Wiley signals and systems e book TLFe BO 185

Figuie 8.16: A systcai in the t imr-tlorrraia (8.40) I h(t) ? X(t) IZ(s).Y(s) c ROC(lz * J ) RO('{.c}fl ROC{h} ~ _ i (8.41) _ It is riot oirly tiiie wlwn h ( t ) is a11 irnpilse rrsponse and biit also gciierally for all pairs of functions ot tirrir wlr The rcgion of ronvcrgt~iceof the coiivolutiorr piotluct tcdiswtion between the iegioiis of coiivergcii~eRUC{ rcgionr oi ( ' o i i ~ r r g e ~miist i c ~ overlap for the Lsplac~.ti product to exist In (8.401 UT('Iixd set oiit the coiitlitio variable s can oiilv take talurs f o r wliirh both X ( s ) aiid H ( s) exist Tlie region ol corixergcenc-r ROC{lz * r} can, howevei actiixllj he I x g w than the intcrscctiorr of RC>C(x) and ROC{h) if, for rxainplc, poles are cancelled oixt by wlmr C { h ( t )* r(t)} does irot exist, it rnc"aris the convoliition prodiict (8.38) t l o ~ sriot rxist itself Expressioii (8.38) is in a r i improper iiitcgral tliet only corivcrgrs iintler ccI tain conditions Thc JtOC{h} of the systeni fiiwtiori U ( s )that we h a w bo far not paid niricli erition to, ran br derivc~lfrom the corinectiori between the system fuiict ion niid impulse r~spoiise H(.s) = L{rh(l)}* (8.42j