Model of square wave noise based on telegraph process

This article addresses the problem of mathematical modeling of square wave noise in electromagnetic signals, particularly in eddy current defectograms, to generate high-quality synthetic samples for training machine learning algorithms to detect and suppress square wave noise in data. A comprehensive study of naive models is conducted: a deterministic square wave signal, a square wave signal with white noise, and a telegraph process with white noise. The telegraph process with white noise serves as the central object of the study. For this model, stationary characteristics are analytically derived: the limiting probability density function and the autocorrelation function. To estimate the model parameters, a fully Bayesian approach is proposed and implemented for the first time, utilizing Gibbs sampling and the Forward Filtering Backward Sampling (FFBS) algorithm to efficiently marginalize the hidden Markov states. The parameter estimation algorithm converges rapidly, reaching an overall variance of $1e{-}6$ value by the 1500th iteration. It is established that classical models possess fundamental limitations due to the unrealistic assumption of a strictly constant period and duty cycle. It is shown that while the telegraph process resolves the issue of stochastic pulse durations, ignoring the continuity of transition fronts leads to a mathematical artifact — a shift in the modes of the theoretical limiting distribution compared to the empirical one. Furthermore, it is demonstrated that the absence of a low-pass filtering mechanism deprives the model's autocorrelation function of its characteristic oscillating component. The experimental confirmation of the significance of these factors justifies the direction for further research: the development of modified stochastic models integrating smooth state-switching mechanisms for the adequate simulation of square wave noise. Eddy current rail defectograms served as the empirical base for testing the models. Nevertheless, the developed mathematical framework can be successfully applied to model square wave noise in other types of electromagnetic signals, such as in ECG and magnetotelluric sounding.

square wave noise, random telegraph noise, burst noise, telegraph process, eddy current testing

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