V.M. Artyushenko¹, V.I. Volovach²

1 Public Budgetary Educational Institution of Higher Education of Moscow Region «Technological University»  (Korolev, Russia)

2 Federal Public Budgetary Educational Institution of Higher Education «Volga Region State University of Service» (Toglyatti, Russia)

Considered and analyzed are the issues of using recurrent methods for detecting single disturbances, first of all, the cumulative sum algorithm (CSA) considered in this article with a reflecting screen when the signal is affected by noise with a non-Gaussian distribution. It is assumed that the time of arrival of the detected signal is unknown, and the value of the signal-to-noise ratio (SNR) has no limitations.

The CSA is described in relation to the case under consideration; the conditions for the absence and presence of discord are determined. It is noted that the value of the upper threshold determines the probability of skipping a disorder, and is also related to the required probability of false detection.

The implementation of CSA in applied problems is considered. It is noted that in real conditions non-Gaussian noise can act at the input of the jitter detector and then the «standard» assumption of the normal nature of the cumulative sum is violated, which leads to inaccurate estimates of the probability of type I and II errors. You should also take into account the growth of the SNR at the detector input. It is shown that under the aforementioned conditions, the calculation of the probability of false alarm errors and correct detection of a disorder can be performed on the basis of the statistical characteristics of the CSA obtained by modeling real processes. The process is described and the results of mathematical modeling both with reflective and absorbing screens and without screens are presented. The histograms of the distribution of the cumulative sum values are obtained, including the case of the presence of a reflecting screen at zero, as well as the numerical values of the parameters of the probability distribution density (PDD) of the cumulative sums. It is noted that the graphical dependence of the PDD of the cumulative sums in the presence of a reflecting screen approaches the logarithmically normal distribution.

The histogram of the distribution of the cumulative sum values without reflecting and absorbing screens is presented. For the latter case, data on the lower boundary of the truncated distribution are given. The growth of the SNR in this case brings the PDD of the cumulative sums closer to the Gaussian distribution. The graphical dependences of the value of the cumulative sum averaged over the realizations and its variance are obtained, which increase in proportion to the SNR. The parameters of linear regression of the mean values of the cumulative sum are obtained. A linear relationship has been established between the average values of the cumulative sum taken in separate sections.

The distribution densities of the cumulative sum obtained in the course of modeling in the absence of a signal and after a disorder make it possible to estimate the probabilities of linear detection errors and signal skipping. Expressions are given that make it possible to determine the named probabilities for a precisely known signal and asymptotic normal statistics of the likelihood ratio describing a quasi-optimal detector.

With the difference of the noise applied in the calculations of the PDD affecting the useful signal from the true form of the PDD, it is proposed to use a coefficient that takes into account the change in the SNR caused by the mismatch of the characteristics of the nonlinear conversion unit and the real PDF of the noise. The graphical dependencies of the comparative efficiency of the CSA acting noise, which is matched and not matched with the PDD are given. It is noted that large values of the SNR lead to the convergence of the above curves. At small values of the SNR, taking into account a priori information about the PDD of the noise affecting the signal allows one to obtain a significant gain, which increases with a greater difference between the PDD of the acting noise from the Gaussian one.

Artyushenko V.M., Volovach V.I. Cumulative sums algorithm modeling in applied problems of detecting signals with a random moment appearance in non-Gaussian noise. Radiotekhnika. 2021. V. 85. № 4. P. 66−76. DOI: https://doi.org/10.18127/j00338486-202104-08 (In Russian)

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