Page 148 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 31 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS 137

and thus, from the convolution property (3.23) we obtain
1
x(t) * ~(t) - -X(S)
5
with the ROC that includes the intersection of the ROC of X(s) and the ROC of the Laplace
transform of u( t 1. Thus,

INVERSE LAPLACE TRANSFORM

3.16. Find the inverse Laplace transform of the following X(s):
1
,
(a) X(s> = - Re(s) > - 1
s+l
1
,
(b) X(s)= - Re(s) < - 1
s+l
S
,
(c) X(s) = - Re(s) > 0
s2+4
s+l
(dl X(s) = , Re(s) > - 1
(s+ 1)'+4
(a) From Table 3-1 we obtain

(b) From Table 3-1 we obtain

~(t) -e-'u(-t)
=
(c) From Table 3-1 we obtain

x(t) = cos2tu(t)

(d) From Table 3-1 we obtain
~(t) e-'cos2tu(t)
=

3.17. Find the inverse Laplace transform of the following X(s):