channel Maximally Decmiated Filter Banks Electrical Computer Engineering University of Maryland College Park Acknowledgment ENEE630 slides were based on class notes developed by Profs KJ Ray Liu and Min Wu The LaTeX slides were made by Prof Min Wu ID: 74206
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7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations Multi-rateSignalProcessing7.M-channelMaximallyDecmiatedFilterBanksElectrical&ComputerEngineeringUniversityofMaryland,CollegeParkAcknowledgment:ENEE630slideswerebasedonclassnotesdevelopedbyProfs.K.J.RayLiuandMinWu.TheLaTeXslidesweremadebyProf.MinWuandMr.Wei-HongChuang.Contact:minwu@umd.edu.Updated:September27,2012. ENEE630LecturePart-11/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) M-channelMaximallyDecimatedFilterBankM-ch.lterbank:Tostudymoregeneralconditionsofalias-free&P.R. Aseachlterhasapassbandofabout2=Mwide,thesubbandsignaloutputcanbedecimateduptoMwithoutsubstantialaliasing.Thelterbankissaidtobe\maximallydecimated"ifthismaximaldecimationfactorisused.[Readings:VaidynathanBook5.4-5.5;TutorialSec.VIII] ENEE630LecturePart-12/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) TheReconstructedSignalandErrorsCreatedRelationsbetween^X(z)andX(z): (details)^X(z)=PM�1l=0A`(z)X(W`z) A`(z),1 MPM�1k=0Hk(W`z)Fk(z),0`M�1. X(W`z)jz=ej!=X(!�2` M),i.e.,shiftedversionfromX(!). X(W`z):`-thaliasingterm,A`(z):gainforthisaliasingterm. ENEE630LecturePart-13/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) ConditionsforLPTV,LTI,andPRIngeneral,theM-channellterbankisaLPTVsystemwithperiodM. Thealiasingtermcanbeeliminatedforeverypossible inputx[n]i A`(z)=0for1`M�1.Whenaliasingiseliminated,thelterbankbecomesanLTIsystem:^X(z)=T(z)X(z),whereT(z),A0(z)=1 MPM�1`=0Hk(z)Fk(z)istheoveralltransferfunction,ordistortionfunction. IfT(z)=cz�n0,itisaperfectreconstructionsystem(i.e.,freefromaliasing,amplitudedistortion,andphasedistortion). ENEE630LecturePart-14/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) TheAliasComponent(AC)MatrixFromthedenitionofA`(z),wehaveinmatrix-vectorform: H(z):MMmatrixcalledthe\AliasComponentmatrix" TheconditionforaliascancellationisH(z)f (z)=t (z);wheret (z)=2664MA0(z)0:03775 ENEE630LecturePart-15/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) TheAliasComponent(AC)MatrixNowexpressthereconstructedsignalas^X(z)=AT(z)X(z)=1 Mf T(z)HT(z)X(z);whereX(z)=2664X(z)X(zW):X(zWM�1)3775: GivenasetofanalysisltersfHk(z)g,ifdetH(z)6=0,wecanchoosesynthesisltersasf (z)=H�1(z)t (z)tocancelaliasingandobtainP.R.byrequiringt (z)=2664cz�n00:03775 ENEE630LecturePart-16/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) DicultywiththeMatrixInversionApproachH�1(z)=Adj[H(z)] det[H(z)] SynthesisltersfFk(z)gcanbeIIReveniffHk(z)gareallFIR. DiculttoensurefFk(z)gstability(i.e.allpolesinsidetheunitcircle) fFk(z)gmayhavehighordereveniftheorderoffHk(z)gismoderate ......)Takeadi erentapproachforP.R.designviapolyphaserepresentation. ENEE630LecturePart-17/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) Type-1PDforHk(z)UsingType-1PDforHk(z):Hk(z)=PM�1`=0z�`Ek`(zM)Wehave E(zM):MMType-1polyphasecomponentmatrixforanalysisbank ENEE630LecturePart-18/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) Type-2PDforFk(z)Similarly,usingType-2PDforFk(z):Fk(z)=PM�1`=0z�(M�1�`)R`k(zM)Wehaveinmatrixform: e TB(z):reverselyorderedversionofe (z)R(zM):Type-2polyphasecomponentmatrixforsynthesisbank ENEE630LecturePart-19/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) OverallPolyphasePresentation Combinepolyphasematricesintoonematrix:P(z)=R(z)E(z)| {z }notetheorder! ENEE630LecturePart-110/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) SimpleFIRP.R.Systems ^X(z)=z�1X(z),i.e.,transferfunctionT(z)=z�1 ExtendtoMchannels: Hk(z)=z�kFk(z)=z�M+k+1;0kM�1)^X(z)=z�(M�1)X(z)i.e.demultiplexthenmultiplexagain ENEE630LecturePart-111/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) GeneralP.R.SystemsRecallthepolyphaseimplementationofM-channellterbank: Combinepolyphasematricesintoonematrix:P(z)=R(z)E(z) IfP(z)=R(z)E(z)=I,thenthesystemisequivalenttothesimplesystem)Hk(z)=z�k,Fk(z)=z�M+k+1 Inpractice,wecanallowP(z)tohavesomeconstantdelay,i.e.,P(z)=cz�m0I,thusT(z)=cz�(Mm0+M�1). ENEE630LecturePart-112/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) DealingwithMatrixInversionTosatisfyP(z)=R(z)E(z)=I,itseemswehavetodomatrixinversionforgettingthesynthesisltersR(z)=(E(z))�1. Question:DoesthisgetbacktothesameinversionproblemwehavewiththeviewpointoftheACmatrixf (z)=H�1(z)t (z)? Solution: E(z)isaphysicalmatrixthateachentrycanbecontrolled.Incontrast,forH(z),only1strowcanbecontrolled(thushardtoensuredesiredHk(z)responsesandf (z)stability) WecanchooseFIRE(z)s.t.detE(z)= z�kthusR(z)canbeFIR(andhasdeterminantofsimilarform). Summary:Withpolyphaserepresentation,wecanchooseE(z)toproducedesiredHk(z)andleadtosimpleR(z)s.t.P(z)=cz�kI. ENEE630LecturePart-113/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) ParaunitaryAmoregeneralwaytoaddresstheneedofmatrixinversion:ConstrainE(z)tobeparaunitary:~E(z)E(z)=dI Here~E(z)=ET(z�1),i.e.takingconjugateofthetransferfunctioncoe .,replacezwithz�1thatcorrespondstotimereverselyordertheltercoe .,andtranspose.Forfurtherexploration:PPVBookChapter6. ENEE630LecturePart-114/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) Relationb/wPolyphaseMatrixE(z)andACMatrixH(z)TherelationbetweenE(z)andH(z)canbeshownas: H(z)=[W]TD(z)ET(zM) whereWistheMMDFTmatrix,andadiagonaldelaymatrixD(z)=266641z�1...z�(M�1)37775 (details)Seealsothehomework. ENEE630LecturePart-115/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations DetailedDerivations ENEE630LecturePart-116/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations TheReconstructedSignalandErrorsCreated A`(z),1 MPM�1k=0Hk(W`z)Fk(z),0`M�1. X(W`z)jz=ej!=X(!�2` M),i.e.,shiftedversionfromX(!). X(W`z):`-thaliasingterm,A`(z):gainforthisaliasingterm. ENEE630LecturePart-117/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations Review:MatrixInversionH�1(z)=Adj[H(z)] det[H(z)]Adjugateorclassicaladjointofamatrix:fAdj[H(z)]gij=(�1)i+jMjiwhereMjiisthe(j;i)minorofH(z)denedasthedeterminantofthematrixbydeletingthej-throwandi-thcolumn. ENEE630LecturePart-118/21 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations AnExampleofP.R.SystemsH0(z)=2+z�1;H1(z)=3+2z�1,E(z)=2132, E�1(z)=AdjE(z) detE(z)=12�1�32. ChooseR(z)=E�1(z)s.t.P(z)=R(z)E(z)=I,)R(z)=2�1�32 F0(z)F1(z)=z�11R(z2)=2z�1�3;�z�1+2 )(F0(z)=�3+2z�1F1(z)=2�z�1 ENEE630LecturePart-119/21