adaptive filterWord下载.docx

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2.Basicconfigurationofanadaptivefilter

Thebasicconfigurationofanadaptivefilter,operatinginthediscrete-timedomaink,isillustratedinFigure2.1.Insuchascheme,theinputsignalisdenotedbyx（k）,thereferencesignald（k）representsthedesiredoutputsignal（thatusuallyincludessomenoisecomponent）,y（k）istheoutputoftheadaptivefilter,andtheerrorsignalisdefinedase（k）=d（k）.y（k）.

Fig.2.1Basicblockdiagramofanadaptivefilter.

Theerrorsignalisusedbytheadaptationalgorithmtoupdatetheadaptivefiltercoefficientvectorw（k）accordingtosomeperformancecriterion.Ingeneral,thewholeadaptationprocessaimsatminimizingsomemetricoftheerrorsignal,forcingtheadaptivefilteroutputsignaltoapproximatethereferencesignalinastatisticalsense.

Fig.2.2Channelequalizationconfigurationofanadaptivefilter:

Theoutputsignaly（k）estimatesthetransmittedsignals（k）.

Fig.2.3Predictorconfigurationofanadaptivefilter:

Theoutputsignaly（k）estimatesthepresent

inputsamples（k）basedonpastvaluesofthissamesignal.Therefore,whentheadaptivefilteroutputy（k）approximatesthereference,theadaptivefilteroperatesasapredictorsystem.

3.Adaptationalgorithm

Severaloptimizationprocedurescanbeemployedtoadjustthefiltercoefficients,including,forinstance,theleastmean-square（LMS）anditsnormalizedversion,thedata-reusing（DR）includingtheaffineprojection（AP）,andtherecursiveleast-squares（RLS）algorithms.AlltheseschemesarediscussedinSection2.3,emphasizingtheirmainconvergenceandimplementationcharacteristics.TheremainingofthebookfocusesontheRLSalgorithms,particularly,thoseemployingQRdecomposition,whichachieveexcellentoverallconvergenceperformance.

3.1ErrorMeasurements

Adaptationofthefiltercoefficientsfollowsaminimizationprocedureofaparticularobjectiveorcostfunction.Thisfunctioniscommonlydefinedasanormoftheerrorsignale（k）.Thethreemostcommonlyemployednormsarethemean-squareerror（MSE）,theinstantaneoussquareerror（ISE）,andtheweightedleast-squares（WLS）,whichareintroducedbelow.

3.2Themean-squareerror

TheMSEisdefinedas

⎩（k）=E[e2（k）]=E[|d（k）−y（k）|2].

WhereRandparetheinput-signalcorrelationmatrixandthecross-correlationvectorbetweenthereferencesignalandtheinputsignal,respectively,andaredefinedas

R=E[x（k）xT（k）],

p=E[d（k）xT（k）].

Note,fromtheaboveequations,thatRandparenotrepresentedasafunctionoftheiterationkornottime-varying,duetotheassumedstationarityoftheinputandreferencesignals.

FromEquation（2.5）,thegradientvectoroftheMSEfunctionwithrespecttotheadaptivefiltercoefficientvectorisgivenby

Theso-calledWienersolutionwo,thatminimizestheMSEcostfunction,isobtainedbyequatingthegradientvectorinEquation（2.8）tozero.AssumingthatRisnon-singular,onegetsthat

3.3Theinstantaneoussquareerror

TheMSEisacostfunctionthatrequiresknowledgeoftheerrorfunctione（k）atalltimek.Forthatpurpose,theMSEcannotbedeterminedpreciselyinpracticeandiscommonlyapproximatedbyothercostfunctions.ThesimplerformtoestimatetheMSEfunctionistoworkwiththeISEgivenby

Inthiscase,theassociatedgradientvectorwithrespecttothecoefficientvectorisdeterminedas

ThisvectorcanbeseenasanoisyestimateoftheMSEgradientvectordefinedinEquation（2.8）orasaprecisegradientoftheISEfunction,which,initsownturn,isanoisyestimateoftheMSEcostfunctionseeninSection2.2.1.

3.4Theweightedleast-squares

AnotherobjectivefunctionistheWLSfunctiongivenby

where0_⎣<

1istheso-calledforgettingfactor.Theparameter⎣k−iemphasizesthemostrecenterrorsamples（wherei≈k）inthecompositionofthedeterministiccostfunction⎩D（k）,givingtothisfunctiontheabilityofmodelingnon-stationaryprocesses.Inaddition,sincetheWLSfunctionisbasedonseveralerrorsamples,itsstochasticnaturereducesintime,beingsignificantlysmallerthanthenoisyISEnatureaskincreases.

2.3AdaptationAlgorithms

Inthissection,anumberofschemesarepresentedtofindtheoptimalfiltersolutionfortheerrorfunctionsseeninSection2.2.Eachschemeconstitutesanadaptationalgorithmthatadjuststheadaptivefiltercoefficientsinordertominimizetheassociatederrornorm.

Thealgorithmsseenherecanbegroupedintothreefamilies,namelytheLMS,theDR,andtheRLSclassesofalgorithms.Eachgrouppresentsparticularcharacteristicsofcomputationalcomplexityandspeedofconvergence,whichtendtodeterminethebestpossiblesolutiontoanapplicationathand.

2.3.1LMSandnormalized-LMSalgorithms

DeterminingtheWienersolutionfortheMSEproblemrequiresinversionofmatrixR,whichmakesEquation（2.9）hardtoimplementinrealtime.OnecanthenestimatetheWienersolution,inacomputationallyefficientmanner,iterativelyadjustingthecoefficientvectorwateachtimeinstantk,insuchamannerthattheresultingsequencew（k）convergestothedesiredwosolution,possiblyinasufficientlysmallnumberofiterations.

TheLMSalgorithmissummarizedinTable2.1,wherethesuperscripts.andHdenotethecomplex-conjugateandtheHermitianoperations,respectively.

TheLMSalgorithmisverypopularandhasbeenwidelyusedduetoitsextremesimplicity.Itsconvergencespeed,however,ishighlydependentontheconditionnumberpoftheinput-signalautocorrelationmatrix[1–3],definedastheratiobetweenthemaximumandminimumEigenvaluesofthismatrix.

IntheNLMSalgorithm,whenυ=0,onehasw（k）=w（k−1）andtheupdatinghalts.Whenυ=1,thefastestconvergenceisattainedatthepriceofahighermisadjustmentthentheoneobtainedfor0<

υ<

1.

2.3.2Data-reusingLMSalgorithms

Asremarkedbefore,theLMSalgorithmestimatestheMSEfunctionwiththecurrentISEvalue,yieldinganoisyadaptationprocess.Inthisalgorithm,informationfromeachtimesamplekisdisregardedinfuturecoefficientupdates.DRalgorithms[9–11]employpresentandpastsamplesofthereferenceandinputsignalstoimproveconvergencecharacteristicsoftheoveralladaptationprocess.

Asageneralizationofthepreviousidea,theAPalgorithm[13–15]isamongtheprominentadaptationalgorithmsthatallowtrade-offbetweenfastconvergenceandlowcomputationalcomplexity.Byadjustingthenumberofprojections,oralternatively,thenumberofdatareuses,oneobtainsadaptationprocessesrangingfromthatoftheNLMSalgorithmtothatofthesliding-windowRLSalgorithm[16,17].

2.3.3RLS-typealgorithms

ThissubsectionpresentsthebasicversionsoftheRLSfamilyofadaptivealgorithms.Importanceoftheexpressionspresentedherecannotbeoverstatedfortheyallowaneasyandsmoothreadingoftheforthcomingchapters.

TheRLS-typealgorithmshaveahighconvergencespeedwhichisindependentoftheEigenvaluespreadoftheinputcorrelationmatrix.Thesealgorithmsarealsoveryusefulinapplicationswheretheenvironmentisslowlyvarying.

ThepriceofallthesebenefitsisaconsiderableincreaseinthecomputationalcomplexityofthealgorithmsbelongingtotheRLSfamily.

ThemainadvantagesassociatedtotheQR-decompositionRLS（QRD-RLS）algorithms,asopposedtotheirconventionalRLScounterpart,arethepossibilityofimplementationinsystolicarraysandtheimprovednumericalbehaviorinlimitedprecisionenvironment.

2.5Conclusion

Itwasverifiedhowadaptivealgorithmsareemployedtoadjustthecoefficientsofadigitalfiltertoachieveadesiredtime-varyingperformanceinseveralpracticalsituations.Emphasiswasgivenonthedescriptionofseveraladaptationalgorithms.Inparticular,theLMSandtheNLMSalgorithmswereseenasiterativeschemesforoptimizingtheISE,aninstantaneousapproximationoftheMSEobjectivefunction.Data-reusealgorithmsintroducedtheconceptofutilizingdatafrompasttimesamples,resultinginafasterconvergenceoftheadaptiveprocess.Finally,theRLSfamilyofalgorithms,basedontheWLSfunction,wasseenastheepitomeoffastadaptationalgorithms,whichuseallavailablesignalsamplestoperformtheadaptationprocess.Ingeneral,RLSalgorithmsareusedwheneverfastconvergenceisnecessary,forinputsignalswithahighEigenvaluespread,andwhentheincreaseinthecomputationalloadistolerable.AdetaileddiscussionontheRLSfamilyofalgorithmsbasedontheQRdecomposition,whichalsoguaranteesgoodnumericalpropertiesinfinite-precisionimplementations,constitutesthemaingoalsofthisbook.Practicalexamplesofadaptivesystemidentificationandchannelequalizationwerepresented,allowingonetovisualizeconvergenceproperties,suchasmisadjustment,speed,andstability,ofseveraldistinctalgorithmsdiscussedpreviously.