Measuring a linear-response, in the time domain one measures a convolution of the to-be-measured response function with the external signal, and in the frequency domain a multiplication of the Fourier transforms of these. If follows that by calculating the time-Fourier transform of the measured signal and dividing it by the time-Fourier transform of the external signal, one obtains the time-Fourier transform of the sought-after linear-response function. For sufficiently weak external signal, the outcome is independent of the shape of the external signal, whether sinusoidal or otherwise. I think nowadays most measuring devises are equipped with some computational devise that is capable of performing Fast Fourier Transformation (FFT). Using a periodic signal, one can achieve the same objective by using band-pass filters.

@Behnam, in many spectroscopies we do not have signals that easily conform with the idea(s) you have mentioned. In Raman scattering in particular, I am not sure how to correctly think about your proposal...

@Vincent, you don't give any hint as to the nature of the material and the excitations you're looking into. That makes it not easy to think about what a good answer could be. Now, you've tagged your question with "strong correlations" and I'll take that as in indicator. My answer (certainly far from being satisfactory) will center around heavy fermion materials.

As far as I know, there is not much experimental material available. There is one paper on CeCoIn5 and/or a related compounds [PRB (2007), see links below]. I mention it because the make allusion to some Fano type profile in their analysis. Zachary, Fisk & Pines et al. do refer to this paper in the context of their developing the "two liquid scenario" for heavy fermions. Personally, I didn't get much out of that paper and as far as I see it, it stands there pretty singular.

On the other hand, we have recently been doing Raman experiments on CePt5 intermetallics, see link below  [paper in review, hopefully out soon:-]. There, we found vibronic and electronic features, we attribute the latter to crystal field excitations of the Ce4f electrons. In evaluating and interpreting these experiments we did not need to recur to Fano type profiles.

On the general side, Fano type profiles are an expression of processes inherent to your material. I would not think therefore, that something like a spectral function "before interaction" actually makes proper sense. The challenge might be to identify the "interaction paths" whose interference give rise to the actual Fano profile. (But then, I have not yet encountered square-like Fano profiles). In this respect, I have not at all understood the contents of the paper I mentioned first. It might nevertheless be of interest to you.

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.75.045108

Article Study of crystal-field splitting in ultrathin CePt5 layers b...

Article Vibrational and electronic excitations in the (Ce, La) MIn_ ...

 @Vincent again,

re-reading your post, I realize that I have probably misunderstood it. Is the following correct?

You have Raman process which would yield a square-like spectral profile weren't it for an additional interaction. Now you think that there is an interaction that should give rise to something like a Fano-type modification of that profile and you are looking for a mathematical expression for the end result.

Spontaneously I have two remarks: In the constexts I have dealt with Fano profiles (resonant photoemission and x-ray absorption) it is important to have two distinct interaction channels leading to the same, indistinguishable final state. the typical Fano profile arises, when one of them has a slow change of phase with energy/frequency and the other one a fast one (typically a resonance). You then wirite down the matrix elements and take the square of the sum, leading to some interference part, yielding the Fano shape. There are probably many instances in the literature exposing the procedure, I have learnt it from studying the article by Almbladh & Hedin, "Beyond the single particle approximation" (or a similar title) in the "Handbook of Synchrotron radiation", Vol. 1b (1983). Though somewhat dated on the more advanced topics now, I still love the style of it.

For me the question would be how the Raman process is adequately formulated in such a context, since anyway it is not the same as a "usual"absorption with Lorentzian shape, is it? My skills end at this point...

Dear Kai, could you please elaborate? Use of band-pass filters requires a signal that is periodic, so that one can select out one of many discrete frequencies and work with it. Use of the Fourier transformation imposes no fundamental limitation on the form of the signal, except that it be sufficiently weak so that contributions of higher-order responses can be neglected. Naturally, FFT should be used with the necessary cautions (aliasing, etc.), but this is what one is also confronted with in using this technique in numerical calculations, where one knows how to avoid the well-known pitfalls.

Well, I may be mistaken, but Raman spectroscopy is about inelastic light scattering, so you record intensity over photon energy or wavelength (scanning a monochromator or using a monochromator to illuminate a 1D-camera) in a limited spectral range [limited towards both small energies by spectral sensitivity or range of interest] and towards high energy by how far you can approach the excitation line without over-illuminating your sensor. So the signal is not periodic in any 'dimension'.

Similar to many other cases of spectroscopy with very limited spectral range actually covered, I'd be very careful to make sure that Fourier techniques do not do more harm than good. I do use them sometimes for the purpose of convolutions but deconvolutions are very sensitive to noise, which may present another issue, depending on the data. If you look at our Raman data in the manuscript I linked then I see a fundamental limitation here (we were pretty happy to have seen what we have seen...)