Page 57 - Schaum's Outline of Theory and Problems of Signals and Systems

SIGNALS AND SYSTEMS [CHAP. 1

(dl Let y,(t) be the output produced by the shifted input x,(t) =x(t -to). Then
y,(t) = T{x(t - to)) =x(t - t,-,)c~s wct
But

Hence, the system is not time-invariant.
(e) Since Icos w,tl s 1, we have

Thus, if the input x(t) is bounded, then the output y(t) is also bounded and the system is
BIB0 stable.

1-35. A system has the input-output relation given by

y = T{x) =x2
Show that this system is nonlinear.

2
T(x, +x2) = (x, +x2) =x: +xS + 2x,x2
2
+ T{x,} +T{x,) =x: +x,
Thus, the system is nonlinear.

1.36. The discrete-time system shown in Fig. 1-36 is known as the unit delay element.
Determine whether the system is (a) memoryless, (b) causal, (c) linear, (d) time-
invariant, or (e) stable.
(a) The system input-output relation is given by

Since the output value at n depends on the input values at n - 1, the system is not
memoryless.
(b) Since the output does not depend on the future input values, the system is causal.
(c) Let .r[n] =cw,x,[n] + a,x2 [n]. Then
y[n] = T{a,x,[n] + a2x2[n]) a,x,[n - 11 + a2x2[n - 11
=
= ~ I Y I + a2y2[nI
I
~
Thus, the superposition property (1.68) is satisfied and the system is linear.
be
(d) Let ~,[n] the response to x,[n] =x[n -no]. Then
yI[n] =T(x,[n]) =x,[n - 11 =x[n - 1 -no]
and y[n -no] =x[n - n, - 11 =x[n - 1 - no] = y,[n]
Hence, the system is time-invariant.

xlnl Unit ylnl = xln-I]
delay b

Fig. 1-36 Unit delay element