Digital Signal Processing Reference
In-Depth Information
Pierre-Simon, Marquis de Laplace
(born March 23, 1749, Beaumount-en-Auge,
Normandy, France; died March 5, 1827) was a mathematician, astronomer, and physi-
cist who is best known for his investigations into the stability of the solar system. The
Laplace transform was invented by Laplace and systematically developed by the
British physicist Oliver Heaviside (1850-1925), to simplify the solution of many differ-
ential equations that describe physical processes.
References:
“Laplace, Pierre-Simon, marquis de.” Encyclopedia Britannica, 2007.
Encyclopedia Britannica Online, 6 Jan. 2007:
http://www.britannica.com/eb/
article-9047167
“Laplace transform.” Encyclopedia Britannica, 2007. Encyclopedia
Britannica Online, 6 Jan. 2007:
http://www.britannica.com/eb/article-9047168
Oliver Heaviside
(born May 18, 1850, London, England; died February 3, 1925)
became a telegrapher, but increasing deafness forced him to retire in 1874. He then
devoted himself to investigations of electricity. In
Electrical Papers
(1892), he dealt
with theoretical aspects of problems in telegraphy and electrical transmission, making
use of an unusual calculatory method called operational calculus, now better known as
the method of Laplace transforms, to study transient currents in networks. His work
on the theory of the telephone made long-distance service practical.
Reference:
“Heaviside, Oliver.” Encyclopedia Britannica, 2007. Encyclopedia
Britannica, Online, 6 Jan. 2007:
http://www.britannica.com/eb/article-9039747
7.1
DEFINITIONS OF LAPLACE TRANSFORMS
We begin by defining the direct Laplace transform and the inverse Laplace trans-
form. We usually omit the term
direct
and call the direct Laplace transform simply
the Laplace transform. By definition, the (
direct
)
Laplace transform F
(
s
) of a time
function is
f(t)
given by the integral
q
f(t)e
-st
dt,
l
b
[f(t)] = F
b
(s) =
L
(7.1)
-
q
l
b
[
#
]
where indicates the Laplace transform. Definition (7.1) is called the
bilateral
,
or
two-sided
,
Laplace transform
—hence, the subscript
b
. Notice that the bilateral
Laplace transform integral becomes the Fourier transform integral if
s
is replaced by
jv
. The Laplace transform variable is complex,
s = s + jv.
We can rewrite (7.1) as
q
q
f(t)e
- (s+ jv)t
dt =
L
(f(t)e
-st
)e
- jv
dt
F(s) =
L
q
-q
to show that the bilateral Laplace transform of a signal can be interpreted as
the Fourier transform of that signal multiplied by an exponential function
f(t)
e
-st
.
The inverse
Laplace taransform
is given by
c+ j
q
1
2pj
L
f(t) =
l
-1
[F(s)] =
F(s)e
st
ds, j =
1
-1,
(7.2)
c- j
q
