convolution terminology

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Fri, 07/17/2015 - 07:29 am

Hi all,

Assuming that we have a Raman envelope and we need to separate ovelapping peaks and then fit to define all necessary parameters (position,height,width). This process is traditionally called deconvolution. However, looking on terminology on internet i am not sure if this is the right term......Are we looking at the convolution of ovelapping peaks or the sum of these peaks? Maybe it is a matter of terminology here but i am a bit confused. Any ideas? I know for example that the voigt profile that can be used to fit one individual peak is the convolution of gaussian and lorentzian and not the sum of these 2. Nevertheless this is probably a different case since we are dealing with the fitting of one peak by using a combined fitting model (voigt), while in the first case i am talking about fitting 2 or more peaks of the actual spectrum (whether we use voigt profile for each peak or gaussian is another issue). I hope i have not confused things. Thank you
In my opinion you have a sum of delta peaks (resonance frequencies of your oscillators) multiplied by an amplitude (their individual strength). This gets convoluted with functions describing lifetime (Lorentz), thermal broadening (Gauss), detector "realness" (Gauss), and a lot of etc's.
Depending on operators and functions you might apply chain rules, commutative rules, distributive rules etc. and readjust the point of view (e.g., to sums of convoluted functions instead of a convolution of a sum).

I use the terms "deconvolution" for reconstructed data and "curve decomposition" for fitted data.
Maybe you want to read:
https://en.wikipedia.org/wiki/Spectral_line_shape#Applications
https://en.wikipedia.org/wiki/Deconvolution
https://en.wikipedia.org/wiki/Curve_fitting

Hope it help,
HJ

July 17, 2015 at 08:20 am - Permalink

Thanks for your prompt reply......Essentially we have two things: 1). Each Raman peak could be the result of of different processes such as: doppler broadening (gaussian profile) and line broadening (lorentzian profile). Due to these effects someone could model a Raman peak using voigt profile which is the convolution of a gaussian and a lorentzian model. 2). My understanding is that when we need to separate two ovelapping peaks that have resulted probably due to spectral resolution limitations of the equipment, the process is called ''decomposition'' of peaks.........Would that make sense? The reason why i am bringing this issue is because i was discussing with a guy who is not familiar with Raman and i was explaining to him that an assymetric feature on my Raman spectrum was the convolution of two individual peaks assigned to certain components of my sample. So, he asked me: is it a convolution or the sum of the peaks? It was then when i realised that probably using ''decomposition'' of these overlapping peaks would have probably been a better term, since i am trying to fit 2 different peaks. I understand that in this case we are dealing with the sum of 2 peaks and we are trying to separate them and fit them. The term ''convolution'' is probably better used to describe the nature of a SINGLE raman peak due to the effects that i mentioned at the beginning. However, i have seen papers where people do not fit voigt profiles on Raman peaks but just Gaussian profiles (especially for solids or powders). I guess it comes down the personal interpretation of each person. I apologise for the long text

July 17, 2015 at 08:45 am - Permalink

Deconvolution involves integrals. Data is usually quantized and integrals technically become sums. Short story long: you get (sums of)^n {calculated, measured} values.

Any other opinions?
HJ

July 17, 2015 at 09:01 am - Permalink

Just to make the issue even more confusing...

If you have a series of impulses (delta functions) that represent the intensity and location of a series of spectral lines, and a peak shape that represents what happens with various line-broadening processes, then the result can be viewed as a convolution of that peak shape with the series of impulses.

If you think about it, the integral that represents this convolution is really just a sum of peaks since the convolution of an impulse with a function is simply that function shifted and scaled. If you have two impulses, then you get a sum of two copies of the function, each shifted and scaled by the two impulses.

The fitting process (hopefully) gets you the locations and intensities as an output.

John Weeks
WaveMetrics, Inc.
support@wavemetrics.com

July 17, 2015 at 09:25 am - Permalink

Ok it makes sense i think.....So basically it is a sum of integrals (convoluted components).......However, i think that the broadening (peak shape) is also a combination (convolution) of doppler broadening coming from the instrumentation (gaussian) and spectral line broadening (lorentzian) for which reason people use voigt profile to fit individual peaks.

July 17, 2015 at 10:07 am - Permalink

I would define the terms rigorously in this way when applied to spectroscopy ...

* Peak fitting is a process where any number of well-defined component peaks are SUMMED to create a single envelope that should best fit the data.
* Convolution is a process where a true signal peak and its transmission function are INTEGRATED JOINTLY over the span of the signal peak to get a new measured signal peak.
* Deconvolution is a process where a true signal (peak) is pulled out of a measured signal (peak) that itself is a convolution of true signal * transmission function.

You are separating overlapping peaks by summing them individually to give a best fit to one peak. You are doing peak fitting not deconvolution.

On the other hand, I think that spectroscopists are generally allowed (albeit by their fellow spectroscopists) to use the term deconvolution to mean pulling out component peaks from a single peak.

Hence the confusion.

I would also note that I was once called out on a poster presentation at a national conference many years ago for using the term deconvolution of component peaks when I should have been saying peak fitting of component peaks.

--
J. J. Weimer
Chemistry / Chemical & Materials Engineering, UAHuntsville

July 17, 2015 at 06:51 pm - Permalink

Ok i am following you now....Just one question because you used the term transmission function......you are talking about the peak shape due to line broadening processes? And the true signal would be the delta function (as you mentioned in your first post) representing intensity and location? Sorry for this but i wanted to correlate terms among your 2 posts

July 18, 2015 at 03:17 am - Permalink

Konstantinos Chatzipanagis wrote:
Ok i am following you now....Just one question because you used the term transmission function......you are talking about the peak shape due to line broadening processes? And the true signal would be the delta function (as you mentioned in your first post) representing intensity and location? Sorry for this but i wanted to correlate terms among your 2 posts


Good catch!

Line broadening functions in a comparable way to transmission function. It convolves the fundamental signal. I would probably be better to state it separately, as in ...

Convolution is the joint integration of a fundamental signal with a line-broadening process or an instrument transmission function.

OTOH, speaking somewhat jokingly, a delta function is only of real interest to a theoretician. All experimentalists realize they will never be able to measure one. So, most experimentalist never talk about their spectra as delta functions convolved with line-broadening. They just say they have a measured signal that is a convolution of the true signal with an instrument transmission function. Consider also that, in most cases of peak-fitting, the component peaks that are used to sum to an envelope peak are already line-broadened and transmission-function broadened because the measured peak is that way already. IOW, most (if not all) experimentalists do not try to "peak fit" delta functions convolved with line-broadening subsequently convolved with a transmission function.

--
J. J. Weimer
Chemistry / Chemical & Materials Engineering, UAHuntsville

July 21, 2015 at 06:43 am - Permalink