Application of Hypergeometric Functions of Several Variables in the Mathematical Theory of Communication: Evaluation of
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Application of Hypergeometric Functions of Several Variables in the Mathematical Theory of Communication: Evaluation of Error Probability in Fading Singlechannel System Yu. A. Brychkov1* and N. V. Savischenko2** (Submitted by A. M. Elizarov) 1
Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Dorodnicyn Computing Center, Moscow, 119333 Russia 2
Military Telecommunications Academy, St. Petersburg, 194064 Russia Received March 6, 2020; revised April 1, 2020; accepted April 10, 2020
Abstract—A method is proposed of evaluation of symbol and/or bit error probabilities for coherent receiving of multipositional signal constructions in communication channel with fadings, which are described with the help of classical and generalized models Multiple-Wave with Diffuse Power (MWDP) fading and of additive white Gaussian noise (AWGN). This method uses the hypergeometric functions of several variables. DOI: 10.1134/S1995080220100066 Keywords and phrases: fading channels, Rayleigh fading, Rician fading, Two Wave with Diffuse Power (TWDP), Multiple-Wave with Diffuse Power (MWDP), B-function, hypergeometric functions, special functions, bit error rate, symbol error rate.
1. INTRODUCTION One of the main problems in mathematical theory of communication is evaluation of the most important characteristics of information transmission system to which in particular include noise immunity (probability of error reception) and information transfer rate. Knowledge of these indicators allows us to evaluate quality and quantity of this information, respectively. We can separate two main types of communication channels for which error probabilities are defined most often: deterministic communication channel with additive white Gaussian noise (AWGN) and communication channel with general fadings (frequency non-selective) and AWGN. Decision of two interrelated problems is required for evaluation of error probability in the communication channel with fadings [1]. The first problem is calculation of the error probability reception of signal constructions in communication channel with deterministic parameters and AWGN. The second problem lies in the choice of mathematic model of fadings which is adequate to the real process in the selected wave range. Supposing that the quadrature components have Gaussian distribution (Gaussian communication channel) which is based on the central limiting theorem we are led to known laws of Rayleigh, Rice, Hoyt, Beckmann and fourparametric law [1]. Another law which is frequently used for description of fadings is the Nakagami m-distribution [2]. Its special cases are one-sided normal distribution (m = 1/2) and the Rayleigh distribution (m = 1). A number of new probability density function is suggested in [3], where an idea from [4] was developed. Physical basis consists in selecting in the multibeam model, in general case, some powerful beams (two beams in [4]) with deterministic amplitudes and homogeneous distributed phase, and a remaining set of less powerful received beams ac
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