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## fourier remodel – Problem with an exponentially modified Gaussian formulation

Sometimes in experimental research, I necessity to reconcile a peak with an exponentially modified Gaussian (EMG) duty. One of the favored formulations of EMG is

$$

y=frac{a_{0}}{2 a_{3}} exp left(frac{a_{2}^{2}}{2 a_{3}^{2}}+frac{a_{1}-x}{a_{3}}privilege)left[operatorname{erf}left(frac{x-a_{1}}{sqrt{2} a_{2}}-frac{a_{2}}{sqrt{2} a_{3}}right)+frac{a_{3}}{left|a_{3}right|}right]
$$

$$

commence{array}{l}

a_{0}=textual content { region }

a_{1}=textual content { focus of the absolute Gaussian }

a_{2}=textual content { width }(>0)

a_{3}=textual content { distortion = time ceaseless }(neq 0)

aim{array}

$$

One level which has been bothering me for some time is that inescapable mixtures of width and distortion result in zeros or senseless outcomes though these mixtures are bodily vivid.

Take for instance, a peak with an region of 1 centered at 30, its Gaussian benchmark aberration is 2 and distortion is 0.9, the above method works very nicely.

However, if we attempt to make the height symmetric, i.e., make the distortion 0.09, the outcome is zero, though a time ceaseless of 0.09 could be very mighty workable. It isn’t a software program mistake (tried Excel or Matlab)- one other unbiased scientist too tried it and the identical factor occurred.

**Alternatively** Numerically, I can employ a discrete Fourier remodel to convolute a Gaussian and an exponential rot with a time ceaseless of 0.09. I get the specified peak. The outcome of DFT convolution with width of two and time ceaseless of 0.9 match completely with the above talked about method.

What is the mathematical intuition for the above equation to fail at inescapable mixtures of width and distortion? Say $a_{2}$=2 and $a_{3}$=0.09, the outcome is zero nevertheless it works for $a_{3}$=0.9.

Thanks.

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