Wiley signals and systems e book TLFe BO 342

Tlreoreins of tlie :-'l'ransform 327 of the Laplace trarrsforrn see Cliaptrr 4 7.. It slioulrl be many properties ol: tlie 1,aplac.e transform also apply to tlic x-transform.. The most i

13.4 Tlreoreins of tlie :-'l'ransform 327

of the Laplace trarrsforrn (see Cliaptrr 4 7 ) T h a t is no coincideticcl ab we saw

in Section 13.3.2 that the ansforni of a series can be thotiglit of as :

( w i t inuous-timc signal con g pmely of delta impulses It slioulrl be

many properties ol: tlie 1,aplac.e transform also apply to tlic x-transform

The most impor tarit tlteorrrri:, of the ;-tuansforrr~ arc snnimarised in Table 13.1 They cair i)ISo be sliowri without rcferrirrg to the Laplace transform by iiiscrting them irrto (13 1) In contrast to tlic tlieoiems 01 the Laplace transform the intlc pcnc-lei~t variable i s deliiied orrly for iiitegei v a l u c ~ In thc square brackets [ 1, no

s h i f t s /t E- Z &re allowed As for the similarity tlieorczn (4.241, for samplt~tl signals

t i w orily scaling of the time axis that i:, pe'rinitted i s cx = -1, so that bccomes the time reversal theorcm for the c-traiisforni Reversing tlie index of a series of values t m i be dorre simply by w d i i i g thcin backwards

rc pclrmit tecl and corrcspondirigly for the shift thcorcm, only in

Shift,

Mult iplicatiorr

by k:

Time :eversal

Tirne-dorrraiii z-domain

x X ( x ) 1- b Y ( z ) R W > RC>C{.cJ? ROC(y)