信號(hào)與系統(tǒng)-chapter2-第3次課

卷積的(Output:Sumofunitimpulse(Output:Sumofunitimpulsex[n]x[k][nkkky[n]=x[n]＊y[n]x[k]h[nk

x(t)x()(t

y(t)

y(t)=x(t)＊(LTI系(LTI系統(tǒng)的輸出響應(yīng)=輸入信號(hào)*系統(tǒng)的單位沖激響應(yīng)CalculationofConvolutionSum/ConvolutionIndependentvariablereplace:x[n]x[k],h[n]h[k]TimeInversalTimeShift:Four積

h[k]h[‐k]Multiplication:的

y[n]x[k]h[nk] Independentvariable

kFour

x(t)x(),h(t)h()Transformationofinh():h()h(‐)h(t‐

x()h(t‐ty(t)x()h(tt(單位沖激信號(hào)*任一信號(hào)=信號(hào)本身[n](單位沖激信號(hào)*任一信號(hào)=信號(hào)本身[n]h[n]A[nk]h[n]Ah[nk

A[nk]x[n]Ax[nk(時(shí)移沖激信號(hào)*任一信號(hào)=信號(hào)的時(shí)移(t(時(shí)移沖激信號(hào)*任一信號(hào)=信號(hào)的時(shí)移(t )h(t)h(t (t)x(t)x(t(t)x(t)x(tConvergenceof

u[n]u[n] Notu[n]1u[n]{u[n1]u[n]} Not

u(t)1u(t){u(t)u(t)1u(t){u(t)u(t)}?NotPropertiesofLTIOutputResponseofLTI y[n]=x[n]y[n]x[n]*h[n] x[k]h[nkk y(t)=x(t)y(t)x(t)*h(t)x()h(tOnlyforLTI(1)Itsoutputresponsecanbeobtainedbyconvolution(2)ItcanbeycharacterizedbyitsUnitImpulseTheCommutativeDiscrete x[n]＊h[n]=h[n]Continuoustime:x(t)＊h(t) h(t)Howtoh y(t)=x(t)hy(t)=h(t)TheDistributive DiscreteDiscretex[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)Howtoy(t)=x(t)y(t)=x(t)xx[n]

nu[n]2nu[n],h[n]2

x[n] 1 x[n]2n小結(jié)

(2) y[n]([nx2[n])*y[n]y1[n]y2[n]Example Exampley1[n]x1[n]*h[n]y2[n]x2[n]*LTI系統(tǒng)對(duì)復(fù)雜和信號(hào)的響應(yīng)是各個(gè)復(fù)雜并聯(lián)系統(tǒng)（LTI）的響應(yīng)是

DiscreteDiscretex[n]＊{h1[n]＊h2[n]}={x[n]＊h1[n]}＊h2[n]Continuoustime:x(t)＊{h1(t)＊h2(t)}={x(t)＊h1(t)}HowtoProvey(t)=x(t)＊{h1(t)y(t)=x(t)＊h1(t)h1(t)LTIsstemwithandwithoutMemorylessDiscretetime: y[n]=kx[n],h[n]=k[n]Continuoustime:y(t)=kx(t),h(t)=k(t)x(t)y(tx(t)y(t)kx(t)x(t)kkImplythat:x(t)＊(t)=x(t)andx[n]k

y[n]kx[n]x[n]k ofLTIsOriginalsystem:h(t)ReverseOriginalsystem:h(t)Reversesystem:h1(t)x(t)x(t)h(t)＊h1(t)=(t)orh[n]Example2.11(時(shí)移系統(tǒng))Example2.12(累加器)Example(積分器) forLTIsDiscreteDiscretetimesystemsatisfythecondition:h[n]=0forn<0Continuoustimesystemsatisfythecondition:h(t)=0fort<0StabilityforLTIDefinitionofstability:Everyboundedinputproducesaboundedoutput.Discretetime y[n] x[k]h[nk],or,x[nk]h[kk k kk|h[k]||y[n]||y[n]||xkk|h[k]||y[n]|

kIf|x[n]|<B,theconditionfor|y[n]|<Ak

|h[k]|

isstableContinuous‐time y(t)x()h(t)d,or,x(t)h( |y(t)||x(t)||h()|dB|h()|d |h()|d , |y(t)|AIf|x(t)|<B,theconditionfor|y(t)|<A

|h()|d

isstableExamplePuretimeshifty(t)x(tt0

h[n][nn0h(t)(tt0Note:(2.89)shouldnotbecommonlyn

y[n]x[k] h[n]kt

y(t)

h(t)u(t)SummaryforPropertieso