信號與系統SignalsandSystems吉林大學TheAnalysisofDiscrete-TimeSystemsinthez-DomainThez-TransformDefinitionofthez-TransformDefinitionofthez-TransformIntuitionontheRelationbetweenZTandLTLT:Let:Definitionofthez-TransformDefinitionBilateral(two-sided)z-Transform:Unilateral(one-sided)z-Transform:Thetransformpairnotation:信號與系統SignalsandSystems吉林大學Thez-TransformCommonz-transformpairsCommonz-transformpairsUnitSampleSequenceCommonz-transformpairsOne-sideExponentialSequencewhereaisarealorcomplexnumber.UnitStepSequenceCommonz-transformpairswhere

aisarealorcomplexnumber.信號與系統SignalsandSystems吉林大學TheRegionofConvergenceforthez-TransformDefinitionTheRegionofConvergenceforthez-TransformThesetofallcomplexnumberszsuchthatthesummationontheright-handside

convergesiscalledtheregionofconvergence(ROC)ofthez-transformF(z).F(z)converges:f(k)z-kisabsolutelysummableFinite-durationsequenceTheRegionofConvergenceforthez-Transformf(k)=0,k

<k1,k>k2,k1<k2k1<0，k2>0：

k1<0，k2

0：k10，k2

>0：0<|z|<

|z|<

|z|>0Example:CausalsequenceTheRegionofConvergenceforthez-Transformf(k)=0,k<0Example:z-planeak

(k)，aisarealorcomplexnumber.AnticausalsequenceTheRegionofConvergenceforthez-TransformExample:f(k)=0,k≥0f(k)=-ak

(-k-1)，aisarealorcomplexnumber.Two-sidedsequenceTheRegionofConvergenceforthez-Transformk=-∞→+∞

0<R1<R2<:R1<|z|<R2

R1>R2

:

ROCdoesnotconvergeTheRegionofConvergenceforthez-TransformROCisboundedbypolesorextendstoinfinity.F(z)isrational:f(k)ROCrightsidedoutsidetheoutermostpole——outsidethecircleofradiusequaltothelargestmagnitudeofthepolesofF(z)leftsidedinsidetheinnermostnonzeropole——insidethecircleofradiusequaltothesmallestmagnitudeofthepolesofF(z)otherthananyatz=0andextendinginwardtoandpossiblyincludingz=0.信號與系統SignalsandSystems吉林大學Propertiesofthez-Transform——LinearityIff1(k)

F1(z),

1<

z

<

1，f2(k)

F2(z),

2<

z

<

2，thenLinearityExample:Iff1(k)

F1(z),

1<

z

<

1，f2(k)

F2(z),

2<

z

<

2，thenLinearityExample:信號與系統SignalsandSystems吉林大學Propertiesofthez-Transform——TimeShiftingTimeShiftingExample:Bilateralz-TransformIff(k)

F(z),

<

z

<

,thenwheremisapositiveinteger.TimeShiftingProof:Unilateralz-Transform——RightshiftIff(k)

F(z)，

z

>

，thenwheremisapositiveinteger.TimeShiftingUnilateralz-Transform——RightshiftIff(k)=0,k<0,thenExample:Iff(k)

F(z)，

z

>

，thenwheremisapositiveinteger.TimeShiftingUnilateralz-Transform——LeftshiftIff(k)

F(z)，

z

>

，thenwheremisapositiveinteger.Proof:TimeShiftingUnilateralz-Transform——LeftshiftIff(k)

F(z)，

z

>

，thenwheremisapositiveinteger.Example:

(k+1)信號與系統SignalsandSystems吉林大學Propertiesofthez-Transform——Scalinginthez-DomainScalinginthez-DomainProof:Iff(k)

F(z),R1<|z|<R2

，thenaisanonzerorealorcomplexnumber.ROCofF(z):ROCof

:Scalinginthez-DomainIff(k)

F(z),R1<|z|<R2

，thenaisanonzerorealorcomplexnumber.Example:

aksin(

k)

(k)，0<a<1Scalinginthez-DomainIff(k)

F(z),R1<|z|<R2

，thenaisanonzerorealorcomplexnumber.Example:(-1)k

(k)信號與系統SignalsandSystems吉林大學Propertiesofthez-Transform——ConvolutionConvolutionProof:Iff1(k)

F1(z),

1<z<

1,f2(k)

F2(z),

2<z<

2,thenConvolutionIff1(k)

F1(z),

1<z<

1,f2(k)

F2(z),

2<z<

2,thenExample:(k+1)

(k)LTIsystems:信號與系統SignalsandSystems吉林大學Propertiesofthez-Transform——DifferentiationandIntegralinthez-DomainDifferentiationinthez-DomainProof:Iff(k)

F(z),

<

z

<

,then

wherekisanypositiveinteger.Differentiationinthez-DomainIff(k)

F(z),

<

z

<

,then

wherekisanypositiveinteger.Example:Ifa=1,thenDifferentiationinthez-DomainIff(k)

F(z),

<

z

<

,then

wherekisanypositiveinteger.Integralinthez-DomainProof:Iff(k)

F(z),

<

z

<

,then

(misaninteger,andk+m>0)Integralinthez-DomainIff(k)

F(z),

<

z

<

,then

(misaninteger,andk+m>0)Example:Integralinthez-DomainIff(k)

F(z),

<

z

<

,then

(misaninteger,andk+m>0)m=0,k>0:信號與系統SignalsandSystems吉林大學Propertiesofthez-Transform——Reflectioninthek-domainReflectioninthek-domainProof:Iff(k)

F(z),

<

z

<

,then

Example:信號與系統SignalsandSystems吉林大學Propertiesofthez-Transform——SummationSummationProof:Iff(k)

F(z),

<

z

<

,then

Example:信號與系統SignalsandSystems吉林大學Propertiesofthez-Transform——Initial-ValueTheoremandFinal-ValueTheoremInitial-ValueTheoremProof:Iff(k)=0,k<0,andf(k)

F(z),then

Example:0Thez-transformofacausalsequencef(k)isfindf(0).Final-ValueTheoremProof:Iff(k)=0,k<0，f(k)

F(z),a<

z<,0≤a<1,then

Final-ValueTheoremIff(k)=0,k<0，f(k)

F(z),a<

z<,0≤a<1,then

Example:f(k)=0,k<0. aisarealnumber,findf(

).Final-ValueTheorem√√××Final-ValueTheoremIff(k)=0,k<0，f(k)

F(z),a<

z<,0≤a<1,then

Example:f(k)=0,k<0. aisarealnumber,findf(

).Final-ValueTheoremIfF(z)isrationalandthepolesof(z-1)F(z)havemagnitudes<1,then

Example:Thez-transformofacausalsequencef(k)is

Poles:信號與系統SignalsandSystems吉林大學TheInversez-TransformTheInversez-Transform（IZT）Integral:DefinitionalongacounterclockwiseclosedcircularcontourthatiscontainedintheROCofF(z).AlternativeproceduresPower-seriesexpansionsPartialfractionexpansionsROCandtheInversez-TransformROCf(k)Causalsequence|z|>af1(k)e

(k)Anticausalsequence|z|<bf2(k)e

(-k-1)Two-sidedsequencea<|z|<b

f1(k)e(k)+

f2(k)e

(-k-1)信號與系統SignalsandSystems吉林大學TheInversez-Transform——PartialfractionexpansionsPartialfractionexpansionsRationalpolynomial:Procedure:PartialfractionexpansionsF(z)f(k)×zIZTPartialfractionexpansions

DistinctPolesSupposethatthepolesz1,z1,…,zNofF(z)aredistinctandareallnonzero.（1）|z|>2；（2）|z|<1；（3）1<|z|<2（1）Example:Partialfractionexpansions

DistinctPolesSupposethatthepolesz1,z1,…,zNofF(z)aredistinctandareallnonzero.（1）|z|>2；（2）|z|<1；（3）1<|z|<2（2）Example:Partialfractionexpansions

DistinctPolesSupposethatthepolesz1,z1,…,zNofF(z)aredistinctandareallnonzero.（1）|z|>2；（2）|z|<1；（3）1<|z|<2（3）Example:Partialfractionexpansions

DistinctPolesz1,2=ae±jbROC：|z|>

Complex

Poles:Partialfractionexpansions

DistinctPolesz1,2=ae±jbComplex

Poles:Example:PartialfractionexpansionsRepeatePolesSupposethatthepolez1isrepeatedrtimes.Matchingcoefficients:Example:PartialfractionexpansionsExample:Step1DividethroughtoobtainwhereF1(z)isstrictlyproper.Step2CarryoutthepartialfractionexpansionofF1(z)and,knowingtheROC,obtaintheinversez-transform.信號與系統SignalsandSystems吉林大學z-DomainAnalysis—TransformoftheInput/outputDifferenceEquationTransformoftheInput/outputDifferenceEquationLTIsystem:Input:f(k)=0,k<0Initialstate:y(-1),y(-2),…,y(-n)z-Transform:Y(z)=Yzi(z)+Yzs(z)IZT:y(k)=yzi(k)+yzs(k)TransformoftheInput/outputDifferenceEquationExample:y(k)-y(k-1)-2y(k-2)=f(k)+2f(k-2),y(-1)=2,y(-2)=-0.5,f(k)=e(k).Findyzi(k),yzs(k),y(k),k≥0.TransformoftheInput/outputDifferenceEquationExample:y(k)-y(k-1)-2y(k-2)=f(k)+2f(k-2),y(-1)=2,y(-2)=-0.5,f(k)=e(k).Findyzi(k),yzs(k),y(k),k≥0.TransformoftheInput/outputDifferenceEquationExample:y(k)-y(k-1)-2y(k-2)=f(k)+2f(k-2),y(-1)=2,y(-2)=-0.5,f(k)=e(k).Findyzi(k),yzs(k),y(k),k≥0.信號與系統SignalsandSystems吉林大學z-DomainAnalysis—TheSystemFunctionTheSystemFunction(TransferFunction)DefinitionDeterminationofthesystemfunction(1)

H(z)=Yzs(z)/F(z)(2)H(z)=Z[h(k)]SystemFunctionofInterconnectionsSeriesconnectionH(z)ParallelconnectionH(z)Parallelconnection

H(z)SystemFunctionforInterconnectionsofLTISystemsExample:Determinethezero-stateoftheLTIsystem.Pole-zeroPlotoftheSystemFunctionPole-zeroplotExample:Aplotofthelocationsinthecomplexplaneofthepolesandzeros.ZerosrootsofN(z)=0——○ZerosrootsofD(z)=0——×zeros:z=0poles:z=1信號與系統SignalsandSystems吉林大學z-DomainAnalysis—BlockDiagramRepresentationofDiscrete-timeSystemsinthez-DomainBlockDiagramRepresentationofDiscrete-timeSystemsinthez-DomainMultiplicationbyacoefficientAdderUnitdelayelement（f(-1)=0）信號與系統SignalsandSystems吉林大學CausalityandStabilityofDiscrete-TimeSystemsCausalityandStabilityofDiscrete-TimeSystemsCausalityk-domain:LTIsystemcausality

h(k)=0,k<0Proof:Necessity:Letf(k)=d(k)

f(k)=0fork<0,theny(k)=h(k).Ifthesystemiscausal,thenh(k)=0fork<0.Sufficiency:

f(k)=0,k<0

k-i<0(i>k)，f(k-i)=0,thenIfh(k)=0,k<0

h(i)=0,i<0,then

yzs(k)=0,k<0CausalityandStabilityofDiscrete-TimeSystemsCausalityk-domain:LTIsystemcausality

h(k)=0,k<0z-domain： ,|z|>R0

AdiscreteLTIsystemiscausalifandonlyifth