Page 133 - Schaum's Outline of Theory and Problems of Signals and Systems

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

I. Causality:
For a causal continuous-time LTI system, we have

h(t) = 0 t<O

Since h(t) is a right-sided signal, the corresponding requirement on H(s) is that the ROC
of H(s) must be of the form

ReW > amax
That is, the ROC is the region in the s-plane to the right of all of the system poles.
Similarly, if the system is anticausal, then
h(t) = 0 t>O

and h(t) is left-sided. Thus, the ROC of H(s) must be of the form

Re( s ) < %in
That is, the ROC is the region in the s-plane to the left of all of the system poles.
2. Stabilio:

In Sec. 2.3 we stated that a continuous-time LTI system is BIB0 stable if and only if
[Eq. (2-2111

The corresponding requirement on H(s) is that the ROC of H(s) contains the jw-axis
(that is, s = jw) (Prob. 3.26).
3. Causal and Stable Systems:

If the system is both causal and stable, then all the poles of H(s) must lie in the left
half of the s-plane; that is, they all have negative real parts because the ROC is of the
form Re(s) >amax, and since the jo axis is included in the ROC, we must have a,, < 0.

C. System Function for LTI Systems Described by Linear Constant-Coefficient Differential
Equations:
In Sec. 2.5 we considered a continuous-time LTI system for which input x(t) and
output y(t) satisfy the general linear constant-coefficient differential equation of the form

Applying the Laplace transform and using the differentiation property (3.20) of the
Laplace transform, we obtain