Tying Convolution with Laplace and Fourier Transforms

[jp1390]

Hi, this post is really to put my understanding to a test as I am not fully confident with my conclusions, so if anyone can point me in the correct direction that would be awesome!

Convolution: the resulting steady-state output of an input passed into a LTI system.

Laplace Transform: general form of Fourier transform, s = σ + jw. Displays frequency spectrum with decay (σ ≠ 0)

Fourier Transform: special case of Laplace transform, s = jw. Non-periodic and periodic signals. Displays frequency spectrum without decay (σ = 0). Phasors?

Fourier Series: Displays frequency spectrum of period signals

Okay, I understand that the Fourier Transform is a special case of the Laplace where there is no σ term, but when would you use either?

For instance, to produce a Bode plot of a system, you end up using the Laplace Transform, but could you not have just used the Fourier Transform? What is the difference in information that you receive for either case?

Also, for periodic signals, the Fourier Series states that it can be broken up into sinusoids, does this just tell which frequencies are going to be affected when passed into a system?

Thanks in advance,
JP

 

[MrAl]

Hi,

Convolution is used in the time domain to compute the output from the input and impulse response.
The Laplace Transform is used in the frequency domain to compute the output from the input and the transfer function.
So both of these compute the output from the input, but one works in the time domain and the other in the frequency domain.
So with F1(s) and F2(s) the Laplace Transforms of f1(t) and f2(t) we would have:
f1(t)*f2(t)=F1(s) F2(s) [the asterisk here stands for convolution, not multiplication which is standard in many texts]
or in words, the convolution of f1 with f2 is equal to the multiplication of F1 and F2.
So if we can express f1 and f2 in the frequency domain, all we have to do is multiply those transformations together. If we want the time solution then we have to use the inverse transform to get the result back into the time domain.
For some functions this is easier said than done, but that's the theory and there are techniques to work with this kind of stuff.

As a simple example, try v1=e^-at and v2=e^-bt. Use the convolution integral on v1 and v2, then transform to V1 and V2 (freq domain) then multiply the two, then transform back to the time domain. Compare results.