Fourier transform for spectroscopy spectra? I was thinking back to my chemistry major days and remembering the bunch of cool spectra we produced by various means (HNMR, IR, UV/Vis) and I remember all the cool 'self-interference' like couplets, triplets doublets, etc. 

My question is basically motivated by the complexity of spectra, and the reductionist instinct to 'break it down' into simpler components. The FFT or DFT is known to separate out some periodic components of signals, as are SVD or tensor decompositions known to isolate independent components. 

I am completely not an expert in any of this but I am interested to hear from any coal-face chemists if they regularly use some 'spectra decomposition' method to aid in interpretation or analysis. I remember being taught how to read various peaks in spectra and analyze them, but I can't help thinking, for some very complex molecules, surely it gets pretty hard. And what if, we don't actually know the structure beforehand? 

So given that Fourier transform is used in NMR, my question is : is there ever a need to take a spectrum and do a computer analysis of it that isn't just about matching peaks to a database of known signals? What other nifty things can be done with a spectrum once it is obtained?
As Mad Scientist said, many spectroscopy methods are already based on Fourier transform of a time-dependent observable. (This is not true of the basic spectrophotometry measurements, where you directly measure the transmittance as a function of frequency.) NMR is a method based on Fourier transform, as is FTIR.

Regarding NMR signal processing, it's a very wide field, which includes interesting research ranging from academic interest to software development. So, it's hard to answer your question other than saying “yes, of course, further processing of NMR signals is sometime possible”. One thing you have to realize is: while you have probably learnt proton NMR during your studies, this is only one NMR technique, and the simplest of them all. There are plenty of different NMR techniques, depending on the type of excitation and sequence of RF pulses, many of them being multidimensional. Each method can use a variety of signal processing techniques.