The article considers the problem of detecting a useful harmonic signal in noise for a small signal-to-noise ratio. A new detection mechanism based on the construction of a stochastic matrix based on a single signal window is proposed. The properties of the considered limit matrices and their limit vectors are investigated and established, for which the corresponding lemmas and theorems are formulated and proved. Formulas are obtained that relate these vectors to the noise envelope with known parameters.
Keywords: signal processing, multiple improper integrals, Fourier transform, signal detection in noise, random stochastic matrix
Received October 8, 2025
Revised December 18, 2025
Accepted December 22, 2025
Andrey Alexeevich Galyaev, Dr. Eng. Sci., Institute of Control Sciences of the Russian Academy of Sciences, Moscow, 117997 Russia, e-mail: galaev@ipu.ru
Leonid Mikhailovich Berlin, Institute of Control Sciences of the Russian Academy of Sciences, Moscow, 117997 Russia, e-mail: berlin.lm@phystech.edu
Pavel Vladimirovich Lysenko, Cand. Sci (Phys.-Math.), Institute of Control Sciences of the Russian Academy of Sciences, Moscow, 117997 Russia, e-mail: pavellysen@ipu.ru
Vladimir Georgievich Babikov, Cand. Sci (Phys.-Math.), Institute of Control Sciences of the Russian Academy of Sciences, Moscow, 117997 Russia, e-mail: babikov@ipu.ru
Anton Pavlovich Potapov, Institute of Control Sciences of the Russian Academy of Sciences, Moscow, 117997 Russia, e-mail: potapov@ipu.ru
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Cite this article as: A.A. Galyaev, L.M. Berlin, P.V. Lysenko, V.G. Babikov, A.P. Potapov. Properties of random stochastic matrices in the detection problem for a small signal-to-noise ratio. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 1, pp. 88–104.
