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The most intuitive explanation I've seen, although a bit abstract, is that Fourier transforms, and the other Fourier-like transforms including the s and z trans
by TeeMassive 5y ago
The most intuitive explanation I've seen, although a bit abstract, is that Fourier transforms, and the other Fourier-like transforms including the s and z transforms, are just about changing the system of coordinates of the data vector using the transform's generative vectors.
Indeed we can easily see that the definition of all those transforms take the form of the sum of a scalar product that computes the projection of the data vector unto the kth dimension in the transformation (e.g. frequency) space.
- alanbernstein 5y agoThis is the interpretation that leads you to imagine what a fractional Fourier transform might be: a rotation other than 90 degrees in the vector space you described. Interesting, if esoteric.
- kortex 5y agoFor those who may not have heard of it: https://en.m.wikipedia.org/wiki/Linear_canonical_transformation https://en.m.wikipedia.org/wiki/Linear_canonical_transformat... Neatly generalizes and ties together all of your favorite integral transforms and has weird connections to virtually every branch of mathematics.
- BlueTemplar 5y agoI was wondering recently whether some kind of 45° rotation transform could be used to represent equally well both "sides" of Heisenberg's unsharpness principle ?