3 ms·

The Fourier transform is also linear, so the same solution should work. No clue if an NN would find it though.

by mananaysiempre 4y ago

perihelions 4y ago
What's an analog implementation of a Fourier transform look like? It sounds interesting!
- bradrn 4y ago
  It looks like Albert Michelson's Harmonic Analyzer: see https://engineerguy.com/fourier/ https://engineerguy.com/fourier/ (and don’t miss the accompanying video series! It’s pretty cool.)
- mananaysiempre 4y ago
  Depends on what kind of “analog” you want. In an NN context, given that you already have “transform with a matrix” as a primitive, probably something very much like sticking a https://en.wikipedia.org/wiki/DFT_matrix https://en.wikipedia.org/wiki/DFT_matrix somewhere. (You are already extremely familliar with the 2-input DFT, for example: it’s the (x, y) ↦ (x+y, x−y) map.) If you want a physical implementation of a Fourier transform, it gets a little more fun. A sibling comment already mentioned one possibility. Another is that far-field (i.e. long-distance; “Fraunhofer”) diffraction of coherent light on a semi-transparent planar screen gives you the Fourier transform of the transmissivity (i.e. transparency) of that screen[1]. That’s extremely neat and covers all textbook examples of diffraction (e.g. a finite-width slit gives a sinc for the usual reasons), but probably impractical to mention in an introductory course because the derivation is to get a gnarly general formula then apply the far-field approximation to it. A related application is any time the “reciprocal lattice” is mentioned in solid-state physics; e.g. in X-ray crystallography, what you see on the CRT screen in the simplest case once the X-rays have passed through the sample is the (continuous) Fourier transform of (a bunch of Dirac deltas stuck at each center of) its crystal lattice[2], and that’s because it’s basically the same thing as the Fraunhofer diffraction in the previous paragraph. Of course, the mammalian inner ear is also a spectral analyzer[3]. [1] https://en.wikipedia.org/wiki/Fourier_optics#The_far_field_approximation https://en.wikipedia.org/wiki/Fourier_optics#The_far_field_a... [2] https://en.wikipedia.org/wiki/Laue_equations https://en.wikipedia.org/wiki/Laue_equations [3] https://en.wikipedia.org/wiki/Basilar_membrane#Frequency_dispersion https://en.wikipedia.org/wiki/Basilar_membrane#Frequency_dis...
  - bradrn 4y ago
    > Another is that far-field (i.e. long-distance; “Fraunhofer”) diffraction of coherent light on a semi-transparent planar screen gives you the Fourier transform of the transmissivity (i.e. transparency) of that screen Ooh, yes, I’d forgotten that! And I’ve actually done this experiment myself — it works impressively well when you set it up right. I even recall being able to create filters (low-pass, high-pass etc.) simply by blocking the appropriate part of the light beam and reconstituting the final image using another lens. Should have mentioned it in my comment…
- ilyt 4y ago
  A prism and CCD sensor. Probably bit more fuckery to get phase info out of that tho.

The Fourier transform is also linear, so the same solution should work. No clue if an NN would find it though.

perihelions 4y ago

What's an analog implementation of a Fourier transform look like? It sounds interesting!

bradrn 4y ago
It looks like Albert Michelson's Harmonic Analyzer: see https://engineerguy.com/fourier/ https://engineerguy.com/fourier/ (and don’t miss the accompanying video series! It’s pretty cool.)
mananaysiempre 4y ago
Depends on what kind of “analog” you want. In an NN context, given that you already have “transform with a matrix” as a primitive, probably something very much like sticking a https://en.wikipedia.org/wiki/DFT_matrix https://en.wikipedia.org/wiki/DFT_matrix somewhere. (You are already extremely familliar with the 2-input DFT, for example: it’s the (x, y) ↦ (x+y, x−y) map.) If you want a physical implementation of a Fourier transform, it gets a little more fun. A sibling comment already mentioned one possibility. Another is that far-field (i.e. long-distance; “Fraunhofer”) diffraction of coherent light on a semi-transparent planar screen gives you the Fourier transform of the transmissivity (i.e. transparency) of that screen[1]. That’s extremely neat and covers all textbook examples of diffraction (e.g. a finite-width slit gives a sinc for the usual reasons), but probably impractical to mention in an introductory course because the derivation is to get a gnarly general formula then apply the far-field approximation to it. A related application is any time the “reciprocal lattice” is mentioned in solid-state physics; e.g. in X-ray crystallography, what you see on the CRT screen in the simplest case once the X-rays have passed through the sample is the (continuous) Fourier transform of (a bunch of Dirac deltas stuck at each center of) its crystal lattice[2], and that’s because it’s basically the same thing as the Fraunhofer diffraction in the previous paragraph. Of course, the mammalian inner ear is also a spectral analyzer[3]. [1] https://en.wikipedia.org/wiki/Fourier_optics#The_far_field_approximation https://en.wikipedia.org/wiki/Fourier_optics#The_far_field_a... [2] https://en.wikipedia.org/wiki/Laue_equations https://en.wikipedia.org/wiki/Laue_equations [3] https://en.wikipedia.org/wiki/Basilar_membrane#Frequency_dispersion https://en.wikipedia.org/wiki/Basilar_membrane#Frequency_dis...
- bradrn 4y ago
  > Another is that far-field (i.e. long-distance; “Fraunhofer”) diffraction of coherent light on a semi-transparent planar screen gives you the Fourier transform of the transmissivity (i.e. transparency) of that screen Ooh, yes, I’d forgotten that! And I’ve actually done this experiment myself — it works impressively well when you set it up right. I even recall being able to create filters (low-pass, high-pass etc.) simply by blocking the appropriate part of the light beam and reconstituting the final image using another lens. Should have mentioned it in my comment…
ilyt 4y ago
A prism and CCD sensor. Probably bit more fuckery to get phase info out of that tho.