Laplace transform - Continuous and Discrete Time Signals and Systems - page 311

Taking the inverse Laplace transform of the above equation leads to the follow-
ing expression for the overall impulse response:

h c ( t ) = h 1 ( t ) ∗ h 2 ( t ) − h 1 ( t ) ∗ h 3 ( t ) + h 4 ( t ) .

6.11 Summary

In this chapter, we introduced the bilateral and unilateral Laplace transforms
used for the analysis of LTIC signals and systems. The Laplace transforms
are a generalization of the CTFT, where the independent Laplace variable,
s = σ + j ω , can take any value in the complex s-plane and is not simply
restricted to the j ω -axis, as is the case for the CTFT. The values of s for
which the Laplace transforms converge constitute the region of convergence
(ROC) of the Laplace transforms. In Section 6.2, we derived the unilateral
Laplace transforms and the associated ROCs for a number of elementary CT
signals; these transform pairs are listed in Table 6.1. Direct computation of
the inverse Laplace transform involves contour integration, which is difficult
to compute analytically. For Laplace transforms, which take a rational form,
the inverse can be easily determined using the partial fraction approach cov-
ered in Section 6.3. The properties of the Laplace transform are covered in
Section 6.4 and listed in Table 6.2. In particular, we covered the linearity, scaling,
shifting, differentiation, integration, and convolution properties, as summarized
below.

(1) The linearity property implies that the Laplace transform of a linear com-
bination of signals is obtained by taking the same linear combination in the
complex s-domain. In other words,

←→

a 1 x 1 ( t ) + a 2 x 2 ( t ) a 1 X 1 ( s ) + a 2 X 2 ( s ) with ROC: at least R 1 ∩ R 2 .

(2) Scaling a signal by a factor of a in the time domain is equivalent to scaling
its Laplace transform by a factor of 1 / a in the s-domain; i.e.

1
a X

s
a

←→

x ( at ) with ROC: aR .

(3) Shifting a signal in the time domain is equivalent to multiplication by a
complex exponential in the s-domain. Mathematically, the time-shifting
property is expressed as follows:

←→

− st 0 X ( s )

x ( t − t 0 ) e with ROC: R .

(4) The converse of the time-shifting property is also true. In other words,
shifting a signal in the s-domain is equivalent to multiplication by a complex
exponential in the time domain:

←→

e s 0 t x ( t )

X ( s − s 0 ) with ROC: R + Re s 0

.