Wiley signals and systems e book TLFe BO 408

16.2. Causal Stable L"1-Systeiiis

393

f 16.22)
For

1. < 1 follows t~ < 0.
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 = 1 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: