Wiley signals and systems e book TLFe BO 408

16.2 Causal Stable L"1-Systeiiis 393 f 16.22) For < follows t~ < The stability of the tliscrek systeiri (16.23) is guaranleed if the zc'r'os of the tfmorninator poiynoniial Q ( x ) lie in the imit circle of the z-plane To d i ~ &this ~ j v f ' f o r m the the ratioid fiinction G ( s ) from Q ( z ) iiiirig tht biliiirar transform (16.24) Zeros of Q ( s ) (and so also of ~ ( z )can ) only arise from the nunrerator polynoniial of @(R),so we test w11t.t her the mimerator of G(s) i s a Humit&polynomid lf it is, the zeros rnrrst lie in the left halt of Ihe s-plane and tlw zeros of Q ( z ) nirist lie in the unit chck of' the 2-plane The biliiiear transfor~nation(16.21) in comparison with &hcr traiisfoorrnacions that i m p the area wi&n thc unit rircle onto tlie left, half of the s-planc has the advantage that @(s) is a rational h i c t i t m and we can use the Hurwitz lest, for examplc On closer ~ ~ i ~ ~ Iiowt.vw, ~ ~ ~ouri procedure o I ~ , is still Raw~d:the point x = is o tltc point s = cx) (bee (16.Zfj), so it is not iricluded in the iftirwitz t t o test t h i s point Q(1) well St,y critcrion for a discrete system with trausfcr function I I ( x ) is now: