信号与系统英文版课件Chapter 10

Content§10.1TheZtransform§10.2TheROCforZT§10.3TheinverseZT§10.4Omit§10.5Omit§10.5PropertiesoftheZT§10.6SomecommonZTpairs§10.7AnalysisandcharacterizationofLTIsystemsusingZT§10.8Systemfunctionalgebraandblockdiagramrepresentation§10.9TheUnilateral(Bilateral)ZTTheZtransformContinuousDiscreteDiscreteak

andX(ej)withperiodof2.whereTheZtransform10.1TheZtransform

Insection3.2,foradiscreteLTIsystemwithh[n],ifinputiszn,theny[n]=H(z)zn(Thedominanceconditionalsomustbesatisfied).Atthattime,H(z)wasreferredtoasthesystemfunction.TheZtransform

Ifz=ej(i.e.|z|=r=1),then

H(z)=H(ej)wasdiscreteFTofh(n),orfrequencyresponseofthediscretesystem.1.DefinitionofZTordenoteitasTheZtransform

i.e.ZTofsequencex[n]isapowerseriesofcomplexvariablez-1,thecoefficientsiscorrespondingvalueofsequencex[n].2.TherelationshipbetweenZTanddiscreteFT∵z=rejThereforeX(z)istheFTofx[n]·r-n,thatisX(z)=F(x[n]r

–n)(10.7)

TheZtransformIfr=1or|z|=1=X(ej)=F(x[n])3.Z–plane(complexZ–plane)js-planeRe[z]Im[z]z-planerunitcircler=1original(=0,=0)r=1,=0

axisjaxisPositiveRe[z]axisunitcirclelefthalfplanerighthalfplaneinunitcircle(r1)outunitcircle(r1)z=rejTheZtransformAstheLTexiststheROC,theZTalso.ButtheROCofZTisaringoroutcircleorincircle.IftheROCincludestheunitcircle,thenFTofx[n]is4.CalculatingexamplesIf|z|>|a|,theseriesconverges,thenTheZtransformIf|z|<|a|,theseriesconverges,thenRe[z]Im[z]1TheZtransformrightshift1(3)ObtainalsoRe[z]Im[z]ROCaoAssignments(P797):10.2TheROCforZTThepropertiesofROCforZT:1)TheROCofX(z)consistsofaringinthez-planecenteredabouttheorigin.

Insomecases,theinnerboundarycanextendinwardtotheorigin.Inothercases,theouterboundarycanextendoutwardtoinfinity.2)TheROCdoesnotcontainanypoles.3)Ifx[n]isoffiniteduration,thentheROCistheentirez-plane,exceptpossiblyz=0

and/orz=∞.TheROCforZT

Example10.5[n]1|z|0(allz)otherOncemore4)IfX(z)isrational,thenitsROCisboundedbypolesorextendstoinfinity.Forexample:X(z)=z2/(z-1)havetwopoles:z=1andz=∞.butX(z)=1/(z-1)hasonlyapole:z=1.TheROCforZT5)IfX(z)isrational,andifx[n]isrightsided,thentheROCisoutsidecircleofoutermostpole.Furthermore,ifx[n]iscausal,theROCalsoincludedz=∞.6)IfX(z)isrational,andifx[n]isleftsided,thentheROCisinsidecircleofinnermostnonzeropoleandpossiblyincludingz=0.

Inparticular,ifx[n]isanti-causal,theROCalsoincludesz=0.Example10.8TheROCforZTTherearethreepossibleROC:If|z|>2,x[n]isrightsequence,If|z|<1/3(z0),x[n]isleftsequence,If1/3<|z|<2,x[n]istwosidedsequence,andthiscasetheROCincludesunitcircle,sotheFTalsoconverges.(seeFigure10.12)Assignments(P798):10.6,10.7.TheinverseZT

ForcalculatingtheinverseZT,havefollowingmethod:

Inversionintegral;Partial–fractionexpansion;Powerseriesexpansion.1)Inversionintegral(contourintegral）

Bothsidesmultiplybyzn-1andintegratealonganyclosedcontourintheROC.TheinverseZTAccordingtocomplexvariableintegralExample:givendeterminex[n].TheinverseZTSolution:X(z)zn-1therearethreepoles:(0,0.5,1)TheinverseZT2)Partial-fractionexpansion.FirstusingPFE,secondusingZTpair(becarefuloftheROC)

TheinverseZTExampleSolution:TheinverseZT3)Power-seriesexpansion

i.e.

X(z)isapowerseriesofz.Thecoefficientsinthisseriesarex[n].TheinverseZTExample10.12considerX(z)=4z2+2+3z-1,|z|:(0,∞)Fromeq.(10.3)obtain

x[n]=4δ[n+2]+2δ[n]+3δ[n-1]ExampleItcanbeexpandedbylongdivision:

i.e.

X(z)=2+0.5z-1+1.25z-2+0.875z-3+….orx[0]=2=1+(-0.5)0

x[1]=0.5=1+(-0.5)1

x[2]=1.25=1+(-0.5)2

….

x[n]=[1+(-0.5)n]u[n]TheinverseZT

IfROCis|z|<0.5(x[n]isleft),thenx[n]canbeexpandedbylongdivision:z-5z2+6z3–….…x[n]={…,6,-5,1,0}n=…,-3,-2,-1,0

TheinverseZTExample：Assignments(p798):10.9,10.10.PropertiesoftheZT10.5.1Linearity(seebyyourself)10.5.2TimeshiftingExceptforthepossibleadditionordeletionoftheoriginor∞.

Compare:x(t-t0)X(s)e-s0tExample:weknow[n]1allz,[n-2]z-2ROCz0,|z|>010.5.3ScalingintheZ–domainPropertiesoftheZTWherez0isanyconstantinz-plane.Compare:es0tx(t)X(s-s0)—shiftingintheSSequencex[n]multiplicationbyexponentialsequencebeequivalenttoscalinginz-domain.Poof:

X(z/z0)==z|z|:|z0|RPropertiesoftheZTExample:weknowInferPropertiesoftheZTRe[z]Im[z]ounitcircleRe[z]Im[z]0unitcircleoPole-zeropatternofx[n]Pole-zeropatternofej0n

x[n]PropertiesoftheZT10.5.4TimeReversal

x[-n]X(1/z),ROC=1/RExample：weknow10.5.5TimeExpansion(1)Definitionoftimeexpansion{x[n/k],Ifnisamultipleofk0,IfnisnotamultipleofkPropertiesoftheZT0123x[n]2134n4321012n364x[n/2]zerosinsertExampleCalculateY(z)PropertiesoftheZTSolution:Compare(a)with(b)Inthesameway,ZTofy[n]=x[2n]isPropertiesoftheZT10.5.7Convolutionproperty

(1)DifferenceintimedomainConsideranLTIsystem:y[n]=h[n]

x[n]ifh[n]=[n]-[n-1](thefirstdifference)theny[n]=x[n]-x[n-1]Lookatz-domainROC:allz,excepttheoriginthen

FromconvolutionpropertyDifferenceandpartialaccumulation.

ProofPropertiesoftheZT(2)Partialaccumulation10.5.8DifferentiationintheZ—DomainWiththepossibledeletionofz=0and/orpossibleadditionofz=1.ExamplePropertiesoftheZTCompare:tx(t)-dX(s)/ds

ROC:RExample:AnotherExamplePropertiesoftheZT10.5.9Theinitialandfinalvaluetheorem

(1)Ifx[n]=0,n<0,andthenumeratororderdenominatororderofX(z),then(2)Ifx[n]X(z),ROC:(1,),and(z-1)X(z),ROC:[1,)10.5.10Summaryofproperties(Table10.1)

ProofExample

ProofSomecommonZTpairsAnalysisLTIsystemsusingZTIftheinputtoaLTIsystemisx[n]=z0n,-<n<andifz0satisfythedominancecondition:|z0|>|pole|maxofH(z),theny[n]=H(z0)z0n,-<n<10.7.1CausalityAcausalLTIsystemhash[n]=0,forn<0.FortheH(z):(1)IftheROC>|a|,includinginfinity,thensystemiscausal.

Dominanceconditionisthatz0mustbelongstotheROCofH(z).AnalysisLTIsystemsusingZT2）LetH(z)isrational,ifandonlyif

(a)theROCis>|a|circleoutsidetheouter-mostpole.

（b）theorderofnumeratortheorderofdenominator.Thensystemiscausal.Example:NotcausalCausalAnalysisLTIsystemsusingZT10.7.2Stability-

equivalenttoh[n]absolutelysummable,i.e.FTofh[n]converges.(1)AnLTIstablesystem,theROCofH(z)mustincludeunitcircle|z|=1.(2)AnLTIstablecausalsystem,itsallofthepolesofrationalH(z)lieinsidetheunitcircle—i.e.|zj|<1.

Re[z]Im[z]r=1Re[z]Im[z]r=1AnalysisLTIsystemsusingZT10.7.3LTIsystemcharacterizedbylinearconstant-coefficientdifferenceequationsFromequationH(z)andh[n]orinverse.GenerallyZTtobothsidesAssignments(P800):10.16ExampleSystemfunctionalgebra10.8.1H(z)forinterconnectionsystemsSeriesandparallelform(omit)Feedbackform:AccordingtoMasonEquationH1(z)H2(z)+x[n]y[n]–+Systemfunctionalgebra10.8.2BlockdiagramrepresentationforcausalLTIsystemsThreebasicoperations:Example10.28(oneordersystem)ThecausalLTIsystemItsdifferenceequationisy[n]–¼y[n-1]=x[n]Systemblockdiagram:aZ-1+SystemfunctionalgebraFormulti-ordersystem,blockdiagramcanrepresentasdirectorcascadeorparallel.Example:GivenWritethedifferenceequation.(b)Drawthreeformsofblockdiagram.z-11/4+x[n]y[n]++SystemfunctionalgebraSolution:(b)FromMasone.q.directform:Systemfunctionalgebra(2)H(z)rewriteas(cascadeform)z–1+x[n]y[n]–0.5+z–1–0.16x[n]0.8z–1+z–1+y[n]–0.5+0.2Systemfunctionalgebra(3)ForH(z),byperformingapartial-fractionexpanding:Parallelform

+x[n]y[n]0.8z–1+0.50.2z–1+0.5Assignments(P800):10.18TheUnilateralZTDefinitionsothat,ifx[n]=0,n<0,UZ{x[n]}=Z{x[n]}ifx[n]0,n<0,UZ{x[n]}Z{x[n]}2.UnilateralZT,ROC:|z|>a.Example10.33Letx[n]=an+1u[n+1]ROCisalwaystheexteriorofacircle.TheUnilateralZTExample10.34Determinex[n]=?Solution:Notes:

(1)UZneednotlabelROC,UZ–1[X(z)]isalwayscausalsignal.(2)TheUnilateralZT

thenthedegreeofp(z)thedegreeofq(z),otherwise,UZ-1[X(z)]doesnotexist.Forexample

Z-1[X(z)]=an+1u[n+1](x[n]0,n=-1)

i.e.theUZ-1oftheX(z)doesnotexist.3.TimeconvolutionofUZT

thenTheUnilateralZT4.TimeshiftingofUZT∵x[n-1]u[n]=