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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!

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  • $\begingroup$ I believe what you are looking for Is detailed in this Agilent app note: volpefirm.com/wp-content/uploads/2017/01/… $\endgroup$ – Dan Boschen Jun 11 '20 at 0:01
  • $\begingroup$ Thank you, Dan! But this is right the problem with all manuals like that one you referenced to. I've collected some experimental data and see no distributions with a negative skewness like it is illustrated in Figure 6. of the Agilent manual referenced by you. As I wrote, the experimental distribution is more resemble to the third expression I gave in my initial question. In order to be sure that I have Rayleigh signal on the input of a lin-log amplifier, I collected cm-wave radar signals from rains. The latter is seen as a classical case of fluctuations with a Rayleigh distribution. $\endgroup$ – Sergey Jun 14 '20 at 11:37
  • $\begingroup$ Interesting, could you share a link to your data? I have a basic understanding of how log-limiting amplifiers work under the hood having worked on those designs in my past which is explained well in this link by ADI : analog.com/en/analog-dialogue/articles/…. Can you look at that and tell me if this is identical to your use of "lin-log" amplifier? $\endgroup$ – Dan Boschen Jun 14 '20 at 13:23
  • $\begingroup$ Note that the output of the log limiting amplifier is the limited signal but the detected output (which is proportional to the log of the input signal) is what would be distributed according to the log-Rayleigh distribution. Perhaps this is what is leading to your discrepancy? $\endgroup$ – Dan Boschen Jun 14 '20 at 14:20
  • $\begingroup$ No, I used not log-limiting amplifier. You can find a short consideration of the scheme here digikey.com/en/articles/… in the chapter "The multistage log amp". The functional scheme was very old-fashioned i.e. a lin-detector (as an inividual separated unit) followed by a lin-log amplifier (also a separated unit). The point of starting taking the logarithm laid -20dB below rms of the channel noise. $\endgroup$ – Sergey Jun 14 '20 at 16:41
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The distribution of the log magnitude output of a log amplifier would be according to the log of the magnitude of the input. Although in implementation the log amp is not simply taking a log of the input signal, the result is indeed the log with reasonable accuracy (typically linear to the log scale within a fraction of a dB). Below shows how this should appear for both Rayleigh and Ricean distributions. Given the OP is not seeing the same skewness as is the case for the Rayleigh at the log magnitude output of the log amp, I would conclude that the input signal (which he does not have access to) was not Rayleigh distributed.

Notice the skewness decreases significantly as the dominant path in the Ricean distribution (A) becomes more prominent, as would be expected.

Comparative Plots

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