Page 210 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 41 THE 2-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS 199

If a discrete-time LTI system is BIBO stable, show that the ROC of its system function
H(z) must contain the unit circle, that is, lzl = 1.

A discrete-time LTI system is BIBO stable if and only if its impulse response h[nl is
absolutely summable, that is [Eq. (2.49)1,

Now

Let z = ejR so that lzl = lejRI = 1. Then

Therefore, we see that if the system is stable, then H(z) converges for z = ei". That is, for a
stable discrete-time LTI system, the ROC of H(z) must contain the unit circle lzl = 1.

Using the z-transform, redo Prob. 2.38.
(a) From Prob. 2.38 the impulse response of the system is

Then

Since the ROC of H(z) is Izl > IaI, z = oo is included. Thus, by the result from Prob. 4.5
we conclude that h[n] is a causal sequence. Thus, the system is causal.
(b) If la1 > 1, the ROC of H(z) does not contain the unit circle lzl= 1, and hence the system
will not be stable. If la1 < 1, the ROC of H(z) contains the unit circle lzl = 1, and hence
the system will be stable.

A causal discrete-time LTI system is described by
y[n] - ;y[n - 11 + iy[n - 21 =x[n] (4.88)

where x[n] and y[n] are the input and output of the system, respectively.

(a) Determine the system function H(z).
(b) Find the impulse response h[n] of the system.
(c) Find the step response s[n] of the system.

(a) Taking the z-transform of Eq. (4.88), we obtain
Y(2) - $z-'~(z) + ;z-~Y(z) =X(z)