On the Distribution Function of the Generalized Beckmann Random Variable and Its Applications in Communications

The Beckmann distribution has a wide range of applications in radio-frequency communications, free-space optical (FSO) communications, and underwater wireless optical communications (UWOC). However, the cumulative distribution function (cdf) of the Beckmann random variable (RV) does not have a closed-form expression, which makes it challenging to derive analytical solutions for the outage probability of systems involving Beckmann RVs. In this paper, we study the generalized Beckmann distribution, which includes the Beckmann, Rayleigh, Rician, Nakagami-m, Hoyt, kappa-mu, eta-mu, single-sided Gaussian, and the Beaulieu-Xie distributions as special cases. Three approaches are proposed to estimate the cdf of the generalized Beckmann distribution, including closed-form upper and lower cdf bounds, single-fold integration based on the closed-form characteristic function, and a left-tail cdf approximation. We compare the three approaches in terms of the ranges of applications and the computation time complexity. Based on the new cdf estimation techniques, one can efficiently evaluate the outage probabilities of pointing-error-limited FSO systems, UWOC systems, and maximum-ratio combining over arbitrarily correlated generalized Beckmann channels.