Page 196 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 41 THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS

4.9. Verify Eq. (4.211, that is,

By definition (4.3)

A pole (or zero) at z = zk in X(z) moves to z = zoz,, and the ROC expands or contracts by
the factor Izol. Thus, we have

4.10. Find the z-transform and the associated ROC for each of the following sequences:

S[n] - all z
(a) From Eq. (4.15)
1

Applying the time-shifting property (4.181, we obtain

(b) From Eq. (4.16)
Z
4.1 IZI> I

Again by the time-shifting property (4.18) we obtain

(c) From Eqs. (4.8) and (4.10)

Z
anu[n] w - Izl> la1
z-a
By Eq. (4.20) we obtain
Z z
an+ 'u[n + I] - z- = - lal< lzl < m (4.73)
z-a z-a

(dl From Eq. (4.16)