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 , the rcgioii of coaxer gelice W t b c m thiiik of (1 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 :k] represent t hc cocfficicnts of tlie ,-transform's Latnent series at the