Page 199 - Schaum's Outline of Theory and Problems of Signals and Systems

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

INVERSE Z-TRANSFORM

4.15. Find the inverse z-transform of

X(z) =z2(l - i2-')(1 -2-')(I + 22-7 0 < lzl< 00 (4.79)
Multiplying out the factors of Eq. (4.79), we can express X(z) as
X(Z) =z2+ tz- 3 +z-I
Then, by definition (4.3)

X(z) =x[-2]z2+x[-1]z +x[o] +x[1]z-'
and we get
x[n] = { ..., O,l,$, - 5,1,0 ,... }
T

4.16. Using the power series expansion technique, find the inverse z-transform of the
following X( 2):

(a) Since the ROC is (z(> la(, that is, the exterior of a circle, x[n] is a right-sided sequence.
Thus, we must divide to obtain a series in the power of z-'. Carrying out the long
division, we obtain

Thus,
1
X(z) = = 1 +a~-'+a~z-~+
1 - az-'
and so by definition (4.3) we have
x[n]=O n<O
x[O]=1 x[l]=a x[2]=a2 *..

Thus, we obtain
x[n] = anu[n]
(6) Since the ROC is lzl < lal, that is, the interior of a circle, x[n] is a left-sided sequence.
Thus, we must divide so as to obtain a series in the power of z as follows, Multiplying
both the numerator and denominator of X(z) by z, we have
z
X(z) = -
z-a