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I have a time series, which I believe should have daily periodicity, the signal is sampled once every 15 minutes, which means that one period is 288 samples.

When I am trying to validate this result based on the FFT of the signal I get different results based on the number of points I sample from the FFT.

I am doing this in python, and first the number of points I sample the FFT is the same as the input array (900).

SEGMENT = np.fft.fft(segment-np.mean(segment))
freqs = np.arange(len(SEGMENT.T))/len(SEGMENT.T)
segment_abs = np.abs(SEGMENT)
fig = plt.figure(); ax = fig.add_subplot(111)
plt.plot(freqs,segment_abs.T,'o--')
plt.ylabel('Mangitude')
plt.xlim((0,0.05)); 
plt.xticks([0, 1/288], labels=['inf','Daily'])

ax.grid(); plt.show()

enter image description here

When I take the inverse of the X-axis I get that the top 2 frequencies are 245 and 332.

If I instead set the number of points to sample from FFT to 288, I get a clean peak which occurs at the 1/288 frequency.

enter image description here

I'm not sure how to interpret this, or how to think here.

If I want to determine if there is a period of 288 samples, which graph is better to look at?

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  • $\begingroup$ Try an FFT length that is an integer multiple of 288. $\endgroup$
    – Hilmar
    Sep 29, 2021 at 21:52
  • $\begingroup$ I'd suggest using the DTFT instead of the DFT (on the entire data) evaluated at 288. $\endgroup$
    – MBaz
    Sep 29, 2021 at 22:12

1 Answer 1

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You need your period to fit an integer times into your DFT length, else you get leakage (not explaining that concept here, it's a very fundamental one and you can find plenty good material on it, here and in other places. Book: any textbook on digital signals).

But.

If I want to determine if there is a period of 288 samples, which graph is better to look at?

If you want to do that, doing a full FFT is the wrong method. Goertzel would be a better choice, or if you know there's only one strong harmonic signal in there, a parametric spectrum estimator (which can estimate the actual frequency instead of "checking" at a fixed raster)

If you want so "tolerance", a band-pass filter might be much lower in complexity (i.e. need fewer samples) than a Goertzel / FFT, too.

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  • $\begingroup$ For me the constraint is that it can be vectorized and fast. I am to do this for around 1000 time series at the same time. Do you have any suggestion on which method might suit best for this constraint? $\endgroup$
    – kspr
    Sep 30, 2021 at 13:55
  • $\begingroup$ yes, that doesn't change my recommendations. $\endgroup$ Sep 30, 2021 at 13:58
  • $\begingroup$ (1000 Time series of a couple hundred samples each: nearly no computational load for modern PCs, at all, so unless you're planning on abandoning python to begin with, you're also barking up the wrong tree, performance-wise ;)) $\endgroup$ Sep 30, 2021 at 14:01
  • $\begingroup$ Thanks for the input! You don't happen to use python and know of any existing libraries for these methods you recommend? $\endgroup$
    – kspr
    Sep 30, 2021 at 14:10
  • $\begingroup$ as said, the math here is so small, your overhead is in python. your bottleneck when processing so few samples is not the math as is - it's python being run on 1000 time series. Is your processing actually too slow? Because, honestly, I don't think it is. $\endgroup$ Sep 30, 2021 at 14:11

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