https://www.gaussianwaves.com/2015/11/interpreting-fft-resul...
Note that complex numbers are merely convenient for working with two dimensional quantities. The square root of -1 is just math geek for orthogonal, and has nothing whatsoever to do with signals
so e^(σ + iω) = e^σ * e^iω is just an exponential growth or decay modulated by a sinusoid.. or, if σ is one, is just a pure oscillation
ω is the usual frequency, but σ + iω is the complex frequency. the fourier transform deals with function that receives ω as input, and the laplace transform deals with functions that receives σ + iω instead.
so the fourier transform is just a special case of the laplace transform with σ = 0
Another useful way of looking at it: Laplace transform is doing many extra Fourier-transform-but-with-decay giving you a map of which "global decay timescale" fits your data best -- since each "slice" is itself sufficient to fully describe the time series
They are all cases of integral transforms with different choices of the set of "primitive fingerprints" -- see chirp transform, wavelet transform, chirplet transform etc -- all taking advantage of the fact that if you choose one set of basis "brushes" that are not redundant with each other (e.g. having red-green-blue brushes is independent, as is magenta-green-yellow but having red-green-blue-yellow is not) then you will be able to describe your signal in terms of a composition of those kernels.
> a map of which "global decay timescale" fits your data best
What do you mean by this?
> -- since each "slice" is itself sufficient to fully describe the time series
But each slice is multiplied by an exponent.. which, okay, becomes a convolution that lets you recover the original function
I referred to the idea that by plugging an imaginary frequency into the Fourier transform [ETA: the grown-up Fourier transform with the complex exponent, not the schoolboy cosine kludge], you get the Laplace transform, and while that changes the inverse Fourier transform in a different way, it’s not hard to work out how specifically and obtain the inverse Laplace transform.
Why you’d want to do that, I actually don’t know how to explain convincingly. The post hoc rationalization is simple and more or less the reason people prefer the Laplace transform in signal processing: you still get a convolution theorem, but are now allowed to work with exponentially increasing functions, which standard Fourier theory (even the tempered distributions version) can’t accomodate. But while that’s useful from a toolbox standpoint, it isn’t satisfying as motivation, I think.
This is not the only way looking at the complex frequency plane turns out to be useful—there’s a whole thing about doing complex analysis to response functions aka propagators—but there too I can’t really say why you’d guess to look in that direction in the first place.
What I mentioned was that this idea of Laplace as imaginary Fourier extends beyond the reals to the group setting at least to some extent, so it’s not entirely an R-specific accident. Again, dunno why, I’ve explored this stuff a bit but am far from an expert.
So you can't take the fourier transform of an exponential? But.. it seems you can? https://proofwiki.org/wiki/Fourier_Transform_of_Exponential_...
Well seen! TFA is certainly inspired by that very old Baez post, where he explained to Oz the many viewpoints of Fourier analysis. I was dismayed to see that all such viewpoints were algebraic in nature, requiring special structure in the base space, thus neglecting the fundamental case of a general manifold without symmetries.
Are those talking about the same thing?
[Information Transmission using the Nonlinear Fourier Transform, Part I: Mathematical Tools](https://arxiv.org/abs/1202.3653)
[Nonlinear Fourier Analysis](https://arxiv.org/abs/1201.5129)
- The idempotent (“tropical”) Fourier transform turns out to be the Legendre transform;
- The fractional Fourier transform, known to physicists as the propagator of the quantum harmonic oscillator, is a pretty fun thing to consider;
- The Fourier-Laplace transform on Abelian groups seems like a fairly straightforward extension of the idea of plugging in a complex frequency, but I haven’t seen a textbook exposition (only an old article);
- The non-linear Fourier transform (with Ki xi + Lij xi xj + ..., finite or infinite sum) seems impressively obscure (I know of a total of one book reference) but occurs in quantum field theory as the “n-loop” or “∞-loop effective action”;
- The odd (in the super sense) Fourier transform turns out to underpin stuff like the Hodge star on differential forms;
- On a finite non-Abelian groups, the duality splits into two: every function on conjugacy classes is a linear combination of irreducible characters; every function on group is a linear combination of irreducible matrix elements; this is probably also doable on Lie groups but I’m too much of a wimp to learn the theory.
(Also, generating functions should by all appearances be a fairly elementary chapter of the Fourier story, as electronic engineers with their “Z-transform” also realize, but I haven’t seen that implemented convincingly in full.)
See as well Baez’s old issue of “This Week’s Finds” where he started with sound and well all the way to spectra of Banach algebras and rings—as in Gelfand duality, algebraic geometry etc. (Can’t seem to locate the specific issue now.) Of course there are also wavelets (there’s even a Fields Medal for those now), but I don’t know that they fit into the representation theory ideology (would be excited to be wrong!).