# Frequency analysis of decaying signal

I am stuck in a simple confusion, please help me with this basic understanding of the concept.

I have an exponential decaying sine signal as a response to a sensor.

This response lasts only for few microseconds. I sample this signal at 50 MSps (max supported). The sensor response is from 8-9 MHz. Then I collect 100 µs worth of data, i.e. $$N=5000$$, samples and compute a 5000-point FFT.

Is it correct to assume my resolution is $$f_s/N$$? I'm considering 100 µs because I send a 2 µs sine pluse to the sensor every 100 µs.

Please let me know how should i find the frequency of the decaying signal?

## 1 Answer

Is it correct to assume my resolution is $$f_s/N$$?

Yes, but it's not useful.

You won't clearly see the frequency of the sine in your FFT, because your signal model is, as far as your description goes,

$$s(t) = A_0\sin(2\pi f_\text{sine}t + \varphi_0) \cdot e^{-\tau t}.\label1\tag1$$

Pay attention to the multiplication of the sine with the exponential envelope! In frequency domain, that's a convolution of the two Fourier transforms of the sine and the exponential. With the latter not being band-limited, you can't even sample this 100% correctly, but more importantly, the things that are "easy" in time domain become smeared and unclear in frequency domain. The FFT is making your life harder here! It's the wrong tool.

Instead, consider Eq. $$\eqref{1}$$ simply as an equation with four unknowns, and solve it by using four or more samples, in time domain. It's not hard to do it generally (it becomes slightly more complicated when you add noise, but let's ignore that for the time being).

First, I'd take two samples that are simple a sampling period $$T_s$$ apart, and divide them:

\begin{align} \frac{s(t_0)}{s(t_0+T_s)} &= \frac {A_0\sin(2\pi f_\text{sine}t_0 + \varphi_0) \cdot e^{-\tau t_0}} {A_0\sin(2\pi f_\text{sine}(t_0+T_s) + \varphi_0) \cdot e^{-\tau (t_0+T_s)}}\\ &=\frac {\sin(2\pi f_\text{sine}t_0 + \varphi_0) } {\sin(2\pi f_\text{sine}(t_0+T_s) + \varphi_0)} \cdot e^{-\tau [t_0-(t_0+T_s)]}\\ % &=\frac {\sin(2\pi f_\text{sine}t_0 + \varphi_0) } {\sin(2\pi f_\text{sine}(t_0+T_s) + \varphi_0)} \cdot e^{\tau T_s}\\ \end{align}

Do the same for the sample one period earlier (at $$t=t_0-T_s$$), and then divide the results. You'll see that the result doesn't contain your exponential term anymore, the $$\sin(2\pi f_\text{sine}t_0 + \varphi_0)$$ itself cancels, and you're just left with an equation that contains the product of two sines. From there it's applying a formula to simplify that product of sines and your observed sample values $$s(t_0\pm T_s)$$; and you'll see how you can solve this!