Page 149 - Schaum's Outline of Theory and Problems of Signals and Systems

LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS [CHAP. 3

Expanding by partial fractions, we have

Using Eq. (3.30), we obtain

Hence,

(a) The ROC of X(s) is Re(s) > - 1. Thus, x(t) is a right-sided signal and from Table 3-1 we
obtain
x(t) = eP'u(t) + e-3'~(t) = (e-' + e-3')~(t)

(b) The ROC of X(s) is Re(s) < -3. Thus, x(t) is a left-sided signal and from Table 3-1 we
obtain
x(t) = -e-'u(-t) - eC3'u( -1) = -(e-' +e-3')u( -1)

(c) The ROC of X(s) is -3 < Re(s) < - 1. Thus, x(t) is a double-sided signal and from
Table 3-1 we obtain

3.18. Find the inverse Laplace transform of

We can write

Then

where