Network Analysis and Synthesis, 1st Edition by Pearson

272 Network Analysis and Synthesis

The introduction of a convergence factor into the Fourier transform creates a new transfor-

mation, which is called Laplace transform. Thus, the Laplace transform is defined as in the

following:

Fs ftee dt fte

tj

tj

t

() () ()

()

==

∫∫

−− −+

∞

00

∞

sw sw

It is to be noted that the lower limit of the integral has been taken as 0 instead of −∞. This is because

with s > 0, the convergence factor e

−s t

will diverge rather than converge when t tends to −∞.

If we substitute s = s + jw, then

Fs ft

ed

t

st

() ()=

∞

−

∫

0

F(s) is called the Laplace transform of f (t).

Laplace transform is a mathematical tool to convert the function from time domain to frequency

domain. Let f(t) be any time domain function, then its Laplace transform is ...