Page 88 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 21 LINEAR TIME-INVARIANT SYSTEMS

(a) Since x2(t) is periodic with period To, we have

x2(t + To - 7) = xZ(t - T)
Then from Eq. (2.70) we have

Thus, f(t) is periodic with period To.
(b) Since both xl(r) and x,(i) are periodic with the same period To, x1(7)x2(t - T) is also
periodic with period To. Then using property (1.88) (Prob. 1.17), we obtain

for an arbitrary a.
(c) We evaluate the periodic convolution graphically. Signals x(r), x(t - T), and x(r)x(t - T)
are sketched in Fig. 2-13(a), from which we obtain
f(t) = I!& - To) 0 < t 5 T0/2 and f(t + To) = f(t)

T0/2 < t I To

which is plotted in Fig. 2-13(b).

PROPERTIES OF CONTINUOUS-TIME LTI SYSTEMS

2.9. The signals in Figs. 2-14(a) and (b) are the input x(t) and the output y(t), respec-
tively, of a certain continuous-time LTI system. Sketch the output to the following
inputs: (a) x(t - 2); (b) ix(t).

(a) Since the system is time-invariant, the output will be y(t - 2) which is sketched in Fig.
2-14(~).
(b) Since the system is linear, the output will be fy(t) which is sketched in Fig. 2-14(d).

2.10. Consider a continuous-time LTI system whose step response is given by
s(t) = ebru(t)

Determine and sketch the output of this system to the input x(t) shown in Fig.
2-15(a).

From Fig. 2-15(a) the input x(t) can be expressed as
x(t) = u(t - 1) - u(t - 3)
Since the system is linear and time-invariant, the output y(t) is given by
y(t)=s(t-1)-s(t-3)
= e-(l- l)u(t - 1) - e-(+3u(t - 3)

which is sketched in Fig. 2-15(b).