Time-domain NMR or: When is the Fourier-Transformation not appropriate? My question has two parts: One is general and has to do with the Fourier-Transformation, one has to do with Time-Domain NMR. Both parts are interlinked, of course.
I tried to find out, why people do Time-Domain (TD) NMR instead of Fourier-Transform (FT) NMR, which is up to my knowledge by far the standard.
Part 1 on TD-NMR
On my search on TD-NMR, I found a vendor that states on his website

...low magnetic field results in a low resolution, which is insufficient
  for obtaining Fourier-transformation frequency spectra.

Certainly, for Fourier-approaches, there has to be a certain number of sample points available to get good spectra. On the other hand, fitting the signal in time-domain with few data points is equally problematic. So, why is it done and is TD-NMR done with some kind of special fitting approach?
Part 2, general
The vendor stated that TD-NMR uses low magnetic fields, which also translates to a low signal-to-noise ratio. Is there a reason why signal processing should be done in TD only because of an SNR issue? I imagine, since the FT is a linear transformation, there should be no gain or loss in terms of SNR, right? Hence, it should not impede the choice of the domain that you would like to work in. 
 A: If you tip $M_0$ to XOY plane and recieve the signal, you'll get some complex data points.
Supposing 1024 points were sampled, there should be an array of complex data with 1024 elements.
The Fourier transform of these complex data is a frequency NMR spectra. The resolution of the spectra is decided by the SNR & # of points and the dwell time.
The Inverse Laplace Transform of the envelope of the complex data (the magnitude of each complex element is usually used) is a Time-Domain distribution map(For FID, ILT would produce a T2* distribution map).
Now we could come to the questions:

*

*Compared with TD NMR, FD-NMR need a much higher SNR and a more uniform B0 to produce high resolution results. If the spectral width are too wide, peaks would be more likely to overlap with each other. The spectra would be useless.
Furthermore, FT is much more robust to noise than ILT. But they both need enough data points.

*Both can suffer with poor SNR.LF-NMR or TD-NMR are compact NMR system, the size are much smaller and they usually use perminate magnet as $B0$. There are no He or N needed. Both price and size are suitable for industrial integration. But using TD-NMR cannot indentify the chemical structure. They can only measue an inexact relaxation distribution map or diffusion map.
Fourier-NMR equipements are expansive and the maintenance cost would be much higher than TD-NMR(which has no maintenance cost). But one can indentify and quantify an certain chemical structure using the frequency domain NMR spectra.

