Page 207 - Schaum's Outline of Theory and Problems of Signals and Systems

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

Setting z = 0 in the above expression, we have

Thus,
z a z
+
Y(z) = - 2 Izl> a
2-a (z-a)
and from Table 4-1 we get

Thus, we obtain the same results as Eq. (2.135).
(b) From Prob. 2.29(b), x[n] and h[nl are given by

From Table 4-1 and Eq. (4.75)

1 z 1
Then Y(z) =X(z) H(z) = - - a<lzl< -
a (z - a)(z - I/a) a
Using partial-fraction expansion, we have

Y( z 1 1 +"-)
-- - -- =-- -
z a (z-a)(z-a) a z-a z-1/a
a 1 a
where C, = -

Thus,
1
1
1
z
Z
Y(z)= -- - a<lzl<-
-
1 -a2 Z-a 1 -a2 Z- 1/a a
and from Table 4-1 we obtain

which is the same as Eq. (2.137).

4.27. Using the z-transform, redo Prob. 2.30.

From Fig. 2-23 and definition (4.3)
~[n] (l,l,l,l] -X(z) = 1 +z-' +z-2z-3
=
h[n] = (l,l, 1) -H(z) = 1 +z-' +z-*