Page 135 - Schaum's Outline of Theory and Problems of Signals and Systems

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

(b)
Fig. 3-9 Two systems in parallel. (a) Time-domain representation; Ib) s-domain representation.

3.7 THE UNILATERAL LAPLACE TRANSFORM
A. Definitions:

The unilateral (or one-sided) Laplace transform X,(s) of a signal x(t) is defined as
[Eq. (3.5)l

The lower limit of integration is chosen to be 0- (rather than 0 or O+) to permit x(t) to
include S(t) or its derivatives. Thus, we note immediately that the integration from 0- to
O+ is zero except when there is an impulse function or its derivative at the origin. The
unilateral Laplace transform ignores x(t) for t < 0. Since x(t) in Eq. (3.43) is a right-sided
signal, the ROC of X,(s) is always of the form Re(s) > u,,, that is, a right half-plane in
the s-plane.

B. Basic Properties:

Most of the properties of the unilateral Laplace transform are the same as for the
bilateral transform. The unilateral Laplace transform is useful for calculating the response
of a causal system to a causal input when the system is described by a linear constant-
coefficient differential equation with nonzero initial conditions. The basic properties of the
unilateral Laplace transform that are useful in this application are the time-differentiation
and time-integration properties which are different from those of the bilateral transform.
They are presented in the following.