It is relatively easy to discover overlaid (additive) signals in natural phenomena using wavelet decomposition, Fourier Transforms, CEEMD, etc, because addition in the time domain is the same as addition in the frequency domain.

However, I cannot find a clear reference on how to determine if amplitude modulation is occurring in a natural phenomena. This type of relationship between known or unknown sources results in convolution in the frequency domain, and I can't figure out how to process a time series of data to elicit possible sources of such a signal.

Let's take for example a commonly available series - monthly sunspots. There's clear peaks at 10 and 12 year intervals that can be seen by taking an FFT, resulting in the commonly known 11 year cycle average, which makes one suspect amplitude modulation somewhere in the natural processes of the sun. How would one determine a suspected carrier frequency or prove that some sort of amplitude modulation is taking place?

  • $\begingroup$ Modulation results in sidebands in the spectrum. A long enough FFT might show such sidebands. $\endgroup$ – hotpaw2 Apr 27 '16 at 4:37
  • $\begingroup$ sure, but the sidebands are mixed in with other noise (natural signals tend to be fairly broadband), so how does one tell if it's just noise or a valid sideband? And, if you identify a sideband, how do you find the modulation frequency? I note by experimentation that CEEMD completely misses the carrier signal. $\endgroup$ – Paul S Apr 27 '16 at 4:48
  • $\begingroup$ You have stated the general problem of statistical induction. $\endgroup$ – hotpaw2 Apr 27 '16 at 5:39
  • $\begingroup$ yes, and statistical induction problems have known solutions for additive signals :-). Now I'm trying to find the same for multiplicative signals. $\endgroup$ – Paul S Apr 27 '16 at 15:17
  • $\begingroup$ Although it might be tedious the two AM sidebands stand in a locked phase relationship. So if you do a two dimensional search (presuming you don't know the frequency) and look for cross-correlation in the phase domain you should find any AM sidebands and can proceed. $\endgroup$ – rrogers May 3 '16 at 21:54

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