... theautocorrelation estimate, since this incorporates prior knowledge of the autocorrelation into theestimation process. The estimation problem then becomes that of estimating the N autocorrelationlags ... LLCTransformDomainAdaptiveFilteringW.KennethJenkinsUniversityofIllinois,Urbana-ChampaignDanielF.MarshallMITLincolnLaboratory 22. 1LMSAdaptiveFilterTheory 22. 2OrthogonalizationandPowerNormalization 22. 3ConvergenceoftheTransformDomainAdaptiveFilter 22. 4DiscussionandExamples 22. 5Quasi-NewtonAdaptiveAlgorithmsAFastQuasi-NewtonAlgorithm•Examples 22. 6The2-DTransformDomainAdaptiveFilter 22. 7Block-BasedAdaptiveFiltersComparisonoftheConstrainedandUnconstrainedFre-quencyDomainBlock-LMSAdaptiveAlgorithms•ExamplesandDiscussionReferencesOneoftheearliestworksontransformdomainadaptivefilteringwaspublishedin1978byDentinoetal.[4],inwhichtheconceptofadaptivefilteringinthefrequencydomainwasproposed.Manypublicationshavesinceappearedthatfurtherdevelopthetheoryandexpandthecurrentunder-standingofperformancecharacteristicsforthisclassofadaptivefilters.InadditiontothediscreteFouriertransform(DFT),otherorthogonaltransformssuchasthediscretecosinetransform(DCT)andtheWalshHadamardtransform(WHT)canalsobeusedeffectivelyasameanstoimprovetheLMSalgorithmwithoutaddingtoomuchcomputationalcomplexity.Forthisreason,thegeneraltermtransformdomainadaptivefilteringisusedinthefollowingdiscussiontomeanthattheinputsignalispreprocessedbydecomposingtheinputvectorintoorthogonalcomponents,whichareinturnusedasinputstoaparallelbankofsimpleradaptivesubfilters.Withanorthogonaltransforma-tion,theadaptationtakesplaceinthetransformdomain,asitispossibletoshowthattheadjustableparametersareindeedrelatedtoanequivalentsetoftimedomainfiltercoefficientsbymeansofthesametransformationthatisusedfortherealtimeprocessing[5,14,17].AdirectformFIRdigitalfilterstructureisshowninFig .22. 1.ThedirectformrequiresN−1delays,Nmultiplications,andN−1additionsforeachoutputsamplethatisproduced.Theamountofhardware(aswellaspower)requiredtoimplementthedirectformstructuredependsonthedegreeofhardwaremultiplexingthatcanbeutilizedwithinthespeeddemandsoftheapplication.AfullyparallelimplementationconsistingofNdelayregisters,Nmultipliers,andatreeoftwo-inputadderswouldbeneededforveryhigh-frequencyapplications.Attheoppositeendoftheperformancespec-trum,asequentialimplementationconsistingofalengthNdelaylineandasingletimemultiplexedmultiplierandaccumulationadderwouldprovidethecheapest(andslowest)implementation.Thisc1999byCRCPressLLCFIGURE ... autocorrelation matrixmust be crudely approximated by the identity matrix.These observations suggest that many of the seemingly distinct adaptive filtering algorithms thatappear scattered about...