V.A. Gluskin1, A.N. Dementiev2, E.А. Gondarenko3, M.Yu. Konyshev4, D.L. Myasnikov5

1 Central Scientific Research Institute of All-Union Combined Arms and Space Forces of the Ministry of Defense
 of the Russian Federation (Moscow, Russia),
2,4,5 MIREA – Russian University of Technology (Moscow, Russia)
 3 Committee of the Navy of the Ministry of Defense of Russia
 5 myasnikov.danil@internet.ru

An approach to obtaining estimates of binary Markov chains by sampling with gaps on the basis of taking into account the properties of binary Markov chains is proposed. The proposed approach allows obtaining suitable estimates in the form of values of elements of transient probability matrices for error sources in discrete communication channels when modeling them by binary Markov chains. It is expedient to use the received estimations at realization of algorithms of decoding of noise-immune codes on the basis of criteria of maximum of a posteriori probability and minimum of average risk. Quality estimations for the received estimations in comparison with the estimations received on the basis of Bernoulli model are presented.

High rates of informatization of modern society have led to a significant increase in demand for telecommunication services. Under the conditions of limitations on frequency and energy resources, the increase in specific speeds of information transmission in information transmission systems (ITS) while ensuring the specified requirements to the reliability of message reception requires an increase in the information efficiency of ITS.

It is known that the most general model for describing random processes (RP) is the so-called general Markov model (GMM), reflecting the dependence of probabilities of states of a random variable on (in general case) a set of previous states. The theory of Markov processes, which is actually the basis of the modern theory of statistical radio engineering, allowed, in conjunction with the Bayesian approach, to obtain theoretically justified optimal decisive rules for synthesizing signal processing devices in noise of different nature. Nevertheless, the results obtained so far in the classical theory of Markov processes do not allow us to give an answer to a number of important for theory and practice questions concerning the distinction of statistical hypotheses under conditions of uncertainty about the parameters of Markov processes. In particular, these include the questions of estimating the parameters of Markov processes, forming realizations of processes with a given distribution of vectors, and analyzing the convergence of processes to the final probability distributions. The application of the theory of Markov processes in statistical radio engineering is related to the Bayesian approach to the synthesis of optimal signal processing devices.

Since the distribution of final probabilities of a one-dimensional BIP influences the distribution of final probabilities of a two-dimensional BIP, the nature of this dependence is of considerable interest. Let us analyze, using expression (2), which determines the equality of the final probabilities of two-dimensional BIPs, to find the limiting values of the final probabilities of two-dimensional BIPs.

The description approach presented in the paper allows not only to fully determine the statistical description of the first-order CM, but also characterizes the chain from the point of view of the "deviation" of correlation relations from the values, the information about which is taken into account at the previous scale of the CM consideration (in this case, when describing a binary random process based on the Bernoulli scheme). The availability of a priori information about one-dimensional BIP allows to limit the range of possible values of two-dimensional BIP. As a result, it becomes possible to obtain reasonable estimates of the two-dimensional BIR.

Since in practice the problems of estimation of CM parameters need to be solved in the presence of a priori uncertainty about the probabilities of a one-dimensional random variable and limited sample size, the problem of obtaining suitable estimates of probabilities of a one-dimensional random variable arises. In this case, a suitable estimate of the one-dimensional random variable is the relative frequency of the event, the accuracy and reliability of which are related to the volume of observations according to the Laplace formula.

Thus, the approach to estimating the parameters of the Markov stationary binary random process model, which takes into account the interrelation of the final probability distributions of binary vectors of different dimensionality, including expressions for the estimates of the elements of the transient probability matrix of a single-link binary Markov chain, which allows us to obtain suitable estimates of the model parameters by sampling with missing data, has been proposed in the aggregate. The proposed approach is based on the preliminary obtaining of probability estimates of a one-dimensional binary quantity (in practice, it is convenient to use the error probability for error sources). The presented dependencies also allow us to solve the problem of simulation modeling of binary vectors when describing them on the basis of CM.

Gluskin V.A., Dementiev A.N., Gondarenko E.А., Konyshev M.Yu., Myasnikov D.L. Estimation of error source parameters in discrete communication channels with error clustering. Dynamics of complex systems. 2023. V. 17. № 4. P. 56−69. DOI: 10.18127/j19997493-202304-06 (in Russian)

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