8.2, due on November 29

I'm excited to see what cool applications there are for the Fast Fourier Transform, since it seems applicable to anything having to do with sensors in a system. I imagine all sorts of uses in algorithms that need use discrete measurements to predict what's going to happen in the future for continuous systems (like self-driving cars?).

Something that doesn't seem intuitive to me is how the discrete Fourier transform is the coefficients of the projection of f onto some subspace. I'd like to see some sort of visualization to better understand it. I also don't understand why the primitive nth roots of unity become zero when the exponent is not a multiple of n. Also, why is the temporal complexity of the FFT O(n log n)?

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