In nuclear magnetic resonance experiments, the raw data is an exponentially decaying sinuoid(s) as a function of time (t $\geq$ 0). In order to see the frequency spectrum the standard protocol is to do a Fast Fourier Transform (FFT) and display one-sided frequency spectrum. The question here initially looked trivial but it has been mentioned in a previous discussion that this is near impossible to predict for FFT [Damped sine wave][1].

a) Simple scenario: We have a non-decaying sine wave of frequency $\omega_0$ and amplitude $A$. FFT of this signal is two peaks (one corresponding to positive frequency and the other one at negative frequency situated at $\omega_0$ and amplitude $A/2$. Note that we have to normalize the FFT with the number of sampled points in order to get the right amplitudes in the frequency spectrum.

b) Real experimental scernaio: Let us say we have an exponentially decaying sine wave $x(t)$, then, 

$$x(t) = A \sin(\omega_0 t) e^{-\alpha t} \, u(t)$$

where $u(t)$ is the Heaviside unit step function, $\alpha$ is positive and real.

In the continuous version of FT:

$$X(\omega) = \int_{0}^{\infty} A \sin(\omega_0 t) e^{-\alpha t} e^{-i\omega t} \, dt$$

The solution is 
$$ X(\omega)=\frac{A\omega_0}{(\alpha+i\omega)^2+\omega_0^2}$$

where $\omega$ is 2$\pi f$ and $f$ is the frequency in Hz. The key question is the how to predict the theoretical peak amplitude of such an exponentially decaying sine wave. In the frequency spectrum we will still see two peaks at $omega_0$, one at negative frequency and one at positive frequency but now their heights are not $A$ but always lesser, because on average the sine wave amplitude is decaying in the time domain. Observationally, the single peak heights are averages of the extreme values of the damped sine wave. However, I am unable to find a mathematical basis for this averaging effect. 

How can we connect this continuous FT result with FFT for predicting the correct theoretical peak height after doing FFT of a damped sine wave?


  [1]: https://dsp.stackexchange.com/questions/93250/predicting-the-amplitude-of-a-damped-sine-wave-from-fft