Regarding hotpaw2's answer to the "Number of FFT points required for a specific frequency resolution for an oversampled signal" question:

Given that the result of an FFT can be interpolated (possibly very accurately using Sinc interpolation), the number of FFT points required to estimate the frequency depends on the signal-to-noise ratio of the data containing your signal, and the type of resolution you require (peak separation, or peak estimation).

In the extreme case of zero noise or other interference, only 3 or 4 non-aliased points are required to exactly reconstruct a pure sinusoid and thus estimate its frequency.

I'm just wondering if you have a reference I can look up for the details about the quantitative relationship between the limit of signal to noise level and allowed data points for the FFT. That is, if I have a data set of 10 which has a signal to noise (S/N) level of 100, is it good enough to reconstruct 3 pure sinusoid and its frequency? and how about the case of S/N level of 10?

  • $\begingroup$ i have some trouble with the semantics you use. regarding the question in the middle, if there is no DC offset, three points positioned at non-aliased times (like they can't all be at zero crossings or at peaks or at the same position on the waveform) suffices to completely define a sinusoid, because three numbers: amplitude, frequency, and phase fully define the sinusoid. if the three points are well placed, then it's three equations and three unknowns. if there is a DC offset, then four points. $\endgroup$ – robert bristow-johnson Jan 12 '17 at 4:49
  • $\begingroup$ with more points, you can try to do something to minimize the additive error. sorta like least-squares fit for a straight line. if the frequency is known (or assumed) in advance, then there is a closed form solution for the amplitude or phase. if the frequency is not known in advance (but you know the possible range) you can guess at various frequencies, calculate the amplitude and phase that best fits, record the minimum square error for that frequency and repeat for other trial frequencies. then pick the frequency with the least mean square error and sorta "zoom in" on it. $\endgroup$ – robert bristow-johnson Jan 12 '17 at 4:54
  • $\begingroup$ to do this well, you should FFT a windowed segment of the alleged sinusoid + noise. use the peak value to determine frequency and tweek the frequency slightly and evaluate using Goertzel to get a lower additive noise component. $\endgroup$ – robert bristow-johnson Jan 12 '17 at 4:57
  • $\begingroup$ Which answer are you quoting by Regarding your answer by ? $\endgroup$ – Gilles Jan 12 '17 at 7:37

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