Page 58 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 11 SIGNALS AND SYSTEMS

(e) Since

ly[n]I = lx[n - 111 I k if Ix[n]l s k for all n

the system is BIB0 stable.

1.37. Find the input-output relation of the feedback system shown in Fig. 1-37.
,-+y-lT

I Y@]
Unit
delay
I I
I I
I I
I I

From Fig. 1-37 the input to the unit delay element is x[n] - y[n]. Thus, the output y[n] of
the unit delay element is [Eq. (1.111)l

Rearranging, we obtain

Thus the input-output relation of the system is described by a first-order difference equation
with constant coefficients.

1.38. A system has the input-output relation given by

Determine whether the system is (a) memoryless, (b) causal, (c) linear, (d) time-in-
variant, or (e) stable.
(a) Since the output value at n depends on only the input value at n, the system is
memoryless.
(b) Since the output does not depend on the future input values, the system is causal.
(c) Let x[n] = a,x,[nl + azx,[n]. Then

Thus, the superposition property (1.68) is satisfied and the system is linear.