Wiley signals and systems e book TLFe BO 330

In Chapter 12 w e got to know cii,
ier transform J,{.c[k]) froin (
xrri for discrt.tc-tinie signals and the Fourier
t r aiisfornr F{r ( f ) }for continuous-i ime signals taii be exp~ewvi,
the rc~lalionship (12.43). For continuous-time signals lioxvcvc1, we also know the
Ltq&,cc trmsform L{,r(t)},wkiich assigns a fiinc-tion X(,s).--or(t) of thc complcx
fiequenq- variahlc s to t,hr tirnc-sigrial .r(t). A c~omparablrt,r;wrsformation for
discrete-(imr) signals is the 2-transform. It is (clearly) iiot named a f t e ~R fariions
I riormally iisrd for i t s tortiplex frequmcy
inathcmatici;.
but, in
v;.~ri;tblr:2 .
Its discussion in this chapter will deal with t l r ~same topics as in Cliap~er4,
w1it.n we tiiscussrti the Laplace transfotorrri. From t lie defiriition of the =-transform,
we first, of all firid tlie rrlationship I.)t>tweenthe ,--transform a n d tlw Foiirirr. transf o ~ m and
.
then the relationship betwc.cn the z-transform arid the Laplace txansform. After that, w r consider coiiwrgcnc.r'. ancl the properlies of the z-transform
and i m m w z-trmn4orm.

The general ciefiriition of the 2-transform call be used with a bilateral sequence
.r[k]whew --xi < k < oc. It, in

I t reprcwnts a scqiierice n [ k ] . whirh inay have comp1t.x ~ l ~ n i c n t bs y, a comp1t.x
i t i n c t ion X(2) iri tlie cornplcx r-plane. The irrfiriite sum in (13.1) usuaIly oiiIy
twnvcrges for certain vahtes of 2 , the rcgioii of coaxer gelice.
W t b c m thiiik of (1 3. I ) in two ways: by coniparison with the Laure
i i ftiiiction of A coinplex nrgiiiricnl, (see (4.15)). we recognise t h a l the> \dues of the
seyimiw 1 :k] represent t hc cocfficicnts of tlie ,-transform's Latnent series at the