Wiley signals and systems e book TLFe BO 342

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-timcsignal 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~tvariable i s deliiied orrly for iiitegei valuc~.In thc square brackets [ 1, no
rc pclrmit tecl. and corrcspondirigly. for the shift thcorcm, only in
shifts /t E- Z &reallowed. As for the similarity tlieorczn (4.241, for samplt~tlsignals
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.

Tirne-dorrraiii

z-domain

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

Shift,

Mult iplicatiorr
by k:

Time :eversal