Laplace Transformation-Detailed Study

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.