Page 156 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 31 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS

Hence, the system function H(s) is

Rewriting H(s) as

and taking the inverse Laplace transform of H(s), we have

Note that h(t) is equal to the derivative of 2e-"dl) which is the step response s(r) of
the system [see Eq. (2.1311.
I
- Re(s)> -1
x(t) =e-'dt) ++
s+l
Thus,

Using partial-fraction expansions, we get

Taking the inverse Laplace transform of Y(s), we obtain
y(t) = (-e-' + 3e-")u(r)

3.26. If a continuous-time LTI system is BIBO stable, then show that the ROC of its system
function H(s) must contain the imaginary axis, that is, s = jo.
A continuous-time LTI system is BIBO stable if and only if its impulse response h(t) is
absolutely integrable, that is [Eq. (2.2111,

By Eq. (3.3)

Let s = jw. Then

Therefore, we see that if the system is stable, then H(s) converges for s = jo. That is, for a
stable continuous-time LTI system, the ROC of H(s) must contain the imaginary axis s = jw.