第十章课件奥本海姆本信号与系统

1TheZ-TransformCH10Theprimaryfocusofthischapterwillbeon:Thez-TransformandtheRegionofConvergenceforthez-TransformTheInversez-TransformGeometricEvaluationoftheFourierTransformfromthePole-ZeroPlotPropertiesofthez-TransformandsomeCommonz-TransformPairsAnalysisandCharacterizationofLTISystemsUsingz-TransformSystemFunctionAlgebraandBlockDiagramRepresentationsTheUnilateralz-Transform210.0INTRODUCTIONz-transformisthediscrete-timecounterpartoftheLaplacetransform,themotivationforandpropertiesofthez-transformcloselyparallelthoseoftheLaplacetransform.However,theyhavesomeimportantdistinctionsthatarisefromthefundamentaldifferencesbetweencontinuous-timeanddiscrete-timesignalsandsystems.z-transformexpandtheapplicationinwhichFourieranalysiscanbeused.

310.1

Thez-TransformThez-transform

ofageneraldiscrete-timesignalx[n]

isdefinedaswherezisacomplexvariable.Wewilldenotethetransformrelationshipbetweenx[n]andX(z)asThedefinationofthe

z-transform

:Therelationshipbetweenx[n]and

X(z)

4Therelationshipbetweenthez-transformandthediscrete-timeFouriertransformExpressingthecomplexvariablezinpolarformas

istheFouriertransformofx[n]multipliedbyarealexponentialSo,thez-transformisanextensionoftheDTFT.Forr=1,orequivalently,|z|=1,z-transformequationreducestotheFouriertransform.5Thez-transformreducestotheFouriertransformforvaluesofzontheunitcircle.z-planeω

1ImReUnitcircleDifferentfromthecontinuous-timecase,thez-transformreducestotheFouriertransformonthecontourinthecomplexz-planecorrespondingtoacirclewitharadiusofunity.Thez-transformreducestothediscrete-timeFouriertransformThez-transformreducestothediscrete-timeFouriertransform6Ingeneral,thez-transformofasequencehasassociatedwithitarangeofvaluesofzforwhichX(z)converges,andthisrangeofvaluesisreferredtoastheregionofconvergence(ROC).Forconvergenceofthez-transform,werequirethattheFouriertransformofconverge.Foranyspecificsequencex[n],itisthisconvergenceforsomevalueofr.IftheROCincludestheunitcircle,thentheFouriertransformalsoconverges.Theregionofconvergence(ROC)―dependsonlyonr=|z|,justliketheROCins-planeonlydependsonRe(s).7ForconvergenceofX(z),werequirethat

Consequently,theregionofconvergenceistherangeofvaluesofzforwhich

Then

Unitcirclez-planeIm

a1RePole-zeroplotandregionofconvergenceforExample10.1for0<α<1Example10.1Considerthesignal8TheROCdoesnotincludetheunitcircle,consequently,itisimpossibletoobtaintheFouriertransformfrom,Z

plate1NowConsiderthestepsignal9Example10.2Determinethez-transformofIf,thissumconvergesandUnitcirclez-planeIma1RePole-zeroplotandregionofconvergenceforExample10.2for0<α<110Example10.3Considerasignalthatisthesumoftworealexponentials:Thez-transformisthenIm

1/31/213/2

Re11Example10.4.Considerthesignal:

Generally，theROCofconsistsofaringinthez-planecenteredabouttheorigin.21/2Z

plateUnitcircle12Example10.5ConsiderthesignalThez-transformofthissignalis,|z|>1/3

Im

1/31

Re13

ifthez-transformisrational,itsnumeratoranddenominatorpolynomialcanbefactarized.

so,thez-transformischacterizedbyallitspolesandzerosexceptaconstantfactor.Thegeometricrepresentationofthez-transform——thePole-Zeroplot：14

x[n]canbeonlydeterminedbyallpolesandzerosofX(z)andtheROCoftheX(z).

Thepole-zeroplot,illustrateallpolesandzerosofthez-transforminzplane,isthegeometricrepresentationofthez-transform.

Thepole-zeroplotisespeciallyusefulfordescribingandanalyzingthepropertiesofthediscrete-timeLTIsystem.1510.2TheRegionofConvergenceforthez-TransformPropertiesoftheROCforz-transform：Property1

TheROCofX(z)consistsofaringinthez-planecenteredabouttheorigin.Property

2TheROCdoesnotcontainanypoles.Property

3Ifx[n]isoffiniteduration,thentheROCistheentirez-plane,exceptpossiblyz=0and/orz=∞.Convergenceisdependentonlyonandnoton.16Example10.6Considertheunitsamplesignalδ[n].withanROCconsistingoftheentirez-plane,includingz=0andz=∞.Ontheotherhand,

theROCconsistsoftheentirez-plane,includingz=∞but

excludingz=0.Similarly,

theROCconsistsoftheentirez-plane,includingz=

0butexcludingz=

∞.17Property4Ifx[n]isaright-sidedsequence,andifthecircleisintheROC,thenallfinitevaluesofzforwhichwillalsobeintheROC.Imz-plane

Re185)PropertyIfx[n]isaleft-sidedsequence,andifthecircleisintheROC,thenallvaluesofzforwhichwillalsobeintheROC.Imz-plane

Re6)Property

Ifx[n]istwosided,andifthecircleisintheROC,thentheROCwillconsistofaringinthez-planethatincludesthecircle.Imz-plane

Re1920poles：（order1）（orderN-1）zeros：0Atz=a,thezerocancelthepole.Consequently,therearenopolesotherthanattheorigin.Example10.7Considethefinitedurationx[n]other21Example10.8ConsideratwosidedsequenceForb>1,thereisnocommonROC,andthusthesequencewillnothaveaz-transform.Forb<1,theROCsoverlap,andthusthez-transformforthecompositesequenceisUnitcircleImb1/bRe22Property

8Ifthez-transformX(z)ofx[n]isrational,andifx[n]isrightsided,thentheROCistheregioninthez-planeoutsidetheoutermostpole―i.e.,outsidethecircleofradiusequaltothelargestmagnitudeofthepolesofX(z).Furthermore,ifx[n]iscausal,thentheROCalsoincludesz=∞.Property

9Ifthez-transformX(z)ofx[n]isrational,andifx[n]isleftsided,thentheROCistheregioninthez-planeinsidetheinnermostnonzeropole―i.e.,insidethecircleofradiusequaltothesmallestmagnitudeofthepolesofX(z)otherthananyatz=0andextendinginwardtoandpossiblyincludingz=0.Inparticular,ifx[n]isanticausal,thentheROCalsoincludesz=0.Property

7Ifthez-transformX(z)ofx[n]isrational,thenitsROCisboundedbypolesorextendstoinfinity.23Example10.9ConsiderallofthepossibleROCsthatcanbeconnectedwiththefunction

ReUnitcircleImTheROCifx[n]isleftsidedTheROCifx[n]isrightsidedTheROCifx[n]istwosidedandistheonlyoneofthethreeforwhichtheFouriertransformconverges.zeros：（order2）poles：24一.Theexpressionoftheinversez-transform：10.3.TheInversez-TtansformMultiplyingbothsidesby,weobtainChangingthevariableofintegrationfromωtozwithrfixedandvaryingoverainterval:Thus,thebasicinversez-transformequationis:25TheformalevaluationoftheintegralforageneralX(z)requirestheuseofcontourintegrationinthecomplexplane.Thesymboldenotesintegrationaroundacounterclockwisecircularcontourcenteredattheoriginandwithradiusr.Therearetwoalternativeproceduresforobtainingasequencefromitsz-transform:oneispartial-fractionexpansion,theotherispower-seriesexpansion.二.Theproceduresforobtainingasequencefromitsz-transform：26Thepartial-fractionexpansionForrathionalz-transformX(z),determineallpolesofit.Then,itcanbeexpandedbythemethodofpartial-fraction,andweobtainexpressionofthez-transformasalinearcombinationofsimplerterms:NowspecifytheROCassociatedwitheachterm.Finally,determinetheinversez-transformofeachoftheseindivialtermsbasedontheROCassociatedwitheach.SothattheinversetransformofX(z)equalsthesunoftheinversetransformsoftheindividualtermsintheequation.27Example10.10Considerthez-transformTherearetwopoles,oneatz=1/3andoneatz=1/4.Performingthepartial-fractionexpansion,weobtain28Power-seriesexpansionThisprocedureismotivatedbytheobservationthatthedefinitionofthez-transformcanbeinterpretedasapowerseriesinvolvingbothpositiveandnegativepowersofz.Thecoefficientsinthispowerseriesare,infact,thesequencevaluesx[n].29Example10.11Considerthez-transformFromthepower-seriesdefinitionofthez-transform,wecandeterminetheinversetransformofX(z)byinspection:Thatis,SomeusefulZTpairs:30Example10.10Considerthez-transform

Thenperforminglongdivision:

FromtheROC,wecanconcludethatthecorrespondingsequencex[n]isright-sided,sothatwearrangethenumeratorpolynomialandthedenominatorpolynomialwithaorderofthepowerofzdecreasing(oraorderofthepowerofincreasing).31

IftheROCis|z|<|α|,beforeperforminglongdivision,wearrangethenumeratorpolynomialandthedenominatorpolynomialwith

aorderofthepowerofzincreasing(oraorderofthepowerofdecreasing).

Thenperforminglongdivision:

32Fortwo-sidedsequence

performingthelongdivisiontothetwotermsrespectively.33theResidueTheoremisapoleoutsidethecontourC

。，

isapoleinsidethecontourC。，Contourintegration34Example10.11Considerthez-transform

Fromthedefinitionoftheinversez-transform,ThiscontourintegrationcanbeevaluatedthroughusingtheResidueTheorem,thusSincetheROCis|z|>1,sothecorrespondingsequenceisrightsided.35For,hasonlytwofirst-orderpoles:ThenFor,hasthreefirst-orderpoles:ThenFor,hastwofirst-orderpoles:andasecond-orderpole:36ThenConsequently,37

Inthediscrete-timecase,theFouriertransformcanbeevaluatedgeometricallybyconsideringthepoleandzerovectorsinthez-plane.Sinceinthiscasetherationalfunctionistobeevaluatedonthecontour|z|=1,weconsiderthevectorsfromthepolesandzerostotheunitcircle.Considerafirst-ordercausaldiscrete-timesystemwithaimpulseresponse:Itsz-transformis

10.4.GeometricEvaluationofTheFourierTransformFromThePole-ZeroPlot38For|a|<1,theROCincludestheunitcircle,andconsequently,theFouriertransformofh[n]convergesandisequaltoH(z)for.Thefrequencyresponseforthefirst-ordersystemis

thepole-zeroplotforH(z),includingthevectorsfromthepole(atz=a)andzero(atz=0)totheunitcircle.39Magnitudeofthefrequencyresponsefora=0.95anda=0.5Phaseofthefrequencyresponsefora=0.95anda=0.5

themagnitudeofthefrequencyresponsewillbemaximumat=0andwilldecreasesmonotonicallyasincreasesfrom0to.40a10Magnitudeofthefrequencyresponsefora=-0.95anda=-0.5Phaseofthefrequencyresponsefora=-0.95anda=-0.5411)LinearityNote:

ROCisatleasttheintersectionofR1andR2,whichcouldbeempty,alsocanbelargerthantheintersection.Forsequencewithrationalz-transform,ifthepolesofaX1(z)+bX2(z)consistofallofthepolesofX1(z)andX2(z)(ifthereisnopole-zerocancellation),thentheROCwillbeexactlytotheoverlapoftheindividualROC.Ifthelinearcombinitionissuchthatsomezerosareintroducedthatcancelpoles,thenROCmaybelarger.Ifandthen10.5.PropertiesofThez-Transform42Exceptforthepossibleadditionordeletionoftheoriginorinfinity.If2)TimeShiftingthen43Example10.12

ConsiderthesignalFromExample7.1,weknowSothat,Infact,Inprocedure(І),onepoleatz=0wasintroduced,anditcanceledazeroatthesamelocation.Andfromthetimeshiftingproperty,(І)Notethechanginginpolesorzeroswhenusingthetime-shiftingproperty.443)Scalinginthez-DomainIfthenGeneralcase:when,thepoleandzerolocationsarerotatedbyandscaledinmagnitudebyafactorof.Specialcase:when,1/2454)TimeReversal5)TimeExpansionConsequence:ifz0isintheROCforx[n],then1/z0isintheROCforx[–n].Ififnisamultipleofkifnisnotamultipleofk466)Conjugation7)TheConvolutionProperty

Consequence:ifx[n]isreal,Thus,ifX(z)hasapole(orzero)atz=z0,itmustalsohaveapole(orzero)atthecomplexconjugatepointz=z0*.Ifandthen47

theROCofX1(z)X2(z)maybelargerifpole-zerocancellationoccursintheproduct.Proof:48SinceFromtheconvolutionproperty,IftheROCofX(z)isR,thentheROCofW(z)mustincludesatleasttheinterconnectionofRwith|z|>1.Consideritsz-transform.(SupposeX(z)isgiven)Example10.13letw[n]betherunningsumofx[n]:498)Differentiationinthez-DomainExample14.

50Example10.15(p773)Determinetheinversez-transformforFromExample10.1,

andhence,519)TheInitial-andFinal-ValueTheoremsInitial-valuetheorem:Final-valuetheorem:Ifx[n]isacausalsequence,i.e.,x[n]=0,forn<0,then52终值定理：若是因果信号，且，除了在可以有一阶极点外，其它极点均在单位圆内，则

证明：在单位圆上无极点

除了在可以有一阶极点外，其它极点均在单位圆内，53

这其实表明：如果有终值存在，则其终值等于在处的留数。54Z平面上极点位置与信号模式的关系示意图5510.6SomeCommonZ-TransformPairsTable10.1,thepropertiesofthez-transform.Table10.2,anumberofusefulz-transformpairs.5610.7.AnalysisandCharacterizationofLTISystemsUsingzTransform

SystemfunctionWeknow,inthetimedomain,theinputandtheoutputofanLTIsystemarerelatedthroughConvolutionbytheimpulseresponseofthesystem.Thus

y[n]=h[n]*x[n]supposeFromConvolutionProperty

Y(z)=H(z)X(z)systemfunction(transferfunction)For,H(z)isthefrequencyresponseoftheLTIsystem.5710.7.1

RelatingCausalitytotheSystemfunctionForacausalLTIsystem,theimpulseresponseiszeroforn<0andthusisrightsided.TheROCassociatedwiththesystemfunctionforacausalsystemistheexteriorofacircleinthez-plane.Becausethepowerseriesdoesnotincludeanypositivepowersofz,theROCincludeinfinity.IfH(z)isrational,forthesystemtobecausal,theROCmustbeoutsidetheoutermostpoleandinfinitymustbeintheROC.Equivalently,thenumeratorofH(z)hasdegreenolargerthanthedenominatorwhenbothareexpressedaspolynomials.58Adiscrete-timeLTIsystemiscausalifandonlyiftheROCofitssystemfunctionistheexteriorofacircle,includinginfinity.Adiscrete-timeLTIsystemwithrationalsystemfunctionH(z)iscausalifandonlyif:(a)theROCistheexteriorofacircleoutsidetheoutermostpole;and(b)withH(z)expressedasaratioofpolynomialsinz,theorderofthenumeratorcannotbegreaterthantheorderofthedenominator.Summarizing,wehavethefollowprinciples:59stableH[n]absolutelysummableh[n]hasFTROCofh[n]’sZTcontainstheunitcircle10.7.2RelatingStabilitytotheSystemfunctionThestabilityofadiscrete-timeLTIsystemisequivalenttoitsimpulseresponsebeingabsolutelysummable,inwhichcasetheFouriertransformoftheimpulseresponseconverges.SinceSummarizing:AnLTIsystemisstableifandonlyiftheROCofitssystemfunctionH(z)includestheunitcircle,|z|=1.AcausalLTIsystemwithrationalsystemfunctionH(z)isstableifandonlyifallofthepolesofH(z)lieinsidetheunitcircle―i.e.,theymustallhavemagnitudesmallerthan1.60ForanLTIsystemwhichisdescribedbyaN-order

linearconstant-coefficientdifferenceequationoftheform10.7.3LTISystemCharacterizedbyLinearConstant-CoefficientDifferenceEquationsThentakingz-transformsofbothsidesoftheaboveequationandusingthelinearityandtime-shiftingproperties,weobtain:61Soitssystemfunction(transferfunction)is:Thus,thesystemfunctionforasystemspecifiedbyadifferenceequationisalwaysrational.的ROC需要通过其它条件确定，如：1.系统的因果性或稳定性。2.系统是否具有零初始条件等。62LTI系统的Z变换分析法：1）由求得及其。2）由系统的描述求得及其。分析步骤：3）由得出并确定它的ROC包括。4）对做反变换得到。63Wecanconcludethatthesystemisnotcausal,becausethenumeratorofH(z)isofhigherorderthanthedenominator.Example10.16(p777)ConsiderasystemwithsystemfunctionwhosealgebraicexpressionisInfact,sinceEvenwedon’tknowtheROCforthissystemfunction.However,fromthepointthattheinversetransformofzisδ[n+1],wecanconcludethissystemisnoncausal.64Example10.17(p777)Considerasystemwithsystemfunction

Wecanconcludethatthesystemiscausal.

FromweseethatH(z)isrationalandthenumeratoranddenominatorofH(z)arebothofdegreetwo.Inaddition,theROCforthissystemfunctionistheexteriorofacircleoutsidetheoutermostpole.65Example10.18(p779)Considerasecond-ordercausalsystemwithsystemfunctionItspoleslocatedatand.

ReUnitcircleImz-planer

θ

1

ReUnitcircleImz-planer

θ

1

r<1

r>1

stable

unstable66Example10.19(p782)Considerastableandcausalsystemwithimpulseresponseh[n]andrationalsystemfunctionH(z).SupposeitisknownthatH(z)containsapoleatz=1/2andazerosomewhereontheunitcircle.Theprecisenumberandlocationofalloftheotherpolesandzerosareunknown.Foreachofthefollowingstatements,letusdeterminewhetherwecandefinitelysaythatitistrue,whetherwecandefinitelysaythatitisfalse,orwhetherthereisinsufficientinformationgiventodetermineifitistrueornot:(a)converges.(b)forsomeω.(c)h[n]hasfiniteduration.(d)h[n]isreal.(e)g[n]=n{h[n]*h[n]}istheimpulseresponseofastablesystem.

╳Insufficient!

67Theuseofthez-transformtoconvertsystemdescriptionstoalgebraicequationsisalsohelpfulinanalyzinginterconnectionsofLTIsystemsandinrepresentingandsynthesizingsystemsasinterconnectionsofbasicsystembuildingblocks.10.8.SystemFunctionAlgebraandBlockDiagramRepresentations68SystemfunctionforinterconnectionsofLTIsystems:ROCcontaining1.SeriesinterconnectionsofLTIsystems

：2.ParallelinterconnectionsofLTIsystems

：ROCcontaining69h1[n]H1(z)x[n]h2[n]H2(z)x(z)y[n]Y(z)e[n]+-E(z)3.Thefeedbackinterconnectionoftwosystems:ROCcontaining70Example10.20ConsiderthecausalLTIsystemwithsystemfunction

ThissystemcanalsobedescribedbythedifferenceequationBlockdiagramrepresentationofthecausalLTIsystemx[n]y[n]1/4

isthesystemfunctionofaunitdelayer.71Ifweusev[n]torepresenttheoutputofthefirstsubsystem,y[n]theoutputoftheoverallsystem,therelationshipbetweenthemisExample10.21ConsideracausalLTIsystemwithsystemfunctionBlockdiagramrepresentationforthesysteminExample10.20y[n]–2x[n]1/4v[n]72Example10.22Considerasecond-orderLTIsystemwithsystemfunction73Direct-formrepresentationforthefirstsystemx[n]1/8v[n]–1/474Direct-formrepresentationforthesysteminExample7.21x[n]1/8y[n]–1/2–1/4–7/4v[n]75Cascade-formrepresentationforthesysteminExample7.21x[n]–1/2

1/4y[n]1/4–276Parallel-formrepresentationforthesysteminExample7.21y[n]x[n]5/3–1/21/4–14/3477由LCCDE描述的LTI系统，其系统函数为有理函数，可将其因式分解或展开为部分分式。Thecascadeformandparallelformblock-diagramrepresentationsoftheLTIsystem：781.thecascadeformrepresentation：

其中是二阶（或一阶）系统函数。由此即可得系统的级联型结构：将因式分解，在无重阶零极点时可得N为偶数时79DDDDCascadeformoftheLTIsystem802.

Theparallelformrepresentation：

将展开为部分分式，在无重阶极点时有N为偶数时81DDDDParallelformoftheLTIsystem82bilateral

z-transform:unilateral

z-transform:10.9.TheUnilateralz-Transform10.9.1

TheUnilateralz-TransformandInverseTransform83Thecalculationoftheinverseunilateralz-transformsisbasicallythesameasforbilateraltransforms,withtheconstraintthattheROCforaunilateraltransformmustalwaysbetheexteriorofacircle.Theunilateralz-transformdiffersfromthebilateraltransforminthatthesummationiscarriedoutonlyovernonnegativevaluesofn.Theunilateralz-transformofx[n]canbethoughtofasthebilateraltransformofx[n]u[n].Thebilateraltransformandtheunilateraltransformofacausalsignalareidentical.TheROCfortheunilateraltransformisalwaystheexteriorofacircle.84Example10.23ConsiderthesignalThebilateraltransformX(z)forthisexa