Page 200 - Schaum's Outline of Theory and Problems of Signals and Systems

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

and carrying out the long division, we obtain
. -
-a-'z -a-2z2-a-3z3-

Thus,

and so by definition (4.3) we have
x[n]=O n20
x[- 11 = -a-1 x[-2] = -a-2 x[-'j] = -0-3
Thus, we get

x[n] = -anu[-n - I]

4.17. Find the inverse z-transform of the following X(z):

(a) The power series expansion for log(1 - r) is given by

Now
1
X(z) = log -az-,) = -lo@ -afl) Izl> lal

Since the ROC is lzl> lal, that is, laz-'I< 1, by Eq. (4.80), X(z) has the power sel
expansion

from which we can identify x[n] as