# Best way to find magnitude and phase of a specific frequency in an empirical time series…

I've a discrete, univariate time series, and I'm interested in to investigate a specific frequency component.

Assume I'm interested in a frequency with a cycle-time of $f$ samples - and I need to get the best understanding I can of its magnitude and phase at the instant of the most recent sample. As I see it, I need to consider $n.f$ samples - where n is a positive integer... and there's a trade-off: with small n, my estimation of the f-frequency will be most adversely affected by noise (other frequencies - higher and lower); with large $n$, the effects of this noise are reduced but I must settle for the average phase over $n$ cycles - which won't account for changes in phase and magnitude over the $n.f$ duration.

There is, of course, another aspect - if I consider a sliding window of $n.f$ - I should expect it to advance by $2.\pi\over f$ with each new sample - and for any change in magnitude to be proportionally small... assuming my analysis of the frequency $f$ component of the signal is meaningful.

My first idea about establishing the phase and magnitude was to do a bunch of FFTs and discard all but the frequency of interest in each. This, however, seems somewhat wasteful.

Are there any well known techniques for addressing this sort of problem? Should I just run with FFTs - or are there more efficient approaches I might adopt?

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I might consider a plain Fourier series (as opposed to an FFT). The Fourier series gives amplitude and phase (ie, sine and cosine coefficients). THis would assume periodicity of the overall signal and knowledge of the fundamental frequency which is often not possible particularly with random stochastic signals.

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