Page 99 - Schaum's Outline of Theory and Problems of Signals and Systems

LINEAR TIME-INVARIANT SYSTEMS [CHAP. 2

To find the constant c, substituting Eq. (2.123) into Eq. (2.120), we obtain

Using Eqs. (1.25) and (1.301, the above equation becomes

so that c = 1. Thus, the impulse response is given by
h(t) = e-"'u(t)

2-24 Consider the system in Prob. 2.20 with y(0) = 0.

(a) Find the step response s(t) of the system without using the impulse response
h(t 1.
(b) Find the step response dr) with the impulse response h(t) obtained in Prob.
2.23.
(c) Find the impulse response h(r from s(t ).
(a) In Prob. 2.20

Setting K = 1, b = 0, we obtain x(t) = u(t) and then y(t) = s(t 1. Thus, setting K = 1,
b = 0, and y(0) =yo = 0 in Eq. (2.109), we obtain the step response

(b) Using Eqs. (2.12) and (2.124) in Prob. 2.23, the step response s(t) is given by

which is the same as Eq. (2.125).
(c) Using Eqs. (2.13) and (2.125), the impulse response h(t) is given by

Using Eqs. (1.25) and (1.30), we have

Thus, h(t) = e-"'u(t)
which is the same as Eq. (1.124).