... frequency realizationfor FIR adaptive digital filters,IEEE Trans. Acoust., Speech, Signal Process.,Vol. ASSP-31(5),1073–1083, Oct. 1983.[2] Bellanger, M.G.,Adpative Digital Filters and Signal Analysis,Marcel ... LLCTransformDomainAdaptiveFilteringW.KennethJenkinsUniversityofIllinois,Urbana-ChampaignDanielF.MarshallMITLincolnLaboratory22.1LMSAdaptiveFilterTheory22.2OrthogonalizationandPowerNormalization22.3ConvergenceoftheTransformDomainAdaptiveFilter22.4DiscussionandExamples22.5Quasi-NewtonAdaptiveAlgorithmsAFastQuasi-NewtonAlgorithm•Examples22.6The2-DTransformDomainAdaptiveFilter22.7Block-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 ... Hadhoud, M.M. and Thomas, D.W., The two-dimensional adaptive LMS (TDLMS) algorithm,IEEE Trans. Circuits Syst.,Vol. 35, 485–494, 1988.[8] Honig, M.L. and Messerschmidt, D.G.,Adaptive Filters:...