Product of Two Envelopes Taken From $\alpha $ - $\mu $ , $\kappa $ - $\mu $ , and $\eta $ - $\mu $ Distributions

This paper presents exact, novel formulations for the probability density function and cumulative distribution function for the product of two independent and non-identically distributed <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$\kappa $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$\eta $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> variates. The expressions are given in terms of both 1) generalized Fox H-function and 2) easily computable series expansions. The formulations derived can be directly used to explore the performance of a number of wireless communication processes, including multihop systems, cascaded channels, radar communications, multiple-input multiple-output links, and others. Due to the high flexibility of the above-mentioned distributions, the results presented here comprise a substantial number of useful product distributions. As application examples, performance metrics for the cascaded fading channel are derived. The validity of the expressions is confirmed via Monte Carlo simulation. It is noteworthy that, because any composite multipath-shadowing fading model is obtained as a particular case of the product of two fading variables, the results given here provide a plethora of composite multipath-shadowing fading scenarios.

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