Page 208 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 41 THE z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS

Thus, by the convolution property (4.26)
Y(Z) =x(z)H(z) =(I +Z-'+Z-~+Z-~)(I +Z-'+Z-~)
+
= 1 + 2 ~ 4 + 3 ~ + - ~ + 24
Hence,
h[nl= {1,2,3,3,2,1)

which is the same result obtained in Prob. 2.30.

4.28. Using the z-transform, redo Prob. 2.32.

Let x[nl and y[nl be the input and output of the system. Then
Z
x[n] = u[n] -X(z) = - lzl > 1
2- 1

Then, by Eq. (4.41)

Using partial-fraction expansion, we have

1 a-1 1 -a
where c, = - C2 = -
a
Thus,
z
1
1-a
H(z)= -- -- lzl> a
a a Z-a
Taking the inverse z-transform of H(z), we obtain

When n = 0,

Then

Thus, h[n] can be rewritten as

which is the same result obtained in Prob. 2.32.