信號與系統SignalsandSystems吉林大學PropertiesofLaplacetransform:RightShiftinTimeProperty3:RightshiftintimeProof:Property3:Rightshiftintime(3)Property3:RightshiftintimeSolution:信號與系統SignalsandSystems吉林大學PropertiesofLaplaceTransform:TimeScalingProperty4:TimescalingProof:信號與系統SignalsandSystems吉林大學PropertiesofLaplaceTransform:ConvolutionTheoremsProperty5:Convolutioninthet-domainProof:Property5:Convolutioninthet-domainContinued:Property

6:Convolutioninthes-domainProof:信號與系統SignalsandSystems吉林大學PropertiesofLaplaceTransform:Differentiationinthet-domainProperty7:Differentiationinthet-domainProof:信號與系統SignalsandSystems吉林大學PropertiesofLaplaceTransform:Integrationinthet-domainProperty8:Integrationinthet-domainProof:Property8:Integrationinthet-domain信號與系統SignalsandSystems吉林大學PropertiesofLaplaceTransform:DifferentiationandIntegrationinthes-DomainProperty9:Differentiationinthes-domainProof:Property10:Integrationinthes-domainProof:信號與系統SignalsandSystems吉林大學PropertiesofLaplaceTransform：InitialandFinal-ValuetheoremsProperty11：Initial-valuetheoremProof:Property12：Final-valuetheoremProof:信號與系統SignalsandSystems吉林大學ComputationoftheInverseLaplaceTransform(Ⅱ)PartialFractionExpansionComputationoftheinverseLaplacetransform(Ⅱ)Partialfractionexpansion(1)Conditions:ComputationoftheinverseLaplacetransform(Ⅱ)Partialfractionexpansion(2)ComputationoftheinverseLaplacetransform(Ⅱ)Partialfractionexpansion(3)信號與系統SignalsandSystems吉林大學SolvingtheDifferentialEquationsinthes-DomainSolvingthedifferentialequationsinthes-domain[Example]Given:Find：

Solvingthedifferentialequationsinthes-domainSolvingthedifferentialequationsinthes-domainSolvingthedifferentialequationsinthes-domain信號與系統SignalsandSystems吉林大學Thes-DomainRepresentationsofCircuits(I)Thes-domainrepresentationsofcircuits(I)1Thes-domainequivalentcircuitelementsThesameresistanceThes-domainrepresentationsofcircuits(I)1Thes-domainequivalentcircuitelementsThes-domainimpedanceThes-domainrepresentationsofcircuits(I)1Thes-domainequivalentcircuitelementsThes-domainimpedanceThes-domainrepresentationsofcircuits(I)2TheformsofKVLandKCLinthes-domain信號與系統SignalsandSystems吉林大學TheBlockDiagramofaSysteminthes-DomainTheblockdiagramofasysteminthes-domainScalarmultiplier1Adder/Subtractor2Theblockdiagramofasysteminthes-domainIntegrator3信號與系統SignalsandSystems吉林大學TheDefinitionofTransferFunctionanditsSolutionsThedefinitionoftransferfunctionanditssolutionsThetransferfunctionⅠHowtofind21.GiventhesystemdifferentialequationThedefinitionoftransferfunctionanditssolutionsHowtofind21.GiventhesystemdifferentialequationThedefinitionoftransferfunctionanditssolutions2.Giventheimpulseresponseh(t)Thedefinitionoftransferfunctionanditssolutions3.GiventhestructureofthecircuitUsingthedefinitioninthes-domainrepresentationofthecircuit.4.Usingthepole-zeroplot信號與系統SignalsandSystems吉林大學TheTransferFunctionandthePole-ZeroPlotThetransferfunctionandthepole-zeroplotPolesandzeros1Zeros:Poles:Thetransferfunctionandthepole-zeroplotThepole-zeroplot2Aplotinthecomplexplaneshowingthelocationsofallthepoles(markedby×)andallthezeros(markedby○)iscalledthepole-zeroplot.Zeros:Poles:信號與系統SignalsandSystems吉林大學ApplicationsofthePole-ZeroPlot:DeterminingtheFormofh(t)Thepoles

beinglocatedintheopen

left-halfcomplexplane1Applicationsofthepole-zeroplot:Determiningtheformofh(t)Thepoles

beinglocatedintheopen

left-halfcomplexplane1Applicationsofthepole-zeroplot:Determiningtheformofh(t)Thepoles

beinglocatedattheorigin2Applicationsofthepole-zeroplot:Determiningtheformofh(t)Thepoles

beinglocatedontheimaginaryaxis3Applicationsofthepole-zeroplot:Determiningtheformofh(t)Thepoles

beinglocatedintheopen

right-halfcomplexplane4Applicationsofthepole-zeroplot:Determiningtheformofh(t)信號與系統SignalsandSystems吉林大學Time-DomainAnalysisofDiscrete-TimeSystemsDiscrete-TimeSignalsDiscrete-TimeSignalsAdiscrete-timesignalf(k)hasvaluesforsomediscontinuouspointwhilehasnotdefinitionforotherpoints.k—integerDefinitionDiscrete-TimeSignalsAnalyticalmethod:Graphicalmethod:Sequencemethod:k=0RepresentationDiscrete-TimeSignalEnergyandPowerEnergy:Power:OperationofDiscrete-TimeSignalsAddition：Multiplication：Difference：forwarddifference:backwarddifferenceRunningsum：OperationofDiscrete-TimeSignalsTimeshift(m>0)RightshiftLeftshiftTransformationsoftheIndependentVariableOperationofDiscrete-TimeSignalsTimereversalTransformationsoftheIndependentVariablef(-k)isobtainedfromthesignalf(k)

byareflectionaboutk=0.BasicDiscrete-TimeSignalsUnitImpulseSequence(UnitSampleSequence)BasicDiscrete-TimeSignalsUnitStepSequenceBasicDiscrete-TimeSignalsRelationshipbetweend(k)ande(k)BasicDiscrete-TimeSignalsRectangularSequenceBasicDiscrete-TimeSignalsUnilateralexponentialsequenceswithrealvalues：f(k)=ak

(k)(aisarealnumber)BasicDiscrete-TimeSignalsUnitrampsequenceSinusoidalSequencesComplexExponentialSequences:Canda:complexnumbers信號與系統SignalsandSystems吉林大學RepresentationsofDiscrete-TimeSystemsRepresentationsofDiscrete-TimeSystemsAdiscrete-timesystemisasystemthattransformsdiscrete-timeinputsintodiscrete-timeoutputs.Definitionf(k):inputy(k):outputInput-outputrelation:f(k)→

y(k)RepresentationsofDiscrete-TimeSystemsnth-orderLinearConstant-CoefficientDifferenceEquation:LTISystemsDescribedbyDifferenceEquatioconstantsRepresentationsofDiscrete-TimeSystemsBlockDiagramRepresentationBasicelementsMultiplicationbyacoefficientAdderUnitDelayElementRepresentationsofDiscrete-TimeSystemsInterconnectionsofSystemsSeries(Cascade)interconnectionParallelinterconnectionFeedbackinterconnection信號與系統SignalsandSystems吉林大學Linearinput/outputdifferenceequationswithconstantcoefficientsInput:f(k)=0fork<0InitialCondition：y(0),y(1),y(2),…,y(n-1)InitialState：y(-1),y(-2),…,y(-n)Linearinput/outputdifferenceequationswithconstantcoefficientsEquation:Solution:Linearinput/outputdifferenceequationswithconstantcoefficientsTheHomogeneousSolutionHomogeneousequation

CharacteristicequationCharacteristicroot

j(j=1,2,3,

,n)HomogeneoussolutionLinearinput/outputdifferenceequationswithconstantcoefficientsTheHomogeneousSolutionExample:y(k)+3y(k-1)+2y(k-2)=f(k),f(k)=2k,k

≥0,y(0)=0,y(1)=2.Findyh(k)

.Characteristicequation:Homogeneousequation

CharacteristicequationCharacteristicroot

j(j=1,2,3,

,n)HomogeneoussolutionLinearinput/outputdifferenceequationswithconstantcoefficientsTheParticularSolutionLinearinput/outputdifferenceequationswithconstantcoefficientsTheParticularSolutionExample:y(k)+3y(k-1)+2y(k-2)=f(k),f(k)=2k,k

≥0,y(0)=0,y(1)=2.Findyp(k),k

≥0.Letyp(k)=P·2k,k

≥0Substitutethesystemequation:信號與系統SignalsandSystems吉林大學TheZero-InputResponse

and

TheZero-StateResponseTheZero-InputResponse

Characteristicequation

j,(j=1,2,3,

,n)CharacteristicrootZero-InputResponse

yzi

(0)，yzi

(1)，…，yzi

(n-1)y(-1)，y(-2)，…，y(-n)

yzi(k)=y(k)-

yzs(k)=y(k)，k<0InitialconditionCharacteristicequation:Characteristicroots:Zero-InputResponse:Example:TheZero-InputResponsey(k)+3y(k-1)+2y(k-2)=f(k),f(k)=2kε(k),y(-1)=0,y(-2)=1/2.Findyzi(k),k

≥0.yzi(k)+3yzi(k-1)+2yzi(k-2)=0TheZero-StateResponseCharacteristicequation

j

(j=1,2,3,

,n)Characteristicroot(distinctroots

j

)Zero-StateResponseyzs(-1)=yzs(-2)=…=yzs

(-n)=0Initialstateyzs(0)，yzs

(1)，…，yzs

(n-1)Initialcondition信號與系統SignalsandSystems吉林大學TheUnitSampleResponse

and

TheUnitStepResponseTheUnitSampleResponseDefinitionTheunitsampleresponseisthezero-stateresponseofthesystemresultingfromtheapplicationoftheunitpulse

(k).Denotedh(k)Initialstateh(-1)=h(-2)=…=h(-n)=0Initialconditionh(0),h(1),h(2),…,h(n-1)HowtofindSolvingadifferenceequationZ-transformTheUnitSampleResponseDeterminationk<0：

(k)

=0，h(k)=0k=0：

(k)

=1，h(0)——recursionk>0：

(k)

=0，h(k)——solutionofahomogeneousequationLTI

system:LetthenCi：determinedbyh1(1),h1(2),…,h1(n)TheUnitStepResponseDefinitionTheunitstepresponseisthezero-stateresponseofthesystemresultingfromtheapplicationoftheunitstepsequencee

(k).Denotedg(k)Initialstateg(-1)=g(-2)=…=g(-n)=0Relationshipbetweenh(k)andg(k)信號與系統SignalsandSystems吉林大學ConvolutionSumConvolutionSum

Ingeneral,twodiscrete-timesignalsf1(k)andf2(k)DefinitionExample1:ConvolutionSumConvolution-SumRepresentationofLTIdiscrete-timesystemsThezero-stateresponse:ConvolutionSum:GraphicalRepresentationGraphicalRepresentationoftheconvolutionsumProcedure:Step1.Drawf1(i)andf2(i)Step2.Reverse

f2(i):f2(i)

f2(-i)Step3.Shift

f2(-i)bykpositiontotheright:f2(-i)

f2(k-i)

Step4.Multiplicationoff1(i)withf2(k-i):

f1(i)f2(k-i)

Step5.Summationoftheproductforallvaluesofi

yieldsonevalueofy(k)Step6.Repeatsteps3and5forallvaluesofk信號與系統SignalsandSystems吉林大學PropertiesoftheConvolutionSumPropertiesoftheConvolutionSumCommutativityProof:PropertiesoftheConvolutionSumAssociativityProof:CascadeinterconnectionofLTIsystemsPropertiesoftheConvolutionSumDistributivitywithadditionProof:ParallelinterconnectionofLTIsystemsPropertiesoftheConvolutionConvolutionwiththeunitpulseProof:Ifk1=0,thenPropertiesoftheConvolutionShiftpropertyProof:Iff(k)=f1(k)*f2(k),then

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.Zerosro