Page 140 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 31 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS

s = -a in the integral in Eq. (3.581, we have

The same result can be obtained by applying L'Hospitalls rule to Eq. (3.58).

3.4. Show that if x(t) is a right-sided signal and X(s) converges for some value of s, then
the ROC of X(s) is of the form

where amax equals the maximum real part of any of the poles of X(s).

Consider a right-sided signal x(t) so that

and X(s) converges for Re(s) = a,. Then

Thus, X(s) converges for Re(s) =a, and the ROC of X(s) is of the form Re($) > a(,. Since the
ROC of X(s) cannot include any poles of X(s), we conclude that it is of the form

Re( s > ~max
where a,,,,, equals the maximum real part of any of the poles of X(s).

3.5. Find the Laplace transform X(s) and sketch the pole-zero plot with the ROC for the
following signals x( t ):
(a) x(t) = e-"u(t) + eP3'u(t)
(b) x(t) = e-"u(t) + e2'u(-t)
(c) x(t) = e2'u(t) + e-3'u(-t)
(a) From Table 3-1