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2.2.LinearTime-InvariantSystems(WHYAbasicfact:IfweknowtheresponseofanLTItosomeinputs,weactuallyknowtheresponsetomanyKeypointsofSignalsdecomposition:basicsignalCHAPTERLinearTime- ResponseResponsesynthesis:basicresponse(impulse2.2.LinearTime-InvariantSystems(WHYAbasicfact:IfweknowtheresponseofanLTItosomeinputs,weactuallyknowtheresponsetomanyKeypointsofSignalsdecomposition:basicsignalResponseResponsesynthesis:basicresponse(impulse2.12.1Discrete-timeLTIsystem:TheconvolutionTheRepresentationofDiscrete-timeSignalsinTermsofImpulses2.12.1Discrete-timeLTIsystem:TheconvolutionTheRepresentationofDiscrete-timeSignalsinTermsofImpulses.1Decompositionofadiscrete-timesignalintoaweightedsumofshifted/.1Decompositionofadiscrete-timesignalintoaweightedsumofshiftedIfx[n]=u[n],x[n][nkkx[n]x[k][nk]k

TheDiscrete-timeUnitImpulseResponseandtheConvolutionSumRepresentationofLTISystems(1)UnitImpulse(Sample)UnitUnitimpulseTheDiscrete-timeUnitImpulseResponseandtheConvolutionSumRepresentationofLTISystemsUnitImpulse(Sample)UnitUnitimpulseGivenGiventheUnitImpulseResponse:

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ExampleExampleExampleExampleCalculationofConvolutionExampleConvolutionSumofLTIInput:Input:x[n]x[k]kk]Output:y[n]=y[n]x[k]h[nk

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x(t)*h(t)

2.3.12.3.1TheCommutativeDiscrete x[n]＊h[n]=h[n]Continuoustime:x(t)＊h(t) h(t)Howtoh y(t)=x(t)hy(t)=h(t)Example2.7(Calculateitbyyourself!)(SimilartoExample2.4)Example2.8(Calculateitbyyourself!)(SimilartoExample2.5)ExperimentExperimentdemonstrationforConvolutionIntegralTheDistributiveProDiscretex[n]＊{h1[n]+h2[n]}=x[n]＊h1[n]+x[n]＊h2[n]Continuoustime:x(t)＊{h1(t)+h2(t)}=x(t)＊h1(t)+x(t)y(t)=x(t)y(t)=x(t)y(t)=x(t)y(t)=x(t)＊h1(t)+x(t)x(Input:SumofunitOutut:Sumofunitimulseres x[n]=

LTIy[n]=

x(t)

y[n]=x[n]＊ConvolutionIy(t)=+￥x(t)h(t-y(t)=x(t)＊ConvolutionExampleExamplex[n]

(12

u[n]

2nu[n],h[n]

n )u[n]n

x2[n]

y[n]

(1[n

x2[n])*y[n]

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Independentvariablereplace: fix[k],h[n] fih[k]TimeInversalTimeShift: fih[-k] fih[n-k] + Four

fix(t),h(t) Transformationoftinh(t): fih(-t) fih(t- x(t)h(t-t)TheAssociativeProDiscretex[n]＊{h1[n]

Continuous

＊{1(t)

2(t)}={x(t)

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Howy(t)=x(t)＊{h1(t)y(t)=x(t)＊h1(t)h1(t)

Integrating:y(t)=-

LTIsstemwithandwithoutMemorylessDiscretetime: y[n]=kx[n],h[n]=k[n]Continuoustime:y(t)=kx(t),h(t)=k(t)x(t)y(x(t)y(t)kx(t)x(t)k(t)(t)kImplythat:x(t)＊(t)=x(t)andx[n]

y[n]

kx[n]

x[n]k

(t)*h(t)=h(tAd[n-k]*x[n]=Ax[n-k(t-)*h(t)=h(t-C-C-

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x(t)

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ConvergenceConvergenceofu[n]*u[-n]= Notu[n]*1=u[n]*{u[n-1]+u[-n]}=?Notu(t)*u(-t)=?Notu(t)*1=u(t)*{u(t)+u(-t)}=?Not

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2.32.3PropertiesofLTIOutputResponseofLTI y(t)=x(t)Convolutiony(t)=x(t)*h(t)=-￥x(t)h(t-

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2.3.12.3.1TheCommutative x[n]＊h[n]=h[n]Continuoustime:x(t)＊h(t) h(t)Howto y(t)=x(t)y(t)=h(t)x[k]u[nx[k]u[nk]kDiscretetimesystemsatisfythecondition:h[n]=0forn<0Continuoustimesystemsatisfythecondition:y[n] h[n]=0for TheDistributiveProDiscretex[n]＊{h1[n]+h2[n]}=x[n]＊h1[n]+x[n]＊h2[n]Continuoustime:x(t)＊{h1(t)+h2(t)}=x(t)＊h1(t)+x(t)Howtoxxh1y(t)=x(t)＊h1(t)+x(t)2.3.72.3.7StabilityforLTIDefinitionofstability:Everyboundedinputproducesaboundedoutput.Discretetime y[n]

k

x[

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k]

or

k

x[n

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|

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|y[n]|Example1Examplex[n]=(21

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u(t)LTIsstemwithandwithoutMemorylessDiscretetime: Continuoustime:y(t)=kx(t),h(t)=kd(t) forProertiesofLTIDiscrete Continuous(1)x[n]＊h[n]=h[n] x(t)＊h(t) h(t)2xn＊{h1n+h2 x(t)

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Identityht=

x[n][n]

y(t)

x(t)(t)

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x[n]K[n]

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x(t)

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yy[n]x[n][nn]x[nn00y(t)x(t)(tt)x(tt y(t)=x(t-t0yy(t)=x(t)*(t-t0)=x(t-t0 h(t)*h1(t)= =d(t-t0)*1t= =y[n]

x[n]h[n]

x[knkn

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x(t)h(t)

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t

x

x(t)*

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h(t)

x(t)

h(t)

u(t)

'(t)[x(t)

'(t)][h(t)

u(t)]'x(t)'

(t)2.3.6 forLTIsDiscretetimesystemsatisfythecondition:h[n]=0forn<0Continuoustimesystemsatisfythecondition:

h[n]=0forhh(t)u(t)u(th(t)1x(t)110t1t101

y(t)

x( )d2.3.72.3.7StabilityforLTIDefinitionofstability:Everyboundedinputproducesaboundedoutput.Discretetime y[n]=x[k]h[n-k],or,x[n-k]h[kk=- k=-

|y[n]

|

x[n-k]||h[k]|<B

②Usingthepropertiesof(t)

h'(t) 2 011

1 h'(t)

(t)

2 y(t)

x(1)

x(1)

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|

|y[n]|<ConvolutionConvolutionIfIf|x[n]|<B,theconditionfor|y[n]|<A|

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y(t)x(t)h(t)y(kt)x(kt)h(t)y(tt0)x(tt0)h(t)x(t)h(t)y(tx(tt)h(tt)y(t2t000x(kt)h(kt)1y(ktk

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CausalLTISystemsDescribedbyLinearConstant-CoefficientDifferentialandDifferenceNMDiscretetimesystem:DifferenceNMk

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x[nkMk0M

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Determinethesystemresponse(output)recursivelyTypicalLTISystemanditsUnitImpulseDiscrete ContinuousIdentity ht=d Gain y(t)=x(t)* (t)=Time sConditionConditionoInitialResty[n]=x[n]*h[n]=x[k

y(t)=x(t)*h(t)=

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y(n)

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h[k]=h(-1)[n],tth()dt=h-1(t), h(t)=ss(t)=-n

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(t)

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Proofxt)h(t)=x(t)*h(t)*u(t)*d'=[x(t)*'(t)]*[h(t)*u(t)]'=x(t)h(1)(t)'HowHowtogettheHomogeneous y(t)SolvethehomogeneousdifferentialNk Nk

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y(N1)(t)21Systemyy(t)y(t)y*(t)NaturalForcedGeneralresponse=Zero-state+Zero-Input=Generalresponse=Zero-state+Zero-Input=Forced+NaturalConvolutionConvolution.. .. BlockDiagramRepresentationsofFirst-orderSystemsDescribedbyDifferentialandDifferenceEquationDiscretetimesystem:DifferenceNNk

bkMkM

BasicelementsforadiscretetimesystemAnMultiplicationbyaC.Anunit ConvolutionBasicMultiplicationMultiplicationbyaaUnitDiscrete Continuous

y(-t)?x(-t)*h(t)y(kt)?x(kt)*h(t)y(t-t0)=x(t-t0)x(-t)*h(-t)=y(-tx(kt)*h(kt)=1y(ktk x(t-t)*h(t-t)=y(t-2t) y[n]+ay[n-KeKewordsforChaterContinuoustime d

y(t)

(t)

dtk

bk

dtkBasicAnMultiplicationbyaAn(differentiator)CausalLTISystemsDescribedbyLinearConstant-CoefficientDifferentialandDifferenceDiscretetimesystem:Difference k

k

Continuoustimesystem:Differential BasicBasicMultiplicationMultiplicationbyax(t)ax(t)aNNkdxk(ConstantHowtogetoutputoftheLTIsystemlike

y(t)

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bx(t)LinearConstant-CoefficientDifference k

k

RecursiveaNy[n-N]+aN-1y[n-(N-1)]+L+a1y[n-1]+a0 (M-1)]+L+ initialReviewReviewforChaptersystemUnitImpulseConvolutionLTIsystemDescribedbyLinearConstant-CoefficientDifferenceandDifferentialEquation(LCCDE)

(ConditionofInitialBlockDiagramy[-1],……,y[-(N- (NvaluesReviewfory[n]=x[n]Reviewfory[n]=x[n]y(t)=x(t)x[n]x[n]x[k][nkkResponsey[n]x[k]h[nkx(t)x()(ty(t)x()h(tk

x[n-kMk=0 Nonrecursiveh[n]

0￡n￡ FiniteImpulse N≥1:y[n]=k

k

x(t)

y(t)

x(t)*h(t)y[n]

x[n]* (t1,t2

[N1,

N2

(t3,t4

[N3

N4

(t1t3,

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Determinethesystemresponse(output)recursivelyPropertiesPropertiesofLTIDiscrete Continuous(1)x[n]＊h[n]=h[n] x(t)＊h(t) h(t)2xn＊{h1n+h2 x(t)

h(t)=k |h[k]|k|h()|d

h[n]=0forn<0h[n]＊h1[n]=[n]

h(t)=0fort<0h(t)＊h1(t)=(t)ConditionConditionoInitialRestSolutionSolutionforDifferenceRecursiveequationyy[n]x[n]1y[n2Initialcondition

=(ConditionofInitial x(n)

n

y(n)

n(ConditionofInitialx(n)=n<y(n)=x(n)=n<y(n)=n<Supposesystemiscausal.Showthat(*)Suppose(*)holds.ShowthatthesystemisHomeworkHomeworkforChapterPartI(卷積和2.3,2.5,PartII(卷積積分)：2.10(a),Part 2.1,2.25(b),PartIV(差分方程求解 , , 入理解ExampleExampleRecursiveequationInitialcondition

2

y[n-(LTI, x(n)= n<(ConditionofInitial y(n)= n<n< y(n)=n=0 n=1

y[0]=121n= 2M

1)222 2

nn1 y[n]=()n2

Theimpulse

2

nfiniteImpulseResponse Recursive k

k

InfiniteImpulseResponse M x[n-kMk=0FiniteImpulseResponse2.4.22.4.2LinearConstant-CoefficientDifferentialcofficientdiffrential dk

dx

=dtdt

dtkaNy(N)(t)+ y(N-1)(t)+L+a1y'(t)+a0=bMx(M)(t)+ x(M-1)(t)+L+bx'(t)+b0 andinitialy(t0),y’(t0),……,y(N- (NvaluesSolutionoftheN-thorderlin