Page 81 - Schaum's Outline of Theory and Problems of Signals and Systems

70 LINEAR TIME-INVARIANT SYSTEMS [CHAP. 2

2.5. Compute the output y(t for a continuous-time LTI system whose impulse response
h(t) and the input x(0 are given by

By Eq. (2.6)
m
y ) ( t ) * h ) = 1 x(r)h(t - r) dr
- m
Functions ~(7) and h(t - 7) are shown in Fig. 2-5(a) for t < 0 and t > 0. From Fig. 2-5(a) we
see that for t < 0, X(T) and h(t - 7) overlap from 7 = - w to 7 = t, while for t > 0, they overlap
from 7 = -01 to 7 = 0. Hence, for t < 0, we have
1
y(r) = j' e'Te-'V-T) dr = e-a'G2a'dr = -eat (2.66~)
- rn 2 a
For t > 0, we have

(4
Fig. 2-5