Page 55 - Schaum's Outline of Theory and Problems of Signals and Systems

SIGNALS AND SYSTEMS [CHAP. 1

Differentiating both sides of the above equation with respect to 1, we obtain

Thus. the input-output relationship is described by another first-order linear differential
equation with constant coefficients.

1.33. Consider the capacitor shown in Fig. 1-33. Let input x(t) = i(t) and output y(t) = i;(t).
(a) Find the input-output relationship.
(b) Determine whether the system is (i) memoryless, (ii) causal, (iii) linear, (i~!) time-
invariant, or (L!) stable.
(a) Assume the capacitance C is constant. The output voltage y(r) across the capacitor and
the input current x(t) are related by [Eq. (1.106)]

(b) (i) From Eq. (1.108) it is seen that the output p(r) depends on the past and the
present values of the input. Thus, the system is not memoryless.
(ii) Since the output y(t) does not depend on the future values of the input, the system
is causal.
(iii) Let x(r) = a,x,(O + a2x2(0. Then

Thus, the superposition property (1.68) is satisfied and the system is linear.
(ir) Let y,(r) be the output produced by the shifted input current x,(l) =x(l - f,,).
Then

Hence, the system is time-invariant.

Fig. 1-33