A Bivariate $\kappa$- $\mu$ Distribution

Abstract

In this paper, a bivariate κ-μ model is presented. Exact expressions for the 1) joint probability density function, the 2) joint cumulative distribution function, 3) joint arbitrary moments, and the 4) normalized envelope correlation coefficient are derived. The joint statistics are given in terms of their respective parameters (κ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , μ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ) and (κ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , μ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ), with μ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> = μ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> = μ > 0 and arbitrary κ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> > 0 and κ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> > 0. The parameter describing the correlation between κ-μ fading channels is then written in terms of the physical instances known to affect it in a wireless medium, namely, Doppler shift, the separation distance between two reception points, frequency, and delay spread. As an application example, the outage probability of a dual-branch selection-combining scheme is presented. The effect of correlation in the various aspects of system performance is then investigated. The validity of the analytical results is supported by reducing them to particular cases, for which results are available in the literature, and by means of simulation for the general cases.