Ratio of Two Envelopes Taken From $\alpha$ -$\mu$ , $\eta$ -$\mu$ , and $\kappa$ -$\mu$ Variates and Some Practical Applications

Abstract

In this paper, new, exact expressions for the probability density functions and the cumulative distribution functions of the ratio of random envelopes involving the $\alpha $ - $\mu $ , $\eta $ - $\mu $ , and $\kappa $ - $\mu $ fading distributions are derived. The expressions are obtained in terms of easily computable infinite series and also in terms of the multivariable Fox H-function. Some special cases of these ratios, namely, Hoyt/Hoyt, $\eta $ - $\mu $ /Nakagami- ${m}$ , $\kappa $ - $\mu $ /Nakagami- ${m}$ , $\kappa $ - $\mu / \eta $ - $\mu $ with an integer $\mu $ for the $\eta $ - $\mu $ variate, $\eta $ - $\mu / \eta $ - $\mu $ with an integer $\mu $ for only one of the $\eta $ - $\mu $ , and their reciprocals are found in novel exact closed-form expressions . In addition, simple closed-form expressions for the asymptotes of the probability density functions and cumulative distribution functions of all ratios, both for the lower and upper tails of the distributions are derived. In the same way, asymptotes for the bit error rate on a binary signaling channel are obtained in closed-form expressions. To demonstrate the practical utility of these new formulations, an application example is provided. In particular, the secrecy capacity of a Gaussian wire-tap channel used for device-to-device and vehicle-to-vehicle communications is characterized using data obtained from field measurements conducted at 5.8 GHz.