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 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_a...

[2] https://en.wikipedia.org/wiki/Laue_equations

[3] https://en.wikipedia.org/wiki/Basilar_membrane#Frequency_dis...