Question

What is the difference between Laplace Transform and fourier Transform? Laplace converts signal to frequency domain in control theory, How?

Answer 1

Fourier transformation and Laplace transformation are related by the equation as F {f(t)}= L{f(t)} s=iw

Answer 2

lets start the explanation with a brief history and that is laplace was a teacher of fourier.. now the difference between these two transform comes at a small place. it was fourier who had first proposed the fourier transform and then laplace corrected a fault in the fourier equation forming the laplace equation.

now firstly lets come to the equation so as fourier proposed first so the fourier transforms is defined as\[F(\omega)=\int\limits_{-\infty}^{\infty}f(t) \exp^-(j \omega t) dt\]
as u know here omega is the frequency spectrum of the signal. this frequency spectrum consist of magnitude spectrum and phase spectrum of \[\left| F(\omega) \right| \]

Answer 3

and the phase spectrum. now the fault in fourier transform comes when we consider a sinusoidal signal. as the limit is from -infinity to infinity this sinusoidal are not finitely integrable.. so here comes laplace who saves us from the difficulty.
laplace puts forward an equation which is quiet similar to fourier except he adds a variable

\[F(s)= \int\limits_{-\infty}^{\infty} f(t) \exp^- st dt\]
here the important variable is the s

Answer 4

where s is complex number and is defined as \[s=\sigma + j \omega \]
here it is where the beauty lies in the equation. here sigma is the attenuation of the signal and omega the angular frequency in radian. now if u consider the attenuation to be zero and apply s in the laplace equation u will see u get back the fourier transform expression. this attenuation factor is what is included by laplace and due to which we get a finite integration of any function... and also laplace is used to convert complex differential equations into algebric equations and this transform performed are more better than fourier