLINDSPARSESIMOCHANNELIDENTIFICATION ............. 55 4.1Introduction ................................... 55 4.2ObservationModel ............................... 56 4.3CombinedIterativeandSVMBasedApproach ................ 58 4.3.1IterativeRegression ........................... 58 4.3.2SVMRegression ............................. 59 4.3.3ImplementationofSupportVectorRegression ............ 60 4.3.3.1TheAdatronAlgorithm ................... 61 5

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................ 63 4.4SimulationResults ............................... 65 5IDENTIFICATIONOFTIMEVARYING&FREQUENCY-SELECTIVEMULTIPATHCHANNEL ...................................... 72 5.1Introduction ................................... 72 5.2BasisExpansionModels ............................ 72 5.3TimeVaryingChannelEstimationusingSVR ................ 75 5.4SimulationResult ................................ 75 6CONCLUSIONSANDFUTUREWORK ...................... 79 6.1Conclusions ................................... 79 6.2FutureWork ................................... 80 APPENDIX AFPGAIMPLEMENTATIONOFADATRONALGORITHMUSINGSYSYEMGENERATOR .................................... 82 A.1Introduction ................................... 82 A.2AdatronEngineinSystemGenerator ..................... 83 A.2.1AddressControlLogic ......................... 84 A.2.2AdatronEngineblock .......................... 85 A.3ImplementationResults ............................ 86 REFERENCES ....................................... 89 BIOGRAPHICALSKETCH ................................ 94 6

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Table page 2-1Dierencebetweenthetrueweights(h)andestimatedweights(^h)withtheparameters(h=0:05;=0:01;=1)whenSNR=10dB. ................. 24 3-1ATTCChannelDPathParameters ......................... 48 A-1Deviceutilizationsummary ............................. 87 7

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Figure page 1-1Asamplesparsechannelimpulseresponse(adaptedfrom[1]) .......... 11 2-1Searchingparameterusingtheproposedmethodbasedonconvolutioninequalityforentropy. ...................................... 22 2-2PerformancecomparisonofproposedmethodwithMP,MP-2andWiener ... 23 3-1Illustrationoftheideaofanoptimalhyperplaneforlinearlyseparablepatterns.Thesupportvectorsareoutlined.(adaptedfrom[2]) ............... 27 3-2Hyperplaneforthenon-separablecase.Theslackvariablepermitsmarginfailure.Datapointiswrongsideofthedecisionsurface.(adaptedfrom[2]) ....... 29 3-3MSEofchannelestimationasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=15dB,200trainingsamples,suggestedvaluesof1:5andC15) ................................ 38 3-4MSEofchannelestimationasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=25dB,200trainingsamples,suggestedvaluesof0:2) ....................................... 39 3-5NumberofsupportvectorsasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=15dB,200trainingsamples) ......... 40 3-6NumberofsupportvectorsasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=25dB,200trainingsamples) ......... 41 3-7SelectionofwhenSNR=15dB ........................... 42 3-8SelectionofwhenSNR=25dB ........................... 43 3-9TrainingPerformanceasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=15dB,200trainingsamples) ............. 44 3-10TrainingPerformanceasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=25dB,200trainingsamples) ............. 45 3-11GeneralizationPerformanceasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=15dB,200trainingsamples) ......... 46 3-12GeneralizationPerformanceasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=25dB,200trainingsamples) ......... 47 3-13SymbolratesampledresponseofATTCchannelD ................ 49 3-14SVMsolutionofsparsechannelestimationforHDTVchannel. ......... 50 3-15MPsolutionofsparsechannelestimationforHDTVchannel. .......... 51 8

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3-1 andchannelresponseshownisobtainedbypulseshapinglterswith11.5%excessbandwidthand5.38MHzsamplingfrequency. ...................................... 52 3-17Performanceofthesparsechannelestimation.Nonzerocoecientsofthesparsechannelsaredrawnfromauniformdistributionon[1;0:2][[0:2;1] ..... 53 4-1Single-inputtwo-outputchannel. .......................... 57 4-2ConvergenceofAdatron ............................... 64 4-3Convergenceofiterativeregression. ......................... 65 4-4Zerosofsubchannelh1;h2 66 4-550trialsoftheSVMbasedmethodandRobust-SSmethodwhenoverestimatedby10tapsandSNR=30dB ............................. 67 4-6Robustnesswhenthechannelorderisexactandoverestimatedby10taps .... 68 4-750trialsoftheSVMbasedmethodusingtheAdatronandrobust-SSmethodwhenthechannelorderisoverestimatedby20tapsandSNR=20dB. ...... 69 4-8Performancecomparisonwhenthechannelorderisoverestimatedby20taps(raised-cosinepulsefollowedbyamultipathchannel). ............... 70 5-1Basisexpansionmodelofatime-varyingsystem .................. 73 5-2MSEComparisonofLSandSVMwhenN=100;L=3,andQ=3. ....... 76 5-3Blindestimationofbasiscoecientsfcq(l)gQq=1whenN=80;L=3,Q=2,andSNR=10dB(50trials)usingLS. ........................ 77 5-4Blindestimationofbasiscoecientsfcq(l)gQq=1whenN=80;L=3,Q=2,andSNR=10dB(50trials)usingSVM. ...................... 78 A-1MatlabimplementationofAdatronalgorithm ................... 82 A-2TopdesignblockofAdatronFPGAimplementationusingXilinxSystemGenerator ............................................. 83 A-3Addresscontrollogicofthememory ........................ 84 A-4MCodeblockcongurationforupdating 85 A-5AdatronEngineblockdetail ............................. 86 A-6TIDSP(TMS320C33)implementationoftheAdatronalgorithm ........ 88 9

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Inthisdissertationsparsechannelestimationisreformulatedassupportvectorregression(SVR)inwhichthechannelcoecientsaretheLagrangemultipliersofthedualproblem.ByemployingtheVapnik's-insensitivitylossfunction,thesolutioncanbeexpandedintermsofareducednumberofLagrangemultipliers(i.e.,thenonzeroltercoecients)andthenasparsesolutionisfound.Furthermore,methodstoextendtheSVRtechniqueareinvestigatedtoderiveaniterativealgorithmforblindestimationofsparsesingle-inputmultiple-output(SIMO)channels.Thismethodcanbealsousedfornon-sparsechannels,inparticularwhenthechannelorderhasbeenhighlyoverestimated.Inthissituation,thestructuralriskminimization(SRM)principlepushesthesmallleadingandtrailingtermsoftheimpulseresponsetozero.ResultsshowthattheSVRapproachoutperformsotherconventionaltechniquesofchannelestimation.Themaindrawbackofthisapproachisthehighcomputationalcostoftheresultingquadraticprogramming(QP)solution.Toreducethecomplexity,weproposeasimpleandfastiterativealgorithmcalledtheAdatrontosolvetheSVRproblemiteratively.Simulationresultsdemonstratetheperformanceofthemethod. 10

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Inmanywirelesscommunicationsystems,thepropagationchannelsinvolvedexhibitalargedelayspread,butasparseimpulseresponseconsistingofasmallnumberofnonzerocoecients.Suchsparsechannelsareencounteredinmanycommunicationsystems.Terrestrialtransmissionchannelofhighdenitiontelevision(HDTV)signalsarehundredsofdatasymbolslongbutthereareonlyafewnonzerotaps[3].Ahillyterraindelayprolehasasmallnumberofmultipathinthebroadbandwirelesscommunication[4]andunderwateracousticchannelsarealsoknowntobesparse[5].AnexampleofthesparsechannelisshowninFig. 1-1 andestimationofsparsechannelswillbemainlyconsideredinthisdissertation. Figure1-1. Asamplesparsechannelimpulseresponse(adaptedfrom[1]) 11

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MallatandZhang[8]haveproposedthematchingpursuit(MP)algorithmwhichbuildsupasequenceofsparseapproximations.Butthealgorithmisgreedy,whenrunformanyinterations,itmightspendingmostofitstimecorrectingforanymistakesmadeintherstfewterms.Alsoitispossibletore-selectapreviouslyselectedvectorinthedictionary. Toavoidthelimitationsofgreedyoptimization,ChenandDonoho[9]havesuggestedamethodofdecompositionbasedonatrueglobaloptimizationwhichiscalledbasispursuit(BP).TheyproposetopickonewhosecoecientshaveminimumL1-normfromthemanypossiblesolutionstoAx=yasinequation( 1{1 ).minkhk1jyAxj2 BothMPandL1-normregularization(BP)canbeviewedastryingtosolveacombinatorialsparsesignalrepresentationproblem.MPprovidesagreedysolution,whileL1-normbasedBPreplacetheoriginalproblemwitharelaxedversion.Sparsesignalrepresentationproblemhasvariousapplicationssuchastime/frequencyrepresentations,speechcoding,spectralestimation,andimagecoding[10{13].Sparsechannelsarefrequentlyencounteredincommunicationapplicationsandwearegoingtofocusonestimationofsparsechannel. Modernestimationtheorycanbefoundattheheartofmanyelectronicsignalprocessingsystemsdesignedtoextractinformation.Thesesystemsincluderadar,sonar, 12

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Channelestimationusingleast-squaresdoesnotexploitthesparsityofchannelsandneedlongtrainingsymbolstoproduceanaccurateestimate.AMPalgorithmbasedsparsechannelestimationmethodisproposedin[1]andthismethodviewedtheestimationproblemasasparserepresentationproblemandexploitedthesparsenatureofthechannelusingMPalgorithm.Itisshownthat,theMPbasedchannelestimationismoreaccurateandoutperformedtheconventionalleastsquarebasedmethodsinrobustnessandlowcomplexityaccuracy.Aparametricmethodforselectingthestructureofsparsemodelhasbeenproposedin[15].Thismethodexploitstheinformationprovidedbythelocalbehaviorofaninformationcriteriontoselectthestructureofsparsemodels. AnadaptiveL1-normregularizationmethod(sparse-LMS)usingtheaugmentedLagrangiantondthesparsesolutionhasbeenproposedin[16].ThismethodaugmentedtheL1-normconstraintofthechannelimpulseresponseasapenaltytermtotheMSEcriterion.However,determinationoftheparametersinregularizationproblems,hereL1-normofthechannelcoecientswhichplayanimportantroleinachievinggoodestimation,hasremainedanopenproblem.In[17],asearchingtoolfortheL1-normofchannelcoecientsusingtheconvolutioninequalityforentropyisdevelopedtoimprovetheperformance[18,19]. 13

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Supportvectormachines(SVMs)areapowerfullearningtechniqueforsolvingclassicationandapproximationwhichcanbederivedfromthestructuralriskminimization(SRM)principle[32].TheSRMprincipleisacriterionthatestablishesatrade-obetweenthecomplexityofthesolutionandtheclosenesstothedata.Thissupportvectormachine(SVM)techniquetypicallyprovidessparsesolutionsintheframeworkofSRM[33].Specically,theSVMsolutioncanbeexpandedintermsofareducedsetofrelevantinputdatasamples(supportvectors). Therstcontributionofthisdissertationistousesupportvectormachines(SVMs)forsparsechannelestimationbecauseSVMisknowntobuildparsimoniousmodelsandresultsinaquadraticproblem[34].Intheproposedformulation,thechannelcoecients 14

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Thesecondcontributionisaniterativealgorithmforblindestimationofsparsesingle-inputmultiple-output(SIMO)channelsisderived.Theworkin[35]wastherstattempttoapplyanSVM-basedapproachtotheblindidenticationofSIMOchannels.However,thesparsityprovidedbytheSVMsolutionwasnotexplicitlyexploitedin[35].InthisdirectionwepresentanewblindidenticationalgorithmbasedonsupportvectorregressionandspecicallytailoredforsparseSIMOchannelsin[34].ThemainideaisthatthesparseSIMOchannelidenticationcanbereformulatedasasetofregressionproblemsinwhichthechannelcoecientsplaytheroleoftheLagrangemultipliers.Thuswecangetasparsesoluton.Thismethodcanbealsousedfornon-sparsechannels,inparticularwhenthechannelorderhasbeenhighlyoverestimated.Inthissituation,thestructuralriskminimization(SRM)principlepushesthesmallleadingandtrailingtermsoftheimpulseresponsetozero. ThemaindrawbackofapplyingthisSVMtechniqueisthehighcomputationalcostoftheresultingquadraticprogramming(QP)problem.Asathirdcontribution,asimpleandfastiterativealgorithmcalledtheAdatronisusedtosolvetheSVRproblem[36,37]. 15

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16

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where 17

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2{1 ),withresidualrp. Thisalgorithmissummarizedasfollows: ^hkp=(sHkprp1) wheretheprojectionontovectorslisdenotedasPsl=slsHl Possibleproblemofthisalgorithmis,thenumberofnonzerotapsshouldbegivenwhichisunrealisticbecauseitisunknown.Thisalgorithmmightspendmostofitstimecorrectingforanymistakesmadeintherstfewtermsbecausethealgorithmisgreedy. 18

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Wherethersttermisthemeansquarederrorbetweenthereceivedsignalsandtheestimatedsignals,andthesecondtermisthepenaltyterm.ThisconstraintxestheL1-normofthechannelimpulseresponsehtoaconstant.ItiswellknownthattheconcavityofthisL1normfunctionyieldsthesparsesolution[13].Thepenaltyfactorcanbeincludedasanadaptiveparameterbymodifyingthecostfunctionas,J(h;)=E(e2k)+"LXi=1jhij#2 whereisapositivestabilizationconstantthatkeepsthepenaltyfactorbounded.ThismodiedcostfunctionisknownastheaugmentedLagrangian[38].Thestochasticgradientsofthecostfunctionaregivenby,@J(h;) (2{7)@J(h;) Thentheadaptationruleofparametersisgivenbyhi(k+1)=hi(k)+h[2ekukiksign(hi)] (2{9)k+1=k+"LXi=1jhij2k# 19

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2LXi=1jhij! wherehistheasymptoticcoecientvectorfromequation( 2{9 ). Notethattheconstantdenotedbyinequation( 2{5 )isunknownsoithastobesearchedcautiouslytogetagoodperformance.Withaninappropriatechoiceof,theperformanceoftheadaptivesystembecomespoor.Thereforeweproposeamethodbasedontheconvolutioninequalityofentropytodeterminebeforelearningsystemparametersusingthestochasticgradientalgorithminequations( 2{7 )and( 2{8 ). wherehissparsechannel. LetHr()denotetheRenyi'sentropy.TheconvolutioninequalityforRenyi'sentropyis[18][19]Hr(r)Hr(s)+logjhkj;8kmeHr(r)Hr(s)maxjhkj Theequalityholdsifandonlyifthelterisapuredelay. Lethmaxdenotethemaxjhij,thenwecanstarttosearchtheadaptiveparameterinequation( 2{5 )usingtheaboveequation( 2{13 ).Ifthevectorhissparse,thentheL1-normofthechannelimpulseresponsewillnotbemuchbiggerthanhmax.Insteadof 20

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2{7 )and( 2{8 ).Aftertraining,wecomputeMSEbetweenthereceivedsignalsrandthelteroutputs.Then,weincreaseandrepeatlearning.WecanstopthetrainingwhentheMSEstarttoincrease.InFig. 2-1 ,wecanstopat=(4hmax). SincewecannotworkdirectlywiththePDF,anonparametricmethodisusedheretoestimatetheentropy.EntropyestimateisobtainedfromRenyi'squadraticentropyestimatorwhichestimatesthePDFbyParzen-windowmethodusingGaussiankernel[39],[40]. Thisproposedmethodcanbesummarizedasfollows: Initial;andh Set=hmaxfromentropyinequalitybyEq.( 2{13 ) 2{9 ),( 2{10 ) ComputeMSEandcheckifit'sreachedminimum. Ifminimum,setag=false. Ifnotminimum,increase

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Searchingparameterusingtheproposedmethodbasedonconvolutioninequalityforentropy. Table 2-1 showsthecomparisonoftheproposedcriterionperformancetotheWiener'sMSEcriterion.Wesearchedtheusingtheproposedmethodbasedonconvolutioninequalityandxthevalueofto(3hmax)fromFig. 2-1 .Here,theestimatedvalueis5.5995whichisveryclosetothevalueofL1-normoftruevectorh,5.6242.Ascanbe 22

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PerformancecomparisonofproposedmethodwithMP,MP-2andWiener 23

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Dierencebetweenthetrueweights(h)andestimatedweights(^h)withtheparameters(h=0:05;=0:01;=1)whenSNR=10dB. overalltap nonzerotap zerotap dierence dierence dierence 0.1746 0.6737 0.2425 0.1520 seeninTable 2-1 ,theproposedmethodestimatesthezerotapcoecientsbetterthanWiener'sMSEsolutionbutthereisapenaltyforthis.MSEisincreasedinthenonzerotapcoecientsestimationwhichcanbeexplainedbytheperformancelossduetothepenaltyterm. WealsocomparedtheperformanceoftheproposedmethodtoWienerlterandmatchingpursuitalgorithmwhichiswidelyusedinsparseapplication.Fig. 2-2 showsthatMPmethodoutperformsothermethodsintermsofthevarianceandMSEofchannelimpulseresponseestimates.Inthissimulation,MPmethodknowsthenumberofnonzerochannelbutinpracticalthenumberofnonzerostapsareunknown.SoweruntheMPmethodwithincorrectnumberofnonzerotaps(here6)anddenoteas'MP-2'.WecanseefromFig. 2-2 thattheperformanceoftheproposedmethodismuchbetterthanWienerandalsobetterthan'MP-2'from10dB.TheMPmethodoutperformsothermethodsbuttheMPalgorithmisgreedyandneedstoknowtheexactnumberofnonzerochannelcoecientswhiletheproposedalgorithmisanadaptivemethodalsoperformsbetterthanMP-2from10dB. 24

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Inthelinearseparablecase,SVMmaximizesthemarginofthehyperplanewhichclassiestheinputdata.Thehyperplanelocationisdeterminedbysomepointsoftheinputdatawhicharetermedsupportvectors.However,SVMshavealsobeenproposedandappliedtoanumberofdierenttypeofproblemssuchasregressionproblem[41,42],detection[43]andinverseproblems[44].Inregression,thegoalofSVMistoconstructahyperplanethatliesclosetoasmanyofthedatapointsaspossible.Wemustchooseahyperplanewithsmallnormwhileminimizingthelossusing-insensitivelossfunction.AremarkablepropertyofSVMisthesparsityofitssolution.Typicallyasmallnumberofsupportvectorsarenonzero.GirosishowedtheequivalencebetweensparseapproximationandSVMs[33]. Inthischapter,supportvectormachinesarebrieyintroducedincludingtheclassicationandtheregressionproblem.ThenwepresentanewSVM-basedsparsesystemidenticationalgorithm.SVMparameterselectionisalsodiscussed.ForadditionalmaterialofSVMsonecanrefertotheworksofV.Vapnik,C.Burges,B.ScholkopfandA.Smola[32,45]. 25

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3.2.1LinearlySeparablePatterns (3{1) Foragivenvectorwandbiasb,theseparationbetweenthehyperplanedenedinequation( 3{1 )andtheclosestdatapointiscalledthemarginofseparation.Thegoalofasupportvectormachineistondtheparticularhyperplaneforwhichthemarginofseparationismaximized.Thiscanbeformulatedasfollows:supposethatallthetrainingdatasatisfytheconstraintswxi+b1;foryi=1 (3{2)wxi+b1;foryi=1 (3{3) wherewisnormaltothehyperplane,jbj kwkistheperpendiculardistancefromthehyperplanetotheoriginandkwkistheEuclideannormofw(Fig. 3-1 ).Theseconstraintscanbecombinedintoyi(wxi+b)108i Thepointsforwhichtheequality( 3{2 )holdslieonthehyperplaneH1:wxi+b=1withnormalwandperpendiculardistancefromtheoriginj1bj kwk.Thepointsforwhichtheequality( 3{3 )holdslieonthehyperplaneH2:wxi+b=1,withnormalwandperpendiculardistancefromtheoriginj1bj kwk.Thereforethemarginisj1b+1+bj kwk=2 26

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Illustrationoftheideaofanoptimalhyperplaneforlinearlyseparablepatterns.Thesupportvectorsareoutlined.(adaptedfrom[2]) Equation( 3{5 )statesthatmaximizingthemarginofseparationbetweenclassesisequivalenttominimizingtheEuclideannormoftheweightvectorw. Theoptimizationproblemndstheoptimalmarginhyperplanebyminw;b1 2kwk2 subjecttoequation( 3{4 ).ByintroducingpositiveLagrangemultipliersi;i=1;2;:::;l,theprimalformulationoftheproblemisLp1 2kwk2lXi=1iyi(wxi+b)+lXi=1i;i0 (3{7)Lpmustbeminimizedwithrespecttow;bsubjecttotheconstraintsi0.Thisisaconvexquadraticprogramming(QP)problemandwecanusethedualwhichisto 27

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(3{9) whichgivestheconditionsw=NXi=1iyixi (3{11) Substitutingtheseconditionsintoequation( 3{7 )givesthedualformulationoftheLagrangianLDlXi=1i1 2lXi;j=1ijyiyj(xixj) (3{12) subjecttoNXi=1iyi=0i0fori=1;2;:::;N ThesolutioncanbefoundbyminimizingLpormaximizingLDandcanbeecientlysolvedby[46]. 28

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Hyperplaneforthenon-separablecase.Theslackvariablepermitsmarginfailure.Datapointiswrongsideofthedecisionsurface.(adaptedfrom[2]) 3-2 .Totreatthisnonlinearlyseparabledatacases,weneedtointroducepositiveslackvariablesi;i=1;2;:::;lintheconstraints.Theconstraints( 3{2 )and( 3{3 )thenbecome:wxi+b1i;foryi=1 (3{16)wxi+b1+i;foryi=1 (3{17)i0;8i Toassignpenaltiesforerrors,theobjectivefunction( 3{6 )ismodiedtominw1 2kwk2+CNXi=1i Minimizingtherstterminequation( 3{19 )isrelatedtominimizingtheVCdimensionofthesupportvectormachine.AsforthesecondtermPii,itisanupperboundon 29

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2lXi;j=1ijyiyj(xixj) (3{20) subjecttoNXi=1iyi=00iC whereNisthenumberofsupportvectors. 3{6 )theKKTconditionsare[47]@ @wvLp=wvlXi=1iyixiv=0v=1;2;:::;d @bLp=lXi=1iyi=0 (3{23)yi(wxi+b)10i=1;2;:::;l wherevrunsfrom1tothedimensiondofthedata.Theequation( 3{22 )toequation( 3{26 )aresatisedatthesolutionofanyconstrainedoptimizationproblem,providedthatthe 30

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3{26 ),bychoosinganyiwherei6=0(i.e.supportvectors)andcomputingb.TheKKTconditionsfortheprimalproblemarealsousedinthenon-separablecase.TheprimalLagrangianisLp=1 2kwk2+CXiiXiifyi(wxi+b)1+igXiii whereiaretheLagrangemultipliersintroducedtoenforcepositivityoftheslackvariablesi.TheKKTconditionsfortheprimalproblemare@Lp (3{29)@Lp (3{30)yi(wxi+b)1+i0 (3{31)i;i;i0 (3{32)fyi(wxi+b)1+ig=0 (3{33)ii=0 (3{34) wherei=1;2;:::;landv=1;2;:::;d. 31

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3{33 )andequation( 3{34 )canbeusedtodeterminetheb.Anytrainingpointforwhich0<>:0ifjxj
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3{36 )withthe-insensitivelossfunctionjxj,theproblemisreplacedbyfollowingminimizationproblem.ClXi=1(i+i)+1 2kwk2 subjecttof(xi)yi+ii=1;:::;lyif(xi)+ii=1;:::;li0i=1;:::;li0i=1;:::;l 3{38 )is:L(f;;;;;;)=ClXi=1(i+i)+1 2wTw+lXi=1i(yif(xi)i)++lXi=1i(f(xi)yii)lXi=1(ii+ii) (3{39) ThesolutionisgivenbyminimizingtheLagrangianequation( 3{39 )withrespecttof(thatisw.r.tw;b),,andmaximizingwithrespectto;;;.@L (3{41)@L 33

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3{35 )withequation( 3{40 ),wecanexpresstheproblem( 3{38 )asbelowsf(x;;)=lXi=1(ii)+b Inthisdualrepresentationequation( 3{44 ),dataappearsonlyinthedotproduct.Substitutingequation( 3{44 )intheLagrangianeq( 3{39 ),weobtainamaximizationproblemwithrespectto;;;,whereiandiarepositiveLagrangemultiplierswhichsolvethefollowingQuadraticProgramming(QP)problemindualrepresentation:max;L(;)=1 2lXi=1lXj=1(ii)(jj)hxi;xjilXi=1(i+i)++lXi=1yi(ii) (3{45) subjecttotheconstraints0;CXli=1(ii)=0 Thisequation( 3{45 )istheQPproblemwhichisconvexandhavingnolocalminimum.ThishastobesolvedtocomputetheSVM.DuetothenatureoftheQPproblem,onlyasmallnumberofcoecientsiiwillbenonzeroandtheassociatedinputdataarecalledsupportvectors.Interpolationerrorofthesupportvectorsiseithergreaterorequaltosoif=0thenalltheinputbecomessupportvectors. 34

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whererisreceivedvector,andSisinputmatrix.PremultiplySTonbothsidesyieldsSTr|{z}y=STSh|{z}x+y=STx TheSVMminimizetheprimalproblemJ(x)=CMXn=1jynxTsnj+1 2xTx 2MXn=1MXm=1(nn)(mm)MXn=1(n+n)+MXn=1yn(nn) Thesolutionisx=MXn=1(nn)sn,Sh TheLagrangemultipliers(nn)intheSVMsolutionofequation( 3{47 )correspondtochannelcoecientshinequation( 3{46 ),yieldingasparsesolution. 35

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whereallinvolvedmatricesandvectorsarereal-valued(thenotation()isRe()and~()isIm()).Letusintroducethefollowingnotationr=264r~r375;h=264h~h375S=264S~SS~S375 3{48 )as:r=Sh 3{48 )isusefultoextendthecomplex-valuednumbertoreal-valuedmatrices. 36

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2xTx TheparameterCdeterminesthetradeobetweenthemodelcomplexityandthedegreetowhichdeviationslargerthanaretolerated.Forexample,ifCistoolarge,thentheobjectiveistominimizetheempiricalriskonly,notconsideringthemodelcomplexitypartintheoptimizationformulationequation( 3{36 ). Theparametercontrolsthewidthofthe-insensitivezone,usedtotthetrainingdata.Thevalueofcanaectthenumberofsupportvectorsusedtoconstructtheregressionfunction.Thebigger,thefewersupportvectorsareselected.Biggervaluesresultinmoreatestimates.Therefore,bothCandaectmodelcomplexitybutindierentway. SomepracticalapproachestothechoiceofCandareasfollows: ToinvestigatetheeectsofparameterCandviasimulation,wecheckedthechannelestimationerror,numberofselectedsupportvectors,trainingerrorandtesterrorwithvariousandCvaluesat15dBand25dB.SparsechanneldataforthissimulationsaretheHDTVchanneldatafromATTC(AdvancedTelevisionTestCenter)tests[3].DependenceofMSE(MeanSquaredError)ofchannelestimationasafunctionofchosenCandvaluesforATTCchannelDdatasetat15dBisshowninFig. 3-3 .Fig. 3-4 alsoshowsthe 37

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OnecanclearlyseethatCvaluesabovecertainthresholdhaveonlyminoreectontheMSEofchannelestimation.Accordingtothesimulationresults,suggestedCis15andsuggestedvalueis1.5forFig. 3-3 andFig. 3-4 Figure3-3. MSEofchannelestimationasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=15dB,200trainingsamples,suggestedvaluesof1:5andC15) 38

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MSEofchannelestimationasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=25dB,200trainingsamples,suggestedvaluesof0:2) 39

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3-5 showsthenumberofsupportvectorsselectedbychosenCandvaluesforATTCchannelDdatasetwith200trainingsamplesat15dBandFig. 3-6 isfor25dB.Wecanseethatsmallvaluescorrespondtohighernumberofsupportvectors,whereasparameterChasnegligibleeectonthenumberofsupportvectors.Fig. 3-7 andFig. 3-7 showsthedependenceofMSEasafunctionofvaluesmoreclearlyat15dBand25dB. Figure3-5. NumberofsupportvectorsasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=15dB,200trainingsamples) 40

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NumberofsupportvectorsasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=25dB,200trainingsamples) 41

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(b)Channelestimationerror SelectionofwhenSNR=15dB 42

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(b)Channelestimationerror SelectionofwhenSNR=25dB 43

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3-9 andFig. 3-10 showstheperformanceintermsoftrainingerrorasafunctionofCandvalues.WecanseethattrainingerrorincreasesasvaluesincreasesandparameterChasnegligibleeectonthenumberofsupportvectors. Figure3-9. TrainingPerformanceasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=15dB,200trainingsamples) 44

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TrainingPerformanceasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=25dB,200trainingsamples) 45

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3-11 andFig. 3-12 showsthegeneralizationperformanceintermsoftesterrorasafunctionofCandvalues.WecanseethatthetesterrorincreasesafteranoptimalvalueswhichcorrespondstothevaluessuggestedfromFig. 3-3 andFig. 3-4 Figure3-11. GeneralizationPerformanceasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=15dB,200trainingsamples) 46

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GeneralizationPerformanceasafunctionofCandvaluesforHDTVsparsechannelDdatafromATTC(SNR=25dB,200trainingsamples) 47

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3-1 andchannelresponseshowninFig. 3-13 whichisobtainedbypulseshapinglterswith11.5%excessbandwidthand5.38MHzsamplingfrequency.Theothersparsechanneldatasethasnonzerocoecientswhicharedrawnfromauniformdistributionon[1;0:2][[0:2;1]sononzerotapsofthischannelaremoredistinctwhereassomecoecientsoftheHDTVchannelareverysmall. Table3-1. ATTCChannelDPathParameters pathdelay(s)delay(T)phaseatten 10.000.00288deg20dB21.809.68180deg0dB31.9510.490deg20dB43.6019.3772deg18dB57.5040.35144deg14dB619.80106.52216deg10dB Fig. 3-14 showstheSVMsolutionoftheestimationforHDTVchannelandFig. 3-15 fortheMPsolution.PerformanceoftheHDTVsparsechannelestimationisshowninFig. 3-16 .WecanseethatSVMoutperformsothermethodsintermsofMSEandvarianceinFig. 3-16 Fig. 3-17 showstheperformanceoftherstsparsechanneldatasetwhichhasnonzerocoecientsdrawnfromauniformdistributionon[1;0:2][[0:2;1].WecanseethatwhenthesparsechannelhasdistinctnonzerocoecientsMPmethodperformsverywell.Butwhenthechanneltapsaresmallwhichcouldbeburiedundernoise,thenMPperformspoorasinFig. 3-16 48

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SymbolratesampledresponseofATTCchannelD 49

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(b)Image SVMsolutionofsparsechannelestimationforHDTVchannel. 50

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(b)Image MPsolutionofsparsechannelestimationforHDTVchannel. 51

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Performanceofthesparsechannel(HDTVchannel)estimation.MultipathparametersofATTCchannelDarelistedinTable 3-1 andchannelresponseshownisobtainedbypulseshapinglterswith11.5%excessbandwidthand5.38MHzsamplingfrequency. 52

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Performanceofthesparsechannelestimation.Nonzerocoecientsofthesparsechannelsaredrawnfromauniformdistributionon[1;0:2][[0:2;1] 53

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54

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Thestructuralriskminimization(SRM)principleisacriterionthatestablishesatrade-obetweenthecomplexityofthesolutionandtheclosenesstothedata.Inparticular,thesupportvectormachine(SVM)technique,whichcanbederivedfromtheSRMprinciple,typicallyprovidesarobustsolution.Theworkin[35]wastherstattempttoapplyanSVM-basedapproachtotheblindidenticationofSIMOchannels[35].However,thesparsityprovidedbytheSVMsolutionwasnotexplicitlyexploitedin[35].Laterworkinthisdirectionwaspresentedin[34],inwhichanewblindidenticationalgorithmbasedonsupportvectorregressionandspecicallytailoredforsparseSIMOchannelswasproposed.Themainideaof[34]isthatthesparseSIMOchannelidenticationcanbereformulatedasasetofregressionproblemsinwhichthechannelcoecientsplaytheroleoftheLagrangemultipliers.Byusingthe-insensitiveVapnik'slossfunctionintheregressionproblem,alargenumberofLagrangemultilpliers(and,therefore,alargenumberofltercoecients)becomezero,thusyieldingasparselterestimate. 55

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4-1 .Inblindchannelidentication,weneedtoidentifytheunknownchannelresponses,h1;h2,fromthereceivedsignalsonly.IftheorderofthechannelsisM,thenthereceivedsignalri(n)fromtheithchannelisri(n)=MXk=0hi(k)s(nk)+ei(n);i=1;2: Whenwecastri(n);hi(k);s(n);ei(n)intovectorsri,hi,sandeirespectively,equation( 4{1 )becomesri=his+ei;i=1;2 (4{2) 1wheredenotesconvolution.AsshowninFig. 4-1 ,usingthechanneloutputs(r1;r2)andthechannelestimates(^h1;^h2),onecanobtainthefollowingmatrix-vectorform,:y1=r1^h2;y2=r2^h1: 56

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Single-inputtwo-outputchannel. whereRi'sareToeplitzmatricesdenedasRi=0BBBBBBB@ri(M)ri(0)ri(M+1)ri(1).........vi(M+N1)ri(N1)1CCCCCCCA; orequivalently,R^h=0; whereR=R2R1;^h=264^h1^h2375: 57

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4.3.1IterativeRegression 4{3 ),wecanformulatethefollowingtwocoupledregressionproblems:R1^h2'y1; Ourgoalistomakey1y2.Thisisanintuitiveandsimplechoice,becauseitdragstheactualoutputsy1andy2closertoeachotherinordertoachievetheequalityin( 4{3 ).Ateachiteration(given^h1and^h2),desiredoutputisconstructedasyd=y1+y2 Thisiterativealgorithmcanbesummarizedasfollowing 1.Initialize^h1=^h2=[nd]: 4.Normalizethesolutionandgotostep2. 58

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CanonicalCorrelationAnalysis(CCA)isawell-knowntechniqueinmultivariatestatisticalanalysistondmaximallycorrelatedprojectionsbetweentwodatasets.CCAwasdevelopedbyH.Hotelling[51]andithasbeenwidelyusedineconomics,meteorologyandinmanymoderninformationprocessingelds,suchascommunicationproblems[52],statisticalsignalprocessing[53],independentcomponentanalysis[54]andblindsourceseparation[55]. 4{8 ),( 4{9 )]:RH1R1^h2| {z }x1=RH1yd| {z }~y1; {z }x2=RH2yd| {z }~y2: TheresultantregressionproblemshaveinputmatricesthataresimplytheconjugatetransposedinputmatrixRHi,andthecorrespondingdesiredoutputvectorsbecomeRHiyd.Moreover,thenewregressorxiadmitsanexpansionintermsoftheltercoecients,which,inthisway,becometheLagrangemultipliersoftheSVMformulation. TheSVMmethodminimizesthefollowingcostfunctionJ(xi)=CMXn=1(n+n)+1 2kxik2;i=1;2 (4{12) 59

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Inequation( 4{12 ),theregularizationparameterCcontrolsthetradeobetweenthetrainingerrorandthecomplexityofthesolution.Ontheotherhand,isaparameterthatdeterminestheprecisionoftheregressionandthereforecontrolsthesparsenessofthenalsolution.Then,thesolutionisalinearcombinationofinputdatax=MXn=1(nn)r wheren;naretwodierentLagrangemultipliers. Inequation( 4{13 ),onlyasmallnumberofLagrangemultipliers(nn),whichcorrespondtothechannelcoecientsh(0);h(1);:::;h(M1)willbenonzero.Accordingly,theoverestimatedchannelcoecientswillbezerosbytheSRMprinciple. 4{12 )isthemaindrawbackofapplyingtheSVMtechniquetopracticalestimationproblems.Severaltechniqueshavebeenproposedtosolvethisproblem,includingtheuseofiterativereweightedleastsquares(IRWLS)techniques[56,57]andtheAdatronalgorithm[2,36,58].TheIRWLSrequiresamatrixinversionateachiterationsothecomputationalburdencouldbeconsiderablyhighevenforamoderatenumberofdata.Ontheotherhand,theAdatronalgorithmisamuchsimplerleastmeansquare(LMS)-likeadaptivealgorithmand 60

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4{10 )and( 4{11 ).Inthedualrepresentation,theoptimizationprobleminequation( 4{12 )canbewrittenasL=1 2MXn=1MXm=1(nn)(mm)hxn;xmiMXn=1(n+n)+MXn=1yn(nn) (4{14) subjectton;n2[0;C]andhidenotesdotproduct. TheAdatronalgorithmmaximizestheabovedualproblemwithgradientascenttechniques[58].Specically,theLagrangemultipliersareupdatedaccordington=NXm=1(mm)hxn;xmi+ynb!; followedbyupdatingnandnwith(n+n)yand(n+n)y,respectively,whereay,maxfa;0gandisthelearningrate.Inaddition,thebiasbisupdatedastob+nnandtheevolutionofthisbiasvaluecanbeusedtochecktheconvergenceofthealgorithm. Aswecanseeinequations( 4{15 )and( 4{16 ),theAdatronalgorithmisverysimpletoimplementespeciallyinDSP/FPGAhardware.Torunthisalgorithm,alloneneedsisjustanadderandamultiplier.Furthermore,thecomputationtimeofAdatronincreaseslinearlywiththenumberofdatawhiletheconventionalQP'sincreasesexponentially.Thissimplicityisitsmainadvantage.Usingthisalgorithm,weproposeablindsparsechannelestimationmethodassummarizedunderAlgorithm3. 61

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1.Set;Candinitializen=0;n=0;8n;andb=0. 2.Foralltrainingsamplesn=1;;M,execute If(n+n)<0thensetn=0;elsenn+n. If(n+n)<0thensetn=0;elsenn+n. 3.Repeatstep2untilconvergence. 4.h(n)=(nn);forn=1;;M. CombiningtheiterativeregressionprocedureandtheAdatronalgorithm,overallalgorithmissummarizedinAlgorithm4. Thecomputationcostoftheproposedmethodisniter1niter2O(M)whereniter1istheiterationnumberofAdatronalgorithm(Algorithm2)andniter2istheoneforiterativeregressionalgorithm(Algorithm3).AdatronconvergenceplotsaredepictedinFig. 4-2 anditshowslargervalueshavefastconvergencespeed.ConvergenceoftheproposedoverallalgorithmisshowninFig. 4-3 62

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1.Intialize;C;and^h1=^h2=[nd]. 2.ObtaintheoutputsX1^h2=y1andX2^h1=y2. 3.Formthedesiredsignalyd=y1+y2 4.ConstructthetransformedSVregressionproblems: 5.SolvetheQPproblemusingAdatron. 6.Normalizethesolutionandgotostep2. 4{14 )is:L=MXm=1n(mm)hxn;xmi+ynnn1 2(n)2hxn;xni=nNXm=1(mm)hxn;xmi+ynb!1 2(n)2hxn;xni=nn 2(n)2hxn;xni=(n)21 Forn,itcanbeshownthatL=(n)21 Fromeqs.( 4{17 )and( 4{18 ),wecangettherelation0
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ConvergenceofAdatron Aswecannoticefrom( 4{17 )and( 4{18 ),boundofthelearningrateisdatadependentandcanbedeterminedfrom0<<2 64

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Convergenceofiterativeregression. IntherstsimulationweconsiderasparseSIMOsystemwhichconsistsofasingletransmitantennaandtworeceiveantennas.Thetwosparsechannelsarerespectively, 65

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Zerosofsubchannelh1;h2 4-4 ,weplotthezerosofh1;h2.Notethattherearepairsofclosezeroswhichimpairsubspacebasedmethodbecauseofabadlyconditionedinputcorrelationmatrix. Fig. 4-5 showstherobustnesstoorderoverestimationwhenSNRis20dB.Itisevidentthattheproposedmethodoutperformsothermethodsinhighlyoverestimatedchannelorderestimate.PerformanceissummarizedinFig. 4-6 andshowstheproposedmethodperformsmuchbetterthanRegaliamethodinidentifyingthecoecientsofzerotapsorverysmalltaps. Inthenextexampleweconsideraraised-cosinepulsewithduration4T(Tisthesymbolperiod)witharoll-ofactor0.1andthemultipathchannelish(t)=(t)0:7(t

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(b)Robust-SSmethod 50trialsoftheSVMbasedmethodandRobust-SSmethodwhenoverestimatedby10tapsandSNR=30dB 67

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Robustnesswhenthechannelorderisexactandoverestimatedby10taps 4-8 depictstheperformanceatdierentSNRs.ThisexampleshowsthatproposedSVMbasedmethodcanbealsousedfornon-sparsechannels,inparticularwhenthechannelorderhasbeenhighlyoverestimated.Inthissituation,thestructuralriskminimization(SRM)principlepushesthesmallleadingandtrailingtermsoftheimpulseresponsetozero.Notethattheperformanceofourproposedmethodismuchbetterthanothermethods,especiallyatlowSNR.InFig. 4-7 ,50trialsofourproposedalgorithmandtherobustmethodproposedin[2],itisclearthattheestimationofourproposedmethodatzerotapcoecientsismuchbetterthantherobust-SSmethod. Inthischapter,ourSVMbasedsparseestimationmethodisextendedinthefollowingdirection:rst,arobustalgorithmfortheblindestimationofanon-sparsechannelwhenthechannelorderhasbeenhighlyoverestimatedisderived;secondly,toavoidthehighcomputationalcostinsolvingaQPproblem,afastandsimplealgorithmcalledthe 68

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(b)Robust-SSmethod 50trialsoftheSVMbasedmethodusingtheAdatronandrobust-SSmethodwhenthechannelorderisoverestimatedby20tapsandSNR=20dB. 69

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Performancecomparisonwhenthechannelorderisoverestimatedby20taps(raised-cosinepulsefollowedbyamultipathchannel). 70

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71

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Inthischapter,weproposetousetheSVMtechniquewiththeexponentialbasisexpansionmodeltoidentifythetime-varyingchannel. (5{1) wheretheTVimpulseresponseh(n;l)dependsontimen.Inthisthesis,adeterministicbasisexpansionisusedtomodeltheTVimpulseresponseh(n;l)[62,63].TVimpulseresponseofrapidlyfadingchannelsisexpandedoverabasisofcomplexexponentialsthatariseduetoDopplereectsencounteredinthemultipathenvironment.TheTVtapsareexpressedasasuperpositionofTVbaseswithtimeinvariant(TI)coecients.By 72

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Basisexpansionmodelofatime-varyingsystem assigningtimevariationstothebases,rapidlyfadingchannelscanbecaptured.NotethatcomplexexponentialsmodeltheDopplereectsin( 5{2 ),andbyusingthem,theestimationoftime-varyingchannelsystemcanbecastintoatime-invariantestimationproblem.Thebasisexpansionmodelisgivenbyx(n)=LXl=0"QXq=1cq(l)ej!qn#| {z }h(n;l)s(nl)+v(n) (5{2) wheres(n)isinput,h(n;l)isnitelyparameterizedforeachlaglviaitsexpansioncoecientscq(l)ontoknownexponentialbasesf1;ej!2n;;ej!Png,asdepictedinFig. 5-1 .ToestimatetheTIparametersfcq(l)g,weassumetheknowledgeofthebasefrequenciesf!igPi=1.Thiscanbyestimatedusingtestsforcyclostationarityoradaptivemaximum-likelihoodmethods[65,66]. From( 5{2 ),wehavex(n)=QXq=1"LXl=0cq(l)s(nl)ej!qn#+v(n) (5{3) 73

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whereSq=266666664s(1)ej!q100s(2)ej!q2s(1)ej!q20............s(N)ej!qNs(N1)ej!qNs(NL+1)ej!qN377777775cq=cq(0)cq(1)cq(L1)Tx=x(1)x(2)x(N)Tv=v(1)v(2)v(N)T 5{5 )canbeestimatedas^c=(SHS)1SHx ThenTVchannelcoecientsh(n;l)areestimatedusingtheestimatedc.Whenthenumberoftrainingsymbolisshort,theperformanceofLSestimate( 5{6 )isbad.In[61],theMPmethodisusedwiththepolynomialbasismodeltondthebestalignedcolumntothereceivedsignal.Inthischapter,weextendthepreviouswork[34,37]totime-varyingenvironmentinadata-aidedmanner. 74

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5{5 ),thesystemmodelisx=S1S2SQ266666664c1c2...cQ377777775+v=Sc+v 2cTc wherennaretheLagrangemultiplierandMisthenumberofsupportvectors. AlsoSVM-basedblindmethoddevelopedinpreviouschaptercanbeeasilyappliedtothisTVcase. whererdenotesrealizationandRisthenumberofrealizations.InFig. 5-3 ,weillustratetheestimatedTIparameterfcq(l)gQq=1ofbasisexpansionmodelusingbothLSandSVMmethod.Theinputsignalisi.i.d.BPSKsignalandSNR=10dB.N=100trainingsymbolswereusedwithachannelorderL=3andtheQ=3baseswerechosen.Allplots 75

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5-2 illustratesthatNMSEofproposedSVMmethodisbetterthanLSmethod.Fig. 5-3 andFig. 5-4 showsthe50trialsofblindestimationofTIparameterfcq(l)gQq=1ofbasisexpansionmodelusingbothLSandSVMmethod. Figure5-2. MSEComparisonofLSandSVMwhenN=100;L=3,andQ=3. Inthischapter,weproposedtousetheSVMtechniquewhichdevelopedinpreviouschapterwiththeexponentialbasisexpansionmodeltoidentifythetime-varyingchannel.SimulationdemonstratedthatBEM-SVMmethodsuccessfullyappliedtotimevaryingchannelestimation. 76

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Blindestimationofbasiscoecientsfcq(l)gQq=1whenN=80;L=3,Q=2,andSNR=10dB(50trials)usingLS. 77

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Blindestimationofbasiscoecientsfcq(l)gQq=1whenN=80;L=3,Q=2,andSNR=10dB(50trials)usingSVM. 78

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Previousworkin[16]formulatessparsechannelestimationastheoptimalmeansquareerror(MSE)estimationofthechannelimpulseresponseregularizedwithaL1-normconstraint.Inchapter3,L1-normofthechannelimpulseresponse,whichplayanimportantroleinachievingthesparsesolution,isestimatedbyusingtheconvolutioninequalityforentropyandthisinformationisexploitedtoimprovetheperformanceofsparse-LMS. Inchapter4,supportvectormachinesarebrieyintroducedincludingtheclassicationandtheregressionproblem.ThenwepresentanewSVM-basedsparsesystemidenticationalgorithm.Wereformulatethesparsechannelestimationproblemasasupportvectorregression(SVR)probleminwhichthechannelcoecientsaretheLagrangemultipliersofthedualproblem.ByemployingVapnik's-insensitivitylossfunction,thesolutionisexpandedintermsofareducednumberofLagrangemultipliers(i.e.,thenonzeroltercoecients)andthenasparsesolutionisfound.ThenthisSVRbasedmethodisalsoappliedtoderiveaniterativealgorithmforblindestimationofsparsesingle-inputmultiple-output(SIMO)channels,inparticularwhenthechannelorderhasbeenhighlyoverestimated. Inchapter5,thepreviouswork[34,35]isextendedinthefollowingdirection:rst,arobustalgorithmfortheblindestimationofanon-sparsechannelwhenthechannelorderhasbeenhighlyoverestimatedisderived;secondly,toavoidthehighcomputationalcostinsolvingaQPproblem,afastandsimplealgorithmcalledtheAdatron[36]isused. 79

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Time-varyingchannelestimationisalsoconsideredbecauseitisamajorobstacletoincreasethecapacityandreliabilityofwirelesscommunicationsystems.Forslowfadingchannels,adaptivealgorithmfortime-invariantsuchasrecursiveleast-squares(RLS)andleastmean-square(LMS)cangivealternativebutthisadaptivealgorithmsdivergewhenchannelvariationsexceedtheconvergencetimeofalgorithm.Inthiscaseadditionalinformationofthetime-varyingchannelisneeded.Mostmodelsoftime-varyingchannelstreatthetapcoecientsasuncorrelatedstationaryrandomprocesses.Byusingbasisexpansionmodel,estimationoftime-varyingchannelsystemcanbecastintoatime-invariantestimationproblem.TVimpulseresponseofrapidlyfadingchannelsisexpandedoverabasisofcomplexexponentialsthatariseduetoDopplereectsencounteredwithmultipathenvironment.TheTVtapsareexpressedasasuperpositionofTVbaseswithTIcoecients.Inchapter6,weproposetousetheSVMtechniquewiththeexponentialbasisexpansionmodeltoidentifythetime-varyingchannel. Inappendix,toinvestigatethefeasibilityoftheFPGAimplementationoftheproposedalgorithm,thecorecomputationofthisAdatronissimulatedinFPGAusingSystemGeneratorwhichisahigh-leveldesigntoolforXilinxFPGAsthatextendsthecapabilitiesofSimulinktoincludeaccuratemodelingofFPGAcircuits.TheperformanceisalsocomparedtotheTITMS320C33DSPimplementation. 80

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ItiswellknownthatSVMperformance(estimationaccuracy)dependsonagoodsettingofmeta-parametersCand.OptimalselectionoftheseparametersaredicultbecauseSVMmodelperformancedependsonalltheseparameterssowepresentedaSVMmeta-parametersettingselectionviathesimulation.Scholkopfetal[68]suggesttocontrolanotherparameterinsteadofsoweshouldinvestigatetheoptimalparametersettingofSVM. Inbasisexpansionmodel(BEM)fortimevaryingchannelestimation,weassumedtheknowledgeofthebasefrequencies.Thiscanbyestimatedusingtestsforcyclostationarityoradaptivemaximum-likelihoodmethods[65,66]soweshouldexplorehowtodecidethebasisofBEM. FPGAimplementationofSVMissimulatedbymappingAdatronontoFPGAbutitwasnotanecientimplementationbecausethewayofsolvingisbatch.MemoryforstoringkernelmatrixwasneededandweshouldwaituntilthecomputationsoflastsamplesoweshouldinvestigateanecientFPGAimplementationalgorithmbasedontheonlinesolutionin[67]. Summarizingthefutureworks, 81

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4{16 canbeimplementedinbelowmatlabscript: FigureA-1. MatlabimplementationofAdatronalgorithm Inthisappendix,weuseSystemGeneratortomaptheAdatronalgorithmshownaboveontoaFPGAandinvestigatethefeasibilityofFPGAimplementationofthealgorithm.AlsotheperformancewillbecomparedtoTITMS320C33DSPimplementation. 82

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A-2 .Addresscontrollogicblockgeneratesthecontrolsignals(address,writeenable)ofthememoryforstoringtheinput,kernelevaluationsandtheLagrangemultipliers.Kernelscanbeprecalculatedinsoftwareandeachrowofkernelscanbefeedthroughtheinpgatewayin.TheLagrangemultipliersarestoredinthesingleportram1,whiletheinputupdateisstoredintheworkspace.TheAdatronengineinFig. A-2 calculatestheupdateoftheLagrangemultipliersusingthedataavailableinthesharedmemory.Twooutputs,out1andout2,fromtheAdaengineblockcorrespondtonandninAdatronalgorithm,andarerestoredinworkspace1thruthegatewayout. FigureA-2. TopdesignblockofAdatronFPGAimplementationusingXilinxSystemGenerator 83

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A-3 showsthedetaillogicsintheaddresscontrolblock.CountersareforgeneratingtheaddressofthememorieswhichstorestheinputdataandtheLagrangemultipliers.SystemGeneratorprovidesaninterfacetotheembeddedprocessorintothedesignsothatSoftwarecanperformread/writeoperationstoasharedmemorythroughnamedassociation.Datawriteenableandalphawriteenablearethesharedmemoriesthatenablesthewritingtomemory. FigureA-3. Addresscontrollogicofthememory 84

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A-5 showsthedetailbuildingblocksofthebelowAdatronalgorithm.n=NXm=1(mm)hxn;xmi+ynb!;If(n+n)<0thensetn=0;elsenn+nn=MXm=1(mm)hxn;xmi+yn+b!;If(n+n)<0thensetn=0;elsenn+n A-4 .WecanseethatconguringtheXillinxMCodeblockisaneasiersolutionthanbuildingthelogictogetherthroughXilinxblocksetlogic. FigureA-4. MCodeblockcongurationforupdating

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AdatronEngineblockdetail A-1 .ThebasicbuildingblockofXilinxVirtexdevicesisthelogiccell(LC).AccordingtotheXilinxVirtexdatasheet[73],anLCincludesa4-inputlookuptable(LUT),carrylogic,andastorageelement.TheXilinxVirtex-Earchitecturecontainscongurablelogicblocks(CLBs).EachVirtex-ECLBcontainsfourLCsandaCLBconsistsoftwoslices. 86

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A-6 showstheimplementationofthebasiccomputationofAdatronalgorithmwithTITMS320C33assemblylanguage.NotethatwecanincreasetheperformanceoftheFPGAimplementationbyparallelizingtheoperation. TableA-1. Deviceutilizationsummary SelectedDevice4vsx35-10 NumberofSlices278NumberofSliceFlipFlops210Numberof4inputLUTs445NumberofFIFO16/RAMB16s2NumberofGCLKs1NumberofDSP48s1 Inthisappendix,weuseSystemGeneratortomaptheAdatronalgorithmshownaboveontoaFPGAresourceincludingaddressgenerationlogic,memory,Adatronengine,andthelogicfabricofVirtexFPGAs.AlsotheperformanceiscomparedtoTITMS320C33DSPimplementation.ResultsshowedthattheFPGAimplementationofAdatronisfeasibleandhasmoredesignmarginthantheDSPimplementation. 87

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TIDSP(TMS320C33)implementationoftheAdatronalgorithm 88

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[46] S.R.Gunn,\MATLABsvmtoolboxavailableathttp://www.isis.ecs.soton.ac.uk/isystems/kernel/,"UniversitySouthampton,ImageSpeechandIntelligentSystemsResearchGroup,Southampton,U.K.,1998. [47] R.Fletcher,PracticalMethodsofOptimization,JohnWileyandSons,2ndedtion,1987. [48] G.P.McCormick,NonlinearProgramming:Theory,AlgorithmsandApplications,JohnWileyandSons,1983. [49] D.MatteraandS.Haykin,\Supportvectormachinesfordynamicreconstructionofachaoticsystem,"inAdvancesinKernelMethods:SupportVectorMachine,B.SchEd.MITPress,1999. [50] A.Smola,N.Murata,B.Scholkopf,andK.Muller,\Asymptoticallyoptimalchoiceof-lossforsupportvectormachine,"inProc.ICANN1998,1998. [51] H.Hotelling,\Relationsbetweentwosetsofvariates,"Biometrika,vol.28,pp.321{377,1936. [52] M.F.Kahn,W.A.Gardner,andM.A.Mow,\Programmablecanonicalcorrelationanalysis:aexibleframeworkforblindadaptivespatialltering,"inTwenty-NinthAsilomarConf.Signals,Syst.,Comput.,Oct1995,pp.351{356. [53] L.L.ScharfandJ.T.Thomas,\Wienerltersincanonicalcoordinatesfortransformcoding,ltering,andquantizing,"IEEETran.onSignalProcessing,vol.46,no.3,pp.647{654,March1998. [54] FrancisR.BachandMichaelI.Jordan,\Kernelindependentcomponentanalysis,"JournalofMachineLearningResearch,vol.3,pp.1{48,2002. [55] O.Friman,M.Borga,P.Lundberg,andH.Knutsson,\AdaptiveanalysisoffMRIdata,"NeuroImage,vol.19,no.3,pp.837{845,2003. [56] F.Perez-Cruz,A.Navia-Vazquez,P.Alarcon-Diana,andA.Artes-Rodrguez,\Anirwlsprocedureforsvr,"inProceedingsoftheEUSIPCO'00,Tampere,Finland,Sept2000. [57] C.C.Gaudes,J.Via,andI.Santamaria,\Anirwlsprocedureforrobustbeamformingwithsidelobecontrol,"inProceedingsoftheThirdIEEESensorArrayandMulti-channelSignalProcessingWorkshop,Sitges,Spain,July2004. [58] J.K.AnlaufandM.Biehl,\TheAdaTron:anadaptiveperceptronalgorithm,"EurophysicsLetters,vol.10,pp.687{692,Dec.1989. [59] M.Visintin,\Karhunen-loeveexpansionofafastrayleighfadingprocess,"IEEElectronicsLetters,vol.32,pp.1712{1713,Aug1996.

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[60] D.K.BorahandB.D.Hart,\Frequency-selectivefadingchannelestimationwithapolynomialtime-varyingchannelmodel,"IEEETran.oncommunications,vol.47,pp.862{873,June1999. [61] Y.LiuandD.K.Borah,\Estimationoftime-varyingfrequency-selectivechannelsusingamatchingpursuittechnique,"inWirelessCommunicationsandNetworking,Mar2003,vol.2,pp.941{946. [62] GeorgiosB.GiannakisandCihanTepedelenlioglu,\Basisexpansionmodelsanddiversitytechniquesforblindidenticationandequalizationoftime-varyingchannels,"inProceedingsoftheIEEE,Oct1998,vol.86,pp.1969{1986. [63] HuiLiuandGeorgiosB.Giannakis,\Deterministicapproachesforblindequalizationoftime-varyingchannelswithantennaarrays,"IEEETran.onSignalProcessing,vol.46,no.11,pp.3003{3013,November1998. [64] M.Martone,\Multiresolutonsequencedetectioninrapidlyfadingchannelsbasedonfocusedwaveletdecomposition,"IEEETran.oncommunications,vol.49,pp.1388{1401,Aug2001. [65] S.BarbarossaandA.Scaglione,\Time-varyingfadingchannels,"2000. [66] GregoryD.Durgin,\Theoryofstochasticlocalareachannelmodelingforwirelesscommunications,". [67] JyrkiKivinen,AlexanderJ.Smola,andRobertC.Williamson,\Onlinelearningwithkernels,"IEEETran.onSignalProcessing,vol.52,no.8,pp.2165{2176,August2004. [68] B.Scholkopf,P.Bartlett,A.Smola,andR.Williamson,\Supportvectorregressionwithautomaticaccuracycontrol,"inProceedingsofICANN98,PerspectivesinNeuralComputing,L.Niklasson,M.Boden,andT.Ziemke,Eds.,pp.111{116.Springer,Berlin,1998,1998. [69] J.Hwang,B.Milne,N.Shirazi,andJ.Stroomer,\Lecturenotesincomputerscience,"inSystemLevelToolsforDSPinFPGAs,pp.534{543.2001. [70] [71] [72] [73] [74] [75] DonghoHan,YadunandanaN.Rao,JoseC.Principe,andKarlGugel,\Real-timepca(principalcomponentanalysis)implementationondsp,"inProceedingsofIJCNN'04,Budapest,Hungary,July2004,pp.2159{2162.

PAGE 94

DonghoHanwasborninSeoul,Korea.HereceivedhisBachelorofSciencedegreesinelectricalengineeringfromtheYonseiUniversity,Seoul,Korea,in1993.From1993to1998,hewasahardwareengineerinSamsungElectronicswherehedevelopedseveralcpuboardsforPBXsystem.HealsoreceivedhisMasterofSciencedegreeinelectricalandcomputerengineeringfromtheUniversityofFloridain2001.HewasintheComputationalNeuroEngineeringLaboratoryintheElectricalandComputerEngineeringDepartmentattheUniversityofFloridaduringhisPh.Dstudy.Hispresentresearchinterestsareintheareasofsignalprocessing,neuralnetwork,communicationalgorithmsandtheirimplementationsinFPGAandDSP.WhileaPh.D.candidate,DonghointernedwithMotorola,wherehereturnedupongraduationforfull-timeemploymentasaDSPengineer. 94


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ANEW CLASS OF SPARSE CHANNEL ESTIMATION METHODS BASED ON
SUPPORT VECTOR MACHINES


















By
DONGHO HAN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2006
































Copyright 2006

by

Dongho Han


































To my family









ACKENOWLED G1\ENTS

I would like to sincerely thank my advisor, Dr. .Josi4 C. Principe, and co-advisor,

Dr. Liuqing Yang, for their support, encouragement and patience in guiding the research.

I would also like to thank Dr. .John Harris and Dr. Douglas Cenzer, for being on my

coninittee and for their helpful advice.

I gratefully acknowledge Dr. Ignacio Santaniarfa and .Javier Via for providing

numerous -II_a----- ~~-R and for helpful coninents on several of my papers that are part

of this dissertation.

I also appreciate the helpful discussions front people in the Computational Neuroengfineeringf

Lah.

Finally, I wish to thank my wife, Hyewon Park, for her support, as well as my dear

princesses, Eujin and .Jiwon, for providing hig motivations to finish my study.











TABLE OF CONTENTS

page

ACK(NOWLEDGMENTS ......... . .. .. 4

LIST OF TABLES ......... ..... .. 7

LIST OF FIGURES ......... .... .. 8

ABSTRACT ......... ...... 10

CHAPTER

1 INTRODUCTION ......... ... .. 11

1.1 Background ........ ... .. 12
1.2 Contribution of this Thesis ....... .. 14
1.3 Organization of the Dissertation . ..... .. 15

2 CONVENTIONAL SPARSE SOLITTION ...... .. 17

2.1 Observation Model ......... . .. 17
2.2 Matching Pursuit (11P') Based Method ...... .. 18
2.3 SPARSE-LMS ... . . .. .. 19
2.3.1 Convolution Inequality For Renyi's Entropy ... .. .. 20
2.4 Simulation Results ......... . 22

:3 SUPPORT VECTOR MACHINE BASED METHOD ... .. .. 25

:3.1 Introduction ......... . .. .. 25
:3.2 Support Vector Classification ........ .. .. 26
:3.2.1 Linearly Separable Patterns . ..... .. 26
:3.2.2 ?-in 1.~I i' t1.1. Patterns ........ .. .. 29
:3.2.3 The K~arush-K~uhn-Tucker Conditions .... ... .. :30
:3.3 Support Vector Regression ......... ... :32
:3.4 Sparse C'I Iall., I Estimation using SVAI ...... .. :35
:3.5 Extension to Complex C'I aIn I . ...... .. :36
:3.6 SVAI Parameter Selection . .. . :37
:3.7 Simulation Results ......... . .. 48

4 BLIND SPARSE SIMO CHANNEL IDENTIFICATION ... .. .. 55

4.1 Introduction ......... . ... .. 55
4.2 Observation Model ......... .. 56
4.3 Combined Iterative and SVAI Based Approach ... . .. 58
4.3.1 Iterative Regression ......... .... 58
4.3.2 SVAI Regression . ...... .. ... 59
4.3.3 Inmplenientation of Support Vector Regression .. .. .. 60
4.3.3.1 The Adatron Algorithm ... .. .. 61









4.3.3.2 Learningf rate for convergence ... ... .. 6:3
4.4 Simulation Results ......... . .. 65

5 IDENTIFICATION OF TIME VARYING & FREQUENCY-SELECTIVE MITLTIPATH
CHANNEL ......... .... 72


Introduction.
Basis Expansion Models
Time Varying C'I .Ill., I Estimation using SVR
Simulation Result.


I I


6 CONCLUSIONS AND FITTIRE WORK .....

6.1 Conclusions
6.2 Future Work ......... ..


APPENDIX


A FPGA IMPLEMENTATION OF ADATRON ALGORITHM
GENERATOR


USING SYSYEM


A.1 Introduction.
A.2 Adatron Engine in System Generator
A.2.1 Address Control Logic
A.2.2 Adatron Engine block.
A.:3 Implementation Results


.


REFERENCES .......... .........

BIOGRAPHICAL SKETCH ...........









LIST OF TABLES
Table pagfe

2-1 Differen~ce between th~e true weights (h) an~d estimated weights (iE) with th~e paramecters
(pah = 0.05, p1 = 0.01, S = 1) when SNVR = 10dB. .... .. 24

3-1 ATTC C'I .Ill., I D Path Parameters . ...... .. .. 48

A-1 Device utilization summary ......... .. .. 87










LIST OF FIGURES

Figure page

1-1 A sample sparse channel impulse response (adapted front [1]) .. .. .. 11

2-1 Searching a~ parameter using the proposed method based on convolution inequality
for entropy. ......... ... . 22

2-2 Performance comparison of proposed method with MP, MP-2 and Wiener .. 2:3

:3-1 Illustration of the idea of an optimal hyperplane for linearly separable patterns.
The support vectors are outlined.(adapted front [2]) .. .. .. 27

:3-2 Hyperplane for the non-separable case. The slack variable ( permits margin failure.
Data point is wrong side of the decision surface. (adapted front [2]) .. .. .. 29

:3-:3 MSE of channel estimation as a function of C and e values for HDTV sparse
channel D data from ATTC (SNR = 15dB, 200 training samples, -II---- -1. Il values
of~ e 1.5 and O n 15) ......... ... :38

:3-4 MSE of channel estimation as a function of C and e values for HDTV sparse
channel D data from ATTC (SNR = 25dB, 200 training samples, -II---- -1. Il values
ofem0.2) ... ......... .......... :39

:3-5 Number of support vectors as a function of C and e values for HDTV sparse
channel D data from ATTC(SNR=15dB, 200 training samples) .. .. .. .. 40

:3-6 Number of support vectors as a function of C and e values for HDTV sparse
channel D data from ATTC(SNR=25dB, 200 training samples) .. .. .. .. 41

:3-7 Selection of e when SNR=15dB ......... ... .. 42

:3-8 Selection of e when SNR= 25dB ......... ... .. 4:3

:3-9 Training Performance as a function of C and e values for HDTV sparse channel
D data from ATTC (SNR=15dB, 200 training samples) .. .. .. 44

:3-10 Training Performance as a function of C and e values for HDTV sparse channel
D data from ATTC (SNR=25dB, 200 training samples) .. .. .. 45

:3-11 Generalization Performance as a function of C and e values for HDTV sparse
channel D data from ATTC (SNR=15dB, 200 training samples) .. .. .. .. 46

:3-12 Generalization Performance as a function of C and e values for HDTV sparse
channel D data from ATTC (SNR=25dB, 200 training samples) .. .. .. .. 47

:3-13 Symbol rate sampled response of ATTC channel D ... .. .. .. 49

:3-14 SVAI solution of sparse channel estimation for HDTV channel. .. .. .. 50

:3-15 MP solution of sparse channel estimation for HDTV channel. .. .. .. 51










:3-16 Performance of the sparse channel (HDTV channel) estimation. Multipath parameters
of ATTC channel D are listed in Table :3-1 and channel response shown is obtained
by pulse shaping filters with 11.5' excess bandwidth and 5.:38MHz sampling
frequency. ......... ... 52

:3-17 Performance of the sparse channel estimation. Nonzero coefficients of the sparse
channels are drawn front a uniform distribution on [-1, -0.2] U [0.2, 1] .. .. 5:3

4-1 Single-input two-output channel. . .. .. .. 57

4-2 Convergence of Adatron ... . .. 64

4-3 Convergence of iterative regression. . ...... .. 65

4-4 Zeros of subchannel hl, h2 -......... ** ** 66

4-5 50 trials of the SVAI hased method and Robust-SS method when overestimated
by 10 taps and SNR=:30dB ...... .. .. 67

4-6 Robustness when the channel order is exact and overestimated by 10 taps .. 68

4-7 50 trials of the SVAI hased method using the Adatron and robust-SS method
when the channel order is overestimated by 20 taps and SNR=20dB. .. .. 69

4-8 Performance comparison when the channel order is overestimated by 20 taps
(raised-cosine pulse followed by a niultipath channel). .. .. .. 70


5-1 Basis expansion model of a tinte-varying system.

5-2 MSE Comparison of LS and SVAI when NV = 100, L = :3, and Q =

5-3 Blind estimation of hasis coefficients {8,(1)}~ when Ni = 80, L
and SNR= 10dB (50 trials) using LS.

5-41 Blind esitimrationl of hasiis coefficients {8,(1)}~ wh~enl N = 80: L
and SNR= 10dB (50 trials) using SVAI.

A-1 Matlah intplenientation of Adatron algorithm .....

A-2 Top design block of Adatron FPGA intplenientation using Xilinx


A-:3 Address control logic of the nienory .....

A-4 MCode block configuration for updating a ...

A-5 Adatron Engine block detail ......

A-6 TI DSP (TM S:320C:33) intplenientation of the Adatron algorithm


:3..

= 3,


=2,


= 2,


System Generator


. 84

. 85

. 86;

. 88









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

A NEW CLASS OF SPARSE CHANNEL ESTIMATION METHODS BASED ON
SUPPORT VECTOR MACHINES

By

Dongho Han

December 2006

C'I I!1-: Josii C. Principe
Co('l! I!1-: Liuqingf Yang
Major Department: Electrical and Computer Engineering

In this dissertation sparse channel estimation is reformulated as support vector

regression (SVR) in which the channel coefficients are the Lagrange nmultipliers of the

dual problem. By employing the Vapnik's e-insensitivity loss function, the solution can

he expanded in terms of a reduced number of Lagrange nmultipliers (i.e., the nonzero filter

coefficients) and then a sparse solution is found. Furthermore, methods to extend the

SVR technique are investigated to derive an iterative algorithm for blind estimation of

sparse single-input nmultiple-output (SIMO) channels. This method can he also used for

non-sparse channels, in particular when the channel order has been highly overestimated.

In this situation, the structural risk nxinintization (SR M) principle pushes the small

leading and trailing terms of the impulse response to zero. Results show that the SVR

approach outperfornis other conventional techniques of channel estimation. The main

drawback of this approach is the high computational cost of the resulting quadratic

progranining (QP) solution. To reduce the complexity, we propose a simple and fast

iterative algorithm called the Adatron to solve the SVR problem iteratively. Simulation

results demonstrate the performance of the method.










CHAPTER 1
INTRODUCTION

In m? Ilw: wireless communication systems, the propagation channels involved exhibit a

large delay spread, but a sparse impulse response consisting of a small number of nonzero

coefficients. Such sparse channels are encountered in many communication systems.

Terrestrial transmission channel of high definition television (HDTV) signals are hundreds

of data symbols long but there are only a few nonzero taps [3]. A hilly terrain delay

profile has a small number of multipath in the broadband wireless communication [4] and

underwater acoustic channels are also known to be sparse [5]. An example of the sparse

channel is shown in Fig. 1-1 and estimation of sparse channels will be mainly considered in

this dissertation.


O
F




-OO OO 00000000 000000 00 00000 0000



a



O


0.4


-0.6

-0.8


10 15 20 25 30 35


Figure 1-1. A sample sparse channel impulse response (adapted from [1])










1.1 Background

The roots of sparse system identification lie deep in signal representation. Instead

of just representing objects as superpositions of sinusoids (the traditional Fourier

representation) we have available alternate dictionaries. Wavelet dictionaries, Gabor

dictonaries, Wavelet Packets, Cosine Packets, and a wide range of other representation [6], [7].

We should obtain the sparsest possible representation of the object (i.e. the one with the

fewest significant coefficients) front a large dictionaries.

Mallat and Zhang [8] have proposed the matching pursuit (j\!P) algorithm which

builds up a sequence of sparse approximations. But the algorithm is greedy, when run

for ]?r Illy interations, it might spending most of its time correcting for any mistakes made

in the first few terms. Also it is possible to re-select a previously selected vector in the

dictionary.

To avoid the limitations of greedy optimization, C'I. l! and Donoho [9] have -II_0-r-- -1.

a method of decomposition based on a true global optimization which is called basis

pursuit (BP). They propose to pick one whose coefficients have nmininiun L1-nornt front

the many possible solutions to Ax = y as in equation (1-1).


nxin ||hl || -Ay -Ax|2 (I


Both MP and L1-nornt regularization (BP) can he viewed as trying to solve a

combinatorial sparse signal representation problem. MP provides a greedy solution, while

Li-nornt based BP replace the original problem with a relaxed version. Sparse signal

representation problem has various applications such as tinte/frequency representations,

speech coding, spectral estimation, and image coding [10-13]. Sparse channels are

frequently encountered in coninunication applications and we are going to focus on

estimation of sparse channel.

Modern estimation theory can he found at the heart of many electronic signal

processing systems designed to extract information. These systems include radar, sonar,










speech, image analysis, biomedicine, communication, control, seismology and so on.

All of these systems share the common problem of needing to estimate the values of a

group of parameters. This problem has a long and glorious history, dating back to Gauss

who in 1975 used least squares data analysis to predict planetary movements [14]. A

salient feature of this least squares method is that no probabilistic assumptions are made

about the data, only a signal model is assumed. The advantage is its broader range of

possible applications. On the negative side, no claims about optimality can he made

and furthermore, the statistical performance cannot he assessed without some specific

assumptions about the probabilistic structure of the data.

C'I .Ill., I estimation using least-squares does not exploit the sparsity of channels and

need long training symbols to produce an accurate estimate. A MP algorithm hased sparse

channel estimation method is proposed in [1] and this method viewed the estimation

problem as a sparse representation problem and exploited the sparse nature of the channel

using MP algorithm. It is shown that, the MP hased channel estimation is more accurate

and outperformed the conventional least square based methods in robustness and low

complexity accuracy. A parametric method for selecting the structure of sparse model

has been proposed in [15]. This method exploits the information provided hv the local

behavior of an information criterion to select the structure of sparse models.

An adaptive L1-norm regularization method (sparse-LMS) using the augmented

Lagrangian to find the sparse solution has been proposed in [16]. This method augmented

the L1-norm constraint of the channel impulse response as a penalty term to the MSE

criterion. However, determination of the parameters in regfularization problems, here

L1-norm of the channel coefficients which pi-li an important role in achieving good

estimation, has remained an open problem. In [17], a searching tool for the L1-norm of

channel coefficients using the convolution inequality for entropy is developed to improve

the performance [18, 19].









Since Sato [20] first proposed the innovative idea of self-recovering (blind) adaptive

identification, blind channel identification and equalization have been studied by many

researchers [21-27]. Since the work of [28], it has been well known that second order

statistics (SOS) are sufficient for blind identification when the input signal is informative

enough and the channels do not share any common roots. Widely used SOS-based

methods include the subspace (SS) approach [29], the least squares (LS) [27] technique

and the linear prediction (LP) methods. However, a common drawback of SS and LS

techniques is their poor performance when the channel order is overestimated. Recently

some robust techniques have been proposed to mitigate this problem (see e.g., [:30, 31]).

Although these methods offer increased robustness, they still fail when the channel order is

highly overestimated.

1.2 Contribution of this Thesis

Before this research had started, L1-norm regularization method (BP) and matching

pursuit (1! P) algorithms are widely used methods in many sparse applications because

it is well known that the concavity of the L1-norm in the parameter space yields sparse

solution. But MP algorithm is greedy and L1-norm regularization method is non-quadratic

optimization problem.

Support vector machines (SVA~s) are a powerful learning technique for solving

classification and approximation which can he derived from the structural risk minimization (SR M)

principle [:32]. The SR M principle is a criterion that establishes a trade-off between

the complexity of the solution and the closeness to the data. This support vector

machine (SVAI) technique typically provides sparse solutions in the framework of

SR M [:33]. Specifically, the SVAI solution can he expanded in terms of a reduced set

of relevant input data samples (support vectors).

The first contribution of this dissertation is to use support vector machines (SVA~s)

for sparse channel estimation because SVAI is known to build parsimonious models and

results in a quadratic problem [:34]. In the proposed formulation, the channel coefficients










pl1 li- the role of the Lagrange multipliers. By using the Vapnik's e-insensitive loss

function only those Lagrange multipliers corresponding to support vectors are nonzero

and therefore a sparse solution is obtained.

The second contribution is an iterative algorithm for blind estimation of sparse

single-input multiple-output (SIMO) channels is derived. The work in [:35] was the first

attempt to apply an SVAI-hased approach to the blind identification of SIMO channels.

However, the sparsity provided by the SVAI solution was not explicitly exploited in [:35].

In this direction we present a new blind identification algorithm hased on support vector

regression and specifically tailored for sparse SIMO channels in [:34]. The main idea is that

the sparse SIMO channel identification can he reformulated as a set of regression problems

in which the channel coefficients plI li- the role of the Lagrange multipliers. Thus we can

get a sparse solution. This method can he also used for non-sparse channels, in particular

when the channel order has been highly overestimated. In this situation, the structural

risk minimization (SR M) principle pushes the small leading and trailing terms of the

impulse response to zero.

The main drawback of applying this SVAI technique is the high computational cost of

the resulting quadratic programming (QP) problem. As a third contribution, a simple and

fast iterative algorithm called the Adatron is used to solve the SVR problem [:36, 37].

1.3 Organization of the Dissertation

The organization of this proposal is as follows: In ('!s Ilter 2, we briefly summarize

the conventional sparse solutions such as MP method and sparse-LMS using the

convolution inequality for entropy. ('!s Ilter :3 introduces support vector machines

including the classification and the regression problem. Then we present a new SVAI-hased

sparse system identification algorithm. SVAI meta-parameter selection is also discussed.

In ('!, Ilter 4, iterative algorithm for blind estimation of sparse SIMO channels will be

derived and Adatron algorithm to reduce the complexity of support vector machine based

method will be also introduced. In C'! Later 5, SVAI technique with the exponential basis










expansion model (BEM) is applied to the time-varying channel identification. Finally,

C'!s Ilter 6 gives conclusions and future work. In appendix, to investigate the feasibility

of the FPGA implementation of the proposed algorithm, the core computation of this

Adatron is simulated in FPGA using System Generator which is a high-level design

tool for Xilinx FPGAs that extends the capabilities of Simulink to include accurate

modeling of FPGA circuits. The performance is also compared to the TI TMS320C33

DSP implementation.









CHAPTER 2
CONVENTIONAL SPARSE SOLUTION

2.1 Observation Model

The sparse channel estimation problem can be stated as follows. Let s(u) be the

training sequence for n = 0,. ., NV 1, that is transmitted through a stationary channel,

h(0), &(1),..., h(M~ 1) which is sparse. The training sequence symbols r(u) for n < 0 can

be obtained from the previous estimates or for the first arriving frame they are assumed

to be zero. White Gaussian noises e(0),...,e(NV 1) are added to this transmitted

signal where NV is the training symbol number and M~ is the length of the channel to be

identified. Then the receive signal r(u) can be expressed in matrix form as


r = Sh + e (2-1)


where




r (0) & (0) e (0)
r= !,h = ,e =

r(Nv 1) h(M~ 1) e(Nv 1)



s(0) s -M 1

S = s (1) a(-l M 2)


s(M+N1) -- s(NV- 1)





Since the channel is sparse, most components of the channel impulse response are

zero. Through this knowledge, a sparse solution to r e Sh is achieved by approximating









r as a linear combination of a small number of columns of S and will be explained in the

following section.

2.2 Matching Pursuit (MP) Based Method

Mallat and Zhang [8] have proposed the use of a greedy algorithm which builds up a

sequence of sparse approximations. The MP algorithm first finds the best fitted column,

sks in the matrix S with the received signal ro = r. Then the projection of the initial

residual ro along with the vector sk, is removed from ro and the residual is rl. Similarly,

sk, is decided that is best aligned to the residual rl. This algorithm keeps finding the

best aligned column to the consecutive residuals until a specified number of taps or small

residuals. After p iterations, one has a representation of the form equation (2-1), with

residual rz>-

This algorithm is summarized as follows:


ki = arg max || Psir,-1|| = arg max l >1 (2-2)

(s Ip1 Sky 1

Tp Fp1-Pkf- 12(2-3)


ak = -l (2 4)


where the projection onto vector St is denoted as





and the channel coefficient at position k, is hkp.

Possible problem of this algorithm is, the number of nonzero taps should be given

which is unrealistic because it is unknown. This algorithm might spend most of its time

correcting for any mistakes made in the first few terms because the algorithm is greedy.









One can give examples of dictionaries and signals where the method gives a solution which

is badly sub-optimal in terms of sparsity.



2.3 SPARSE-LMS

An adaptive method using the augmented Lagrangian to find the sparse solution has

been proposed by Rao et. al [16]. This method augmented the L1-norm constraint of the

channel impulse response as a penalty term to the MSE criterion. The cost function is


J(h) = E(e ) + ~ A |i| (2-5)

Where the first term is the mean squared error between the received signals and the

estimated signals, and the second term is the penalty term. This constraint fixes the

Li-norm of the channel impulse response h to a constant a~. It is well known that the

concavity of this L1 norm function yields the sparse solution [13]. The penalty factor A

can be included as an adaptive parameter by modifying the cost function as,


J(h, A) = E(ea ) +n AS~ |hs| -A2 (2-6)

where p is a positive stabilization constant that keeps the penalty factor A bounded.

This modified cost function is known as the augmented Lagrangian [38]. The stochastic

gradients of the cost function are given by,

8 J(h, A)
=-2ekhki + 9 sin~i)] (2-7)

di=


Then the adaptation rule of parameters is given by


As(k + 1) = As(k) + rlh[2ekhks k X8 98 ~i)] (2-9)

Ak+1 = k~ rlA i| a 2A (2-10)









where rlh and rl are step size.

Ak COnVergeS to a value A* [16]





where h* is the .I- n ei ni ic~ coefficient vector from equation (2-9).

Note that the constant denoted by a~ in equation (2-5) is unknown so it has to be

searched cautiously to get a good performance. With an inappropriate choice of a~, the

performance of the adaptive system becomes poor. Therefore we propose a method based

on the convolution inequality of entropy to determine a~ before learning system parameters

using the stochastic gradient algorithm in equations (2-7) and (2-8).



2.3.1 Convolution Inequality For Renyi's Entropy

Recall the system model is


r = Sh + e (2-12)


where h is sparse channel.

Let ,(-) denote the Renyi's entropy. The convolution inequality for Renyi's entropy

is [18] [19]


H,(r) > H,(s) +log |hk|, Vk



eH,(r)-H,(s) > maX Ihk | (2-13)


The equality holds if and only if the filter is a pure delay.

Let hmax denote the max |hi|, then we can start to search the adaptive parameter

a~ in equation (2-5) using the above equation (2-13). If the vector h is sparse, then the

L1-norm of the channel impulse response will not be much bigger than hmax. Instead of










searching a~ in an arbitrary range, we can simplify searching hased on this information.

Therefore, we start searching with a~ = hmax, and train the adaptive filter with the

algorithm given by equations (2-7) and (2-8). After training, we compute MSE between

the received signals r and the filter outputs. Then, we increase c0 and repeat learning.

We can stop the training when the MSE start to increase. In Fig. 2-1, we can stop at

a~ = (4 hmax).

Since we cannot work directly with the PDF, a nonparametric method is used

here to estimate the entropy. Entropy estimate is obtained from Renyi's quadratic

entropy estimator which estimates the PDF hy Parzen-window method using Gaussian

kernel [39], [40].

This proposed method can he summarized as follows:

Algorithm 1 Summary of sparse channel estimation with regularization method

Initial A, /3 and h

Calculate input, output entropy

Set a~ = hm,, from entropy inequality by Eq. (2-13)

while flag is true do

Train adaptive filter by Eq. (2-9), (2-10)

Compute MSE and check if it's reached minimum.

If minimum, set flag= false.

If not minimum, increase a~

end while






















0.2-



0.1-


0.0-


S1 .5 2 2 .5 3 3 .5 4
max

Figure 2-1. Searching a~ parameter using the proposed method based on convolution
inequality for entropy.


2.4 Simulation Results

Nonzero coefficients of the sparse channels are drawn from a uniform distribution on

[-1, -0.2] U [0.2, 1] and the number of the nonzero coefficients is 8 which is 10'; of the

channel length, M~ = 80. These nonzero coefficients are positioned randomly over 80 taps.

A white Gaussian training data with zero mean and unit variance is transmitted over the

sparse channel and a white Gaussian noise with zero mean and variance of ag is added

to the received data. Variance a~ is varied to change the SNR from 5dB to 30dB. The

number of training data is NV = 200. 1000 Monte Carlo simulations with different input

signals, SNR, and the nonzero channel coefficients are performed.


Table 2-1 shows the comparison of the proposed criterion performance to the Wiener's

MSE criterion. We searched the a~ using the proposed method based on convolution

inequality and fix the value of a~ to (3 kmax) from Fig. 2-1. Here, the estimated a~ value

is 5.5995 which is very close to the value of L1-norm of true vector h, 5.6242. As can be






































LL
(1 0.02


0.015*


0.01






0 5 10 15 20
SNR(dB)


Figure 2-2. Performance comparison of proposed method with MP, MP-2 and Wiener










Table 2-1. Difference between? the true weigh~ts (h) anld estimated weights (G) with t~he
parameters (pal = 0.05, p~ = 0.01, P = 1) when SNVR = 10dB.

overall tap nonzero tap zero tap
difference difference difference
| -hene | 0.6960 0. 1746 0.6737
|h hmse,+L1 | 0 0.2425C 0. 1520



seen in Table 2-1, the proposed method estimates the zero tap coefficients better than

Wiener's MSE solution but there is a penalty for this. MSE is increased in the nonzero tap

coefficients estimation which can be explained by the performance loss due to the penalty

term.

We also compared the performance of the proposed method to Wiener filter and

matching pursuit algorithm which is widely used in sparse application. Fig. 2-2 shows

that MP method outperforms other methods in terms of the variance and MSE of channel

impulse response estimates. In this simulation, MP method knows the number of nonzero

channel but in practical the number of nonzeros taps are unknown. So we run the MP

method with incorrect number of nonzero taps (here 6) and denote as 'MP-2'. We can see

from Fig. 2-2 that the performance of the proposed method is much better than Wiener

and also better than 'MP-2' from 10dB. The MP method outperforms other methods

but the MP algorithm is greedy and needs to know the exact number of nonzero channel

coefficients while the proposed algorithm is an adaptive method also performs better than

MP-2 from 10dB.









CHAPTER :3
SUPPORT VECTOR MACHINE BASED METHOD

3.1 Introduction

Support Vector Machines (SVAI) proposed by V. Vapnik are a new method to solve

pattern recognition problems which are based on the Structural Risk Minintization (SR M) [:32].

While the Empirical Risk Minintization (ERM) principle is used by neural networks to

nminintize the error on the training data, SR M nminintizes a bound on the test error.

In the linear separable case, SVAI naxintizes the margin of the hyperplane which

classifies the input data. The hyperplane location is determined by some points of the

input data which are termed support vectors. However, SVA~s have also been proposed

and applied to a number of different type of problems such as regression problem [41, 42],

detection [4:3] and inverse problems [44]. In regression, the goal of SVAI is to construct a

hyperplane that lies close to as many of the data points as possible. We must choose a

hyperplane with small norm while nxinintizing the loss using e-insensitive loss function. A

remarkable property of SVAI is the sparsity of its solution. Typically a small number of

support vectors are nonzero. Girosi showed the equivalence between sparse approximation

and SVA~s [:33].

In this chapter, support vector machines are briefly introduced including the

classification and the regression problem. Then we present a new SVAI-hased sparse

system identification algorithm. SVAI parameter selection is also discussed. For additional

material of SVA~s one can refer to the works of V. Vapnik, C. Burges, B. Schoilkopf and A.

Smola [:32, 45].









3.2 Support Vector Classification

3.2.1 Linearly Separable Patterns

Consider the training sample (xi, yi) 7, where xi is the input pattern for the ith

example and di is the corresponding desired response and which are linearly separable.
The equation of a decision surface in the form of a hyperplane that does the separation is

w'x +b = (3-1)


For a given vector w and bias b, the separation between the hyperplane defined in

equation (3-1) and the closest data point is called the margin of separation. The goal
of a support vector machine is to find the particular hyperplane for which the margin of

separation is maximized. This can be formulated as follows: suppose that all the training

data satisfy the constraints

w xi + b > 1, Ior yi = 1 (3-2)

w xi + b < 1, Ior yi = -1 (3-3)


where w is normal to the hy;perp~lane, ~i is the perpendicular distance from th~e

hyperplane to the origin and ||w|| is the Euclidean norm of w (Fig. 3-1). These constraints
can be combined into


y(w xi + b) 1 '> 0 Vi3-4

The points for which the equality (3-2) holds lie on the hyperplane H1 : w xi + b = 1

with normal w and perpendicular distance from the origin ~. The points for which
the equality (3-3) holds lie on the hyperplane H2 : W Xi + b = -1, with normal w and

perpendicular distance from thle origin ~. Tlherefor~e thle madrginl is

|1- b +1+ b| 2
(3-5)
||w|| ||w||
































Figure 3-1. Illustration of the idea of an optimal hyperplane for linearly separable
patterns. The support vectors are outlined.(adapted from [2])


Equation (3-5) states that maximizing the margin of separation between classes is

equivalent to minimizing the Euclidean norm of the weight vector w.

The optimization problem finds the optimal margin hyperplane by


Psi~tiv-e Examples


O~rigi pace ~f possible inputs a


min |w |2
w,b 2


(36)


subject to equation (3-4). By introducing positive Lagrange multipliers asi, i = 1, 2, .. ,1,

the primal formulation of the problem is


L,-2 ||w||2 0494w W-Xi + b) + as asi > 0
i= 1 i= 1


(3-7)


L, must be minimized with respect to w, b subject to the constraints ai '> 0. This is

a convex quadratic programming (QP) problem and we can use the dual which is to


maramn










maximize L, with respect to a i, subject to the constraints


iiL,
8w
8 L,
= 0


(3-8)

(3-9)


which gives the conditions


i= 1


i= 1

Substituting these conditions into equation (3-7) gives the dual formulation of the

Lagfrangfian


(3-10)


(3-11)


1
LD ti -
i= 1


1
tit]#i#/Xi xy)
i, j= 1


(3-12)


i= 1


asi > 0 for i


1,2,...,N


New data x can be classified using


f= asyxe-+b





i= 1


(3-13)


(3-14)


(3-15)


The solution can be found by minimizing L, or maximizing LD and can be efficiently

solved by [46].


subject to










B"1
d= I ~o6~n~uEw~Les I


~II

I h~eear~e E;ca~Fb I



~ Q
I,

~i~nit;s P~sc~ od~ irp~ms r ~o~


Figure 3-2. Hyperplane for the non-separable case. The slack variable ( permits margin
failure. Data point is wrong side of the decision surface. (adapted from [2])

3.2.2 Nonseparable Patterns

Consider the case where some data points (xi, yi) fall on the wrong side of the

decision surface as illustrated in Fig. 3-2. To treat this nonlinearly separable data cases,

we need to introduce positive slack variables (4, i = 1, 2, in the constraints. The

constraints (3-2) and (3-3) then become:

w xi + b > 1 (4, Ior yi = 1 (3-16)

w xi + b < 1 + (4, Ior yi = -1 (3-17)

i> 0, Vi3-18)

To assign penalties for errors, the objective function (3-6) is modified to


mi-n 1|w||2C (;19)
i= 1

Minimizing the first term in equation (3-19) is related to minimizing the VC dimension

of the support vector machine. As for the second term Ci 4, it is an upper bound on










the number of test errors. The parameter C controls the tradeoff between complexity

of the machine and the number of nonseparable points. It may be viewed as a form of a

regularization parameters. A larger C corresponds to assigning a higher penalty to errors.

the dual problem for nonseparable patterns becomes:


LD- i t :Cit]yiy(Xi xj) (3-20)
i= 1i,=1

subject to



i= 1




The nonseparable case differs from the separable case in that the constraint asi > 0 is

replaced with the more stringent constraint 0 < asi < C. The solution is again given by


w = aiyixi (3-21)
i= 1

where NV is the number of support vectors.

3.2.3 The Karush-Kuhn-Tucker Conditions

For the primal problem equation (3-6) the KKET conditions are [47]


Lp = wv ) :' I .. = 0 v = 1, 2, .. ,d (3-22)
i= 1


L, = a sc~yi = 0 (3-23)
i= 1

ye (w xi + b) 1 > 0 ,2 (3-24)

asi> 0 Vi3-25

a~(yi(w xi + b) 1} = 0 V (3-26)


where v runs from 1 to the dimension d of the data. The equation (3-22) to equation (3-26)

are satisfied at the solution of any constrained optimization problem, provided that the









intersection of the set of feasible directions with the set of descent directions coincides

with the intersection of the set of feasible directions for linearized constraints with the

set of descent directions [47, 48]. This regularity assumption holds for all support vector

machines, since the constraints are ahr-l-w linear. In addition, the problem for SVMs

is convex and for convex problems the KKET conditions are necessary and sufficient for

w, b, a~ to be a solution [47]. Thus, finding a solution to the KKET conditions is equivalent

to solving the SVM problem. The threshold b is found by using the KKET condition,

equation (3-26), by choosing any i where asi / 0 (i.e. support vectors) and computing b.

The KKET conditions for the primal problem are also used in the non-separable case. The

primal Lagfrangfian is


L, = |w||2CC s 04yiW Xi+ b) -iZ 1+i (4} p( (327)

where pi are the Lagrange multipliers introduced to enforce positivity of the slack

variables (4. The KKET conditions for the primal problem are

dL,
= we one.. = 0 v =1,2, .. ,d (3-28)
i= 1
dL,
asy = 0(3-29)
i= 1
iiL,
=C asi p-i = 0 (3-30)

y (w -xi + b) 1 + (4 > 0 (3-31)

as, 4, p > 0(3-32)

a {yi(w -xi + b) 1 + (4} = 0 (3-33)

Ops = 0(3-34)


where i = 1, 2, .. ,1 and v = 1, 2, .. ., d.









The KKET complementary conditions equation (3-33) and equation (3-34) can be used

to determine the b. Any training point for which 0 < asi < C, that is not penalized can be

taken to compute b.



3.3 Support Vector Regression

The structural risk minimization (SRM) principle is a criterion that establishes a

trade-off between the complexity of the solution and the closeness to the data. We apply

the SRM principle to the sparse channel estimation problem. In the proposed formulation,

the channel coefficients pIIl w the role of the Lagrange multipliers. To solve this sparse

channel estimation, we will next reformulate it as a set of support vector regression (SVR)

problem.

Consider the problem of approximating the set of data { (xi, yi)l ) with a linear

function f(x).


f (x) = (w, x) +b (3-35)


I, b are parameters to be estimated from the data. The method of SVM regression

corresponds to the following minimization:


J(w) =~ C ys- (xi)|e+~ || w||2 _36)
i= 1

The parameter C controls the tradeoff between training error and regfularization terms in

equation (3-36) and the e-insensitive loss function is defined as


V(x) = |XIe__0if|| (3-37)
|X| otherwise









Since it is difficult to solve equation (3-36) with the e-insensitive loss function |xl, the

problem is replaced by following minimization problem.


cC~t (( +() + |w ||


(338)


f (xi) yi < e + (4

yi f(xi)
i> 0

e*>0


i = 1, 1

i=1,...,l

i= 1,...,1

i=1... 1


Notice that the penalty is paid only when the absolute value of the interpolation error

exceeds e. To solve the above constrained minimization problem, we use Lagrange

multipliers. The Lagrangian corresponding to equation (3-38) is:


"C((4 + () +Wr w + af(y, f(xi) E -(f)
i= 1 i= 1


L( f, I, I*, a~, a~*, r, r*)


I> i= 1 l


i=


(339)


The solution is given by minimizing the Lagrangian equation (3-39) with respect to f (that

is w.r.t w, b), (, (* and maximizing with respect to a~, a~*, y, y*.


iBL


iBL
db
dL

iiLi


i= 1


i= 1

0 y 7 = C as


(3-40)


(3-41)


(3-42)

(3-43)


0 y 7 = C: af


subject to










Substituting w in equation (3-35) with equation (3-40), we can express the problem

(3-38) as belows


f(x, a, a*) = a -a) < x, x, > +b (3-44)
i= 1

In this dual representation equation (3-44), data appears only in the dot product.

Substituting equation (3-44) in the Lagrangian eq (3-39), we obtain a maximization

problem with respect to a~, a~*, y, y*, where of5 and asi are positive Lagrange multipliers

which solve the following Quadratic Programming(QP) problem in dual representation:

m11 1/cc~)

i= 1 j= 1 i= 1

+ yi~a- ai)(3-45)
i= 1

subject to the constraints


0 < a~*,c a

i= (af i) = 0

This equation (3-45) is the QP problem which is convex and having no local minimum.

This has to be solved to compute the SVM. Due to the nature of the QP problem, only

a small number of coefficients of5 asi will be nonzero and the associated input data are

called support vectors. Interpolation error of the support vectors is either greater or equal

to a so if a = 0 then all the input becomes support vectors.










3.4 Sparse Channel Estimation using SVM

The system model is


r = Sh (3-46)


where r is received vector, and S is input matrix. Premultiply ST on both sides yields


S~r = ST Sh
y x



y = STx


where y is a new output vector, and ST is a new input matrix.

The SVM minimize the primal problem





or maximize the dual.



n= 1 m= 1
M M



The solution is





a Sh (3-47)


The Lagrange multipliers (a;~ an,) in the SVM solution of equation (3-47) correspond to

channel coefficients h in equation (3-46), yielding a sparse solution.









3.5 Extension to Complex Channel

To handle the complex-valued channel and data, we use the isomorphism between

complex numbers and real-valued matrices. Consider the system model

r = Sh


where S is a complex-valued matrix of dimension m x n, h is a complex-valued vector

of length n and r is a complex-valued vector of dimension m. This system can be

equivalently expressed as follows:

r S-S h

r SS h

where all involved matric~es and vectlors are real-valued (the notation (-) is Re(-) and (-) is

Im(-) ). Let us introduce the following notation


r, h
r=I ,h=


S= -


Then we can write equation (3-48) as:




The representation in equation (3-48) is useful to extend the complex-valued number to

real-valued matrices.










3.6 SVM Parameter Selection

Recall that SVAI minimizes


J(x) = xs x| xx


It is well known that SVAI performance (estimation accuracy) depends on a good setting

of meta-parameters C and e. Optimal selection of these parameters are difficult because

SVAI model performance depends on all these parameters.

The parameter C determines the trade off between the model complexity and the

degree to which deviations larger than e are tolerated. For example, if C is too large, then

the objective is to minimize the empirical risk only, not considering the model complexity

part in the optimization formulation equation (:336).

The parameter e controls the width of the e-insensitive zone, used to fit the training

data. The value of e can affect the number of support vectors used to construct the

regression function. The bigger e, the fewer support vectors are selected. Bigger e values

result in more flat estimates. Therefore, both C and e affect model complexity but in

different way.

Some practical approaches to the choice of C and e are as follows:

Selecting parameter C equal to the range of output values [49].

'!~~ Ch.--- e so that the percentage of support vectors in the SVAI model is around 501'

Optimal e values are proportional to noise variance [50].

To investigate the effects of parameter C and e via simulation, we checked the channel

estimation error, number of selected support vectors, training error and test error with

various e and C values at 15dB and 25dB. Sparse channel data for this simulations are the

HDTV channel data from ATTC (Advanced Television Test Center) tests [:3]. Dependence

of AISE (11. as! Squared Error) of channel estimation as a function of chosen C and e

values for ATTC channel D data set at 15dB is shown in Fig. :3-:3. Fig. :3-4 also shows the










dependence of AISE of channel estimation error as a function of chosen C and e values for

ATTC channel D data set at 25dB.

One can clearly see that C values above certain threshold have only minor effect on

the MSE of channel estimation. According to the simulation results, -II_ -- -blIIC is 15 and

-II---- -r. I1 e value is 1.5 for Fig. :3-:3 and Figf. :3-4,


Estimation Error


1.5 1

1-C
0.5


I C


Figure :3-:3. MSE of channel estimation as a function of C and e values for HDTV sparse
channel D data from ATTC (SNR = 15dB, 200 training samples, -II---- -1. *1
values of e m 1.5 and C a 15)






















x1-5






6s 4









20
15 `~2

11
0.5
C 00



Figure 3-4. MSE of channel estimation as a function of C and a values for HDTV sparse
channel D data from ATTC (SNR = 25dB, 200 training samples, -II---- -1h Il
values of e m 0.2)









Fig. 3-5 shows the number of support vectors selected by chosen C and a values for

ATTC channel D data set with 200 training samples at 15dB and Fig. 3-6 is for 25dB.

We can see that small e values correspond to higher number of support vectors, whereas

parameter C has negligible effect on the number of support vectors. Fig. 3-7 and Fig. 3-7

shows the dependence of MSE as a function of a values more clearly at 15dB and 25dB.


Number of Support Vectors


30O


Figure 3-5. Number of support vectors as a function of C and a values for HDTV sparse
channel D data from ATTC(SNR=15dB, 200 training samples)
























120

110

100

S90

S 80

70

60

50
20
40
0 10
0.5
1.
1.5 2 0



Figure 3-6. Number of support vectors as a function of C and a values for HDTV sparse
channel D data from ATTC(SNR=25dB, 200 training samples)













Number of Support Vectors


1 1.5


2 2.5


(a) Nilliter of support vectors

Estimation Error


x10-4

2.6


1.4
0.5


1 1.5 2 2.5


(b) Channel estimation error


Figure 3-7. Selection of e when SNR=15dB




































0.5 1 1.5


(a) Number of support vectors


x 10-






















0 0.5 1 1.5


(b) Channel estimation error


Figure 3-8. Selection of a when SNR= 25dB










Fig. 3-9 and Figf. 3-10 shows the performance in terms of training error as a function

of C and a values. We can see that training error increases as a values increases and

parameter C has negligible effect on the number of support vectors.


Training Performance





0.035


0.03


.c 0.025


r 0.02

0.015


0.01,
30

20
2.5

10 1.5

C 0 0.5



Figure 3-9. Training Performance as a function of C and a values for HDTV sparse
channel D data from ATTC (SNR=15dB, 200 training samples)



















Training Performance


x 10-3




8~ 4










20
15 1 -. 2

11
0.5
9 00


Figure 3-10. Training Performance as a function of C and a values for HDTV sparse
channel D data from ATTC (SNR=25dB, 200 training samples)










Fig. 3-11 and Figf. 3-12 shows the generalization performance in terms of test error as

a function of C and a values. We can see that the test error increases after an optimal e

values which corresponds to the values -II__- -rh I1 from Fig. 3-3 and Fig. 3-4.


Generalization


Performance


0.03


0.025


0.02


0.015


0.01
30


C 0 0.5


Figure 3-11. Generalization Performance as a function of C and a values for HDTV sparse
channel D data from ATTC (SNR=15dB, 200 training samples)



















Generalization Performance


x 10-3


Figure 3-12. Generalization Performance as a function of C and a values for HDTV sparse
channel D data from ATTC (SNR=25dB, 200 training samples)









3.7 Simulation Results

To perform the sparse channel estimation, we use the two data set. One data set is

the HDTV channel data from ATTC (Advanced Television Test Center) [3]. These data

sets are commonly used in literature for sparse channel estimation. Multipath parameters

of ATTC channel D are listed in Table 3-1 and channel response shown in Fig. 3-13 which

is obtained by pulse shaping filters with 11.5' excess bandwidth and 5.38MHz sampling

frequency. The other sparse channel data set has nonzero coefficients which are drawn

from a uniform distribution on [-1, -0.2] U [0.2, 1] so nonzero taps of this channel are more

distinct whereas some coefficients of the HDTV channel are very small.


Table 3-1. ATTC C'I .Ill., I D Path Parameters

path delay(ps) dilli-(T) phase atten
1 0.00 0.00 288 deg 20 dB
2 1.80 9.6;8 180 deg 0 dB
3 1.95 10.49 0 deg 20 dB
4 3.60 19.37 72 deg 18 dB
5 7.50 40.35 144 deg 14 dB
6 19.80 106.52 216 deg 10 dB




Fig. 3-14 shows the SVM solution of the estimation for HDTV channel and Fig. 3-15

for the MP solution. Performance of the HDTV sparse channel estimation is shown in

Fig. 3-16. We can see that SVM outperforms other methods in terms of MSE and variance

in Fig. 3-16.

Fig. 3-17 shows the performance of the first sparse channel data set which has nonzero

coefficients drawn from a uniform distribution on [-1, -0.2] U [0.2, 1]. We can see that

when the sparse channel has distinct nonzero coefficients MP method performs very

well. But when the channel taps are small which could be buried under noise, then MP

performs poor as in Fig. 3-16.

































ATTC Channel D Response
0.6

0.4-

0.2-



-0.2-

-0.4-

-0.6-

-0.8
0 20 40 60 80 100 120



1.2



0.8-

m 0.6-

P 0.4-

0.2-



-0.2
0 20 40 60 80 100 120


Figure 3-13. Symbol rate sampled response of ATTC channel D












































(a) Real







0.8-



0.6-



0.4-



0.2-







-0.2
0 20 40 60 80 100 120



(b) Image


Figure 3-14. SVM solution of sparse channel estimation for HDTV channel.






































(a) Real


0.6 t


-0.2
0


20 40 60 80 100 120


(b) Image

Figure 3-15. MP solution of sparse channel estimation for HDTV channel.


- true
SMPI























x 10-3
12
-4- MP
-#- SVM
10 Wiener


8 -
















-2
0 5 10 15 20
SNR(dB)


Figure :3-16. Performance of the sparse channel (HDTV channel) estimation. Multipath
parameters of ATTC channel D are listed in Table :3-1 and channel response
shown is obtained by pulse shaping filters with 11.5' excess bandwidth and
5.:38MHz sampling frequency.































































Figure 3-17. Performance of the sparse channel estimation. Nonzero coefficients of the
sparse channels are drawn from a uniform distribution on [-1, -0.2] U [0.2, 1]


I I I I


I


0.1


-4- MP
-#- SVM
Wiener


0.08


0.06


'


-0.02


5 10 15
SNR(dB)










In this chapter, support vector machines are briefly introduced including the

classification and the regression problem. Then we present a new SVM-based sparse

system identification algorithm. SVM parameter selection is also discussed. Simulations

demonstrated SVM-based sparse channel estimation method outperforms other methods

even when the sparse channel contains very small nonzero taps.









CHAPTER 4
BLIND SPARSE SIMO CHANNEL IDENTIFICATION

4.1 Introduction

Blind estimation of single-input multiple-output (SIMO) channels is a widely studied

problem with many signal processing applications. Since the work of [28], it has been

well known that second order statistics (SOS) are sufficient for blind identification

when the input signal is informative enough and the channels do not share any common

roots. Widely used SOS-based methods include the subspace (SS) approach, the least

squares (LS) technique and the linear prediction (LP) methods. However, a common

drawback of SS and LS techniques is their poor performance when the channel order

is overestimated. Recently some robust techniques have been proposed to mitigate this

problem (see e.g., [30, 31]). Although these methods offer increased robustness, they still

fail when the channel order is highly overestimated.

The structural risk minimization (SRM) principle is a criterion that establishes

a trade-off between the complexity of the solution and the closeness to the data. In

particular, the support vector machine (SVM) technique, which can be derived from

the SRM principle, typically provides a robust solution. The work in [35] was the

first attempt to apply an SVM-based approach to the blind identification of SIMO

channels [35]. However, the sparsity provided by the SVM solution was not explicitly

exploited in [35]. Later work in this direction was presented in [34], in which a new blind

identification algorithm based on support vector regression and specifically tailored for

sparse SIMO channels was proposed. The main idea of [34] is that the sparse SIMO

channel identification can be reformulated as a set of regression problems in which the

channel coefficients ph-li the role of the Lagrange multipliers. By using the e-insensitive

Vapnik's loss function in the regression problem, a large number of Lagrange multilpliers

(and, therefore, a large number of filter coefficients) become zero, thus yielding a sparse
filter estimate.










In [37] the previous work [34, 35] is extended in the following direction: first, a robust

algorithm for the blind estimation of a non-sparse channel when the channel order has

been highly overestimated is derived, secondly, to avoid the high computational cost in

solving a QP problem, a fast and simple algorithm called the Adatron [36] is used. We

propose to combine iterative regression and SRM principle to solve blind sparse SIMO

channel identification problem.

4.2 Observation Model

Without loss of generality, in this work we focus on the one-input, two-output SIMO

system shown in Fig. 4-1. In blind channel identification, we need to identify the unknown

channel responses, hl, h2, from the received signals only. If the order of the channels is M~,

then the received signal ri(u) from the ith channel is


rsu)= Ask~~ -k)+ei(u), i =1, 2. (4-1)


When we cast rs(u), As(k), s(u), ei(u) into vectors ri, hi, s and ei respectively, equation (4-1)

becomes


ri = hi s + ei, i = 1, 2 (4-2)


1 where denotes convolution. As shown in Fig. 4-1, using the channel outputs (rl, r2

and the channel estimates (hl, h2), One CRI1 Obtain the following matrix-vector form,:


Y1 = ri h2,

y2 = 2 hi


This relationship can be re-expressed in a matrix-vector form as:


Y1 = Rih2 2 Rhl = y2, (4




















Figure 4-1. Single-input two-output channel.


where Ri's are Toeplitz matrices defined as





Re = r(M + 1) -


r(0)

r(1)


1) -- iiv- rsN1)


(4-4)


or equivalently,


(4-5)


where


R = 2 1, h ='

If we solve (??) by minimizing hH HRh with the constraint || h ||= 1, then yr is the

LS solution which is the eigfenvector corresponding to the minimum eigfenvalue of RHR

Based on (??), we will next develop a SVM based robust blind identification method even

when the channel order is highly overestimated.


Rh = 0,










4.3 Combined Iterative and SVM Based Approach

4.3.1 Iterative Regression

Fr-om equation (4-3), we can formulate the following two coupled regression problems:


Rih2 1, (4-6)




Our goal is to make yl a y2. This is an intuitive and simple choice, because it drags the

actual outputs yl and y2 ClOSer to each other in order to achieve the equality in (4-3). At

ea~ch iteration? (given hr anld h2), desired output is constructed as

y1 + y2
yd =


and we end up with following two new uncoupled regression problems:


Rih2 Yd, (8

Rahl ~ yd, (4-9)


This iterative algorithm can be summarized as following

Algorithm 2 Iterative regression [35]


1. Initialize hi = h2 [ ]

2. Obtain the outputs Rla h2 Y1 and Rahi ~ y2 and form the desired signal yd =
Y1+Y2


3. Solve two new LS regression problems with yd as desired output.

4. Normalize the solution and go to step 2.









Note that this algorithm converges to the CCA (Canonical Correlation Analysis)

solution


arg max p= h R12h2 subject to h Rllhl Tah = hf2h
hi,hz

Canonical Correlation Analysis (CCA) is a well-known technique in multivariate

statistical analysis to find maximally correlated projections between two data sets. CCA

was developed by H. Hotelling [51] and it has been widely used in economics, meteorology

and in many modern information processing fields, such as communication problems [52],

statistical signal processing [53], independent component analysis [54] and blind source

separation [55].

4.3.2 SVM Regression

To fully exploit the sparse approximation characteristics provided by SVMs, each of

the regression problems is premultiplied by its conjugate transposed input matrix RH to

yield [c.f. (4-8),(4-9)]:


RF R hi2 d, (4-10)


RF R~h = R ye (4-11)


The resultant regression problems have input matrices that are simply the conjugate
transposed input matrix R andl the correpondirngr desi~red~f~ outputvecor become R yg

Moreover, the new regressor xi admits an expansion in terms of the filter coefficients,

which, in this way, become the Lagrange multipliers of the SVM formulation.

The SVM method minimizes the following cost function


J~e)= (a () | x |2,i= 1, 2 (4-12)










subject to


94 (u) xrir (u) < e + (n, a=1

xyri(u) 5i(u) Ie+ *, 1,.M

,n > 0, =1..M

(* > 0 =1,.,


where ( and (* are positive slack variables introduced by SVM procedure and r(u) denotes

the n-th column of Ri for i = 1, 2.

In equation (4-12), the regularization parameter C controls the tradeoff between the

training error and the complexity of the solution. On the other hand, e is a parameter that

determines the precision of the regression and therefore controls the sparseness of the final

solution. Then, the solution is a linear combination of input data


x = ~(as an)r (4-13)

where at~, an, are two different Lagrange multipliers.

In equation (4-13), only a small number of Lagrange multipliers (at~ an,), which

correspond to the channel coefficients h(0),h(1),..., h(M~-1) will be nonzero. Accordingly,

the overestimated channel coefficients will be zeros by the SRM principle.


4.3.3 Implementation of Support Vector Regression

The computational cost in solving the QP problem in equation (4-12) is the main

drawback of applying the SVM technique to practical estimation problems. Several

techniques have been proposed to solve this problem, including the use of iterative

reweighted least squares (IRWLS) techniques [56, 57] and the Adatron algorithm [2, 36, 58].

The IRWLS requires a matrix inversion at each iteration so the computational burden

could be considerably high even for a moderate number of data. On the other hand, the

Adatron algorithm is a much simpler least mean square (LMS)-like adaptive algorithm and










its convergence rate is exponential with the number of iterations. However, it is a memory

intensive method because all the kernel products need to be precomputed and saved.

4.3.3.1 The Adatron Algorithm

We can use the Adatron algorithm to solve the re-formulated regression in equations (4-10)

and (4-11). In the dual representation, the optimization problem in equation (4-12) can

be written as



n= 1 m= 1
M M



subject to as,, at~ E [0, C] and () denotes dot product.

The Adatron algorithm maximizes the above dual problem with gradient ascent

techniques [58]. Specifically, the Lagrange multipliers are updated according to


6ca, =' ($ (am a ~) (x, x) + Un e b ,(4-15)



ba =9 (a a) x, m +Un-e (4-16)

followed by updating as, and a~ with (as, + 6can)t and (ag~ + 6cat)t, respectively, where

at -^ max~a, 0} and rl is the learning rate. In addition, the bias b is updated as to

b + 6ca, 6cag and the evolution of this bias value can be used to check the convergence of

the algorithm.

As we can see in equations (4-15) and (4-16), the Adatron algorithm is very simple

to implement especially in DSP/FPGA hardware. To run this algorithm, all one needs is

just an adder and a multiplier. Furthermore, the computation time of Adatron increases

linearly with the number of data while the conventional QP's increases exponentially. This

simplicity is its main advantage. Using this algorithm, we propose a blind sparse channel

estimation method as summarized under Algorithm 3.









Algorithm 3 Adatron algorithm


1. Set e, C and initialize an, = 0, at~ = 0, Vu, and b = 0.

2. For all training samples n = 1, M, execute




If (as, + 6cad < 0 then set an, = 0, else an, <- an + 6ca,.


Ha:, = 9- E (am, a,) (x,, x,) + y, a- b .

If (at* + bad < 0 then set at* = 0, else at~ <- an + 6can.

3. Repeat step 2 until convergence.

4. h(u) = (as, a,,),fo =1, ,M.,



Combining the iterative regression procedure and the Adatron algorithm, overall

algorithm is summarized in Algorithm 4.

The computation cost of the proposed method is niter1 x niter2 x O(M~) where

niter1 is the iteration number of Adatron algorithm (Algorithm 2) and niter2 is the one

for iterative regression algorithm (Algorithm 3). Adatron convergence plots are depicted

in Fig. 4-2 and it shows larger rl values have fast convergence speed. Convergence of the

proposed overall algorithm is shown in Fig. 4-3.









Algorithm 4 Overall algorithm


1. Intialize e, C, and hi = h2 = TLn d].

2. Obtain the outputs Xlh2 = 1 and X~hl = y2*

3. Form the desired signal yd =

4. Construct the transformed SV regression problems:

XTw1 = X yd and XTW2 = X yd.

5. Solve the QP problem using Adatron.
-> the Lagrange multipliers are the filter coefficients.

6. Normalize the solution and go to step 2.


4.3.3.2 Learning rate for convergence

The change in the Lagrangian (4-14) is:


sL = Ja(om a zz)(x,, xm) + yban
m= 1









For be s, it can be shown that


e1,- (a )


ib)


(sC~n>2(XnXn>


(417)


6L1 = (a)2 8 )


From eqs. (4-17)and (4-18), we can get the relation 0 < rl(x,, ~) < 2 for a positive

change of 6L.













1


0.9


0.8


0.7

E
a, 0.6


a5 0.5


0 0.4


- 0.3
O


x 10-3


-


-


-


-


0.2


0.1


20 40 60 80 100
niter1



Figure 4-2. Convergfence of Adatron


120


As we can notice from (4-17)and (4-18), bound of the learning rate rl is data

dependent and can be determined from


0 < rl < ~


(4-19)


COnvergence of Adatron



































5 10


15 20 25 30


niter2


Figure 4-3. Convergfence of iterative regression.


4.4 Simulation Results

Several simulations have been conducted to test the performance of our proposed

algorithm. The performance is measured in terms of the normalized mean squared

error (NMSE) defined as [31]:


11~
NVIfSE = min ch-h
||h ||2 ask>0



where At > Af is the estimated channel order.

In the first simulation we consider a sparse SIMO system which consists of a single

transmit antenna and two receive antennas. The two sparse channels are respectively,


Convergence of iterative method






































-1 -0.5 0 0.5 1


Zero plot
-0

Oc


0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8


O


O


O


yO


tO


CA O



o'


O


X O


O
A O


O


Figure 4-4. Zeros of subchannel h h2


Hl(z) = 1 0.62z-s 0.33z-14 + 0.08z-24, H2(z) = 0.91 + 0.56z-11 0.28z-".~ Input of

this system is NV = 100, i.i.d. BPSK( signals. In Fig.4-4, we plot the zeros of hl, h2. NOte

that there are pairs of close zeros which impair subspace based method because of a badly

conditioned input correlation matrix.

Figf.4-5 shows the robustness to order overestimation when SNR is 20dB. It is evident

that the proposed method outperforms other methods in highly overestimated channel

order estimate. Performance is summarized in Figf.4-6 and shows the proposed method

performs much better than Regalia method in identifying the coefficients of zero taps or

very small taps.

In the next example we consider a raised-cosine pulse with duration 4T (T is the

symbol period) with a roll-off factor 0.1 and the multipath channel is h(t) = 6(t) 0.76(t -


O h,

X h2
























0 5 10 15 20 25 30 35


(a) SVM based liethod


.c -0.5-

-1
0 5 10 15 20 25 30 35 40





~j0.5-





-0.5
0 5 10 15 20 25 30 35 40



(b) Robust-SS niethod


Figure 4-5. 50 trials of the SV1\ based method and Robust-SS method when
overestimated by 10 taps and SNR=30dB










Overestimated Channel Order


0.3-

0.2-

0.1-

10 15 20 25 30
SNR(dB)


Figure 4-6. Robustness when the channel order is exact and overestimated by 10 taps


~). The input signal is i.i.d. BPSK; signal and the received data is sampled at twice the

symbol rate to obtain a SIMO system. Fig. 4-8 depicts the performance at different SNRs.

This example shows that proposed SVM based method can be also used for non-sparse

channels, in particular when the channel order has been highly overestimated. In this

situation, the structural risk minimization (SRM) principle pushes the small leading

and trailing terms of the impulse response to zero. Note that the performance of our

proposed method is much better than other methods, especially at low SNR. In Fig. 4-7,

50 trials of our proposed algorithm and the robust method proposed in [2], it is clear that

the estimation of our proposed method at zero tap coefficients is much better than the

robust-SS method.

In this chapter, our SVM based sparse estimation method is extended in the following

direction: first, a robust algorithm for the blind estimation of a non-sparse channel when

the channel order has been highly overestimated is derived, secondly, to avoid the high

computational cost in solving a QP problem, a fast and simple algorithm called the











Adatron, SNR =20


0 5 10 15 20 25 30 35

(a) SVM based method using the Adatron

Robust-SS, SNR =20
5


10 15 20

(b) Robust-SS method


25 30 35


Figure 4-7. 50 trials of the SV1\ based method using the Adatron and robust-SS method
when the channel order is overestimated by 20 taps and SNR=20dB.















































0.4-



0.3-



0.2-



0.1
10 12 14 16 18 20 22 24 26 28 30
SNR(dB)

Figure 4-8. Performance comparison when the channel order is overestimated by 20 taps
(raised-cosine pulse followed by a multipath channel).










Adatron [36] is used. We proposed to combine iterative regression and SRM principle to

solve blind sparse SIMO channel identification problem. Also we shows that proposed

SVM based method can be also used for non-sparse channels, in particular when the

channel order has been highly overestimated.









CHAPTER 5
IDENTIFICATION OF TIME VARYING & FREQITENCY-SELECTIVE MITLTIPATH
CHANNEL

5.1 Introduction

Tinte-varying channel estimation is all in r~ obstacle to increase the capacity

and reliability of wireless coninunication systems. For slow fading channels, adaptive

algorithm for tinte-invariant such as RLS and LMS can provide a reasonable alternative

but these adaptive algorithms do not track fast channel variations. For fast fadingf

channels explicit incorporation of the channel's tinte-varying (TV) characteristics is

needed. Recently, a deterministic hasis expansion model is widely used for cellular

radio applications, especially when the nmultipath is caused by a few strong reflectors.

Receivers that explicitly model the channel variation in time are more successful than the

LMS or RLS. The K~arhunen-Loibve (K(L) expansion [59], the polynomial model [60, 61],

the exponential model [62, 63], and wavelets [64] have been used for modeling the TV

channel. The tinte-varying taps are expressed as a superposition of tinte-varying hases

with tinte-invariant coefficients. By assigning time variations to the bases, rapidly fading

channels with coherence time as small as a few tens of symbols can he captured.

In this chapter, we propose to use the SVAI technique with the exponential basis

expansion model to identify the tinte-varying channel.

5.2 Basis Expansion Models

The TV system is modeled by



l=0

where the TV impulse response h(n; 1) depends on time n. In this thesis, a deterministic

hasis expansion is used to model the TV impulse response b (n; 1) [62, 63]. TV impulse

response of rapidly fading channels is expanded over a basis of complex exponentials that

arise due to Doppler effects encountered in the nmultipath environment. The TV taps

are expressed as a superposition of TV hases with time invariant (TI) coefficients. By




























Figure 5-1. Basis expansion model of a time-varying system


assigning time variations to the bases, rapidly fading channels can be captured. Note

that complex exponentials model the Doppler effects in (5-2), and by using them, the

estimation of time-varying channel system can be cast into a time-invariant estimation

problem. The basis expansion model is given by

x u)=o c, (1) e""' s (n 1) + v (u) (5-2)


h (n; 1)

where s(u) is input, h(n; 1) is finitely parameterized for each lag 1 via its expansion

coefficients c,(1) onto known exponential bases {1, e "2", -, e "P"}, as depicted in

Fig.5-1. To estimate the TI parameters {cq(1)}, we assume the knowledge of the base

frequencies {w,)}(7. This can by estimated using tests for cyclostationarity or adaptive

maximum-likelihood methods [65, 66].

Fr-om (5-2), we have



q(> V1=1 l=0 (3









This relationship can be re-expressed in a matrix-vector form as:


(5-4)


Sc + v


(5-5)


where


S(1)e "al

S(2)ejWq2


S (N)ejw4 N


0

S 1)ejWq2


s(NV-1l)ej"e"


S, =




c =


c, (0) c, (1) c, (L 1)


x = x(1) x (2) - x (N




Using Least Squares, c in (5-5) can be estimated as

c =(SHS)- SHx (5-6)

Then TV channel coefficients h(n, 1) are estimated using the estimated c. When the

number of training symbol is short, the performance of LS estimate (5-6) is bad. In [61],

the MP method is used with the polynomial basis model to find the best aligned column

to the received signal. In this chapter, we extend the previous work [34, 37] to time-varying
environment in a data-aided manner.


x = SI S2 SP i


--- s(NV- L +1)ej4

Ti










5.3 Time Varying Channel Estimation using SVR

Fr-om equation (5-5), the system model is


C1



C2



=Sc + v


Then we can estimate time-invariant parameter c using SVR technique by minimizing



n= 1

and the solution is


c = (s a) S,(5-7)
n= 1

where as~ as, are the Lagrange multiplier and M is the number of support vectors.

Also SVM-based blind method developed in previous chapter can be easily applied to

this TV case.

5.4 Simulation Result

Several simulations have been conducted to test the performance of the proposed

algorithm. Normalized mean square error (NMSE) between a h and its estimate h is

computed as follows:

1 |h h||2lll
NMSE R |h||2(5-8)
r=1

where r denotes realization and R is the number of realizations. In Fig. 5-3, we illustrate

the estimated TI parameter {c,(1)} ~, of basis expansion model using both LS and SVM

method. The input signal is i.i.d. BPSK( signal and SNVR = 10dB. NV = 100 training

symbols were used with a channel order L = 3 and the Q = 3 bases were chosen. All plots









are an average of 100 Monte Carlo runs. Here we can see that SVAI performs better than

the LS method. Fig. 5-2 illustrates that NMSE of proposed SVAI method is better than

LS method. Fig. 5-3 and Fig. 5-4 shows the 50 trials of blind estimation of TI parameter

{8,(1)} 1 of basis expansion model using both LS and SVAI method.


0.09


0
10 15 20 25
SNR(dB)


Figure 5-2. MSE Comparison of LS and SVAI when N


100, L = 3, and Q


In this chapter, we proposed to use the SVAI technique which developed in previous

chapter with the exponential basis expansion model to identify the tinte-varying channel.

Simulation demonstrated that BEM-SVAI method successfully applied to time varying

channel estimation.




















LS, SNR =10


1 2 3 4 5 6


h2


2 3 4 5 6


Figure 5-3. Blind estimation of basis coefficients {c:,(1)},_ when N
and SNR= 10dB (50 trials) using LS.


80, L = 3, Q = 2,






















SVM, SNR =10


1 2 3 4 5 6


r

CI
O


2 3 4 5 6


Figure 5-4. Blind estimation of basis coefficients {c:,(1)},_ when N
and SNR= 10dB (50 trials) using SVM.


80, L = 3, Q = 2,









CHAPTER 6
CONCLUSIONS AND FUTURE WORK(

6.1 Conclusions

This dissertation aims at exploring normal signal processing methods to estimate

sparse channels. Before this research had started, L1-norm regularization method (BP)

and matching pursuit (jl\!) algorithms are widely used methods in many sparse

applications. In chapter 2, the MP method and L1-norm regularization are briefly

summarized to understand the conventional sparse solutions.

Previous work in [16] formulates sparse channel estimation as the optimal mean

square error (j\!SE) estimation of the channel impulse response regularized with a

Li-norm constraint. In chapter 3, L1-norm of the channel impulse response, which pi i

an important role in achieving the sparse solution, is estimated by using the convolution

inequality for entropy and this information is exploited to improve the performance of

sparse-LMS.

In chapter 4, support vector machines are briefly introduced including the classification

and the regression problem. Then we present a new SVM-based sparse system identification

algorithm. We reformulate the sparse channel estimation problem as a support vector

regression (SVR) problem in which the channel coefficients are the Lagrange multipliers

of the dual problem. By employing Vapnik's e-insensitivity loss function, the solution

is expanded in terms of a reduced number of Lagrange multipliers (i.e., the nonzero

filter coefficients) and then a sparse solution is found. Then this SVR based method is

also applied to derive an iterative algorithm for blind estimation of sparse single-input

multiple-output (SIMO) channels, in particular when the channel order has been highly

overestimated.

In chapter 5, the previous work [34, 35] is extended in the following direction: first,

a robust algorithm for the blind estimation of a non-sparse channel when the channel

order has been highly overestimated is derived, secondly, to avoid the high computational

cost in solving a QP problem, a fast and simple algorithm called the Adatron [36] is used.










Combined iterative regression and SR M principle is proposed to solve blind sparse SIMO

channel identification problem.

Tinte-varying channel estimation is also considered because it is all is r~ obstacle to

increase the capacity and reliability of wireless coninunication systems. For slow fading

channels, adaptive algorithm for tinte-invariant such as recursive least-squares (RLS) and

least nican-square (LMS) can give alternative but this adaptive algorithms diverge when

channel variations exceed the convergence time of algorithm. In this case additional

information of the tinte-varying channel is needed. Most models of tinte-varying

channels treat the tap coefficients as uncorrelated stationary random processes. By

using hasis expansion model, estimation of tinte-varying channel system can he cast into

a tinte-invariant estimation problem. TV impulse response of rapidly fading channels

is expanded over a basis of complex exponentials that arise due to Doppler effects

encountered with nmultipath environment. The TV taps are expressed as a superposition

of TV hases with TI coefficients. In chapter 6, we propose to use the SVAI technique with

the exponential basis expansion model to identify the tinte-varying channel.

In appendix, to investigate the feasibility of the FPGA intplenientation of the

proposed algorithm, the core computation of this Adatron is simulated in FPGA using

System Generator which is a high-level design tool for Xilinx FPGAs that extends the

capabilities of Siniulink to include accurate modeling of FPGA circuits. The performance

is also compared to the TI T:\S:320C:33 DSP intplenientation.



6.2 Future Work

Support vector machine approach was successfully applied to sparse channel

estimation problem including blind SIMO channel estimation in this research. Higfh

computational cost of the resulting quadratic progranining problem is reduced by using

Adatron algorithm to solve the SVR problem iteratively. The Adatron is a way of solving

a batch problem in a saniple-by-saniple basis but it's not an on-line technique. Recently










an online learning method in reproducing Hilbert space by considering classical stochastic

gradient descent in [67] so we should consider an on-line solution.

It is well known that SVM performance (estimation accuracy) depends on a good

setting of meta-parameters C and e. Optimal selection of these parameters are difficult

because SVM model performance depends on all these parameters so we presented a SVM

meta-parameter setting selection via the simulation. Schoilkopf et al [68] -11---- -r to control

another parameter v instead of a so we should investigate the optimal parameter setting of

SVM.

In basis expansion model (BEM) for time varying channel estimation, we assumed the

knowledge of the base frequencies. This can by estimated using tests for cyclostationarity

or adaptive maximum-likelihood methods [65, 66] so we should explore how to decide the

basis of BEM.

FPGA implementation of SVM is simulated by mapping Adatron onto FPGA but

it was not an efficient implementation because the way of solving is batch. Memory for

storing kernel matrix was needed and we should wait until the computations of last sample

so we should investigate an efficient FPGA implementation algorithm based on the online

solution in [67].

Summarizing the future works,

Investigate the on-line algorithm

V-SVM and optimal selection of SVM meta-parameters

Estimation of the basis frequencies in BEM model.

Optimized FPGA implementation of SVM










APPENDIX A
FPGA IMPLEMENTATION OF ADATRON ALGORITHM USING SYSYEM
GENERATOR

A.1 Introduction

There has been considerable recent progress in tool development to support

DSP applications in FPGAs. System Generator is a high-level design tool for Xilinx

FPGAs that extends the capabilities of Simulink to include accurate modeling of FPGA

circuits [69]. System Generator provides Simulink libraries for arithmetic and logic

functions, memories, and DSP functions. System Generator provides abstractions for these

resources. As we saw in chapter 4, calculations for the Adatron algorithm 4-16 can he

implemented in helow matlah script:


for i=1:1ength (X)
templ=alpha-alpha_ast;
delta_alpha (i)=eta*(-K (i, :)*templ+y (i) -epslnba d)
if alpha(i)+delta_alpha(i)<0
alpha (i) =0;
else
alpha(i)=alpha(i)+delta_alpha(i);
end

delta_alpha_ast(i) =eta* (K(i, :)*templ-y (i) eslnba d)
if alpha_ast (i)+delta_alpha_ast (i) 0
alpha_ast (i)=0;
else
alpha~_ast(i)=alpha_ast (i)+delta_alpha_ast()
end

bias_ada=bias_ada+delta_alpha (i) -delta_alpaat()

end


Figure A-1. Alatlah implementation of Adatron algorithm



In this appendix, we use System Generator to map the Adatron algorithm shown

above onto a FPGA and investigate the feasibility of FPGA implementation of the

algorithm. Also the performance will be compared to TI TMS:320C:33 DSP implementation.










A.2 Adatron Engine in System Generator

A System Generator model of a Adatron engine block is shown in Fig. A-2. Address

control logic block generates the control signals (address, write enable) of the nienory

for storing the input, kernel evaluations and the Lagrange nmultipliers. Kernels can he

precalculated in software and each row of kernels can he feed through the inp gateway

in. The Lagrange nmultipliers are stored in the single port ran11, while the input update

is stored in the workspace. The Adatron engine in Fig. A-2 calculates the update of the

Lagrange nmultipliers using the data available in the shared nienory. Two outputs, out1

and out2, front the Ada engine block correspond to ~,z and n,*, in Adatron algorithm, and

are restored in workspacel thru the gateway out.


Gateway Out1 Signal To
Workspacel


Ada engine


Signal From Gae yIn
Workspacel


Figure A-2. Top design block of Adatron FPGA intplenientation using Xilinx System
Generator










A.2.1 Address Control Logic

Fig. A-3 shows the detail logics in the address control block. Counters are for

generating the address of the memories which stores the input data and the Lagrange

multipliers. System Generator provides an interface to the embedded processor into the

design so that Software can perform read/write operations to a shared memory through

named association. Data write enable and alpha write enable are the shared memories

that enables the writing to memory.


data addr


alpha_counter


dout

From Register1 lhaw
<< 'alpha_we' >>


Delay


Figure A-3. Address control logic of the memory


data counter


dout

From Register daaw
<< 'data we' >>










A.2.2 Adatron Engine block

Fig. A-5 shows the detail building blocks of the below Adatron algorithm.






If (as, + 6cad < 0 then set an, = 0, else an, <- an + 6ca,





If (as + a,), < 0 then set as=0 es s s a


Mult and accumulator calculates the sum of multiplications of input and Lagrange

multipliers and result is provided to MCode block. Other calculations such as .II1I1;1!

subt 1 Ilriin and comparing to compute be~ is executed in calalpha MCode block and

is shown in Fig. A-4. We can see that configuring the Xillinx MCode block is an easier

solution than building the logic together through Xilinx blockset logic.



function alpha = calalpha(palpha, eta, xLag, y, en,
epsilon, b)

delta = eta*(palpha + xLag + y epsilon b);
if delta < 0
alpha = 0;
else
alpha = palpha + delta;
end




Figure A-4. MCode block configuration for updating a~


























MCode


SFrom Rebg ister4
Register1


MCodel


Figure A-5. Adatron Engine block detail


A.3 Implementation Results

All results were obtained using the System Generator V8.1 [70], Xilinx ISE 8.1i [71,

72], and XST synthesis tool to target a Virtex4 xc4vsx35-10 part. All models use

dedicated multipliers, block memory for storage, and 16-bit precision. FPGA device

implementation results are summarized in table A-1. The basic building block of Xilinx

Virtex devices is the logic cell (LC). According to the Xilinx Virtex data sheet [73], an LC

includes a 4-input look up table (LUT), carry logic, and a storage element. The Xilinx

Virtex-E architecture contains configurable logic blocks (CLBs). Each Virtex-E CLB

contains four LCs and a CLB consists of two slices.










If we have an NV input data sample, we will have an NVx NV kernel matrix and NV

Lagrange nmultipliers. Suppose adaptation needs 50 iterations for the Adatron to converge

and 100 input data samples. Then it takes (NV + 7) x 50(= 5:350) clock cycles to evaluates

the updates of the Lagrange nmultipliers while it takes (2NV+ 25) x 50(= 11250) clock cycles

with TI TiS:320CW:3 DSP intplenientation [74, 75]. 5:350 cycles corresponds to 26.75p~s at

200MHz clock in the Xilinx ISE 8.1i. Figf A-6 shows the intplenientation of the basic

computation of Adatron algorithm with TI TM S:320CW:3 assembly language. Note that we

can increase the performance of the FPGA intplenientation by parallelizing the operation.
Table A-1. Device utilization suninary

Selected Device 4vsx:35ff668-10
Number of Slices 278
Number of Slice Flip Flops 210
Number of 4 input LUTs 445
Number of FIFO16/R AMB16s 2
Number of GCLEKs 1
Number of DSP48s 1


In this appendix, we use System Generator to nmap the Adatron algorithm shown

above onto a FPGA resource including address generation logic, nienory, Adatron

engine, and the logic fabric of Virtex FPGAs. Also the performance is compared to TI

TM S:320CW:3 DSP intplenientation. Results showed that the FPGA intplenientation of

Adatron is feasible and has more design margin than the DSP intplenientation.



























* compute K(i,:)*templ
Idi @y, rO
Idf Epsilon, r1
Idf bias, r2
Idf eta, r3
Idf 0.0, r4
Idi @alpha, r5
Idi @kaddr, ar1
Idi @taddr, ar2


Idi
rptb
mpyf3
eloop addf

addf
subf
subf
mpyf
addf
cmpf
bnz
Idf
next stf


N, rc
eloop
*arl++%,*ar2++,r7
r7, r4

rO, r4
rl, r4
r2, r4
r3, r4
r5, r4
0, r4
next
0, r4
r4, @delta


Figure A-6. TI DSP (TM\ S320C33) implementation of the Adatron algorithm









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BIOGRAPHICAL SKETCH

Dongho Han was born in Seoul, K~orea. He received his Bachelor of Science degrees

in electrical engineering from the Yonsei University, Seoul, K~orea, in 1993. From 1993

to 1998, he was a hardware engineer in Samsung Electronics where he developed

several cpu boards for PBX system. He also received his Master of Science degree in

electrical and computer engineering from the University of Florida in 2001. He was in the

Computational NeuroEngineering Laboratory in the Electrical and Computer Engineering

Department at the University of Florida during his Ph.D study. His present research

interests are in the areas of signal pro, f -in neural network, communication algorithms

and their implementations in FPGA and DSP. While a Ph.D. candidate, Dongho interned

with Motorola, where he returned upon graduation for full-time employment as a DSP

engineer.