Page 212 - Schaum's Outline of Theory and Problems of Signals and Systems

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

4.33. Using the z-transform, redo Prob. 2.41.
As in Prob. 2.41, from Fig. 2-30 we see that
q[n] = 2q[n - 11 +x[n]

~[nl= s[nl+ 34[n - 11
Taking the z-transform of the above equations, we get
Q(z) = 22-'Q(z) + X(z)

Y(z) = Q(z) + ~z-'Q(z)
Rearranging, we get
(1 - 22-')Q(L) =X(z)

(1 + 3.2-')Q(z) = Y(z)
from which we obtain

Rewriting Eq. (4.89), we have
(1 - 22-')Y(2) = (1 + 32-')x(~)
or

Y(z) - 22-'Y(z) =X(z) + 32-'X(z) (4.90)
Taking the inverse z-transform of Eq. (4.90) and using the time-shifting property (4.181, we
obtain
y[n] - 2y[n - 11 =x[n] + 3x[n - 11
which is the same as Eq. (2.148).

4.34. Consider the discrete-time system shown in Fig. 4-8. For what values of k is the
system BIB0 stable?