CHAP. 11
(e)
SIGNALS AND SYSTEMS
Since l y [ n ] I = l x [ n - 111 I k
if I x [ 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 Unit delay
I I I I
I
Y@]
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-invariant, 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 , [ n l + a z x , [ n ] . Then
Thus, the superposition property (1.68) is satisfied and the system is linear.