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
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