Page 129 - Schaum's Outline of Theory and Problems of Signals and Systems

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

then x(-t) *X(-S) R' = -R (3.19)

'Thus, time reversal of x(t) produces a reversal of both the a- and jw-axes in the s-plane.
Equation (3.19) is readily obtained by setting a = - 1 in Eq. (3.18).

F. Differentiation in the Time Domain:

If

~(t) ++X(S) ROC = R

then

Equation (3.20) shows that the effect of differentiation in the time domain is multiplication
of the corresponding Laplace transform by s. The associated ROC is unchanged unless
there is a pole-zero cancellation at s = 0.

G. Differentiation in the s-Domain:
If

41) ++X(S) ROC = R

then -

H. Integration in the Time Domain:
If

then

Equation (3.22) shows that the Laplace transform operation corresponding to time-domain
integration is multiplication by l/s, and this is expected since integration is the inverse
operation of differentiation. The form of R' follows from the possible introduction of an
additional pole at s = 0 by the multiplication by l/s.

I. Convolution:
1 f

xdl) *w) ROC= R,

~2(4 ++XZ(S) ROC = R2