Page 191 - Schaum's Outline of Theory and Problems of Signals and Systems

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

From Eq. (4.3) and given x[n] we have

=5~~+3~-2+4z-~-3z-~
For z not equal to zero or infinity, each term in X(z) will be finite and consequently X(z) will
converge. Note that X(z) includes both positive powers of z and negative powers of z. Thus,
from the result of Prob. 4.2 we conclude that the ROC of X(z) is 0 < lzl < m.

4.4. Consider the sequence

OsnsN-l,a>O
otherwise
Find X(z) and plot the poles and zeros of X(z).

By Eq. (4.3) and using Eq. (1.90), we get
N- I N- I I - (az-~)~ 1 zN-aN
X(Z) = C anz-"= C (az-I)"= =-
1 -az-~ z ~ - z-a (4.55)
~
n=O n=O
From Eq. (4.55) we see that there is a pole of (N - 1)th order at z = 0 and a pole at z =a.
Since x[n] is a finite sequence and is zero for n < 0, the ROC is IzI > 0. The N roots of the
numerator polynomial are at
Zk = aei(2rk/N) k=0,1, ..., N- 1 (4.56)
The root at k = 0 cancels the pole at z = a. The remaining zeros of X(z) are at

The pole-zero plot is shown in Fig. 4-4 with N = 8.

z-plane
.--4b-- .
*
(N - 1)th
order pole ,a' *@. Pole-zero cancel
'\#; /
\

Y I F
I I Re(z)

Fig. 4-4 Pole-zero plot with N = 8.