L.I. Dvoiris – Dr.Sc.(Eng.), Professor

I.N. Kryukov – Dr.Sc.(Eng.), Professor

Yu.V. Buglak – Post-graduate student

Problem statement. Stationary random processes (processes whose probabilistic characteristics do not depend on time t and depend only on the interval t2- t1), whose time averages coincide with the set averages, are called ergodic, and this property of processes is called ergodicity. When comparing a theory with an experiment, it is necessary to compare the calculated values of certain physical quantities with the experimental data. Usually, only the average values of phase functions for all States corresponding to a given energy (so-called phase averages) are easily determined theoretically. On the other hand, since the measurement of any physical quantity takes a finite time, and is large in terms of the speed of processes, the result of any measurement is the average time (i.e. along the trajectory) of the corresponding phase function. Thus, to compare experimental data with theoretical data, it is necessary to justify the replacement of time averages with phase ones. A system in which the phase averages coincide with the time averages is called an ergodic system. Finding out the conditions under which the system is ergodic is the main task of the Ergodic theory. Attempts to establish the conditions of ergodicity of the physical system were made by L. Boltzmann, but the first mathematically strict result was obtained only in 1931. Birkhoff, who proved that the system is ergodic if and only if its phase space cannot be divided into the sum of two invariant (i.e., consisting of entire trajectories) sets, each of which has a positive volume. Ergodicity of processes is important because the observation of a large number of implementations of a random process can be replaced by the observation of a single, but rather long, implementation. The obtained characteristics of seismic signal implementations (mathematical expectation, variance, correlation function, spectral density, etc.) will coincide with those obtained by processing a large number of implementations with sufficient accuracy. Building hardware platforms for detecting and recognizing signals from potential violators involves investigating the stationary and ergodic nature of signals. The results of such studies depend on the model of digital signal processing, in particular the reduction to stationarity and ergodicity. An increase in the number of studied properties of interference signals contributes to an increase in the number of detection models. At the same time, each property of the General population of the studied signals is used as a component of a multidimensional feature space with an assessment of the most informative components. For the General set of seismic signals reduced to stationarity and ergodicity, the next step is to search for a method for forming an invariant feature space. Invariance will ensure the independence from the type of soil, weather conditions and other identified factors. Detection and classification models based on invariant feature space can ensure high quality of operation regardless of weather conditions and time of year, which will reduce the cost of financial and human resources for servicing such systems as part of existing (prospective) border systems and complexes. Goal: To develop and test a mathematical apparatus for preliminary research of signals from objects of detection and interference on stationarity and ergodicity. Results: A technique for evaluating the probabilistic properties of passive detection signals has been developed. Estimates of signals and interference for belonging to the class of stationary and ergodic processes are obtained. Practical significance: The proposed signal research technique is used to create quasi-optimal detectors at the stage of conceptual and outline design of digital signal processing models.

1. Tikhonov V.I. Statisticheskaya radiotekhnika. Izd. 2-e, pererab. i dop. M.: Radio i svyaz. 1982. 624 s. (in Russian)
2. Levin B.R., Shvarts V. Veroyatnostnye modeli v sistemakh svyazi i upravleniya. M: Radio i svyaz. 1985. 312 s. (Statisticheskaya teoriya svyazi. № 24). (in Russian)
3. Sergienko A.B. Tsifrovaya obrabotka signalov. SPb.: Piter. 2002. 608 s. (in Russian)
4. Boks Dzh., Dzhenkins G. Analiz vremennykh ryadov. Prognoz i upravlenie. M: Mir. 1974. 406 s. (in Russian)