On a matrix method for calculating the probability of adjacent errors in a channel with Rayleigh fades

Abstract

A new numerical method for finding the probability of adjacent errors in a channel with Rayleigh fades is proposed. Within the framework of the Markov model of normal independent random processes forming a complex random process that simulates signal fading, it is shown that the joint probability density of the module of the complex envelope (CO) of the signal at adjacent time points can be represented as the product of the probability density of the module of the CO signal at the initial time and the conditional probability densities of the module of the CO signal at adjacent time points. The probability of occurrence of two or more consecutive erroneous symbols (adjacent errors) in a channel with fading is defined as the averaging of the corresponding probability in a channel with white Gaussian noise over the joint probability density of the CO signal module at adjacent time points. By approximating a multidimensional integral with finite integral sums and dividing the joint probability density into multipliers, we can find the desired probability of adjacent errors in the form of a multiple product of several typical matrices. These matrices can be calculated in advance using conditional probability densities of the signal modulus at adjacent time points. The proposed algorithm has no restrictions on the type of modulation. The results of the calculation and numerical experiment to determine the probabilities of the occurrence of adjacent errors are presented.