LAPLACE TRANSFORM FOR
SIGNALS
Basically
Laplace Transform, in mathematics, is an integral transform which takes a real
variable t ( usually time ) to a complex variable s ( frequency ). It is
named after its discoverer Pierre Simon Laplace ( French Mathematician &
Scientist).
Laplace
transform can be defined as a
frequency domain approach for continuous time domain signals irrespective of
the stability of the system.
Consider
a function f (t)
which is to be continuous and defined for all values of t ≥0.
Then
Laplace transform is
That
is, Laplace transform is very much similar to the Fourier transform. But
Fourier transform of a function is complex function of real variable
(frequency) unlike Laplace transform which is a complex function of complex
variable.
If
the lower limit is 0, then the transformation is referred to as one sided or
unilateral Laplace Transform while in the two sided or bilateral transform
lower limit is -∞.
INVERSE LAPLACE TRANSFORM
Inverse Laplace transformation converts a frequency domain function
F(s) to the time domain function f (t).
Inverse Laplace transform is determined as follows
So, we can denote the transformation relationship between f (t) and F[S] as
While analysing problems Laplace transform can be applied easily if
we consider it as two categories,
that is as functional and Operational transforms.
A functional Laplace transform is the Laplace transform for a
specific function such as sin〖θ , t,e^(-at)… 〗,
while an operational transform defines a general mathematical
property of the Laplace transform.
(Note that we can
attain the same result by applying direct equation also.)
FUNCTIONAL LAPLACE
TRANSFORM
A
functional transform is simply the Laplace transform of a specified function of
t. On unilateral Laplace transform of the following functions the pairs
obtained are:
OPERATIONAL LAPLACE
TRANSFORM
Operational
transform indicate how mathematical operations performed on either f (t)
or F[S] are converted into the other.
