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# how are laplace and fourier transforms related?

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#### treehugger

##### Member level 2
the subject makes itself clear. i need a thorough discussion of how these 2 important transformations are related. when do we use laplace transform when do we use fourier?

regards.

Laplace will converge on more functions because of the added e^-st term.

LAPLACE you use when you have an initial state in your system. It covers both transient and steady state responses.

FOURIER you use when you care only for the steady state.

This is a great topic which we have talked about on and on.

Look at it. Everybody was getting excited and .... we ended up drawing pistols ...

The link to the discussion by steve10 is really interesting.I was about to post my thoughts but thought that the forum link would answer much better

the factor s=σ+jω... ie σ refers to attenuation and ω stands for angular frequency..
if basis function used is e**(-st) then it is laplace transform..if transform relinqushes attenuation(a nondamping system) then basis function becomes
e**-jωt.. this is called as fourier transform..

Hi,
To visualise that look at the attached figure, the RHS is Laplace transform , by putting sigma = 0; we will get Fourier transform.
Regards,

### treehugger

Points: 2
thuvu said:
LAPLACE you use when you have an initial state in your system. It covers both transient and steady state responses.

FOURIER you use when you care only for the steady state.

i dont think this distinction is true.

how can you justify that Fourier transform cannot deal with transients?

tantoun2004 said:
Hi,
To visualise that look at the attached figure, the RHS is Laplace transform , by putting sigma = 0; we will get Fourier transform.
Regards,

dear tantoun2004 i could not make any significant inference from your image that you added..could you plz explain it..so that will usefull for me..

Dear electronics_kumar,
When u get Laplace Transform of a fn as a function in s , You can substitute
s=sigma + j*omega.
and then plot this as a function of both sigma and omega which is shown as the surface on the RHS. If you put sigma =0 (cutting the surface by a plane) I get Fourier Transform of the function...This is of course only valid if we use Laplace Transform by integrating from -infinity to infinity (Generalised case),
or considering functions which are zero for t<0 (such as causal systems impulse response).
So we imagine that Laplace Transform is a generalised Fourier Transform in which poles and zeros are shown...
https://www.dspguide.com/
chapter 32
Best wishes,

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