If your underlying signal is at frequency that is not a harmonic of the sampling length, then you get "ringing" and it's completely unclear how to deal with it (something something Bessel functions)
Actually using DFTs is a nightmare ..
- If I have several dominant frequencies (not multiples of the sampling rate) and I want to know them precisely, it's unclear how I can do that with an FFT
- If I know the frequency a priori and just want to know the phase shift.. also unclear
- If I have missing values.. how do i fill the gaps to distort the resulting spectrum as little as possible?
- If I have samples that are not equally spaced, how am I supposed to deal with that?
- If my measurements have errors, how do I propagate errors through the FFT to my results?
So outside of audio where you control the fixed sample rate and the frequencies are all much lower than the sample rate... it's really hard to use. I tried to use it for a research project and while the results looked cool.. I just wasn't able to backup my math in a convincing way (though it's been a few years so I should try again with ChatGPT's hand-holding)
I recommend people poke around this webpage to get a taste of what a complicated scary monster you're dealing with