Page 153 - Schaum's Outline of Theory and Problems of Signals and Systems

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

3.21. Find the inverse Laplace transform of

2 + 2se-" + 4eP4'
Re(s) > - 1
X(s) =
s2 + 4s + 3
We see that X(s) is a sum

where

2 2 s 4
X2(s) = XAs) =
XI($) = s2 + 4s + 3 st + 4s + 3 s2 + 4s + 3
If
xl(f )-Xds) x2(t) #XZ(S) ~3(f -X3(s)
then by the linearity property (3.15) and the time-shifting property (3.16) we obtain

~(t) =xl(t) +x2(t -2) +x3(t-4) (3.85)
Next, using partial-fraction expansions and from Table 3-1, we obtain

3.22. Using the differentiation in s property (3.211, find the inverse Laplace transform of

We have

and from Eq. (3.9) we have
1
e-"u(t) o - Re(s) > -a
s+a
Thus, using the differentiation in s property (3.21), we obtain
X(I) = te-"'u(t)