On the statistics of the ratio of nonconstrained arbitrary α‐μ random variables: A general framework and applications

Abstract

In this paper, we derive closed‐form exact expressions for the main statistics of the ratio of two squared α‐μ random variables, which are of interest in many scenarios for future wireless networks where generalized distributions are more suitable to fit with field data. Importantly, different from previous proposals, our expressions are general in the sense that are valid for nonconstrained arbitrary values of the parameters of the α‐μ distribution. Thus, the probability density function, cumulative distribution function, moment generating function, and higher‐order moments are given in terms of both (i) the Fox H‐function for which we provide a portable and efficient Wolfram Mathematica code and (ii) easily computable series expansions. Our expressions can be used straightforwardly in the performance analysis of a number of wireless communication systems, including either interference‐limited scenarios, spectrum sharing, full‐duplex, or physical‐layer security networks, for which we present the application of the proposed framework. Moreover, closed‐form expressions for some classical distributions, derived as special cases from the α‐μ distribution, are provided as byproducts. The validity of the proposed expressions is confirmed via Monte Carlo simulations.

Approximations for the product, ratio, and sum ofα‐μrandom variables with application in the analysis of cognitive radio networks

“…More recently, several works have obtained expressions for the main statistics of operations between 𝛼-𝜇 RVs and highlighted the applicability of these expressions. [25][26][27] In all three cases, the derived expressions were the probability density function (PDF), cumulative distribution function (CDF), and generalized moment generating function (MGF). In Badarneh et al, 25 it was analyzed the ratio of the products of fluctuating two-ray RVs with practical applications, such as wireless multihop transmission, multiple scattering channels, spectrum sharing, and interference-limited networks.…”

“…More recently, several works have obtained expressions for the main statistics of operations between α ‐ μ RVs and highlighted the applicability of these expressions 25‐27 . In all three cases, the derived expressions were the probability density function (PDF), cumulative distribution function (CDF), and generalized moment generating function (MGF).…”

Sum of Fisher-Snedecor F Random Variables and Its Applications

“…Although the numerical evaluation for multivariate Foxs H-function is not available in popular mathematical packages such as MATLAB and Mathematica, its efficient implementations have been reported. For example, two Mathematica implementations of the single Foxs H-function are provided in [14] and [15], a Python implementation for the multivariable Foxs H-function is presented in [16], and an efficient GPUoriented MATLAB routine for the multivariate Foxs H-function is introduced in [17]. In the following, we will utilize these novel implementations to evaluate our results.…”

Safeguarding MTC at the Physical Layer: Potentials and Challenges

“…A wide variety of scenario-specific 5G channel models have been already proposed in the literature (an excellent survey can be found in [46]). Then, it is essential to revise PHY security techniques and metrics regarding these new channel models [47][48][49]. Indeed, various PHY security techniques are invalidated in poor scattering environments where a strong correlation between legitimate and wiretap channels exists.…”