# Recovering Decay Constant from Fourier Transformed Exponential Decay in NMR

I'm currently in a NMR lab for an undergraduate physics class, and I am attempting to determine the decay constant $$\tau$$ (e.g. $$T_2$$) associated with a free induction decay signal. However, our instrument has a periodic background noise present, which limits my ability to determine the decay constant. An approximation of the signal would be $$cos(\omega_0t)M_0e^{-t/\tau}+ N(t)$$ where N represents the noise and $$\omega_0$$ is the larmor frequency / the frequency at which the FID oscillates.

In an effort to work around this, I am attempting to remove the background from my signal by applying a Fourier transform to both signal and background, then subtracting the two. This (ideally) produces a Lorentzian, centered at the Larmor frequency ($$\omega_0$$), with a width parameter ($$\Gamma$$) that corresponds to the decay constant of the exponential before the Fourier transform. I've attempted to research a numerical method to determine the original $$\tau$$, but all sources I've encountered stick to merely qualitative descriptions (shorter decay times mean larger FWHM values).

Below, I provide an example of my process so far, in case there are any obvious errors that are producing my difficulties.

Example of a signal with no noise present. In this example, $$\omega_0$$ is 2*$$\pi$$*50, and $$\tau$$ = 625.

After applying a Fourier transform, I take the magnitude of the resulting data, square it, and fit a Lorentzian to it (equation below), which produces the following figure. Note that data curve is not visible because the fitted curve overlays it nearly perfectly. $$L(x) = \frac{1}{\pi} \frac{a*\frac{1}{2}\Gamma}{(x-x_0)^2+ (\frac{1}{2}\Gamma)^2}$$ where a is a scaling parameter, $$x_0$$ is the center of the curve, and $$\Gamma$$ controls the width of the curve. Source

However, at this point, I am unable to determine how to go from the parameters of the fitted Lorentzian back to determining the decay constant of the original signal. Any help would be greatly appreciated.

Perhaps the actual calculation of the Fourier transform is challenging because the function blows up toward $$-\infty$$, which means it would not have a well defined Fourier transform. In that case you can "cheat" a little by defining a function that decays in both directions away from 0. You can the break the Fourier integral into two parts making it easier to solve. It will give you a real valued spectrum, but you are not interested in the phase anyway, because you are going to compute the modulus square of the spectrum.