$\alpha$-$\mu$, $\kappa$-$\mu$, and $\eta$-$\mu$ Mixed Products: Exact and Simple-Approximate Solutions, and Practical and RIS Applications

The received signal in numerous practical wireless transmission scenarios, such as multi-hop systems, cascaded channels, keyhole multiple-input-multiple-output, and radar systems, can be realistically modeled as a product of random envelopes. In this context, the present contribution derives novel analytic expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the mixed product of an arbitrary number of random variates taken from the $\alpha$-$\mu$, $\kappa$-$\mu$, and Extended $\eta$-$\mu$ distributions. These models are known to characterize generalized multipath fading effects accurately. The derived analytic results are exact and expressed in relatively simple, easily computable algebraic representations. The associated formulations are subsequently used in deriving tractable asymptotic expressions for the lower tail of both corresponding PDFs and CDFs. These asymptotes are particularly useful since they provide meaningful insights into the behavior of the channel statistics within the high signal-to-noise ratio regime. In addition, a single $\alpha$-$\mu$ distribution is employed as a valuable and practical close approximation to the mixed product distribution with an arbitrary number of variates. The usefulness of such an approximation is demonstrated in the performance analysis of a reconfigurable intelligent surface (RIS) assisted communication system, for which the mathematical treatment is usually very intricate. The accuracy and utility of the derived expressions are also validated through comparisons with Monte Carlo simulations. Finally, an important practical application is demonstrated by applying the derived results to a realistic multi-hop communication scenario in the context of body area networks. This verifies that the proposed results greatly adhere to the target environment, whilst useful theoretical and practical insights on the corresponding impact of generalized multipath fading are developed.