computing fft of damped waves - matlab

Question

I have a time series which is a linear combination of damped waves. The data is real.
Y(t) =SUM_w exp(- gamma t) sin(omega t)
There is no analytic form but this is a closest guess. I want to fourier analyze (FFT) such data and get the real frequencies and damping rates.
I am using matlab but any tool would be fine
Thanks!

Answer 1

Your question would be a better fit for http://math.stackexchange.com, where LaTeX rendering is available for formula. Instead, you have to use e.g. this bookmarklet for proper display:
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First of all, I assume your function is more precisely of the kind
$\sum\limits_k e^{-\gamma_k t}\sin(\omega_k t)$ for $t>0$ and $0$ for $t<0$ (otherwise the function would tend to infinity for $t\to-\infty$). Since a damped function is not periodic, you cannot use Fourier analysis but have to use the Fourier transform, which yields an amplitude for continuous frequencies instead of discrete ones. Using the complex representation $\sin(x) = \frac{e^{ix}-e^{-ix}}{2i}$, each term in the sum can be Fourier transformed individually, yielding
$\frac1{2\pi}\int_0^\infty e^{-\gamma_k t}\sin(\omega_k t)e^{-i\omega t}\,dt = \frac{\omega_k}{2\pi[(\gamma+i\omega)^2+\omega_k^2]} = \frac{\omega_k}{2\pi}\frac{(\gamma^2-\omega^2+\omega_k^2) + 2i\gamma\omega\omega_k}{[\gamma^2-\omega^2+\omega_k^2]^2+4\gamma^2\omega^2}$
Since you state the damped sine is only a guess, you have to discretize this unlimited integral somehow, though due to the damping a cutoff at some sufficiently large time $t$ should be in order. If the damping $\gamma_k$ is actually the same for all summands, your life becomes easier: Multiply the data by $e^{+\gamma t}$ to obtain a periodic signal which you can now really FFT on.

Answer 2

First digitize your function:
t=0:dt:T; % define sampling interval dt and duration T according to your needs
Y=sum(exp(-gamma*t).*sin(omega*t));
Then do fft and plot:
Y_f=fft(Y);
plot(abs(Y_f));