Page 213 - Schaum's Outline of Theory and Problems of Signals and Systems

THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS [CHAP. 4

From Fig. 4-8 we see that

Taking the z-transform of the above equations, we obtain

k
Y(z) = Q(z) + ?z-'Q(z)

Rearranging, we have

from which we obtain

which shows that the system has one zero at z = -k/3 and one pole at z = k/2 and that the
ROC is Jzl> lk/2l. Thus, as shown in Prob. 4.30, the system will be BIB0 stable if the ROC
contains the unit circle, lzl= 1. Hence the system is stable only if IkJ < 2.

UNILATERAL Z-TRANSFORM

4.35. Find the unilateral z-transform of the following x[n]:

(a) x[n] = anu[n]
(b) x[n] = an + 'u[n + 11
(a) Since x[nl = 0 for n < 0, X,(z) = X(z) and from Example 4.1 we have

(b) By definition (4.49) we have

1 az
=a =- Izl > la1
1-az-' z-a

Note that in this case x[n] is not a causal sequence; hence X,(z) + X(z) [see Eq. (4.73) in
Prob. 4.101.