Comment by krackers
12 hours ago
> FFT has periodic boundary conditions
FFT is simply an algorithm to efficiently compute the DFT. The fact that the article makes no mention of DFT vs fourier series vs DTFT is going to end up creating more confusion that it solves. For some reason introductory tutorials always start with the DFT (usually mistakenly using FFT and DFT interchangeably), even though to me the continuous fourier transform is far easier to conceptually understand. Going from continuous fourier transform to the DTFT, is just applying the FT to a dirac-combed (sampled) function. Then from DTFT to DFT you introduce periodic boundary condition. Fourier series is just applying FT to a function that happens to already be periodic, resulting in a finite set of discrete frequencies.
There is a connection between fourier series and DFT in that if the fourier series is computed for the periodic resummation of a signal, and then the DFT is computed for the original signal (which implicitly involves applying a periodic boundary condition), the DFT is just the periodic resummation of the fourier series.
I spent ages meditating on this image https://en.wikipedia.org/wiki/Discrete_Fourier_transform?#/m... before everything finally clicked, it's a shame that introductions never once mention DTFT
Completely agree the transition between continuous and discrete domains is often glossed over and people use DFT, FT, DTFT and FFT almost interchangeably (I certainly have been guilty of that myself, and you are correct the FFT and DFT are equivalent for this discussion).
An interesting fact (somewhat related to your mentioning of the DTFT) is that one can consider the DFT as a filter with a sinc transfer function. That's essentially how you can understand the spectrum of an OFDM signal. You perform a block based FFT on your input bit/symbol stream, so you have waves at different carriers. However, because the stream is timevarying you essentially get sinc shaped spectra spaced at the symbol rate (excluding cycling prefixes etc.). So your OFDM spectrum is composed of many sincs spaced at fb, which is very squarish which is one of the reasons why OFDM is so advantageous.
Had a professor go through this and distinguish DTFT vs DFT, etc.
Sadly that wasn’t my linear systems class, which omitted this in both the lectures and textbook.