diy spectroscopy w/o diffraction

Exploring a cheapo diy spectrometry possibility.

The standard spectro meter uses eg. reflected light bounced off a diff. grating. One modern twist is to put this light onto a ccd, then diferent parts of ccd get different colors, and the resultant intensity vs. color is the spectrum.  No moving parts.

However another poss. I am trying to grok is doing this without a grating,  instead using eg rgb leds. It would require de convoluting a signal that is due to known combo of led spectra * unknown reflection spectrum. I cant figure if the information is 'inthere' or hopelessly smeared out.  Take for example 2 leds and vary their percent contribution to illumination,  call this percent p,  freq. is w, and the individual led spectra are led1(w) and led2(w) . The combined illumination is then

L(w,p)= p led1(w) + (1-p) led2(w)

Assuming the sample is 'nice' and doesnt play any frequency shifting tricks (i believe this is quite rare and limited eg to some crystals that can do frequency doubling, anyway it seems this must be an assumption of optical spectroscopy ) then it has an unknown  reflection spectrum R(w) to be found.

Finally assume that the detector has flat spectrum - this can be done by normalizing the actual response, i dont think it helps anything to include the detector sensitivity curve in this analysis. Use two or more detectors e.g. Rgb values from a ccd if that helps matters, I dont think it does. So the problem is to use the detected signal D(p) from the light detector/ccd, which is a weighted average of the intensities at each freq., to find R.

======== ============= ========== ==============

D(p)= integ over w of  (L(w,p)*R(w))

= integ over w of ( p led1(w) R(w) + (1-p) led2(w) R(w))

Knowing D(p) (and led1, led2) can R be found?

======== ============= =========== =============

Alternatively / maybe equivalently, is there only one solution for R ( in which case numeric solution is possible if nothing else). Use 3 leds if necessary, etc. Differentiating wrt p does not seem to help matters. Any maths whizzes able to shed some spectrum on this?

Take a gander at [see link] which does something close, I am not convinced he extracts a spectrum however.