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If I'm analyzing a signal in the frequency domain, I know about the well known Nyquist criteria that the sample frequency must be > 2x of the highest component present in the signal.

However, there are very few references related to the lowest frequency I can analyze. This problem comes up in a lot of natural data where we have limited windows of observation because the instruments to do so were only invented or in wide spread use in the last 50-100 years, and natural cycles can be decades, centuries, or longer.

One would think that you could analyze a component where at least one period was in the sample window, but I find because of required windowing to prevent edge effects that the signal often disappears (it depends on the phase). I then thought it was a mirror of Nyquist and I could resolve any signal with at least 2 periods in a window, but actual analysis shows that to not be true when there's a another overlying signal whose period is similar.

I found through experiment that I needed at least 5 periods to resolve a signal where there's another signal near the same frequency.

Is there a formal way of representing this problem or a definitive reference I can use?

Would the same issue apply to wavelet analysis in the same manner?

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  • $\begingroup$ I think you may need to study your signal in the DFT domain and increase the resolution (the number of points per sample). Have a look to this question: electronics.stackexchange.com/questions/12407/… $\endgroup$ – Behind The Sciences Mar 13 '16 at 6:26
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    $\begingroup$ I forgot to mention that I padded like crazy. It just makes the FFT smoother, it doesn't add any information. There's a lower frequency information limit. You can't use a 10 second window to detect a 1/100000000000000 HZ signal for example, even if you pad with billions of zeroes. But what is that formal limit, formally? is the question here $\endgroup$ – Paul S Mar 13 '16 at 14:00
  • $\begingroup$ From the answers in that thread "EDIT: There is also a trick to pad the input with zeros and taking a bigger FFT. It will not improve your differentiation ability but may make the spectrum more readable. It is basically a trick similar to antialiasing in vector graphics." $\endgroup$ – Paul S Mar 13 '16 at 14:01
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The "limit" depends on the signal-to-noise ratio, and the nearest interfering frequency(s).

If the nearest interfering frequency carrier is tuned 1% away, you might need (not just 6, but) 250 cycles or more to resolve the two separated frequency peaks, depending on the window and the S/N. Whereas, a single pure unmodulated sinusoid in zero noise might be determined in as little as 3 or 4 non-aliased samples.

Also, there may be better low frequency estimators than depending on just a single non-rectangularly-windowed peak DFT magnitude bin, as using just that magnitude bin might be throwing some information about the phase of interference from the (nearby) conjugate negative frequency image that could be useful in improving frequency estimation and/or detection.

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  • $\begingroup$ Okay, as a general statement you are probably right, but I've run empirical experiments in matlab to pretty much conclude this. What I want is some formal mathematical proof or reference to a proof or at least math based discussion. $\endgroup$ – Paul S Mar 14 '16 at 14:15
  • $\begingroup$ I bolded the actual question I need answered, sorry if there's some confusion about that. $\endgroup$ – Paul S Mar 14 '16 at 21:24

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