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

TL;DR

This work tackles the problem of accurately modeling IRS-assisted channels under Nakagami-$m$ fading by characterizing the sum-product of i.n.i.d. Nakagami-$m$ RVs. It develops exact PDFs and MGFs using Meijer $G$-functions, provides exact series for double-Nakagami sums, and introduces simple asymptotic forms to enable tractable outage probability and ASER analyses. The framework is applied to IRS scenarios, including SISO and MISO downlink with and without direct links, with diversity-order analyses showing $div= N m_{I,D}$ (and extensions to $NM m_{I,D}$ or $m_{S,D}+N m_{I,D}$ when a direct link is present). Validation through Monte Carlo simulations and comparisons with CLT and gamma-based approximations demonstrate high accuracy and practical utility for system design and benchmarking in realistic fading environments.

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.

Figures (4)

• Figure 1: IRS-assisted communication system model.
• Figure 2: Comparison of probability density function of the sum of double-Nakagami RVs for different values of $N$, when the average channel power gain is $\rho=1$.
• Figure 3: An average symbol error rate comparison for different values of $N$ by letting $\alpha=1$ and $g=1$ for BPSK modulation.
• Figure 4: Outage probability versus average signal-to-noise ratio with $\gamma_{\text{th}}~=~5 \, \text{dB}$ for different values of $N$.

Theorems & Definitions (11)

• Theorem 1: Moments Generating Function
• Proof 1
• Lemma 1: Probability Density Function
• Theorem 2: MGF of Double-Nakagami RVs
• Proof 2
• Lemma 2: PDF of Double-Nakagami RVs
• Theorem 3: Asymptotic MGF
• Proof 3
• Lemma 3: Asymptotic PDF
• Theorem 4: Upper Bound for MGF of Nakagami-$m$ RVs