Page 56 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 11 SIGNALS AND SYSTEMS

0
Fig. 1-34 Unit ramp function.

(c) Let x(t) = klu(t), with k, # 0. Then

where r(t) = tu(t) is known as the unit ramp function (Fig. 1-34). Since y(t) grows
linearly in time without bound, the system is not BIB0 stable.

1.34. Consider the system shown in Fig. 1-35. Determine whether it is (a) memoryless, (b)
causal, (c) linear, (d) time-invariant, or (e) stable.

From Fig. 1-35 we have
y(t) = T{x(t)} =x(t) cos w,t
Since the value of the output y(t) depends on only the present values of the input x(t),
the system is memoryless.
Since the output y(t) does not depend on the future values of the input x(t), the system
is causal.
Let x(t) = a,x(t) + a2x(t). Then

y(t) = T(x(t)} = [a,xl(t) + a2x2(t)] cos w,t
= a,x,(t) cos w,t + a2x2(t) cos w,t
= "l~,(t) +ff2~2(t)
Thus, the superposition property (1.68) is satisfied and the system is linear.

Fig. 1-35