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

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L$</tex-math></inline-formula> independent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">exact</i> solutions for the sum statistics of ratios of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> distribution, whereas shadowing fading is governed by an inverse <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> 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.