# Mathematical Resolution of Overlapping Signals

I would like to take the opinion of physicists (my area is chemistry). During instrumental analysis, the output is a instrument response vs. time. Each peak corresponds to a molecule which is absorbing light. For example, in the attached figure a signal marked VII, it tells me that there are three components because there are three overlapping peaks. Here is the mathematical problem:

As you can see, actually, all roman numerals correspond to 3 signals, but in III,IV, V the three peaks are merged. I am interested in the area under each peak if it were separated to the baseline. Area under the curve is proportional to the number of molecules, hence this quantity is of interest.

One common option is that we assume a function such as exponentially modified Gaussian or just simple Gaussian and fit a curve by iterative curve fitting and minimization of residuals. This would assist in finding the area under each peak. The only drawback of this approach is that we need to apply a model function and fit the data.

Are there any other methods with which can mathematically resolve the peaks and be able to find the area under them. For examples, let us take IV, there are three peaks and one would like to know the area of the individual peak. Is there any method which would reduce the width of the peaks, hence resolve them, yet maintain the area corresponding to each?

Thanks.

• Did you try a FFT? What result do you get? – PackSciences Feb 23 '19 at 18:55
• How would FFT help in recovering "peak areas" of the three peaks under say I, II or IV? – M. Farooq Feb 23 '19 at 19:21
• "width" of the peak can be diminished by a FFT by using a frequency mask. – PackSciences Feb 23 '19 at 19:26
• I was not aware of it. Do you have a link for FFT being used for reducing peak width? I only know about the Fourier self-deconvolution in which we divide the FFT of the raw data by a single peak of desired width. This approach works to some extent but noise in the real data is magnified like crazy. Thanks. – M. Farooq Feb 23 '19 at 19:30
• The signal that you measure is a convolution of the true signal and the instrument response function. A way to get rid of the latter is a deconvolution procedure. However, deconvolution is cumbersome in practice, and it can only be performed if you have a model for the instrument response function. Therefore you always come back to the point that you have to apply a model. I see no better way than to do some kind of peak fit, which is much less cumbersome than a deconvolution. – flaudemus Feb 23 '19 at 19:44