Signals and Systems - Electrical Engineering

3.3 The One-Sided Laplace Transform 185

o

o

o

o

1 t

− 1

1 s-plane

w(t)

j 4 π

j 2 π

−j 2 π

−j 4 π

jΩ

σ

FIGURE 3.9
The Laplace transform of triangular signal3(t)has as ROC the wholes-plane, since it has no poles but an
infinite number of double zeros at±j 2 πk, fork=±1,±2,....

We will consider next the basic properties of the one-sided Laplace transform—many of these prop-
erties will be encountered in the Fourier analysis, presented in a slightly different form, given the
connection between the Laplace and the Fourier transforms. Something to observe is the duality that
exists between the time and the frequency domains. The time and the frequency domain represen-
tations of continuous-time signals and systems are complementary—that is, certain characteristics of
the signal or the system can be seen better in one domain than in the other. In the following, we
consider the properties of the Laplace transform of signals but they equally apply to the impulse
response of a system.

3.3.1 Linearity

For signalsf(t)andg(t), with Laplace transformsF(s)andG(s), and constantsaandb, we have the Laplace

transform is linear:

L[af(t)u(t)+bg(t)u(t)]=aF(s)+bG(s)

The linearity of the Laplace transform is easily verified using integration properties:

L[af(t)u(t)+bg(t)u(t)]=

∫∞

0

[af(t)+bg(t)]u(t)e−stdt

=a

∫∞

0

f(t)u(t)e−stdt+b

∫∞

0

g(t)u(t)e−stdt

=aL[f(t)u(t)]+bL[g(t)(t)]

We will use the linearity property to illustrate the significance of the location of the poles of the
Laplace transform of causal signals. As seen before, the Laplace transform of an exponential signal