A Unified Framework for Mixed Fading Sums

Abstract

This paper introduces a unified analytical framework that yields exact, compact, and tractable solutions for the probability density function (PDF) and cumulative distribution function (CDF) of the sum of a broad class of mixed independent positive random variables. To demonstrate its breadth and practical value, the proposed framework is applied to the longstanding problem of deriving the exact statistics of mixed sums of fading variates. Specifically, we derive novel expressions for the PDF and CDF of sums of Gaussian-class and non-Gaussian-class fading distributions, thereby covering a plethora of conventional, generalized, and recently introduced fading models. The framework accommodates independent and identically distributed (i.i.d.) sums, independent but not necessarily identically distributed (i.n.i.d.) sums, and mixed-type sums. Furthermore, we conduct an exact performance analysis of equal-gain combining and maximal-ratio combining diversity receivers operating under mixed fading conditions. The proposed framework provides an exact and versatile analytical foundation for channel modeling and performance evaluation across a wide range of wireless fading scenarios. Finally, we introduce the novel $\alpha$-$\mu$ \textit{Mixture distribution} that unifies all Gaussian-class fading models.