Basic multi rate operations decimation and interpolation - PDF document

(a.k.a. Maximally Decimated Filter Bank)(a.k.a. Maximally Decimated Filter Bank) M. Wu: ENEE630 Advanced Signal Processing Review: Quadrature Mirror Filter (QMF) BankReview: Quadrature Mirror Filter (QMF) Bank M. Wu: ENEE630 Advanced Signal Processing MM-ch. Maximally Decimated Filter Bank. Maximally Decimated Filter Bank M. Wu: ENEE630 Advanced Signal Processing PolyphaseImplementation d by M.Wu © 1 Slides (creat e MCP ENEE63 U M. Wu: ENEE630 Advanced Signal Processing 1BasicMultirateOperations2InterconnectionofBuildingBlocks 2.1Decimator-ExpanderCascades2.2NobleIdentities TheNobleIdentitiesConsideraLTIdigitallterwithatransferfunctionG(z): Question:WhatkindofimpulseresponsewillalterG(zL)have?Recall:thetransferfunctionG(z)ofaLTIdigitallterisrationalforpracticalimplementation,i.e.,aratioofpolynomialsinzorz�1.Thereshouldnotbetermswithfractionalpowerinzorz�1. ENEE630LecturePart-136/37 1BasicMultirateOperations2InterconnectionofBuildingBlocks 2.1Decimator-ExpanderCascades2.2NobleIdentities Conditionfory1[n]=y2[n]Equiv.toexaminetheconditionofWkM M�1k=0WkLM M�1k=0: iMandLarerelativelyprime .Question:Proveit.(seehomework).Equivalenttoshow:f0;1;:::;M�1gf0;L;2L;:::(M�1)LgmodMiMandLarerelativelyprime . )Thustheoutputsofthetwodecimator-expandercascades,Y1(z)andY2(z),areidenticaland(a)(b)iMandLarerelativelyprime. ENEE630LecturePart-134/37 1BasicMultirateOperations2InterconnectionofBuildingBlocks 2.1Decimator-ExpanderCascades2.2NobleIdentities Decimator-ExpanderCascades Questions: 1 Isy1[n]alwaysequaltoy2[n]? Notalways. E.g.,whenL=M,y2[n]=x[n],buty1[n]=x[n]
cM[n]6=y2[n],wherecM[n]isacombsequence 2 Underwhatconditionsy1[n]=y2[n]? ENEE630LecturePart-131/37 1BasicMultirateOperations2InterconnectionofBuildingBlocks 2.1Decimator-ExpanderCascades2.2NobleIdentities InterconnectionofBuildingBlocks:BasicProperties Basicinterconnectionproperties: )bythelinearityof#M&"L Readings:VaidyanathanBookx4.2;tutorialSec.IIB ENEE630LecturePart-130/37 1BasicMultirateOperations2InterconnectionofBuildingBlocks 1.1DecimationandInterpolation1.2DigitalFilterBanks UniformDFTFilterBankAlterbankinwhichtheltersarerelatedby Hk(z)=H0(zWk) iscalledauniformDFTlterbank . TheresponseofltersjHk(!)jhavealargeamountofoverlap. ENEE630LecturePart-126/37 1BasicMultirateOperations2InterconnectionofBuildingBlocks 1.1DecimationandInterpolation1.2DigitalFilterBanks DFTFilterBankConsiderpassingx[n]throughadelaychaintogetMsequencesfsi[n]g:si[n]=x[n�i] i.e.,treatfsi[n]gasavectors [n],thenapplyWs [n]togetx [n].(WinsteadofWduetonewestcomponentrstinsignalvector)Question:Whataretheequiv.analysislters?Andifhavingamultiplicativefactoritothesi[n]? ENEE630LecturePart-123/37 1BasicMultirateOperations2InterconnectionofBuildingBlocks 1.1DecimationandInterpolation1.2DigitalFilterBanks DigitalFilterBanksAdigitallterbankisacollectionofdigitallters,withacommoninputoracommonoutput. Hi(z):analysislters xk[n]:subbandsignals Fi(z):synthesislters SIMOvs.MISO Typicalfrequencyresponseforanalysislters: Canbe marginallyoverlapping non-overlapping (substantially)overlapping ENEE630LecturePart-121/37 1BasicMultirateOperations2InterconnectionofBuildingBlocks 1.1DecimationandInterpolation1.2DigitalFilterBanks Frequency-DomainIllustrationofDecimation InterpretationofYD(!) Step-1:stretchX(!)byafactorofMtoobtainX(!=M) Step-2:createM�1copiesandshifttheminsuccessiveamountsof2 Step-3:addallMcopiestogetherandmultiplyby1=M. ENEE630LecturePart-112/37 1BasicMultirateOperations2InterconnectionofBuildingBlocks 1.1DecimationandInterpolation1.2DigitalFilterBanks Transform-DomainAnalysisofDecimatorsYD(z)=P1n=�1yD[n]z�n=P1n=�1x[nM]z�n Puttingalltogether: YD(z)=1 MPM�1k=0X(WkMz1 M) (details) YD(!)=1 MPM�1k=0X�!�2k M
(details) ENEE630LecturePart-111/37 1BasicMultirateOperations2InterconnectionofBuildingBlocks 1.1DecimationandInterpolation1.2DigitalFilterBanks M-foldDecimator yD[n]=x[Mn];M2N Correspondingtothephysicaltimescale,itisasifwesampledtheoriginalsignalinaslowerratewhenapplyingdecimation. Questions: Whatpotentialproblemwillthisbring? Underwhatconditionscanweavoidit? Canwerecoverx[n]? ENEE630LecturePart-15/37 1BasicMultirateOperations2InterconnectionofBuildingBlocks 1.1DecimationandInterpolation1.2DigitalFilterBanks Input-OutputRelationontheSpectrum YE(z)=X(zL) (details) Evaluatingontheunitcircle,theFourierTransformrelationis: YE(ej!)=X(ej!L))YE(!)=X(!L) i.e.L-foldcompressedversionofX(!)along! ENEE630LecturePart-18/37 1BasicMultirateOperations2InterconnectionofBuildingBlocks 1.1DecimationandInterpolation1.2DigitalFilterBanks L-foldExpander yE[n]=(x[n=L]ifnisintegermultipleofL2N0otherwise Question:Canwerecoverx[n]fromyE[n]? !Yes.Theexpanderdoesnotcauselossofinformation. Question:Are"Land#Mlinear andshiftinvariant ? ENEE630LecturePart-16/37 1BasicMultirateOperations2InterconnectionofBuildingBlocks 1.1DecimationandInterpolation1.2DigitalFilterBanks BasicMulti-rateOperations:DecimationandInterpolation Buildingblocksfortraditionalsingle-ratedigitalsignalprocessing:multiplier(withaconstant),adder,delay,multiplier(of2signals) Newbuildingblocksinmulti-ratesignalprocessing: M-folddecimator L-foldexpander Readings:VaidyanathanBookx4.1;tutorialSec.IIA,B ENEE630LecturePart-14/37 4MultistageImplementations5SomeMultirateApplications 5.1ApplicationsinDigitalAudioSystems5.2SubbandCoding/Compression5.AWarm-upExercise FilterBankforSubbandCoding RoleofFk(z): Eliminatespectrumimagesintroducedby"2,andrecoversignalspectrumoverrespectivefreq.range IffHk(z)gisnotperfect,thedecimatedsubbandsignalsmayhavealiasing. fFk(z)gshouldbechosencarefullysothatthealiasinggetscanceledatthesynthesisstage(in^x[n]). ENEE630LecturePart-123/24 4MultistageImplementations5SomeMultirateApplications 5.1ApplicationsinDigitalAudioSystems5.2SubbandCoding/Compression5.AWarm-upExercise SubbandCoding 1 x0[n]andx1[n]arebandlimitedandcanbedecimated 2 X1(!)hassmallerpowers.t.x1[n]hassmallerdynamicrange ,thuscanberepresentedwithfewerbits Supposenowtorepresenteachsubbandsignal,weneedx0[n]:16bits/samplex1[n]:8bits/sample)1610k 2+810k 2=120kbps ENEE630LecturePart-122/24 4MultistageImplementations5SomeMultirateApplications 4.1InterpolatedFIR(IFIR)Design4.2MultistageDesignofMultirateFilters MultistageDecimation/ExpansionSimilarly,forinterpolation, Summary Byimplementinginmultistage,notonlythenumberofpolyphasecomponentsreduces,butmostimportantly,thelterspecicationislessstringentandtheoverallorderoftheltersarereduced. Exercises: Closebookandthinkrsthowyouwouldsolvetheproblems. Sketchyoursolutionsonyournotebook. ThenreadV-bookSec.4.4. ENEE630LecturePart-16/24 6QuadratureMirrorFilter(QMF)BankAppendix:DetailedDerivations 6.1ErrorsCreatedintheQMFBank6.2ASimpleAlias-FreeQMFSystem6.ALookAhead Summary:T(z)=2z�1E0(z2)E1(z2)Case-1H0(z)isFIR: P.R.:requirepolyphasecomponentsofH0(z)tobepuredelays.t.H0(z)=c0z�2n0+c1z�(2n1+1)[cons]H0(!)responseisveryrestricted. Formoredesirablelterresponse,thesystemmaynotbeP.R.,butcanminimizedistortion:{eliminatephasedistortion:chooselterorderNtobeodd,andh0[n]besymmetric(linearphase) {minimizeamplitudedistortion:jH0(!)j2+jH1(!)j21 Case-2H0(z)isIIR: E1(z)=1 E0(z)cangetP.R.butrestrictthelterresponses. eliminateamplitudedistortion:choosepolyphasecomponentstobeallpass,s.t.T(z)isall-pass,butmayhavesomephasedistortion ENEE630LecturePart-130/38 6QuadratureMirrorFilter(QMF)BankAppendix:DetailedDerivations 6.1ErrorsCreatedintheQMFBank6.2ASimpleAlias-FreeQMFSystem6.ALookAhead SummaryMany\wishes"toconsidertowardachievingalias-freeP.R.QMF:(0)aliasfree,(1)phasedistortion,(2)amplitudedistortion,(3)desirablelterresponses. Can'tsatisfythemallatthesametime,sooftenmeetmostofthemandtrytoapproximate/optimizetherest. Aparticularrelationofsynthesis-analysislterstocancelalias:(F0(z)=H1(�z)F1(z)=�H0(�z)s.t.H0(�z)F0(z)+H1(�z)F1(z)=0. Weconsideredaspecicrelationbetweentheanalysislters:H1(z)=H0(�z)s.t.responsesymmetricw.r.t.!==2(QMF) Withpolyphasestructure:T(z)=2z�1E0(z2)E1(z2) ENEE630LecturePart-129/38 6QuadratureMirrorFilter(QMF)BankAppendix:DetailedDerivations 6.1ErrorsCreatedintheQMFBank6.2ASimpleAlias-FreeQMFSystem6.ALookAhead PolyphaseRepresentation:MatrixForm Inmatrixform:(withMIMOtransferfunctionforintermediatestages)E1(z)00E0(z)| {z }synthesis111�1111�1| {z }24200235E0(z)00E1(z)| {z }analysis =2E0(z)E1(z)002E0(z)E1(z) �Note:Multiplicationisfromleftforeachstagewhenintermediatesignalsareincolumnvectorform. ENEE630LecturePart-119/38 3ThePolyphaseRepresentationAppendix:DetailedDerivations 3.1BasicIdeas3.2EcientStructures3.3CommutatorModel3.4Discussions:MultirateBuildingBlocks&PolyphaseConcept FractionalRateConversion TypicallyLandMshouldbechosentohavenocommonfactorsgreaterthan1(o.w.itiswastefulaswemaketheratehigherthannecessaryonlytoreduceitdownlater) H(z)lterneedstobefastasitoperatesinhighdatarate. ThedirectimplementationofH(z)isinecient:(thereareL�1zerosinbetweenitsinputsamplesonlyoneoutofMsamplesisretained ENEE630LecturePart-113/25 3ThePolyphaseRepresentationAppendix:DetailedDerivations 3.1BasicIdeas3.2EcientStructures3.3CommutatorModel3.4Discussions:MultirateBuildingBlocks&PolyphaseConcept GeneralCasesIngeneral,forFIRlterswithlengthN: M-folddecimation:MPU=N M,APU=N�1 M L-foldinterpolation:MPU=N,APU=N�L lteringisperformedatalowerdatarate APU=(N L�1)L ENEE630LecturePart-112/25 3ThePolyphaseRepresentationAppendix:DetailedDerivations 3.1BasicIdeas3.2EcientStructures3.3CommutatorModel3.4Discussions:MultirateBuildingBlocks&PolyphaseConcept AlternativePolyphaseRepresentationIfwedeneR`(z)=EM�1�`(z);0`M�1,wearriveatthe Type-2polyphaserepresentation H(z)=PM�1`=0z�(M�1�`)R`(zM) Type-1:Ek(z)isorderedconsistentlywiththenumberofdelaysintheinput Type-2:reverselyorderthelterRk(z)withrespecttothedelays ENEE630LecturePart-17/25 3ThePolyphaseRepresentationAppendix:DetailedDerivations 3.1BasicIdeas3.2EcientStructures3.3CommutatorModel3.4Discussions:MultirateBuildingBlocks&PolyphaseConcept ExtensiontoMPolyphaseComponentsForagivenintegerMandH(z)=P1n=�1h[n]z�n,wehave:H(z)=P1n=�1h[nM]z�nM+z�1P1n=�1h[nM+1]z�nM+:::+z�(M�1)P1n=�1h[nM+M�1]z�nM Type-1PolyphaseRepresentation H(z)=PM�1`=0z�`E`(zM) wherethe`-thpolyphasecomponentsofH(z)givenMisE`(z),P1n=�1e`[n]z�n=P1n=�1h[nM+`]z�nNote:0`(M�1);strictlywemaydenoteasE(M)`(z). ENEE630LecturePart-15/25 3ThePolyphaseRepresentationAppendix:DetailedDerivations 3.1BasicIdeas3.2EcientStructures PolyphaseRepresentation:Denition Wehave TheserepresentationsholdwhetherH(z)isFIRorIIR,causalornon-causal. ENEE630LecturePart-13/25 8GeneralAlias-FreeConditionsforFilterBanks9TreeStructuredFilterBanksandMultiresolutionAnalysisAppendix:DetailedDerivations BriefNoteonSubbandvsWaveletCoding Theoctave(dyadic)frequencypartitioncanre ectthelogarithmiccharacteristicsinhumanperception. Waveletcodingandsubbandcodinghavemanysimilarities(e.g.fromlterbankperspectives) Traditionallysubbandcodingusesltersthathavelittleoverlaptoisolatedierentbands Wavelettransformimposessmoothnessconditionsontheltersthatusuallyrepresentasetofbasisgeneratedbyshiftingandscaling(dilation)ofamotherwaveletfunction Waveletcanbemotivatedfromovercomingthepoortime-domainlocalizationofshort-timeFT)ExploremoreinProj#1.SeePPVBookChapter11 UMdECE ENEE630LecturePart-114/23 8GeneralAlias-FreeConditionsforFilterBanks9TreeStructuredFilterBanksandMultiresolutionAnalysisAppendix:DetailedDerivations Discussions(3)If2-chQMFwithG(z),F(z),Gs(z),Fs(z)hasP.R.withunit-gain andzero-delay ,wehavex[n]=x[n].(4)Forcompressionapplications:canassignmorebitstorepresentthecoarseinfo,andtheremainingbits(ifavailable)tonerdetailsbyquantizingtherenementsignalsaccordingly. UMdECE ENEE630LecturePart-113/23 8GeneralAlias-FreeConditionsforFilterBanks9TreeStructuredFilterBanksandMultiresolutionAnalysisAppendix:DetailedDerivations Discussions(1)Thetypicalfrequencyresponseoftheequivalentanalysisandsynthesisltersare: (2)Themultiresolutioncomponentsvk[n]attheoutputofFk(z): v0[n]isalowpassversionofx[n]ora\coarse"approximation; v1[n]addssomehighfrequencydetailssothatv0[n]+v1[n]isanerapproximationofx[n]; v3[n]addsthenestultimatedetails. UMdECE ENEE630LecturePart-112/23 8GeneralAlias-FreeConditionsforFilterBanks9TreeStructuredFilterBanksandMultiresolutionAnalysisAppendix:DetailedDerivations Multi-resolutionAnalysis:SynthesisBank UMdECE ENEE630LecturePart-111/23 8GeneralAlias-FreeConditionsforFilterBanks9TreeStructuredFilterBanksandMultiresolutionAnalysisAppendix:DetailedDerivations Multi-resolutionAnalysis:AnalysisBankConsiderthevariationofthetreestructuredlterbank(i.e.,onlysplitonesubbandsignals) H0(z)=G(z)G(z2)G(z4))H0(!)=G(!)G(2!)G(22!) UMdECE ENEE630LecturePart-110/23 8GeneralAlias-FreeConditionsforFilterBanks9TreeStructuredFilterBanksandMultiresolutionAnalysisAppendix:DetailedDerivations (Binary)Tree-StructuredFilterBank Cananalyzetheequivalentltersbynobleidentities. Ifa2-channelQMFbankwithH(K)0(z),H(K)1(z),F(K)0(z),F(K)1(z)isalias-free,thecompletesystemaboveisalsoalias-free. Ifthe2-channelsystemhasP.R.,sodoesthecompletesystem.[Readings:PPVBook5.8] UMdECE ENEE630LecturePart-19/23 8GeneralAlias-FreeConditionsforFilterBanks9TreeStructuredFilterBanksandMultiresolutionAnalysisAppendix:DetailedDerivations (Binary)Tree-StructuredFilterBankAmulti-stagewaytobuildM-channellterbank:Splitasignalinto2subbands)furthersplitoneorbothsubbandsignalsinto2)


Question:Underwhatconditionsistheoverallsystemfreefromaliasing?HowaboutP.R.? UMdECE ENEE630LecturePart-18/23 8GeneralAlias-FreeConditionsforFilterBanks9TreeStructuredFilterBanksandMultiresolutionAnalysisAppendix:DetailedDerivations MostGeneralP.R.Conditions NecessaryandSucientP.R.Conditions P(z)=cz�m00IM�rz�1Ir0forsomer20;:::;M�1: Whenr=0,P(z)=I
cz�m0,asthesucientconditionseeninxI.7.3. (Details) UMdECE ENEE630LecturePart-17/23 8GeneralAlias-FreeConditionsforFilterBanks9TreeStructuredFilterBanksandMultiresolutionAnalysisAppendix:DetailedDerivations OverallTransferFunctionTheoveralltransferfunctionT(z)afteraliasingcancellation:^X(z)=T(z)X(z),where T(z)=z�(M�1)fP0;0(zM)+z�1P0;1(zM)+


+z�(M�1)P0;M�1(zM)g (Details)Forfurtherexploration:SeePPVBook5.7.2forderivations. UMdECE ENEE630LecturePart-16/23 8GeneralAlias-FreeConditionsforFilterBanks9TreeStructuredFilterBanksandMultiresolutionAnalysisAppendix:DetailedDerivations InsightsoftheTheorem DenoteP(z)=[Ps;`(z)]. (Details)Forfurtherexploration:SeePPVBook5.7.2fordetailedproof.Examinetherelationbetween^X(z)andX(z),andevaluatethegaintermsonthealiasedversionsofX(z). UMdECE ENEE630LecturePart-15/23 8GeneralAlias-FreeConditionsforFilterBanks9TreeStructuredFilterBanksandMultiresolutionAnalysisAppendix:DetailedDerivations CirculantandPseudoCirculantMatrix (right-)circulantmatrix24P0(z)P1(z)P2(z)P2(z)P0(z)P1(z)P1(z)P2(z)P0(z)35Eachrowistherightcircularshiftofpreviousrow. pseudocirculantmatrix24P0(z)P1(z)P2(z)z�1P2(z)P0(z)P1(z)z�1P1(z)z�1P2(z)P0(z)35Addingz�1toelementsbelowthediagonallineofthecirculantmatrix. Bothtypesofmatricesaredeterminedbythe1strow. Propertiesofpseudocirculantmatrix(orasanalternativedenition):Eachcolumnasup-shiftversionofitsrightcolumnwithz�1tothewrappedentry. UMdECE ENEE630LecturePart-14/23 8GeneralAlias-FreeConditionsforFilterBanks9TreeStructuredFilterBanksandMultiresolutionAnalysisAppendix:DetailedDerivations GeneralAlias-freeConditionRecallfromSection7:TheconditionforaliascancellationintermsofH(z)andf (z)isH(z)f (z)=t (z)=2664MA0(z)0:03775 Theorem AM-channelmaximallydecimatedlterbankisalias-freei thematrixP(z)=R(z)E(z)ispseudocirculant. [Readings:PPVBook5.7] UMdECE ENEE630LecturePart-13/23 7M-channelMaximallyDecimatedFilterBankAppendix:DetailedDerivations 7.1TheReconstructedSignalandErrorsCreated7.2TheAliasComponent(AC)Matrix7.3ThePolyphaseRepresentation7.4PerfectReconstructionFilterBank7.5RelationbetweenPolyphaseMatrixE(z)andACMatrixH(z) DealingwithMatrixInversionTosatisfyP(z)=R(z)E(z)=I,itseemswehavetodomatrixinversionforgettingthesynthesisltersR(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 Perfect Reconstruction Filter Bank Perfect Reconstruction Filter Bank 2010) d by M.Wu © equivalent to thesimplePRsystem ontheleft 0 Slides (creat e the P . R If allowing P(z) to have some constant delay in practicaldesign:ie P () m0 I MCP ENEE63 0 practical e = c z– I T(z) = c z–(M m0 + M –1) M. Wu: ENEE630 Advanced Signal Processing10/13/2010 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) TheAliasComponent(AC)MatrixFromthedenitionofA`(z),wehaveinmatrix-vectorform: H(z):MMmatrixcalledthe\AliasComponentmatrix" TheconditionforaliascancellationisH(z)f (z)=t (z);wheret (z)=2664MA0(z)0:03775 ENEE630LecturePart-15/21