Product and Ratio of Two $α-κ-μ$ Shadowed Random Variables and its Application to Wireless Communication

TL;DR

This work derives the exact statistical characterizations for the product $Y=X_1X_2$ and ratio $Z=X_1/X_2$ of two independent, non-identically distributed $\alpha-\kappa-\mu$ shadowed fading RVs using Mellin transforms. It provides closed-form series expressions for PDFs, CDFs, and MGFs, along with Laplace-type integral representations, and develops asymptotic and simple moment-based approximations (Gamma for the product and Beta-prime for the ratio). The results are applied to cascaded wireless links, physical-layer security metrics, and IRS-assisted communications, illustrating broad applicability and unifying several prior fading models. The methods enable quantitative performance analysis in generalized fading environments and support efficient resource allocation in next-generation wireless systems.

Abstract

This work studies the product and ratio statistics of independent and non-identically distributed (i.n.i.d) $ α-κ- μ$ shadowed random variables. We derive the series expression for the probability density function (PDF), cumulative distribution function (CDF), and moment generating function (MGF) of the product and ratio of i.n.i.d $ α- κ- μ$ shadowed random variables. We then give the single integral representation for the derived PDF expressions. Further, as application examples, 1) outage probability has been derived for cascaded wireless systems, and 2) physical-layer security metrics like secrecy outage probability and strictly positive secrecy capacity are derived for the classic three-node model with $α-κ-μ$ shadowed fading. Next, we discuss an intelligent reflecting surface-assisted communication system over $α-κ-μ$ shadowed fading.

Figures (15)

• Figure 1: Simulated and Approximate PDF of $Y$
• Figure 2: Simulated and Approximate PDF of $Z$
• Figure 3: PDF of $Y$ for various values of $\alpha_{1}$, $\alpha_{2}$ in (a), and for various values of $\kappa_{1}$, $\kappa_{2}$ in (b).
• Figure 4: PDF of $Y$ for various values of $\mu_{1}$, $\mu_{2}$ in (a), and for various values of $m_{1}$, $m_{2}$ in (b).
• Figure 5: PDF of $Z$ for various values of $\alpha_{1}$, $\alpha_{2}$ in (a), and for various values of $\kappa_{1}$, $\kappa_{2}$ in (b).