Re: Laplace vs. Fourier Transform



Firebird skrev:
> I understand the mathematical differences between the two - e.g. -
>
> a) LT is more general b/c it is a function of a complex variable 's',
> whereas FT is a function of an imaginary variable (real part = 0)

Agreed

> b) LT converges for a larger range of functions

I don't know, you could well be right.

> But when to use which? I have used LT for solving differential eqns and
> thus for anything that is derived from diff eqs, and FT for frequency
> analysis. But can it be the other way round? LT used for frequency
> analysis

Yep. With s = sigma + jw, set sigma = 0 and you have yor frequency
analysis.

> and FT for diff eqs?

Sure. That's done all the time. Fourier himself developed what we
know as the "Fourier series" while studing the differential equation
governing heat transfer.

> When to use which?

I would say that the LT is used when time-domain responses are
important, while the FT is used for frequency-domain analysis.
I have noted that the FT is used for signals with constant-mean
signals, while the LT can be used for, say, ramp signals.

> Why is FT preferred for frequency analysis?

Because it is specialized for the case sigma = 0, as you mentioned
above. It gets to the point with far less work than the LT. I don't
think there exists such a thing as the Fast Laplace Transform.
The Fast Fourier Transform, FFT, is used all over the place.

> Why not just have one transform that works for
> everything? The FT and LT look so similar - it certainly should be possible
> to combine the two into a more general transform (maybe LT would be enough)
> that works for everything. If this is not possible, why not?

There are two different uses. It's like cars. Some drive a general-
purpose vehicle, like an SUV, where you can move people, move cargo,
drive on the road, drive off the road. It does everything, but none
of it particularly well. Then you have the specialized vehicles,
like the F1 cars, that does one thing, go fast, extremely well, but
is utterly useless for anything else one might consider a car to do.

Perhaps not a very good comparision, but as far as I can see, the LT
is the general-purpose tool that does everything but requires lots of
work, while the FT is a more specialized tool that is more efficient
with what it does, not least because of the FFT algorithm.

> Are there any cases where only one of them works?

I don't know. If it is, I'll bet the LT works where the FT doesn't.
The reason would be that, as you said, there exists some function
where the LT converges and the FT does not.

Rune

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