On the Sum of Ratios of \alpha- \mu Variates With Application to Composite Fading Channels

• Published in: IEEE transactions on vehicular technology 2024-11, p.1-6
• Main Authors: Jimenez, Lenin Patricio Jimenez, Garcia, Fernando Dario Almeida, Alvarado, Maria Cecilia Luna, Filho, Jose Candido S. Santos, Yacoub, Michel Daoud, Fraidenraich, Gustavo
• Format: Article
• Language: English
• Subjects: <inline-formula xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> <tex-math notation="LaTeX"> alpha</tex-math> </inline-formula>-<inline-formula xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> <tex-math notation="LaTeX"> mu</tex-math> </inline-formula> distribution; Composite fading channels; Diversity reception; equal-gain combining; Fading channels; Mathematical models; maximal-ratio combining; performance analysis; Probability density function; Random variables; Receivers; Shadow mapping; sum of random variables; Symbols; Voltage; Wireless communication
• ISSN: 0018-9545
• DOI: 10.1109/TVT.2024.3509132

Sums of random variables play a fundamental role in assessing the performance of wireless communication systems. This work introduces novel and exact formulations for the probability density function and the cumulative distribution function of the sum of ratios of L independent \alpha-\mu variates. These formulations are presented in terms of tractable fast-converging series. To the best of the author's knowledge, these are the first reported exact solutions for the sum statistics of ratios of \alpha-\mu variates. Based upon these expressions, we analyze the performance of equal-gain combining and maximal-ratio combining diversity receivers operating over composite fading channels, i.e., in the presence of both multipath and shadowing fading effects. For the analysis, we consider that multiphath fading follows an \alpha-\mu distribution, whereas shadowing fading is governed by an inverse \alpha-\mu distribution. Exact and asymptotic expressions are obtained for the key performance metrics of the system, specifically the outage probability and the average-symbol error rate. Interestingly, the derived performance metrics demonstrated that the system's diversity gain solely depends on the multipath distribution parameters and the number of branches of the combining receivers. The validity of all our findings is confirmed via Monte Carlo simulations.