Signals estimated by the FFT have two parameters: magnitude and phase. FFT results evade intuition because complex numbers are cartesian. If you convert them to polar coordinates they make more sense as magnitude and phase
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
From a physics perspective, complex frequency results in “evanescent waves” - ie, waves that decay rather than oscillate (technically a fully complex frequency of the form a+ib will both oscillate and decay)
Yep: e^iω (with ω real) is an oscillation, but e^σ (with σ real) is an exponential decay when σ < 1, an exponential growth with σ > 0 and constant with σ = 1
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
> 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.
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.
> but are now allowed to work with exponentially increasing functions, which standard Fourier theory (even the tempered distributions version) can’t accomodate.
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