2023学年完整公开课版chapter3

1Chapter3:

z-Transform

2Contents3.1Thez-Transform3.1.1PolesandZerosonthez-PlaneandStability3.1.2TheROCofz-Transform3.1.3ThePropertiesofz-Transform3.2TheInversez-Transform3.2.1GeneralExpressionofInversez-Transform3.2.2Inversez-TransformbyPartial-FractionExpansionDSP23ZTransformZTofthesequencex(n)isdefinedas:Z,complexvariableInverseZT:34ZTransformWehavetoaskifX(z)converge?Foranyx(n),theregionofconverge(ROC)ofZTisthecomplexplanemakingX(z)converge.i.e.:{z:X(z)exists}Differentx(n)withdifferentROCmayhavethesameZT.So,theROCofeachX(z)shouldbedefined.45ZTransform:Pole-zeroPlotSystemhasZTas:56ZTransform:Pole-zeroPlotROC:ROCisdeterminedby|z|=r,intermsofthetheoryofcomplexvariablefunction,itcanbeacircularband:r1<|z|<r2IntheROC,X(z)isananalyticfunction,andthepoleofX(z)isoutofROC,withthepoleastheedge.r1canbezero,r2canbe∞.Ifr2<r1，itmeansROCisnotexist,neithertheZT.6ZTransformRight-sideSequence:Whenn<0,x(n)=0;Usuallycausalsequence;X(z)onlycontainsthenegativeindexesofz.TheROC:|z|>r1,outsideofradiusr1.778ZTransformExample1:DeterminetheZTofx(n):89ZTransformSolution:9ThePoleZTransformLeft-sideSequence:Whenn≥0,x(n)=0;X(z)onlycontainsthepositiveindexesofz.TheROC:|z|<r2,insideofradiusr2.101011ZTransformExample2:DeterminetheZTofx1(n):1112ZTransformSolution:12ThePole13ZTransformAboutthesetwoexamples,ifb=a:But:13=≠14ZTransformIfb=a:X1(z)hasthesameformwithX(z),exceptfortheROC.ThatimpliesthattheROCinsuresonlyoneZTofx(n).DifferentROCmeansdifferentZT.ROCplaysanimportantroleinsystemanalysis.14ZTransformTwo-sideSequence:Containsright-sidesequenceandleft-sidesequence.SotheROCisdefinedas:r1<|z|<r2ornotexistifr2<r1.151516ZTransformExample3:Define:DeterminetheZTofx2(n).1617ZTransformSolution:1718ZTransformConclusion:=Right-sidesequence+Left-sidesequence(1)Theconvergenceconditionisdecidedbytheamplitudeof|z|,soitconvergesintheboundaryofacircle.(2)Right-sideSequence(n≥0):theROCis|z|>|a|,whereaisthepole.(3)Left-sideSequence(n<0):theROCis|z|<|b|,wherebisthepole.18ThePropertiesofZTLinear2.Timeshifting3.Frequencyshifting(scalinginthez-domain)1919ThePropertiesofZT4.Differential5.Conjugation2020ThePropertiesofZT6.InitialValueTheoremIfn<0,x(n)=0,andthen:7.Convolutioninz-domainTheconvolutioninthediscretetimedomainequalstothemultiplicationinzdomain.212122CommonZTPairsTheTableofcommonZTpairs:2223TheInverseZTDefinitionofInverseZT:23(ROC)24InverseZTCalculationofInverseZT:TheinverseZTneedstocalculatetheintegralinthecomplexcontourC,usuallyitiscomplexanddifficult.Normalmethods:GeneralExpressionofInversez-TransformPartfractionalmethod2425InverseZT-GeneralExpressionofInversez-TransformThegeneralexpressionfortheinverseZTisgivenby:isaclosedcurveintheROC,containingtheorigin.2526InverseZT-GeneralExpressionofInversez-TransformGeneralExpressionofInversez-Transform:UsingtheResiduemethod,theintegralbecomes:2627InverseZT-GeneralExpressionofInversez-TransformThecalculationofResidue:For:TheResidueis:For1orderPole:2728InverseZT-GeneralExpressionofInversez-TransformNote:IfROCisoutofacircle,weusuallyuseformulaNo.1:IfROCisinsideofacircle,weusuallyuseformulaNo.2:IfROCisaring,usebothofthem.2829InverseZT-GeneralExpressionofInversez-TransformNote:ThepolesofX(z)zn-1includetwoparts:PolesfromX(z):usuallyhavelimitednumbersandordersPolesfromzn-1:usuallyexistatz=0andz=∞Usually,wewanttochoosetheregionthatX(z)zn-1havelimitednumbersandorders’polestocalculatetheresidueeasilyandtrytoavoidtheresidueatz=∞.2930InverseZT-GeneralExpressionofInversez-TransformNote:IftheROCisoutsideofacircle,usuallywecalculatex(n)atn>0andchooseformulaNo.1.BecauseX(z)havelimitedpolesinsideofCandzn-1isanalyticatz=0,buttherearehighorderpolesatz=∞forzn-1whennislarge.3031InverseZT-GeneralExpressionofInversez-TransformNote:IftheROCisinsideofacircle,usuallywecalculatex(n)atn<0andchooseformulaNo.2.BecauseX(z)havelimitedpolesoutsideofCandzn-1isanalyticatz=∞,buttherearehighorderpolesatz=0forzn-1whennislarge.IftheROCisaring,usuallyweusebothofformulaNo.1andNo.2.3132InverseZT-GeneralExpressionofInversez-TransformNote:Actually,n=0isnottheonlyedgeforresiduemethod.Now,wecanreachonemoregeneralmethod:ExtractX(z)=X0(z)zm,misaninteger.Therefore,X0(z)isanalyticatbothn=0andn=∞.Then:X(z)zn-1=X0(z)zmzn-1=X0(z)zn+m-1=X1(z)AfterdeterminingtheROCofX(z)andC:Wecangetx(n)atn≥1-mbycalculatetheresidueofX1(z)insideofCandthenget-x(n)atn<1-moutsideofC.3233InverseZT-GeneralExpressionofInversez-TransformExample:If:Pleasegiveitstimedomainsignalx(n)withResidueMethod.3334InverseZT-GeneralExpressionofInversez-TransformSolution:3435InverseZT-GeneralExpressionofInversez-TransformSolution: andm=1.Evidently,thefunctionhastwopolesat1/3and1,anditisanalyticoutofthepoles.3536InverseZT-GeneralExpressionofInversez-TransformSolution:(1)IfROC>1:BothofthetwopolesareinsideofC,useformulaNo.1.Whenn<1-m=0,x(n)=0.Whenn≥1-m=0:3637InverseZT-GeneralExpressionofInversez-TransformSolution:(2)IfROC<1/3:BothofthetwopolesareoutsideofC,useformulaNo.2.Whenn≥1-m=0,x(n)=0.Whenn<1-m=0:3738InverseZT-GeneralExpressionofInversez-TransformSolution:(3)If1/3<ROC<1:Polez=1/3isinsideofC,useformulaNo.1.Whenn≥0:Polez=1isoutsideofC,useformulaNo.2.Whenn<0:3839InverseZT-GeneralExpressionofInversez-TransformSolution:(3)If1/3<ROC<1:39InverseZT-GeneralExpressionofInversez-TransformConclusionoftheResidueMethod:1)Drawthezero-poleplot,findtheROC,anddrawtheclosedcurveC,containingtheorigin.2)CalculatetheresiduenumbersinandoutofC,andgetthevaluesofx(n).404041InverseZT-PartfractionalmethodWehavealreadygotsomesequences’ZT:Therefore,wecouldextractX(z)tothesumofmanyfractionalparts.Then,usethealreadygotZTformtogetx(n).ROCisveryimportant!4142InverseZT-PartfractionalmethodSuchas:ForROC(|z|>r1):ForROC(|z|<r2):4243InverseZT-PartfractionalmethodForROC(r1<|z|<r2):Two-sidesequence:4344Inve