數(shù)字信號(hào)處理與控制課件：Chapter6 Z-Transform

1Chapter6Z-TransformDefinitionROC(RegionofConverges)z-TransformPropertiesTransferFunction

2Z-TransformIncontinuoussignalsystem,weuseS-TransformandFTasthetoolstoprocessproblemsinthetransformdomain;soindiscretesignalsystem,weuseZ-TransformandDFT.Z-Transformcanmakethesolutionfordiscretetimesystemsverysimple.3Z-TransformTheDTFTprovidesafrequency-domainrepresentationofdiscrete-timesignalsandLTIdiscrete-timesystemsBecauseoftheconvergencecondition,inmanycases,theDTFTofasequencemaynotexist.46.1DefinitionandPropertiesDTFTdefinedby：leadstothez-transform。

z-transformmayexistformanysequencesforwhichtheDTFTdoesnotexist。5DefinitionandPropertiesForagivensequenceg[n],itsz-transformG(z)isdefinedas:wherez=Re(z)+jIm(z)isacomplexvariable.6DefinitionandPropertiesIfweletz=rej,thenthez-transformreducesto:Forr=1(i.e.,|z|=1),z-transformreducestoitsDTFT,providedthelatterexists。Thecontour|z|=1isacircleinthez-planeofunityradiusandiscalledtheunitcircle。7DefinitionandPropertiesLiketheDTFT,thereareconditionsontheconvergenceoftheinfiniteseries：

Foragivensequence,thesetRofvaluesofzforwhichitsz-transformconvergesiscalledtheregionofconvergence(ROC)8DefinitionandPropertiesFromourearlierdiscussionontheuniformconvergenceoftheDTFT,itfollowsthattheseries：convergesif{g[n]r-n}isabsolutelysummable,i.e.,if：9DefinitionandPropertiesIngeneral,theROCofaz-transformofasequenceg[n]isanannularregionofthez-plane:where

Note:Thez-transformisaformofaLaurentseriesandisananalyticfunctionateverypointintheROC。10DefinitionandPropertiesExample-Determinethez-transformX(z)ofthecausalsequencex[n]=n[n]

anditsROC.NowTheabovepowerseriesconvergesto:ROCistheannularregion|z|>|α|.11DefinitionandPropertiesExample-Thez-transformm(z)oftheunitstepsequencem[n]canbeobtainedfrom:

mROCistheannularregion.bysettinga=1,12DefinitionandPropertiesNote:TheFouriertransformofasequenceconvergesuniformlyifandonlyiftheROCofthez-transfromofthesequenceincludestheunitcircle.Note:Onlywayanuniquesequencecanbeassociatedwithaz-transformisbyspecifyingitsROC.13CommonlyUsed

z-transform146.2Rational

z-TransformInthecaseofLTIdiscrete-timesystemsweareconcernedwithinthiscourse,allpertinentz-transformsarerationalfunctionsofz-1.Thatis,theyareratiosoftwopolynomialsinz-1:15Rational

z-TransformArationalz-transformcanbealternatelywritteninfactoredformas:16Rational

z-TransformAtarootz=lofthenumeratorpolynomialG(l)=0andasaresult,thesevaluesofzareknownasthezerosofG(z).Atarootz=lofthedenominatorpolynomialG(l),andasaresult,thesevaluesofzareknownasthepolesofG(z).17Rational

z-TransformConsider:NoteG(z)hasMfinitezerosandNfinitepoles:IfN>MthereareadditionalN-Mzerosatz=0(theorigininthez-plane)IfN<MthereareadditionalM-Npolesatz=0.18Rational

z-TransformExample-Thez-transformmhasazeroatz=0andapoleatz=1.19Rational

z-TransformAphysicalinterpretationoftheconceptsofpolesandzeroscanbegivenbyplottingthelog-magnitude20log10|G(z)|asshownonnextfigurefor:20Rational

z-Transformpolesz=0.4±j0.6928,zerosz=1.2±jROCofaRational

z-transformROCofaz-transformisanimportantconcept.WithoutthediscussionoftheROC,thereisnouniquerelationshipbetweenasequenceanditsz-transform.Hence,thez-transformmustalwaysbespecifiedwithitsROC.22ROCofaRational

z-transformMoreover,iftheROCofaz-transformincludestheunitcircle,theDTFTofthesequenceisobtainedbysimplyevaluatingthez-transformontheunitcircle.RelationshipbetweenROCofz-transformoftheimpulsesequenceofacausalLTIDTsystemanditsBIBOstability.23ROCofaRational

z-transformTheROCofarationalz-transformisboundedbythelocationsofitspoles.TounderstandtherelationshipbetweenthepolesandtheROC,itisinstructivetoexaminethepole-zeroplotofaz-transform.24ROCofaRational

z-transformConsideragainthepole-zeroplotofthez-transformm(z).25Asequencecanbeoneofthefollowingtypes:finite-length,right-sided,left-sidedandtwo-sided.Ingeneral,theROCdependsonthetypeofthesequenceofinterest.ROCofaRational

z-transform26ROCofDifferentSequencesExample-Aright-sidedsequencewithnonzerosamplevaluesforn0issometimescalledacausalsequence.Consideracausalsequenceu1[n].Itsz-transformisgivenby:27ROCofDifferentSequencesExample-Aleft-sidedsequencewithnonzerosamplevaluesforn0issometimescalledaanticausalsequence.Considerananticausalsequencev1[n].Itsz-transformisgivenby:28ROCofaRational

z-transformThezerosandpolesofarationalz-transformcanbeeasilydeterminedusingMATLAB[z,p,k]=tf2zp(num,den) determinesthezeros,poles,andthegainconstantofarationalz-transformwiththenumeratoranddenominatorcoefficientsspecifiedbythevectornumandden,respectively.29ROCofaRational

z-transformThepole-zeroplotisdeterminedusingthefunctionzplane.Thez-transformcanbeeitherdescribedintermsofitszerosandpoles:zplane(zeros,poles).or,itcanbedescribedintermsofitsnumeratoranddenominatorcoefficients:zplane(num,den).PleaselookatP259exampleaboutz-transformusingMATLAB.306.4TheInversez-TransformGeneralExpressionInversez-TransformbyTableLook-upMethodInversez-TransformbyPartial-FractionExpansionInversez-TransformviaLongDivisionMATALBComputation316.4.2TableLook-upMethodIllustratethismethodinExample6.12:Fromtable6.1,weget:So:326.4.3Partial-FractionExpansionInpracticalapplication,theX(z)istherationalfractionofz.WhereA(z),B(z)arepolynomialsofz.So:Note:youmustmakeeachXk(z)easytosolveitsIZT,andpayattentiontoit’sROC.33Partial-FractionExpansionIf:ThenX(z)isexpandedaspartial-fraction:whereziisar-orderpolesofX(z),eachzk

isasinglepoleofX(z)(k=1,2,…,N-r).34Partial-FractionExpansionExample-ConsiderBylongdivisionwearriveat:35Partial-FractionExpansionExample-Letthez-transformH(z)ofacausalsequenceh[n]begivenby:

Apartial-fractionexpansionofH(z)isthenoftheform:36Partial-FractionExpansionNow:and:37Partial-FractionExpansionHenceTheinversetransformoftheaboveisthereforegivenby:386.4.5

Inversez-TransformviaLongDivisionBecauseincommoncondition,X(z)isarationalfraction,wecandividenumeratorpolynomialbydenominatorpolynomialtogettheexpansionofthepowerseries.Note:Whenweuselongdivisionmethods,wemustfirstestimatetheROCofx(n),thenexpandX(z)intoappropriatex(n).

39Inversez-TransformviaLongDivisionExample

–ConsideracausalH(z):Longdivisionofthenumeratorbythedenominatoryields:Asaresult:40Inversez-Transform

UsingMATLABThefunctionimpzcanbeusedtofindtheinverseofarationalz-transformG(z).ThefunctioncomputesthecoefficientsofthepowerseriesexpansionofG(z).Thenumberofcoefficientscaneitherbeuserspecifiedordeterminedautomatically.416.5z-TransformProperties42z-TransformProperties1.InitialvaluetheoremForacausalsequencex(n),exist:2.TerminalvaluetheoremForacausalsequencex(n),andthepoleofX(z)=Z[x(n)]insidetheunitcycle(inunitcyclethereareatbestsinglepoles),then:43z-TransformPropertiesExample-Considerthetwo-sidedsequence:Letx[n]=n[n]andy[n]=-n

[-n-1]withX(z)andY(z)denoting,respectively,theirz-transforms:Now:44z-TransformPropertiesUsingthelinearitypropertywearriveat:TheROCofV(z)isgivenbytheoverlapregionsof|z|>||and|z|<||:If||<||,thenthereisanoverlapandtheROCisanannularregion||<|z|<||.If||>||,thenthereisnooverlapandV(z)doesnotexist.

45z-TransformPropertiesExample-Determinethez-transformanditsROCofthecausalsequenceWecanexpressx[n]=v[n]+v*[n]whereThez-transformofv[n]isgivenby:46z-TransformPropertiesUsingtheconjugationpropertyweobtainthez-transformofv*[n]as:Finally,usingthelinearitypropertyweget:47z-TransformPropertiesOr:Example-Determinethez-transformY(z)andtheROCofthesequence:

Wecanwritey[n]=nx[n]+x[n]wherex[n]=n[n]

48z-TransformPropertiesNow,thez-transformX(z)ofx[n]=n[n]isgivenby:Usingthedifferentiationproperty,wearriveatthez-transformofnx[n]as:49z-TransformPropertiesUsingthelinearitypropertywefinallyobtain:506.6ComputationoftheConvolutionSumofFinite-LengthSequencesTabularmethodsforthecomputationofthelinearandcircularconvolutionhavebeenoutlined.Now,wedescribealternatemethodsforthecomputationofthelinearandcircularconvolutionthatbasedonthemultiplicationoftwopolynomial.516.6.1LinearConvolutionLetx[n]andh[n]betwosequencesoflengthsL+1andM+1,respectively.Theirz-transforms,X(z)andH(z)be:So,thez-transformoflinearconvolutionsequencey[n]isY(z):Anditscoefficientisy[n].526.6.2CircularConvolutionLetx[n]andh[n]bothbetwosequencesofdegreeN-1,Theirz-transforms,X(z)andH(z)be:So,thez-transformoflinearconvolutionsequencey[n]isY(z):53CircularConvolutionLetYc(z)denotethepolynomialofdegreeN-1whosecoefficientsisyc[n],itcanbeshownthat(Problem6.17):Themodulooperationwithrespecttoz-N=1inaboveequationofYL(z).PleaselookatP279abouttheprocessexample.546.7TheTransferFunctionThez-transformoftheimpulseresponseofanLTIsystem,calledthetransferfunction,isapolynomialinz-1.Inmostpracticalcases,theLTIdigitalfilterofinterestischaracterizedbyalineardifferentequationwithconstantandrealcoefficient,soit’stransferfunctionisrationalz-transform.556.7.1DefinitionOrigin:Intimedomain,useunitsampleresponseh[n]torepresentaLTIsystem:TodoZTforbothsides,weget:Then:It’sthetransferfunctionofaLTIsystem.566.7.2TransferFunctionExpressionFIRDigitalFilterInthecaseofanLTIFIRdigitalfilter,withit’simpulseresponseh[n]definedforN1≤n≤N2,andthus,h[n]=0forn＜N1andn＞N2.Therefore,thetransferfunctionisgivenby:57TransferFunctionExpressionFinite-DimensionLinearTime-InvariantIIRDiscrete-TimeSystemConsideranLTIdiscrete-timesystemcharacterizedbyadifferenceequation

Itstransferfunctionisobtainedbytakingthez-transformofbothsidesoftheaboveequation,Thus:58TransferFunctionExpression1.arethefinitezeros,andarethefinitepolesofH(z).2.IfN>M,thereareadditional(N-M)zerosatz=0;ifN＜M,thereareadditional(M-N)polesatz=0.59TransferFunctionExpression3.ForacausalIIRfilter,the