I struggle with two simple questions.
First. What is the distribution of the signal on the output of the lin-log amplifier if Rayleigh noise is driven to the input of it.
Using an approach with the inverse function it is possible to obtain the expression for the distribution of the output signal. It looks like the First Extreme Value Distribution (i.e. with $\exp(\exp(x))$ pattern). In other words, if $f_{in}$ is the input of the lin-log amplifier: $$ f_{in} = \frac{r}{D}e^{\frac{-r^{2}}{2D}} $$ one can get $f_{out}$ - the distribution of the output signal: $$ f_{out} = \frac{e^{-\frac{e^{2x}}{2D}-2x}}{D} $$
But I found a statement somewhere in engineers forums that a lin-log amplifier works more trickily and "takes a logarithm of an envelope" and "there were a lot of problems with erroneous simulations of lin-log signal transformation". It was just statements without explanations that gave me feed for thoughts. I found experimental data. It looks more like: $$ f_{out} = \frac{\log(x)}{K_{1}}e^{\frac{-\log(r)^{2}}{K_{2}}} $$ I'm afraid that it is very similar to the First Extreme Value Distribution and they are indistinguishable in an experimental environment. For example, the additional logarithm of both of them results in Gaussian-like distributions and the difference between them are again almost completely indistinguishable.
Second. How to simulate the lin-log amplifier (detector) in R or MATLAB. The problem is that taking the logarithm of a random variable with Rayleigh's distribution results in a distribution with a negative skewness which does absolutely not resemble the above-mentioned alternatives expected. The same distribution with a negative skewness one can get using normally distributed in-phase and quadrature components of the signal.
Is there anybody who understands the problem and could share the idea on how to represent the output of the lin-log detector mathematically and simulate it numerically?
Thanks!