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1、DiscreteLTIsystems:theconvolutionsumConsidersignalx[n]:Anarbitrarysequencecanberepresentedasalinearcombinationofshiftedunitimplulses?[n-k],wheretheweightsinthislinearcombinationarex[k].Writeas:……-4-3-2-101234nx[n]DiscreteLTIsystems:theconvolutionsum(s
2、iftingproperty—篩選性)DiscreteLTIsystems:theconvolutionsumLeth[n]denotetheresponseoflinearsystemto?[n].i.e.h[n]—theunitimpulseresponse.then,each?[n]ofx[n]?response:….….DiscreteLTIsystems:theconvolutionsum….….Thisresultisreferredtoastheconvolutionsum,andt
3、heoperationontheright-handsideofEQ.isknownastheconvolutionsumofx[n]andh[n].DiscreteLTIsystems:theconvolutionsumWerepresenttheoperationasy[n]=x[n]?h[n](2.7)Thesame,isreferredtoastheconvolutionsumofx[n]and?[n].Somenotes:AnLTIsystemiscompletelycharacteri
4、zedbyitsh[n].DiscreteLTIsystems:theconvolutionsumThegraphofconvolutionsum.Transformindependentvariable:x[n],h[n]?x[k],h[k]andh[k]?h[-k]Shifth[-k]nsteps?h[n-k].Foranyn,x[k]multipliedbyh[n-k]and?x[k]·h[n-k].DiscreteLTIsystems:theconvolutionsumExample2.2
5、determiney[n]=x[n]?h[n].Answer:(a)DiscreteLTIsystems:theconvolutionsumandh[k]?h[-k](b)Shifth[-k]totheright(n>0)ortotheleft(n<0).n<0,y[n]=0n=0,y[0]=x[0]?h[0]=0.5?1=0.5n=1,y[1]=x[0]h[1]+x[1]h[0]=0.5+2=2.5DiscreteLTIsystems:theconvolutionsum(c)Foranypart
6、icularvalueofn,wemultiplythesetwosignalsandsumoverallvaluesofk.n=2,y[2]=x[0]h[2]+x[1]h[1]=0.5+2=2.5n=3,y[3]=x[1]h[2]=2n?4,y[n]=0DiscreteLTIsystems:theconvolutionsumory[n]={0.5,2.5,2.5,2}n=0,1,2,3,Themainstepsofconvolutionsumgraph:reversalshiftmultiply
7、sum(3)Calculationofconvolutionsumcanbedonebyuprightmultiplication.DiscreteLTIsystems:theconvolutionsumh[n]{0.52}(0)x[n]{111}(0)0.520.520.52y[n]{0.52.52.52}(0)DiscreteLTIsystems:theconvolutionsumExample2.3x[n]andh[n]givenbydeterminey[n].Answer:forn?0(S
8、eeproblem1.54)ContiuousLTIsystems:theconvolutionintegralWebeginbyconsideringapulseor“staircase”approximation,x(t)toacontinuoussignalx(t).IfdefineContiuousLTIsystems:theconvolutionintegralThen,,andtheshadepulseis?,isAs,writtenas……tk?x(k?)Contiu
