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There is an interesting observation using the first derivative to deconvolve an exponentially modified Gaussian:

The animation is here, https://terpconnect.umd.edu/~toh/spectrum/SymmetricalizationAnimation.gif

The main idea is that if we have an Exponentially Modified Gaussian (EMG) function, and we add a small fraction of first derivative to the original EMG, it results in recovering the original Gaussian while preserving the original area. The constant multiplier is the 1/time constant of the EMG. This is a very useful property.

Has anyone seen this deconvoluting property of the first derivative mentioned elsewhere in mathematical literature? An early reference from the 1960s from a Chemistry paper shows a picture a similar picture. This observation was just by chance, I am looking for a fundamental connection and if the first derivative can be used to deconvolute other types of convolutions besides the exponential ones.

Thanks. Sharpening by using first derivative

Ref: J. W., and Charles N. Reilley. "De-tailing and sharpening of response peaks in gas chromatography." Analytical Chemistry 37, (1965), 626-630.

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Recent literature on the exponential deconvolution technique includes:

Reconstruction of exponentially modified functions (2019)
"We prove that any function, convolved with exponent, can be reconstructed by adding of its own derivative, multiplied by exponential decay time constant."

Exponentially Modified Peak Functions in Biomedical Sciences and Related Disciplines (2017)

Comparison of integration rules in the case of very narrow chromatographic peaks (2018)

Reconstruction of chromatographic peaks using the exponentially modified Gaussian function (2011)

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  • $\begingroup$ Dear Carlo, The author of the first link Reconstruction of exponentially modified functions (2019) posted my private communication on ResearchGate without my permission, which was quite inappropriate of him. I had brought this idea to his attention for exponentially modified Gaussians privately. Now it is known that the original idea appeared in 1960s (see original figure). The main point I am interested is that can the derivative be used to deconvolve any other type of convolution besides the exponential decay? $\endgroup$ – M. Farooq Jun 29 at 12:25

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