On the Product of Two <inline-formula> <tex-math notation="LaTeX">$\kappa$ </tex-math> </inline-formula>–<inline-formula> <tex-math notation="LaTeX">$\mu$ </tex-math> </inline-formula> Random Variables and its Application to Double and Composite Fading Channels

Abstract

In this paper, we perform a systematic investigation of the statistics associated with the product of two independent and non-identically distributed $\kappa$ – $\mu$ random variables. More specifically, we develop novel analytical formulations for many of the fundamental statistics of interest, namely, the probability density function, cumulative distribution function, and moment-generating function. Using these new results, closed-form expressions are obtained for the higher order moments, amount of fading and channel quality estimation index, while analytical formulations are obtained for the outage probability, average channel capacity, average symbol error probability, and average bit error probability. These general expressions can be reduced to a number of fading scenarios, such as the double Rayleigh, double Rice, double Nakagami- $m$ , $\kappa$ – $\mu$ /Nakagami- $m$ , and Rice/Nakagami- $m$ , which all occur as special cases. Additionally, as a byproduct of the work performed here, formulations for the $\kappa$ – $\mu /\kappa$ – $\mu$ composite fading model can also be deduced. To illustrate the efficacy of the novel expressions proposed here, we provide useful insights into the outage probability of a dual-hop system used in body area networks, and demonstrate the suitability of the $\kappa$ – $\mu /\kappa$ – $\mu$ composite fading for characterizing shadowed fading in device-to-device channels.