# How to detect decaying oscillations in a signal

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.

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

• What do you mean by dominating timescales? Do you mean the time points at which the peaks occur? – Amal May 8 '16 at 13:54
• Welcome to DSP.SE! Interesting question. You will probably also need to include a phase term in the signal model: $f(t) = e^{\sigma t + i\omega t + i \phi}$. – Peter K. May 8 '16 at 13:55
• 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. – Blazej May 8 '16 at 14:17
• 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? – LJSilver May 8 '16 at 14:45
• I'm sorry, I don't understand the question "batch or online analysis?". – Blazej May 8 '16 at 15:07

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.
• Take FFT of $ft$, it should peak at $\omega0$. And it should fall on both sides of the peak by factor the $\frac{1}{\sigma+j2\pi f}$. – Sujeet May 10 '16 at 1:29