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I have a set of N time series, each of length T, that describe separate realisations of a single physical system. For each series, I can compute an FFT to find the Fourier spectrum up to a period 2/T, and this is slightly different for each realisation.

What is the best way to estimate the true Fourier spectrum of the system up to period 2/T using these samples?

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If you suspect that your system may not have a simple harmonic oscillation behaviour, but have more periodicities at different frequencies, then a Fourier transform may not even be the optimal method to find those frequencies – some corrections are needed.

For the optimal method, and also for quantifying the uncertainty about your results, I recommend Bretthorst's great text, Bayesian Spectrum Analysis and Parameter Estimation (Springer 1998), freely available on his website. The introductory chapter of that text gives you a complete answer, more complete than could be written here.

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    $\begingroup$ I have a chaotic system so there will indeed be many frequency components. My main aim is to find the best estimate of the frequency spectrum in order to generate synthetic random data using phase randomisation. This is to use as a null hypothesis for further analysis. $\endgroup$ – JoshD Jun 10 at 12:30
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    $\begingroup$ Cool problem! Bretthorst will be a great guide. Check also his other works at bayes.wustl.edu/glb/bib.html . There are some misconceptions around spectrum inference. For example, some people believe that the uncertainty about the value of a signal's frequency is given by the width of the Fourier-transform peak, but that's not the case. Bretthorst explains why. $\endgroup$ – pglpm Jun 10 at 12:34

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