Multi-position signal coherent reception error probability in a channel with generalized gamma or K fading and white noise

Abstract

Introduction:One of the main problems in communication theory is giving definitions to such characteristics of an information transmission system as noise immunity (error probability) and transfer rate. Their knowledge allows you to determine the transmitted information quality and quantity, respectively. The calculation of the error probability for a communications channel (for example, with fading) allows you to estimate the loss or gain in noise immunity with modems of various signal designs.Purpose:Developing a technique for calculating the probability of a bit error with coherent reception of signals in a communication channel with additive white Gaussian noise and general (non-selective in terms of frequency) fading described by gamma distribution orKdistribution (called gamma fading andKfading respectively).Results:The obtained relations allow you to calculate the reception noise immunity in arbitrary twodimensional signaling designs for a communication channel with gamma orKfading and additive white Gaussian noise. Examples are given of bit error probability calculation for phase modulation and quadrature-amplitude modulation signals. The calculation of error probability in a channel with fading for two different distribution laws was reduced to a new special function which is an integral of the product of Tricomi function and an algebraic function. This allowed us to develop a universal mathematical approach valid for both variants of fading. To calculate the new special function introduced in the article and its representation through the known functions, we used both classical hypergeometric functions and a generalized hypergeometric functions of two variables which is Kampé de Fériet function.Practical relevance:When using multi-position signal structures in new telecommunication standards, this technique allows you to obtain an estimate of their potential noise immunity, objectively comparing signal constructions and correctly choosing the best options for new modem developer.