Page 139 - Schaum's Outline of Theory and Problems of Signals and Systems

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

Thus, we obtain
1
ealu( - t) H - - Re(s) <a
s-a

3.2. A finite-duration signal x(t) is defined as
t, It It,
= 0 otherwise

where I, and I, are finite values. Show that if X(s) converges for at least one value of
s, then the ROC of X(s) is the entire s-plane.
Assume that X(s) converges at s = a,; then by Eq. (3.3)

Since (u, - 0,) > 0, e-(ul-"~)l is a decaying exponential. Then over the interval where x(t) + 0,
the maximum value of this exponential is e-("l-"o)'l, and we can write

Thus, X(s) converges for Re(s) = a, > u,,. By a similar argument, if a, < u,, then

/I2 ( " 1 1 dl < e(w~-~~)l~ l''l~(r)le-~~'dt <m (3.57)
' 1 'I
and again X(s) converges for Re(s) = u, <u,. Thus, the ROC of X(s) includes the entire
s-plane.

3.3. Let

OItlT
~(t)
=
otherwise
Find the Laplace transform of x(t).
By Eq. (3.3)

- - - [ 1 -e-(s+u)T~ ( 3.58)
-
e
s+a , s+a
Since x(f is a finite-duration signal, the ROC of X(s) is the entire s-plane. Note that from Eq.
(3.58) it appears that X(s) does not converge at s = -a. But this is not the case. Setting