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581
CHAPTER
32
The Laplace Transform
The two main techniques in signal processing, convolution and Fourier analysis, teach that a
linear system can be completely understood from its impulse or frequency response.
This is a
very generalized approach, since the impulse and frequency responses can be of nearly any shape
or form.
In fact, it is
too
general for many applications in science and engineering.
Many of the
parameters in our universe interact through
differential equations
.
For example, the voltage
across an inductor is proportional to the derivative of the current through the device.
Likewise,
the force applied to a mass is proportional to the derivative of its velocity.
Physics is filled with
these kinds of relations.
The frequency and impulse responses of these systems cannot be
arbitrary, but must be consistent with the solution of these differential equations.
This means that
their impulse responses can only consist of
exponentials
and
sinusoids
.
The Laplace transform
is a technique for analyzing these special systems when the signals are
continuous
.
The z-
transform is a similar technique used in the
discrete
case.
The Nature of the s-Domain
The Laplace transform is a well established mathematical technique for solving
differential equations.
It is named in honor of the great French mathematician,
Pierre Simon De Laplace (1749-1827).
Like all transforms, the Laplace
transform changes one signal into another according to some fixed set of rules
or equations.
As illustrated in Fig. 32-1, the Laplace transform changes a
signal in the time domain into a signal in the
s-domain
, also called the
s-
plane
.
The time domain signal is continuous, extends to both positive and
negative infinity, and may be either periodic or aperiodic.
The Laplace
transform allows the time domain to be
complex
; however, this is seldom
needed in signal processing.
In this discussion, and nearly all practical
applications, the time domain signal is completely real.
As shown in Fig. 32-1, the s-domain is a complex plane, i.e., there are real
numbers along the horizontal axis and imaginary numbers along the vertical
axis.
The distance along the real axis is expressed by the variable,
F
, a lower

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582
X
(
T
)
’
m
4
&
4
x
(
t
)
e
&
j
T
t
dt
X
(
F
,
T
)
’
m
4
&
4
[
x
(
t
)
e
&
F
t
]
e
&
j
T
t
case Greek sigma.
Likewise, the imaginary axis uses the variable,
T
, the
natural frequency.
This coordinate system allows the location of any point to
be specified by providing values for
F
and
T
.
Using complex notation, each
location is represented by the complex variable,
s
, where:
.
Just as
s
’
F
%
j
T
with the Fourier transform, signals in the s-domain are represented by capital
letters.
For example, a time domain signal,
, is transformed into an s-
x
(
t
)
domain signal,
, or alternatively,
.
The s-plane is continuous, and
X
(
s
)
X
(
F
,
T
)
extends to infinity in all four directions.

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