《信號與線性系統(tǒng)分析基礎 》 課件 6.3 常系數(shù)線性差分方程的解

信號與系統(tǒng)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(tǒng)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(tǒng)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(tǒng)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(tǒng)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(tǒng)SignalsandSystems吉林大學Propertiesofthez-Transform——Reflectioninthek-domainReflectioninthek-domainProof:Iff(k)

F(z),

<

z

<

,then

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

F(z),

<

z

<

,then

Example:信號與系統(tǒng)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.