m07 - The Laplace Transform
1.1
2006/08/01 13:01:33.302 GMT-5
2006/09/16 22:44:17.777 GMT-5
C.
Sidney
Burrus
csb@rice.edu
C.
Sidney
Burrus
csb@rice.edu
Doug
Kochelek
kochelek@rice.edu
complex variable
Fourier Transform
Laplace Transform
The Laplace Transform can be viewed as a generalization of the Fourier Transform giving a function of a complex variable. The Laplace Transform converts a linear, constant coefficient differential equation into an algebraic equation
The Laplace Transform
The Laplace transform can be thought of as a generalization of the Fourier
transform in order to include a larger class of functions, to allow the use of
complex variable theory, to solve initial value differential equations, and to
give a tool for input-output description of linear systems. Its use in system
and signal analysis became popular in the 1950's and remains as the central
tool for much of continuous time system theory. The question of convergence
becomes still more complicated and depends on values of
s
used in the inverse transform which must be in a ``region of convergence"
(ROC).
Definition of the Laplace Transform
The definition of the Laplace transform (LT) of a real valued function defined
over all positive time
t
is
F
(
s
)
=
∫
−
∞
∞
f
(
t
)
e
−
s
t
ⅆ
t
and the inverse transform (ILT) is given by the complex contour integral
f
(
t
)
=
1
2
π
j
∫
c
−
j
∞
c
+
j
∞
F
(
s
)
e
s
t
ⅆ
s
where
s
=
σ
+
j
ω
is a complex variable and the path of integration for the ILT must be in the
region of the
s
plane where the Laplace transform integral converges. This definition is often
called the bilateral Laplace transform to distinguish it from the unilateral
transform (ULT) which is defined with zero as the lower limit of the forward
transform integral
(). Unless stated
otherwise, we will be using the bilateral transform.
Notice that the Laplace transform becomes the Fourier transform on the
imaginary axis, for
s
=
j
ω
.
If the ROC includes the
j
ω
axis, the Fourier transform exists but if it does not, only the Laplace
transform of the function exists.
There is a considerable literature on the Laplace transform and its use in
continuous-time system theory. We will develop most of these ideas for the
discrete-time system in terms of the z-transform later in this chapter and
will only briefly consider only the more important properties here.
The unilateral Laplace transform cannot be used if useful parts of the signal
exists for negative time. It does not reduce to the Fourier transform for
signals that exist for negative time, but if the negative time part of a
signal can be neglected, the unilateral transform will converge for a much
larger class of function that the bilateral transform will. It also makes the
solution of initial condition differential equations much easier.
Examples can be found in
and are similar to those of the z-transform presented later in these notes.
Indeed, note the parallals and differences in the Fourier series, Fourier
transform, and Z-transform.
A. Papoulis
The Fourier Integral and Its Applications
McGraw-Hill
1962
R. N. Bracewell
The Fourier Transform and Its Applications
McGraw-Hill
1985
New York
Third