Chapter6

z-Transform

MainContentsinthisbook

TheDiscrete-TimeFourierTransform(DTFT)TheDiscreteFourierTransform(DFT)Thez-Transform(ZT)MainContentsWeknowtheDiscrete-timesignalscanberepresentedin

time-domain:(aweightedlinearcombinationofdelayedunitsamplesequences.)MainContentsInthischapterWe’llstudytherepresentationoftheDiscrete-timesignalsinTransform-domain

(

b.Zdomain(asequenceintermsofcomplexexponentialsequencesoftheform{Z-n})a.frequencydomain(asequenceintermsofcomplexexponentialsequencesoftheform{}or{WNn})MainContentsThereare3kindsoftransform-domainrepresentation:DTFT:DFT:ZT:z-TransformAgeneralizationoftheDTFTdefinedbyleadstothez-transformMoreover,useofz-transformtechniquespermitssimplealgebraicmanipulationsz-transformmayexistformanysequencesforwhichtheDTFTdoesnotexist§6.1DefinitionandPropertiesConsequently,z-transformhasbecomeanimportanttoolintheanalysisanddesignofdigitalfiltersForagivensequenceg[n],itsz-transformG(z)isdefinedas wherez=Re(z)+jIm(z)isacomplexvariable★§6.1DefinitionandPropertiesTheabovecanbeinterpretedastheDTFTofthemodifiedsequence{g[n]r-n}Forr=1(i.e.,|z|=1),z-transformreducestoitsDTFT,providedthelatterexistsIfweletz=rej,thenthez-transformreducesto§6.1DefinitionandPropertiesThecontour|z|=1isacircleinthez-planeofunityradiusandiscalledtheunitcircleLiketheDTFT,thereareconditionsontheconvergenceoftheinfiniteseriesForagivensequence,thesetRofvaluesofzforwhichitsz-transformconvergesiscalledtheregionofconvergence(ROC)★§6.1DefinitionandPropertiesFromourearlierdiscussionontheuniformconvergenceoftheDTFT,itfollowsthattheseriesconvergesif{g[n]r-n}isabsolutelysummable,i.e.,if§6.1DefinitionandPropertiesIngeneral,theROCofaz-transformofasequenceg[n]isanannularregionofthez-plane: Note:Thez-transformisaformofaLaurentseriesandisananalyticfunctionateverypointintheROCwhere§6.1DefinitionandPropertiesExample6.1-Determinethez-transformX(z)ofthecausalsequencex[n]=n[n]anditsROCNowROCistheannularregion|z|>|a|Theabovepowerseriesconvergesto§6.1DefinitionandPropertiesExample-Thez-transformm(z)oftheunitstepsequencem[n]canbeobtainedfrom mROCistheannularregionbysettinga=1:§6.1DefinitionandPropertiesNote:Theunitstepsequencem[n]isnotabsolutelysummable,andhenceitsDTFTdoesnotconvergeuniformlyExample6.2-Considertheanti-causalsequence§6.1DefinitionandPropertiesItsz-transformisgivenbyROCistheannularregion§6.1DefinitionandPropertiesNote:Thez-transformsofthetwosequencesn[n]and-n[-n-1]

areidenticaleventhoughthetwoparentsequencesaredifferentOnlywayauniquesequencecanbeassociatedwithaz-transformisbyspecifyingitsROC§6.1DefinitionandPropertiesTheDTFTG(ej)ofasequenceg[n]convergesuniformlyifandonlyiftheROCofthez-transformG(z)ofg[n]includestheunitcircleTheexistenceoftheDTFTdoesnotalwaysimplytheexistenceofthez-transform§6.1DefinitionandPropertiesExample-Thefiniteenergysequence whichconvergesinthemean-squaresensehasaDTFTgivenby §6.1DefinitionandPropertiesHowever,hLP[n]doesnothaveaz-transformasitisnotabsolutelysummableforanyvalueofrSomecommonlyusedz-transformpairsarelistedonthenextslideTable6.1CommonlyUsed

z-transform§6.2Rational

z-transformInthecaseofLTIdiscrete-timesystemsweareconcernedwithinthiscourse,allpertinentz-transformsarerationalfunctionsofz-1

Thatis,theyareratiosoftwopolynomialsinz-1:§6.2Rational

z-transformThedegreeofthenumeratorpolynomialP(z)isMandthedegreeofthedenominatorpolynomialD(z)isNAnalternaterepresentationofarationalz-transformisasaratiooftwopolynomialsinz:§6.2Rational

z-transformArationalz-transformcanbealternatelywritteninfactoredformas§6.2Rational

z-transformAtarootz=lofthenumeratorpolynomialG(l)=0andasaresult,thesevaluesofzareknownasthezerosofG(z)Atarootz=lofthedenominatorpolynomialG(l),andasaresult,thesevaluesofzareknownasthepolesofG(z)§6.2Rational

z-transformNoteG(z)hasMfinitezerosandNfinitepolesIfN>MthereareadditionalN-Mzerosatz=0(theorigininthez-plane)IfN<MthereareadditionalM-Npolesatz=0Consider§6.2Rational

z-transformExample-Thez-transformmhasazeroatz=0andapoleatz=1§6.2Rational

z-transformAphysicalinterpretationoftheconceptsofpolesandzeroscanbegivenbyplottingthelog-magnitude20log10|G(z)|asshownonnextslidefor§6.2Rational

z-transformpolesz=0.4±j0.6928,zerosz=1.2±j1.2§6.2Rational

z-transformObservethatthemagnitudeplotexhibitsverylargepeaksaroundthepointsz=0.4±j0.6928whicharethepolesofG(z)Italsoexhibitsverynarrowanddeepwellsaroundthelocationofthezerosatz=1.2±j1.2§6.3ROCofaRational

z-transformROCofaz-transformisanimportantconceptWithouttheknowledgeoftheROC,thereisnouniquerelationshipbetweenasequenceanditsz-transformHence,thez-transformmustalwaysbespecifiedwithitsROC§6.3ROCofaRational

z-transformMoreover,iftheROCofaz-transformincludestheunitcircle,theDTFTofthesequenceisobtainedbysimplyevaluatingthez-transformontheunitcircleThereisarelationshipbetweentheROCofthez-transformoftheimpulseresponseofacausalLTIdiscrete-timesystemanditsBIBO(Bounded-Input,Bounded-Output)stability

§6.3ROCofaRational

z-transformTheROCofarationalz-transformisboundedbythelocationsofitspolesTounderstandtherelationshipbetweenthepolesandtheROC,itisinstructivetoexaminethepole-zeroplotofaz-transformConsideragainthepole-zeroplotofthez-transformm(z)§6.3ROCofaRational

z-transformInthisplot,theROC,shownastheshadedarea,istheregionofthez-planejustoutsidethecirclecenteredattheoriginandgoingthroughthepoleatz=1§6.3ROCofaRational

z-transformExample-Thez-transformH(z)ofthesequenceh[n]=(-0.6)n[n]isgivenbyHeretheROCisjustoutsidethecirclegoingthroughthepointz=-0.6§6.3ROCofaRational

z-transformAsequencecanbeoneofthefollowingtypes:finite-length,right-sided,left-sidedandtwo-sidedIngeneral,theROCdependsonthetypeofthesequenceofinterest§6.3ROCofaRational

z-transformExample-Considerafinite-lengthsequenceg[n]definedfor-MnN,whereMandNarenon-negativeintegersand|g(n)|<Itsz-transformisgivenby§6.3ROCofaRational

z-transformNote:G(z)hasMpolesatz=andNpolesatz=0AscanbeseenfromtheexpressionforG(z),thez-transformofafinite-lengthboundedsequenceconvergeseverywhereinthez-planeexceptpossiblyatz=0and/oratz=

§6.3ROCofaRational

z-transformExample-Aright-sidedsequencewithnonzerosamplevaluesforn0issometimescalledacausalsequenceConsideracausalsequenceu1[n]Itsz-transformisgivenby§6.3ROCofaRational

z-transformItcanbeshownthatU1(z)convergesexteriortoacircle|z|=R1,includingthepointz=

Ontheotherhand,aright-sidedsequenceu2[n]withnonzerosamplevaluesonlyforn3-MwithMnonnegativehasaz-transformU2(z)withMpolesatz=

TheROCofU2(z)isexteriortoacircle|z|=R1,excludingthepointz=§6.3ROCofaRational

z-transformExample-Aleft-sidedsequencewithnonzerosamplevaluesforn0issometimescalledaanticausalsequenceConsiderananticausalsequencev1[n]Itsz-transformisgivenby§6.3ROCofaRational

z-transformItcanbeshownthatV1(z)convergesinteriortoacircle|z|=R3,includingthepointz=0Ontheotherhand,aleft-sidedsequencewithnonzerosamplevaluesonlyfornNwithNnonnegativehasaz-transformV2(z)withNpolesatz=0TheROCofV2(z)isinteriortoacircle|z|=R4,excludingthepointz=0§6.3ROCofaRational

z-transform

Example-Thez-transformofatwo-sidedsequencew[n]canbeexpressedascanbeinterpretedasthez-transformofaright-sidedsequenceanditthusconvergesexteriortothecircle|z|=R5ThefirsttermontheRHS,§6.3ROCofaRational

z-transformcanbeinterpretedasthez-transformofaleft-sidedsequenceanditthusconvergesinteriortothecircle|z|=R6IfR5<R6,thereisanoverlappingROCgivenbyR5<|z|<R6

IfR5>R6,thereisnooverlapandthez-transformdoesnotexistThesecondtermontheRHS,§6.3ROCofaRational

z-transformExample-Considerthetwo-sidedsequenceu[n]=n whereacanbeeitherrealorcomplexItsz-transformisgivenbyThefirsttermontheRHSconvergesfor|z|>||,whereasthesecondtermconvergesfor|z|<||§6.3ROCofaRational

z-transformThereisnooverlapbetweenthesetworegionsHence,thez-transformofu[n]=ndoesnotexist§6.3ROCofaRational

z-transformTheROCofarationalz-transformcannotcontainanypolesandisboundedbythepolesToshowthatthez-transformisboundedbythepoles,assumethatthez-transformX(z)hassimplepolesatz=α

andz=βAssumethatthecorrespondingsequencex[n]isaright-sidedsequence§6.3ROCofaRational

z-transformThenx[n]hastheformx[n]=(r1αn+r2βn)μ[n-N0],|α|<|β|Now,thez-transformoftheright-sidedsequenceγnμ[n=N0]existsifforsomez§6.3ROCofaRational

z-transformTheconditionholdsfor|z|>|γ|

butnotfor|z|≤|γ|x[n]=(r1αn+r2βn)μ[n-N0],|α|<|β|hasanROCdefinedby|β|<|z|≤∞§6.3ROCofaRational

z-transformLikewise,thez-transformofaleft-sidedsequencex[n]=(r1αn+r2βn)μ[-n-N0],|α|<|β|hasanROCdefinedby0≤|z|<|α|Finally,foratwo-sidedsequence,someofthepolescontributetotermsintheparentsequence,n<0andtheotherpolescontributetotermsn>0§6.3ROCofaRational

z-transformTheROCisthusboundedontheoutsidebythepolewiththesmallestmagnitudethatcontributesforn<0andontheinsidebythepolewiththelargestmagnitudethatcontributesforn≥0TherearethreepossibleROCsofarationalz-transformwithpolesatz=α

and

z=β(|α|<|β|)§6.3ROCofaRational

z-transform§6.3ROCofaRational

z-transformIngeneral,iftherationalz-transformhasNpoleswithRdistinctmagnitudes,thenithasR+1ROCsThusthereareR+1distinctsequenceswiththesamez-transformHence,arationalz-transformwithaspecifiedROChasauniquesequenceasitsinversez-transorm★§6.3ROCofaRational

z-transformTheROCofarationalz-transformcanbeeasilydeterminedusingMATLAB[z,p,k]=tf2zp(num,den) determinesthezeros,poles,andthegainconstantofarationalz-transformwiththenumeratorcoefficientsspecifiedbythevectornumandthedenominatorcoefficientsspecifiedbythevectorden

§6.3ROCofaRational

z-transform

[num,den]=zp2tf(z,p,k)implementsthereverseprocessThefactoredformofthez-transformcanbeobtainedusingsos=zp2sos(z,p,k)Theabovestatementcomputesthecoefficientsofeachsecond-orderfactorgivenasanL6matrixsos§6.3ROCofaRational

z-transform§6.3ROCofaRational

z-transformThepole-zeroplotisdeterminedusingthefunctionzplaneThez-transformcanbeeitherdescribedintermsofitszerosandpoles:zplane(zeros,poles)or,itcanbedescribedintermsofitsnumeratoranddenominatorcoefficients:zplane(num,den)§6.3ROCofaRational

z-transformExample-Thepole-zeroplotof -poleO-zeroobtainedusingMATLABisshownbelow-4-3-2-101-2-1012RealPartImaginaryPart§6.4

Inversez-Transform ismerelytheDTFTofthemodifiedsequenceg[n]r-nAccordingly,theinverseDTFTisthusgivenbyGeneralExpression:Recallthat,forz=rej,thez-transformG(z)givenby§6.4

Inversez-TransformBymakingachangeofvariablez=rej,thepreviousequationcanbeconvertedintoacontourintegralgivenbywhereC’isacounterclockwisecontourofintegrationdefinedby|z|=r§6.4

Inversez-TransformButtheintegralremainsunchangedwhenC’isreplacedwithanycontourCencirclingthepointz=0intheROCofG(z)ThecontourintegralcanbeevaluatedusingtheCauchy’sresiduetheoremresultinginTheaboveequationneedstobeevaluatedatallvaluesofnandisnotpursuedhere§6.4.3

InverseTransformby

Partial-FractionExpansionArationalz-transformG(z)withacausalinversetransformg[n]hasanROCthatisexteriortoacircleHereitismoreconvenienttoexpressG(z)inapartial-fractionexpansionformandthendetermineg[n]bysummingtheinversetransformoftheindividualsimplertermsintheexpansion§6.4.3

InverseTransformby

Partial-FractionExpansionArationalG(z)canbeexpressedaswherethedegreeofP1(z)islessthanNIfMNthenG(z)canbere-expressedas§6.4.3

InverseTransformby

Partial-FractionExpansionTherationalfunctionP1(z)/D(z)iscalledaproperfractionExample-ConsiderBylongdivisionwearriveat§6.4.3

InverseTransformby

Partial-FractionExpansionSimplePoles:Inmostpracticalcases,therationalz-transformofinterestG(z)isaproperfractionwithsimplepolesLetthepolesofG(z)beatz=k,1kNApartial-fractionexpansionofG(z)isthenoftheform§6.4.3

InverseTransformby

Partial-FractionExpansionTheconstantslinthepartial-fractionexpansionarecalledtheresiduesandaregivenbyEachtermofthesuminpartial-fractionexpansionhasanROCgivenby|z|>|l|and,thushasaninversetransformoftheforml(l)n[n]§6.4.3

InverseTransformby

Partial-FractionExpansionTherefore,theinversetransformg[n]ofG(z)isgivenbyNote:Theaboveapproachwithaslightmodificationcanalsobeusedtodeterminetheinverseofarationalz-transformofanoncausalsequence§6.4.3

InverseTransformby

Partial-FractionExpansionExample-Letthez-transformH(z)ofacausalsequenceh[n]begivenbyApartial-fractionexpansionofH(z)isthenoftheform§6.4.3

InverseTransformby

Partial-FractionExpansionNow and§6.4.3

InverseTransformby

Partial-FractionExpansionHenceTheinversetransformoftheaboveisthereforegivenby§6.4.3

InverseTransformby

Partial-FractionExpansionMultiplePoles:IfG(z)hasmultiplepoles,thepartial-fractionexpansionisofslightlydifferentformLetthepoleatz=nbeofmultiplicityLandtheremainingN-Lpolesbesimpleandatz=l,1lN-L§6.4.3

InverseTransformby

Partial-FractionExpansionThenthepartial-fractionexpansionofG(z)isoftheformTheresidueslarecalculatedasbeforewheretheconstantsiarecomputedusing§6.4.4

Partial-FractionExpansionUsingMatlab[r,p,k]=residuez(num,den)developsthepartial-fractionexpansionofarationalz-transformwithnumeratoranddenominatorcoefficientsgivenbyvectorsnumanddenVectorrcontainstheresiduesVectorpcontainsthepolesVectorkcontainstheconstants§6.4.4

Partial-FractionExpansionUsingMatlab[num,den]=residuez(r,p,k)convertsaz-transformexpressedinapartial-fractionexpansionformtoitsrationalform§6.4.5Inversez-TransformviaLongDivisionThez-transformG(z)ofacausalsequence{g[n]}canbeexpandedinapowerseriesinz-1Intheseriesexpansion,thecoefficientmultiplyingthetermz-1isthenthen-thsampleg[n]Forarationalz-transformexpressedasaratioofpolynomialsinz-1,thepowerseriesexpansioncanbeobtainedbylongdivision§6.4.5Inversez-TransformviaLongDivisionExample-ConsiderAsaresultLongdivisionofthenumeratorbythedenominatoryields§6.4.6InverseTransform

UsingMATLABThefunctionimpzcanbeusedtofindtheinverseofarationalz-transformG(z)ThefunctioncomputesthecoefficientsofthepowerseriesexpansionofG(z)Thenumberofcoefficientscaneitherbeuserspecifiedordeterminedautomatically§6.5z-TransformProperties

Table6.2Someusefulpropertiesofthez-transformp.321§6.5z-TransformPropertiesLetx[n]=n[n]andy[n]=-n

[-n-1]withX(z)andY(z)denoting,respectively,theirz-transforms

Example6.24-Considerthetwo-sidedsequenceandNow§6.5z-TransformPropertiesTheROCofV(z)isgivenbytheoverlapregionsof|z|>||and|z|<||If||<||,thenthereisanoverlapandtheROCisanannularregion||<|z|<||If||>||,thenthereisnooverlapandV(z)doesnotexistUsingthelinearitypropertywearriveat§6.5z-TransformPropertiesExample-Determinethez-transformanditsROCofthecausalsequenceThez-transformofv[n]isgivenbyWecanexpressx[n]=v[n]+v*[n]where§6.5z-TransformPropertiesUsingtheconjugationpropertyweobtainthez-transformofv*[n]asFinally,usingthelinearitypropertyweget§6.5z-TransformPropertiesExample-Determinethez-transformY(z)andtheROCofthesequenceor,Wecanwritey[n]=nx[n]+x[n]wherex[n]=n[n]

§6.5z-TransformPropertiesNow,thez-transformX(z)ofx[n]=n[n]isgivenbyUsingthedifferentiationproperty,wearriveatthez-transformofnx[n]as§6.5z-TransformPropertiesUsingthelinearitypropertywefinallyobtain§6.7TheTransferFunctionAgeneralizationofthefrequencyresponsefunctionTheconvolutionsumdescriptionofanLTIdiscrete-timesystemwithanimpulseresponseh[n]isgivenby§6.7TheTransferFunctionTakingthez-transformsofbothsidesweget§6.7TheTransferFunctionThus,Y(z)=H(z)X(z)

Or,Therefore,§6.7TheTransferFunctionHence,H(z)=Y(Z)/X(z)ThefunctionH(z),whichisthez-transformoftheimpulseresponseh[n]oftheLTIsystem,iscalledthetransferfunctionorthesystemfunctionTheinversez-transformofthetransferfunctionH(z)yieldstheimpulseresponseh[n]§6.7TheTransferFunctionItstransferfunctionisobtainedbytakingthez-transformofbothsidesoftheaboveequationThus

ConsideranLTIdiscrete-timesystemcharacterizedbyadifferenceequation§6.7TheTransferFunctionOr,equivalentlyas

Analternateformofthetransferfunctionisgivenby§6.7TheTransferFunction1,2,…,

Marethefinitezeros,and1,2,…,

NarethefinitepolesofH(z)IfN>M,thereareadditional(N-M)zerosatz=0IfN<M,thereareadditional(M-N)polesatz=0

Or,equivalentlyas§6.7TheTransferFunctionForacausalIIRdigitalfilter,theimpulseresponseisacausalsequenceTheROCofthecausaltransferfunctionThustheROCisgivenbyisthusexteriortoacirclegoingthroughthepolefurthestfromtheorigin§6.7TheTransferFunctionExample-ConsidertheM-pointmoving-averageFIRfilterwithanimpulseresponse

Itstransferfunctionisthengivenby§6.7TheTransferFunctionThetransferfunctionhasMzerosontheunitcircleatz=ej2k/M,0kM-1ThereareM-1polesatz=0andasinglepoleatz=1Thepoleatz=1exactlycancelsthezeroatz=1TheROCistheentirez-planeexceptz=0M=8§6.7TheTransferFunctionExample-AcausalLTIIIRdigitalfilterisdescribedbyaconstantcoefficientdifferenceequationgivenbyy[n]=x[n-1]-1.2x[n-2]+x[n-3]+1.3y[n-1]-1.04y[n-2]+0.222y[n-3]Itstransferfunctionisthereforegivenby§6.7TheTransferFunctionAlternateforms:ROC:

Note:Polesfarthestfromz=0haveamagnitude★§6.7.3

FrequencyResponsefromTransferFunctionIftheROCofthetransferfunctionH(z)includestheunitcircle,thenthefrequencyresponseH(ej)oftheLTIdigitalfiltercanbeobtainedsimplyasfollows:

ForarealcoefficienttransferfunctionH(z)itcanbeshownthat★§6.7.3

FrequencyResponsefromTransferFunctionForastablerationaltransferfunctionintheformthefactoredformofthefrequencyresponseisgivenby§6.7.3

FrequencyResponsefromTransferFunctionItisconvenienttovisualizethecontributionsofthezerofactor(z-k)andthepolefactor(z-k)fromthefactoredformofthefrequencyresponseThemagnitudefunctionisgivenby§6.7.4

FrequencyResponsefromTransferFunctionThephasefunctionisgivenbyRezjImz1-1-jk§6.7.4

GeometricInterpretationofFrequency