I found myself in working with an FFT that does not have sufficient frequency resolution and I would like to increase this sensitivity. Since I have to perform this FFT on multiple noise acquisitions (which in first approximation should have all the same spectrum) I would like to combine all these noise "segments" to produce one very long noise acquisition to make the FFT. I have been researching how to perform such a task, but I have not anything yet. In particular, I guess I have to insert some zeros in between the single noise windows when I combine then in order to account for discontinuities and phase shifts. Do you know how to perform such a task ?
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$\begingroup$ maybe the raw unparameterized spectrum estimator that is the FFT-based periodogram then isn't the optimal choice? Either way, to help you, aside from more or, most likely, less efficiently building a larger FFT out of small FFTs, it would enable us to help you much better if you could define what you're looking for when analysing that noise. Is there a specific signal in there? Or is it about noise powers only? If there is a signal, can you describe it, mathematically or else? $\endgroup$– Marcus MüllerCommented Apr 12, 2022 at 21:42
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The frequency resolution is directly related to the duration of the FFT (nominally $1/T$ and then only gets worst when windowing is applied). The challenge when combining multiple captures is that phase synchronization from capture to capture is necessary- otherwise the resulting combining of multiple asynchronous captures will be identical to the Welch and other periodogram methods which serve to reduce the noise in the result (as a power spectral density result) but the resolution bandwidth in that case is set by the single capture duration $T$.
More significantly with combining very long captures we will run into a limitation on stationarity of the signal, as well as the long term stability of any clock used for synchronization. Any gap between captures will further diminish the amount of usable time available for a capture. For this purpose, I recommend using the Allan Deviation as I describe here to assess the stability of the synchronization process and the signal itself (given the description of the problem in computing a long FFT, it is then assumed that the signal would be stationary over that period since the FFT itself will only provide the average). From the plot of the Allan Deviation, we can easily determine the maximum time duration where stationarity can be assumed, and this would be the maximum possible capture time for any experiment up to which an improvement in resolution bandwidth can be achieved (Any longer duration would just result in a worst result so there is no benefit to doing that).
answered Apr 12, 2022 at 22:25