On the Phase Statistics of the κ-μ Process

A comprehensive study on the $\kappa $ - $\mu $ phase statistics is conducted. A substantial number of new formulas, exact and approximate, closed form, and integral form, are presented. In particular, a tight close-form approximation for the phase probability density function is found that yields results almost indistinguishable from the exact integral formulation. Most strikingly, the approximate formulation comprises the exact Nakagami- $m$ phase as well as the exact Von Mises (Tikhonov) densities. Joint statistics involving combinations of the envelope, phase, and their time derivatives are derived in an exact manner. The exact phase crossing rate is then obtained in an integral form. A closed-form approximation is proposed that yields very good results as compared to the exact formulation. A Monte Carlo simulation plot is used to validate the formula of the exact phase crossing rate. The formulations presented here drastically facilitate the use of the phase statistics of the $\kappa $ - $\mu $ fading model.

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