第9章-Z变换v101

Chapter6z-TransformMainContentsinthisbook•TheDiscrete-TimeFourierTransform(DTFT)•TheDiscreteFourierTransform(DFT)•Thez-Transform(ZT)MainContentsWeknowtheDiscrete-timesignalscanberepresentedintime-domain:(aweightedlinearcombinationofdelayedunitsamplesequences.)][][knanxkkMainContentsInthischapterWe’llstudytherepresentationoftheDiscrete-timesignalsinTransform-domain(b.Zdomain(asequenceintermsofcomplexexponentialsequencesoftheform{Z-n})a.frequencydomain(asequenceintermsofcomplexexponentialsequencesoftheform{}or{WNn})njeMainContentsThereare3kindsoftransform-domainrepresentation:)(][jDTFTeXnxDTFT:DFT:)(][NDFTWXnxZT:)(][ZXnxZTz-Transform•AgeneralizationoftheDTFTdefinedbynnjjenxeX][)(leadstothez-transform•Moreover,useofz-transformtechniquespermitssimplealgebraicmanipulations•z-transformmayexistformanysequencesforwhichtheDTFTdoesnotexist§6.1DefinitionandProperties•Consequently,z-transformhasbecomeanimportanttoolintheanalysisanddesignofdigitalfilters•Foragivensequenceg[n],itsz-transformG(z)isdefinedasnnzngzG][)(wherez=Re(z)+jIm(z)isacomplexvariable★§6.1DefinitionandProperties•TheabovecanbeinterpretedastheDTFTofthemodifiedsequence{g[n]r-n}•Forr=1(i.e.,|z|=1),z-transformreducestoitsDTFT,providedthelatterexistsnnjnjerngerG][)(•Ifweletz=rej,thenthez-transformreducesto§6.1DefinitionandProperties•Thecontour|z|=1isacircleinthez-planeofunityradiusandiscalledtheunitcircle•LiketheDTFT,thereareconditionsontheconvergenceoftheinfiniteseriesnnzng][•Foragivensequence,thesetRofvaluesofzforwhichitsz-transformconvergesiscalledtheregionofconvergence(ROC)★§6.1DefinitionandProperties•FromourearlierdiscussionontheuniformconvergenceoftheDTFT,itfollowsthattheseriesnnjnjerngerG][)(nnrng][convergesif{g[n]r-n}isabsolutelysummable,i.e.,if§6.1DefinitionandProperties•Ingeneral,theROCofaz-transformofasequenceg[n]isanannularregionofthez-plane:ggRzRggRR0•Note:Thez-transformisaformofaLaurentseriesandisananalyticfunctionateverypointintheROCwhere§6.1DefinitionandProperties•Example6.1-Determinethez-transformX(z)ofthecausalsequencex[n]=n[n]anditsROC•Now0][)(nnnnnnzznzX1for,11)(11zzzXROCistheannularregion|z|||Theabovepowerseriesconvergesto§6.1DefinitionandProperties•Example-Thez-transform(z)oftheunitstepsequence[n]canbeobtainedfrom1for,11)(11zzzX1for1111zzz,)(z1ROCistheannularregionbysetting=1:§6.1DefinitionandProperties•Note:Theunitstepsequence[n]isnotabsolutelysummable,andhenceitsDTFTdoesnotconvergeuniformly•Example6.2-Considertheanti-causalsequence]1[][nnyn§6.1DefinitionandProperties•Itsz-transformisgivenby11)(mmmnnnzzzYzzzzmmm110111for,1111zzzROCistheannularregion§6.1DefinitionandProperties•Note:Thez-transformsofthetwosequencesn[n]and-n[-n-1]areidenticaleventhoughthetwoparentsequencesaredifferent•Onlywayauniquesequencecanbeassociatedwithaz-transformisbyspecifyingitsROC§6.1DefinitionandProperties•TheDTFTG(ej)ofasequenceg[n]convergesuniformlyifandonlyiftheROCofthez-transformG(z)ofg[n]includestheunitcircle•TheexistenceoftheDTFTdoesnotalwaysimplytheexistenceofthez-transform§6.1DefinitionandProperties•Example-ThefiniteenergysequencennnnhcLP,sin][ccjLPeH,00,1)(whichconvergesinthemean-squaresensehasaDTFTgivenby§6.1DefinitionandProperties•However,hLP[n]doesnothaveaz-transformasitisnotabsolutelysummableforanyvalueofr•Somecommonlyusedz-transformpairsarelistedonthenextslideTable6.1CommonlyUsedz-transform§6.2Rationalz-transform•InthecaseofLTIdiscrete-timesystemsweareconcernedwithinthiscourse,allpertinentz-transformsarerationalfunctionsofz-1•Thatis,theyareratiosoftwopolynomialsinz-1:NNNNMMMMzdzdzddzpzpzppzDzPzG)()(........)()()(1111011110§6.2Rationalz-transform•ThedegreeofthenumeratorpolynomialP(z)isMandthedegreeofthedenominatorpolynomialD(z)isN•Analternaterepresentationofarationalz-transformisasaratiooftwopolynomialsinz:NNNNMMMMMNdzdzdzdpzpzpzpzzG11101110........)()(§6.2Rationalz-transform•Arationalz-transformcanbealternatelywritteninfactoredformasNMzdzpzG11011011)()()(NMMNzdzpz1010)()()(§6.2Rationalz-transform•Atarootz=lofthenumeratorpolynomialG(l)=0andasaresult,thesevaluesofzareknownasthezerosofG(z)•Atarootz=lofthedenominatorpolynomialG(l),andasaresult,thesevaluesofzareknownasthepolesofG(z)§6.2Rationalz-transformNoteG(z)hasMfinitezerosandNfinitepoles•IfNMthereareadditionalN-Mzerosatz=0(theorigininthez-plane)•IfNMthereareadditionalM-Npolesatz=0NMMNzdzpzzG1010)()()()(Consider§6.2Rationalz-transform•Example-Thez-transform1for111zzz,)(hasazeroatz=0andapoleatz=1§6.2Rationalz-transform•Aphysicalinterpretationoftheconceptsofpolesandzeroscanbegivenbyplottingthelog-magnitude20log10|G(z)|asshownonnextslidefor121212.42.88G(z)=10.80.64zzzz§6.2Rationalz-transformpolesz=0.4±j0.6928,zerosz=1.2±j1.2§6.2Rationalz-transform•Observethatthemagnitudeplotexhibitsverylargepeaksaroundthep