I have N measurements of an under-damped cosine signal, represented by the equation $$ x(t) = Ae^{-\alpha t} \cos(2 \pi ft) $$ where $ A $ is a known constant. My objective is to estimate the remaining parameters $ \alpha $ and $ f $ using the Fast Fourier Transform (FFT).
The Fourier transform that I have computed is given by: $$ A \frac{1}{2\pi} \alpha \left[ \frac{1}{(\omega - 2\pi f)^2 + \frac{\alpha^2}{4\pi^2}} + \frac{1}{(\omega + 2\pi f)^2 + \frac{\alpha^2}{4\pi^2}} \right] $$
As I recall, this expression corresponds to a sum of two Lorentzian functions, where $ \omega = 2\pi f $
It's important to note that my sampling frequency is $ \frac{1}{0.01665} $ Hz.
I'm currently facing challenges in finding an appropriate fitting curve in the frequency domain, from which I'll be able to extract $\alpha$ and $f$. Could someone please help me with this matter?