A Novel Stochastic Model for IRS-Assisted Communication Systems Based on the Sum-Product of Nakagami-$m$ Random Variables

TL;DR

This paper develops exact formulas and asymptotic expressions for the sum-product of Nakagami-m random variables, enabling precise modeling of IRS-assisted communication channels and improving outage and error rate predictions.

Contribution

It introduces new exact formulas and series representations for the sum-product of Nakagami-m RVs, enhancing modeling accuracy for IRS-assisted fading channels.

Findings

Exact formulas for PDF and MGF of Nakagami-m sum-product

Asymptotic expressions for diversity order

Validation through Monte-Carlo simulations

Abstract

This paper presents exact formulas for the probability distribution function (PDF) and moment generating function (MGF) of the sum-product of statistically independent but not necessarily identically distributed (i.n.i.d.) Nakagami-$m$ random variables (RVs) in terms of Meijer's G-function. Additionally, exact series representations are also derived for the sum of double-Nakagami RVs, providing useful insights on the trade-off between accuracy and computational cost. Simple asymptotic analytical expressions are provided to gain further insight into the derived formula, and the achievable diversity order is obtained. The suggested statistical properties are proved to be a highly useful tool for modeling parallel cascaded Nakagami-$m$ fading channels. The application of these new results is illustrated by deriving exact expressions and simple tight upper bounds for the outage probability (OP) and average symbol error rate (ASER) of several binary and multilevel modulation signals in intelligent reflecting surfaces (IRSs)-assisted communication systems operating over Nakagami-$m$ fading channels. It is demonstrated that the new asymptotic expression is highly accurate and can be extended to encompass a wider range of scenarios. To validate the theoretical frameworks and formulations, Monte-Carlo simulation results are presented. Additionally, supplementary simulations are provided to compare the derived results with two common types of approximations available in the literature, namely the central limit theorem (CLT) and gamma distribution.