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I have a sampled signal which contains (among other things) some damped oscillations. From the underlying theory I have reasons to believe that it can be characterized by some specific frequencies of oscillation $\omega$ and rates of decay $-\sigma$. I believe the signal contains components of the form $f(t)=e^{\sigma t+i \omega t}$. It might be that these are "there" only in some part of the signal, because at late times other components win over (due to exponential damping of these oscillations). I am interested in finding these dominating timescales. I need a hint how to proceed. There are plenty of tools available, like Fourier analysis, wavelets etc. However, I don't know which is the most appropriate for this task, as I have no experience with any of them, except maybe the Fourier transform. Below I include a graph of the signal that I want to analyze.

One of my datasets

http://i.imgur.com/fZwhdkq.png

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  • $\begingroup$ What do you mean by dominating timescales? Do you mean the time points at which the peaks occur? $\endgroup$ – Amal May 8 '16 at 13:54
  • $\begingroup$ Welcome to DSP.SE! Interesting question. You will probably also need to include a phase term in the signal model: $f(t) = e^{\sigma t + i\omega t + i \phi}$. $\endgroup$ – Peter K. May 8 '16 at 13:55
  • $\begingroup$ Ok, to be more precise: What I am looking for are complex frequencies $\sigma + i \omega t$ which are somehow "dominant" in the signal. In the language of Fourier transform, I suspect there should be some visible peak in the spectrum of the signal. But this is only analogy, because I am interested in complex frequencies and traditional Fourier analysis doesn't apply. $\endgroup$ – Blazej May 8 '16 at 14:17
  • $\begingroup$ For what I can see in the signal the frequency of the dumped oscillation is not constant, it is slightly decrescent. Which kind of analysis do you need? Batch or online? $\endgroup$ – LJSilver May 8 '16 at 14:45
  • $\begingroup$ I'm sorry, I don't understand the question "batch or online analysis?". $\endgroup$ – Blazej May 8 '16 at 15:07
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Your signal model lends itself particularly well for so called parametric frequency analysis. The basic idea is that you find the autocovariance matrix of your signal and identify its eigenstructure. The eigenvalues correspond to the exponents of a sum-of-exponentials model. In your case you will find conjugate pairs of eigenvalues with non-zero imaginary part describing the exponential decay.

This thesis looks like a good introduction to the topic: http://zet10.ipee.pwr.wroc.pl/record/11/files/Leon_mon.pdf

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If you take fourier transform $f_t=e^{\sigma t}\times e^{i\omega_0 t}$ it will be convolution of the two individual signal spectrums. $e^{i\omega_0 t}$ will result into impulse at $\omega_0$ and $e^{\sigma t}$ will give decaying spectrum. You can get some insight about these parameters from simple Fourier transform.

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  • $\begingroup$ Sorry, I don't understand what do you suggest I do. $\endgroup$ – Blazej May 8 '16 at 19:59
  • $\begingroup$ Take FFT of $ft$, it should peak at $\omega0$. And it should fall on both sides of the peak by factor the $\frac{1}{\sigma+j2\pi f}$. $\endgroup$ – Sujeet May 10 '16 at 1:29

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