Page 201 - Schaum's Outline of Theory and Problems of Signals and Systems

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

Since the ROC is JzJ < lal, that is, la-'zl < 1, by Eq. (4.801, X(z) has the power series
expansion

from which we can identify x[n] as

4.18. Using the power series expansion technique, find the inverse z-transform of the
following X( z):

Z 1
(a) X(z)= IzI< -
2z2 - 3z + 1 2

(a) Since the ROC is (zl < i, x[n] is a left-sided sequence. Thus, we must divide to obtain a
series in power of z. Carrying out the long division, we obtain
z + 3z2 + 7z3 + 15z4 + - . .

15z4 -
Thus,

and so by definition (4.3) we obtain

(b) Since the ROC is lzl> 1, x[n] is a right-sided sequence. Thus, we must divide so as to
obtain a series in power of z- ' as follows:

Thus,