Wiley signals and systems e book TLFe BO 186

8.4, Goiivoliit ion 171 Tlie iioriiicll rules governing the regioii of the Laplace trazisforro apply in this ( AYC, and as thev are all give11 in Chapter 1.5.3 we will not repeat tlierii 1ier~ Aii iittei (-'sthg consqiieiice can bc obtained from the coiivolutiovi tlieorein ( U ) , i l w delta iiripidst: is input as ~ j t ) (8.43) As h ( t )M ( t ) = izjt), however, (8.11) TIiis caii be irnrnrtliatr4y confr iried with 1., elective propcrty x C{cT(f)} = &(f) P- fit = I , ROC: =B: I (8.45) the entire ccrinplex plane, as the delta, impulse) The rrAgiori of cmriwrgciwc eiiclo is a signal of firiit,e ditration Just as thc number is the unity clement ol in1dtiplication is the unity eleriieiit of mnvohttiori Figitrc 8.17 illiistratcs this Figitrc #.I7 l,T[-sy tn with a delta irnpulsc a input function With the convohitioii theorem ~e tan d s o easily confirm that e z l is thc~eigeizfunction 0.t LTI-systems 111C3aptc.r we derived (,his p ~ o p ~ r from t y the cuncepts of hiearitty a n d time-invariaiice Convohttion for s(i)= esr gives J -x J - x> (8.46) Tlie response t o an exponciitial function is likewise an exponential luiirtion mnltiplied by t h f hys~cmi l'urictioii N (5) 'I'Cic~ corivoliitioii integral clcarly converges when the coniplex fieqiiciiry of the the inpiit i g r d r ( l ) lies in the rtygoii of couivergcnce of the byst erii lunction