## Exponential Deconvolution Using the First Derivative

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

## Product of independent random variables and tail deconvolution

Suppose $$X, Y$$ are two independent non-negative random variables. The conditions

(i) $$\mathbb{P}(X > t) = \frac{C}{t^p} + o(t^{-p})$$

(ii) $$\mathbb{P}(Y > t) = o(t^{-q})$$ for any $$q > 0$$

imply

(iii) $$\mathbb{P}(XY > t) = \frac{C \mathbb{E}[Y^p]}{t^p} + o(t^{-p})$$.

(Of course here I am talking about the asymptotic behaviour as $$t \to \infty$$ and $$p > 0$$.)

My question concerns a converse of this statement: if I know (ii) and (iii), does that imply (i)?

(While I would very much love to see that this is true, I have the impression that this claim is false but just haven’t come up with a counter-example.)

I am aware that the converse holds at the exponential level, i.e. $$\lim_{t \to \infty} \frac{\log \mathbb{P}(X > t)}{\log t} = -p.$$

One may consider random variable $$Y$$ with a density (which is sufficient for my purpose) if that helps. In case a counter-example for the “full” converse can be found, I would like to know if the “full” converse can still hold when (a) $$Y$$ is a lognormal random variable or slightly more generally (b) $$Y$$ has a tail upper bounded by that of some lognormal.

Update: to clarify, $$Y$$ is a given random variable, the distribution of which is hence given and cannot be chosen freely. In particular $$Y$$ is not a constant (otherwise the converse is trivially true unless $$Y = 0$$ a.s., in which case the converse is trivially false).

## Deconvolution using Fourier transforms

I have a 2D signal in the form of a function $$g(x_m,y_m)$$ given as

\begin{aligned} g(x_m,y_m) = \ & \int^{\infty}_{-\infty} \mathrm{d}x_o \ \mathrm{d}y_o \ \frac{1}{\varepsilon^2} P(x_o – x_m,y_o -y_m) \ & \quad \times \int^{\infty}_{-\infty} \mathrm{d}x_i \ \mathrm{d}y_i \ R(x_o – x_i,y_o -y_i)L(x_i,y_i) \end{aligned} \tag{1}

The integrals in equation $$(1)$$ can be seen as a convolution of $$P,R$$ and $$L$$ as $$g(x_m,y_m)= ((P/\varepsilon^2)*R*L)(x_m,y_m) \tag{2}$$

I want to find $$L$$ when $$g$$,$$R$$ and $$P$$ are given.

I tried to use Fourier transforms to find $$L$$ as:

$$L = \mathcal{F}^{-1} \left[ \frac{\mathcal{F}[g]}{\mathcal{F}[R] \cdot \mathcal{F}[P/\varepsilon^2]} \right]$$

R[x_, y_] := 0.609739 E^(-444.116 (-0.00244704 + x)^2 - 444.116 (-0.0322566 + y)^2) +  0.0691803 E^(-48.9858 (-0.0105298 + x)^2 -  48.9858 (0.0054951 + y)^2) +  0.66442 E^(-426.449 (0.00315949 + x)^2 -  426.449 (0.0248433 + y)^2);   g[x_, y_] := 3.06909 E^(-18.585 x^2 + x (13.6144 - 27.7795 y) + (12.2542 - 18.1432 y) y) +  0.402245 E^(-7.37814 x^2 + x (9.61481 - 5.57202 y) + (6.46554 - 6.35048 y) y) + 120.468 E^(-0.0245919 x^2 + x (-0.00325668 + 0.00197362 y) + (-0.00103919 - 0.0281421 y) y) + 0.773818 E^(-3.79704 x^2 + (-15.0351 - 16.3606 y) y + x (1.75472 + 2.86666 y)) + 0.0833316 E^(-38.7396 x^2 + (-10.6949 - 14.1731 y) y + x (32.7513 + 18.7984 y));  epsilon = 0.048; P[x_, y_] := E^(-(2/epsilon)^2 (x^2 + y^2));  FTR = FourierTransform[R[x, y], {x, y}, {u, v}, FourierParameters -> {0, -2 \[Pi]}]; FTg = FourierTransform[g[x, y], {x, y}, {u, v}, FourierParameters -> {0, -2 \[Pi]}]; FTP = FourierTransform[P[x, y], {x, y}, {u, v}, FourierParameters -> {0, -2 \[Pi]}];  InverseFourierTransform[FTg/((1/epsilon^2)FTP*FTR),{u, v}, {xi, yi}, FourierParameters -> {1,-2*Pi}] 

However, the InverseFourierTransform takes a lot of time and does not return any result. How do I find $$L(x_i,y_i)$$?, am I doing something wrong here?.

Posted on Categories cheapest proxies