A c~omparablr t,r;wrsformation for discrete- imr signals is the 2-transform.. Its discussion in this chapter will deal with t l r ~ same topics as in Cliap~er 4, w1it.n we tiiscussrti t
Trang 1In Chapter 12 w e got to know cii,
ier transform J,{.c[k]) froin ( xrri for discrt.tc-tinie signals and th e Fourier
t r aiisfornr F{ r ( f ) } for continuous-i ime signals taii be exp~ewvi,
the rc~lal ionship (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~omparablr t,r;wrsformation for discrete-( imr) signals is the 2-transform It is (clearly) iiot named a f t e ~ R fariions inathcmatici; but, in I riormally iisrd for i t s tortiplex frequmcy v;.~ri;tblr: 2
Its discussion in this chapter will deal with t l r ~ same topics as in Cliap~er 4, w1it.n we tiiscussrti the Laplace transfotorrri From t lie defiriition of the =-transform,
we first, of all firid tlie rrlationship I.)t>tween the , transform a n d tlw Foiirirr trans-
f o ~ m and then the relationship betwc.cn the z-transform arid the Laplace txans- form After that, w r consider coiiwrgcnc.r' ancl the properlies of the z-transform and i m m w z-trmn4orm
T h e 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 s , b 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
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
W t b c m thiiik of (1 3 I ) in two ways: b y coniparison with the Laure