On the Exact Sum of Products of Rayleigh Envelopes With Application to RIS

The sum of products of random variables arises in many applications, particularly in wireless communications. However, the analytical derivation of essential statistics, such as the probability density function and cumulative distribution function, still purveys significant challenges. The objective of this work is to derive exact formulations for the density and distribution of the sum of an arbitrary N instances, each comprising
$L_{n}, n\in \{1,2,\ldots ,N\}$
products of independent Rayleigh-distributed variates. This renders our exact formulations rather flexible to comprise different fading scenarios. Based on our findings, we investigate the performance of a communication system employing a reconfigurable intelligent surface (RIS) by deriving key metrics, such as outage probability and average bit error rate. Also, we employ the central limit theorem to approximate the sum of envelope products and performance metrics. In addition, we conduct an asymptotic analysis to deepen our understanding of the system’s performance. Specifically, we obtain simple closed-form expressions for all the metrics derived here. Our analytical results are validated through Monte Carlo simulations.