**In the previous post, **I outlined the reasoning steps taken in my last paper [1] which claimed that random modulation (RanM) is not a valid EMI reduction technique. One of the most important arguments was that the technique is on average not better than deterministic modulation (DetM) in the context of their influence on probability of error in systems with serial communication devices. In other words: if the system is vulnerable, and the communication may be corrupted (i.e. flipped bits) due to the presence of switching power converters, then it does not matter whether the converter is switching with a fixed frequency or several randomly changing frequencies.

Okay, but how can we show that? One possibility is of course to **devise experiments** which compare the two modulation techniques. This is what we did, in fact. However, experiments, being the "true" source of data (bearing in mind what Popper [2], Sady [3] or other philosophers of science think about such statement) are very limited in their usefulness. Sometimes there are **too many parameters** (and the reviewers often want you to incorporate even more) or some of them are very **difficult to control**. If we were to construct an experiment taking into account all these, we would have spent a year not leaving the lab, just to perform this simple experiment.

What comes next is therefore **modeling**. In this context, there are two main ways of modelling that I can think of:

1) *a posteriori*: perform the experiment with certain known set of parameters (i.e. at specific frequencies) and use the existing data for interpolation or approximation;

2) *a priori*: construct the model separately from the experiment, trying to mimic the phenomenon and making it as simple as possible (but not simpler, as Einstein? [4] would have rightly pointed out) to incorporate the parameters of interest.

In the paper we have adhered to the second option. In fact, we used existing models from [5] and [6], and enhanced them a bit - for instance simplified them for specific cases. I this post I want to present the second of these:** the diophantine equation-based model for deterministic modulation**,

Here, I don't want to dwelve too much into the details of the models, which are thoroughly explained in the paper.** **Instead, I want to show you a simple web application, which I programmed to understand better and play around with parameters of the model.

Below you can find the web application for the exploration of diophantine model.

If the application is frozen, all you have to do is to click Reload on the lower left part of the window. The application is created with R programming language [7], using the Shiny library, and published on the shinyapps server. The source code is available on my GitHub page.

** **

After you have played around a bit, it's time to explain the parameters so that you can play more. As you may recall from the previous post, the important parameters of both models are: the switching frequency (or the reciprocal of it - switching period), the data transmission frequency (or the reciprocal bit checking frequency) and the width of the interference, i.e. time the signal is above a predefined threshold.

In the diophantine model, all of the above parameters are discrete. In other words, when we refer to "time", we are actually referring to an integer *T = 0, 1, ...,* which can be thought of as a discrete event. In the above application, the switching period *M *refers to the discrete period between the next switch-ON or switch-OFF event of a transistor in the converter. Similarly, the distance between bit checking *N *refers to a discrete period between the successive bit checking times. The interference width *dt *is now discretized and represented by the parameter *D. *Furthemore,* t*he number of bits *n *is obviously an integer number (unless you think you can send 2,5 bits). The converter start moment *t0 *refers to the moment at which the converter starts switching. In our frame of reference the moment *0* is the moment of sending the first bit, therefore *t0 *can take any value from *-M + 1 *to *0. *One can easily see that for our case it suffices that the converter starts after *-M+1* and before the moment *0*, because other possibilites are redundant from the perspective of the final aim, which is to predict the probability of encountering an error in the reading.

The switching times are represented by bold vertical segment on the top axis. The successive bits are shown on the bottom axis with arrows. If the bit finds itself in the shaded region (the interference zone) it is colored red (WARNING: possible interference!), otherwise it is blue.

This tool allowed me to reason about the diophantine model, based on very simple small examples.

I hope this will spark interest of some of you as well. If you have any further queries or the app is not working I would be much obliged for contacting me.

**References:**

[1] K. Niewiadomski, R. Smolenski, P. Lezynski, J. Bojarski, D. W. P. Thomas and F. Blaabjerg, "Comparative Analysis of Deterministic and Random Modulations Based on Mathematical Models of Transmission Errors in Series Communication," in* IEEE Transactions on Power Electronics*, vol. 37, no. 10, pp. 11985-11995, Oct. 2022, doi: 10.1109/TPEL.2022.3175737.

[2] K. Popper, The Logic of Scientific Discovery. Basic Books, 1959.

[3] W. Sady, Spór o racjonalność naukową od Poincarégo do Laudana, Wrocław: FNP, 2000.

[4] https://quoteinvestigator.com/2011/05/13/einstein-simple/

[5] R. Smolenski, J. Bojarski, A. Kempski and P. Lezynski, "Time-domain-based assessment of data transmission error probability in smart grids with electromagnetic interference", *IEEE Trans. Ind. Electron.*, vol. 61, no. 4, pp. 1882-1890, May 2014.

[6] R. Smolenski, J. Bojarski, P. Lezynski and Z. Sadowski, "Diophantine equation based model of data transmission errors caused by interference generated by DC-DC converters with deterministic modulation", *Bull. Polish Acad. Sci. Tech. Sci.*, vol. 64, no. 3, pp. 575-580, Sep. 2016.

[7] R Core Team (2020). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.