Signals and Systems - Electrical Engineering

196 C H A P T E R 3: The Laplace Transform

FIGURE 3.13

Full-wave rectified causal signal.

− 1 0 1 2 3 4 5 6 7

−0.2

0

0.2

0.4

0.6

0.8

1

t

x(

t)

Solution

The first period of the full-wave rectified signal can be expressed as

x 1 (t)=sin( 2 πt)u(t)+sin( 2 π(t−0.5))u(t−0.5)

and its Laplace transform is

X 1 (s)=

2 π( 1 +e−0.5s)

s^2 +( 2 π)^2

And the train of these sinusoidal pulses

x(t)=

∑∞

k= 0

x 1 (t−0.5k)

will then have the following Laplace transform:

X(s)=X 1 (s)[1+e−s/^2 +e−s+···]=X 1 (s)

1

1 −e−s/^2

=

2 π( 1 +e−s/^2 )

( 1 −e−s/^2 )(s^2 + 4 π^2 ) n

3.3.5 Convolution Integral.................................................................

Because this is the most important property of the Laplace transform we provide a more extensive

coverage later, after considering the inverse Laplace transform.

The Laplace transform of the convolution integral of a causal signalx(t), with Laplace transformsX(s), and a

causal impulse responseh(t), with Laplace transformH(s), is given by

L[(x∗h)(t)]=X(s)H(s) (3.16)