Page 160 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 31 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS

(b) Applying the above property to signal xt(t) = du(t)/dt, we obtain

Note that Eq. (3.46) can be obtained by continued application of the above proce-
dure.

3.33. Verify Eqs. (3.47) and (3.481, that is,
1
(a) L-X(T) d~ ++ -X,(S)
S

(a) Let

Then

Now if

dl) ++G,(s)
then by Eq. (3.44)

X1(s) =sG1(s) -g(O-) =sGI(s)
Thus,

(b) We can write

Note that the first term on the right-hand side is a constant. Thus, taking the unilateral
Laplace transform of the above equation and using Eq. (3.47), we get

3.34. (a) Show that the bilateral Laplace transform of x(t) can be computed from two
unilateral Laplace transforms.
(b) Using the result obtained in part (a), find the bilateral Laplace transform of
e-21rl.