Page 205 - Schaum's Outline of Theory and Problems of Signals and Systems

194 THE 2-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS [CHAP.
4

4.24. Find the inverse z-transform of

1 Z
-
-
X(2) = 2 2 lzl> lal
(1 a z ) (z - a)
From Eq. (4.78) (Prob. 4.12)

Now, from Eq. (4.85)

and applying the time-shifting property (4.20) to Eq. (4.86), we get
~[n] (n + l)anu[n + 1] = (n + l)anu[n]
=
since x[-1]=0at n= -1.

SYSTEM FUNCTION

4.25. Using the z-transform, redo Prob. 2.28.

From Prob. 2.28, x[n] and h[n] are given by
x[n] = u[n] h[n] = anu[n] O<a<1
From Table 4-1

L
h[n] = anu[n] - H(z) = - lzl> la1
Z-a
Then, by Eq. (4.40)

Using partial-fraction expansion, we have

where

Thus,