On the Exact Sum PDF and CDF of α-μ Variates

The sum of random variables (RVs) appears extensively in wireless communications, at large, both conventional and advanced, and has been subject of longstanding research. The statistical characterization of the referred sum is crucial to determine the performance of such communications systems. Alth...

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Vydáno v:	IEEE transactions on wireless communications Ročník 22; číslo 8; s. 1
Hlavní autoři:	Garcia, Fernando Dario Almeida, Parente, Francisco Raimundo Albuquerque, Yacoub, Michel Daoud, Filho, Jose Candido Silveira Santos
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York IEEE 01.08.2023 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:	Computation Convergence cumulative distribution function Distribution functions Diversity reception Fading channels Formulations Probability density function Probability density functions Random variables Receivers Statistical analysis sums of random variables Wireless communication Wireless communications α-μ distribution
ISSN:	1536-1276, 1558-2248
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Abstract	The sum of random variables (RVs) appears extensively in wireless communications, at large, both conventional and advanced, and has been subject of longstanding research. The statistical characterization of the referred sum is crucial to determine the performance of such communications systems. Although efforts have been undertaken to unveil these sum statistics, e.g., probability density function (PDF) and cumulative distribution function (CDF), no general efficient nor manageable solutions capable of evaluating the exact sum PDF and CDF are available to date. The only formulations are given in terms of either the multi-fold Brennan's integral or the multivariate Fox H -function. Unfortunately, these methods are only feasible up to a certain number of RVs, meaning that when the number of RVs in the sum increases, the computation of the sum PDF and CDF is subject to stability problems, convergence issues, or inaccurate results. In this paper, we derive new, simple, exact formulations for the PDF and CDF of the sum of L independent and identically distributed α-μ RVs. Unlike the available solutions, the computational complexity of our analytical expressions is independent of the number of summands. Capitalizing on our unprecedented findings, we analyze, in exact and asymptotic manners, the performance of L -branch pre-detection equal-gain combining and maximal-ratio combining receivers over α-μ fading environments. The coding and diversity gains of the system for both receivers are analyzed and quantified. Moreover, numerical simulations show that the computation time reduces drastically when using our expressions, which are arguably the most efficient and manageable formulations derived so far.
AbstractList	The sum of random variables (RVs) appears extensively in wireless communications, at large, both conventional and advanced, and has been subject of longstanding research. The statistical characterization of the referred sum is crucial to determine the performance of such communications systems. Although efforts have been undertaken to unveil these sum statistics, e.g., probability density function (PDF) and cumulative distribution function (CDF), no general efficient nor manageable solutions capable of evaluating the exact sum PDF and CDF are available to date. The only formulations are given in terms of either the multi-fold Brennan’s integral or the multivariate Fox [Formula Omitted]-function. Unfortunately, these methods are only feasible up to a certain number of RVs, meaning that when the number of RVs in the sum increases, the computation of the sum PDF and CDF is subject to stability problems, convergence issues, or inaccurate results. In this paper, we derive new, simple, exact formulations for the PDF and CDF of the sum of [Formula Omitted] independent and identically distributed [Formula Omitted]-[Formula Omitted] RVs. Unlike the available solutions, the computational complexity of our analytical expressions is independent of the number of summands. Capitalizing on our unprecedented findings, we analyze, in exact and asymptotic manners, the performance of [Formula Omitted]-branch pre-detection equal-gain combining and maximal-ratio combining receivers over [Formula Omitted]-[Formula Omitted] fading environments. The coding and diversity gains of the system for both receivers are analyzed and quantified. Moreover, numerical simulations show that the computation time reduces drastically when using our expressions, which are arguably the most efficient and manageable formulations derived so far. The sum of random variables (RVs) appears extensively in wireless communications, at large, both conventional and advanced, and has been subject of longstanding research. The statistical characterization of the referred sum is crucial to determine the performance of such communications systems. Although efforts have been undertaken to unveil these sum statistics, e.g., probability density function (PDF) and cumulative distribution function (CDF), no general efficient nor manageable solutions capable of evaluating the exact sum PDF and CDF are available to date. The only formulations are given in terms of either the multi-fold Brennan's integral or the multivariate Fox H -function. Unfortunately, these methods are only feasible up to a certain number of RVs, meaning that when the number of RVs in the sum increases, the computation of the sum PDF and CDF is subject to stability problems, convergence issues, or inaccurate results. In this paper, we derive new, simple, exact formulations for the PDF and CDF of the sum of L independent and identically distributed α-μ RVs. Unlike the available solutions, the computational complexity of our analytical expressions is independent of the number of summands. Capitalizing on our unprecedented findings, we analyze, in exact and asymptotic manners, the performance of L -branch pre-detection equal-gain combining and maximal-ratio combining receivers over α-μ fading environments. The coding and diversity gains of the system for both receivers are analyzed and quantified. Moreover, numerical simulations show that the computation time reduces drastically when using our expressions, which are arguably the most efficient and manageable formulations derived so far.
Author	Parente, Francisco Raimundo Albuquerque Garcia, Fernando Dario Almeida Filho, Jose Candido Silveira Santos Yacoub, Michel Daoud
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SubjectTerms	Computation Convergence cumulative distribution function Distribution functions Diversity reception Fading channels Formulations Probability density function Probability density functions Random variables Receivers Statistical analysis sums of random variables Wireless communication Wireless communications α-μ distribution
Title	On the Exact Sum PDF and CDF of α-μ Variates
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