Laplace & Fourier: When to Use?

[amaresh92]

greetings,

Can anyone tell me when we should use Laplace transform and Fourier transform? It seems both of them are equal except σ .

thanks in advanced.

 

[Baluncore]

The Laplace transform resolves a function into moments in the complex plane. It is best used in studies that involve the poles and zeros of a response, such as the exponential responses to an impulse.

The Fourier transform resolves a function into oscillations in the frequency domain. It is best used for the study of harmonic content and phase in repeating signals.

Laplace maps to the complex plane while Fourier maps to the frequency domain. You use the one that takes you from the time domain to your preferred domain for analysis or manipulation.

 

[rbj]

Balun, you can just as well (or better) say that "Laplace maps to the entire complex plane while Fourier maps to the j \omega axis of the complex plane."

it is true that the double-sided Laplace transform is identical to the continuous Fourier transform with the substitution that s = j \omega.

there are some signals that, without handwaving, are hard to F.T. while they are easy to L.T. e.g. the unit step function, u(t). there are issues of convergence and the conditions to get either the Laplace integral or the Fourier integral to converge to a finite result. but where these issues are identical for both the F.T. and L.T. (such as for x(t) = e^{-\alpha t} u(t)), then the transformed result must be the same between the F.T. and L.T. with the substitution of s = j \omega.

a similar relationship exists between the Z Transform and the Discrete-Time Fourier Transform. one exists in the entire complex z plane while the other exists on the unit circle of the same z plane, where z = e^{j \omega}.

there is a common mathematical underpinning to all of these transforms. it depends on whether your signal in the time or frequency domain is discrete or not. whether you can get the integral to converge or not. whether you define your signals and linear systems to exist before t=0 or not (that affects the double or single-sidedness of the integral or summation).

 

[Baluncore]

rbj; I can agree with everything you write.
I am not a mathematician so it is mostly beyond me.
After three days, amaresh92's question deserved a reply.
So I threw in an answer in the hope that it would stimulate some response.
Now 4 days later, I am enlightened by your interesting reply. Thank you.