Oh yikes yikes yikes. Yes, I wrote that question way too fast and mixed things up. You're totally right, Bessel functions are related to FM and not this ringing

Oh yikes yikes yikes. Yes, I wrote that question way too fast and mixed things up. You're totally right, Bessel functions are related to FM and not this ringing. I got things confused. Sorry about that "Aligning the signal's period with the window length M allows reproducing the whole infinite duration signal from a N coefficient slice because all slices are identical." Yeah but this isn't really a solution. I mean, yes.. you can cut your data to be a multiple of the frequency you're looking for, do an FFT, look at the phase. If you didn't know the exact frequency you could also do a "search" truncating your data at different values and looking for a peak. But that's.. ugly. The problem is you end up throwing away data. In my usecase I have maybe 2-3 periods of the frequency I'm interested in. Truncating my data throws away a lot of information. I'd like to do an FFT of all the data I have (it's actually not evenly spaces, so I'd probably have to interpolate it) then I'd like to look at the ringing directly and estimate the original frequency and phase. Ideally after that I'd somehow calculate some uncertainties on those values