Asymptotically Exact Approximations to Generalized Fading Sum Statistics

Abstract

We propose a unified approach to approximate the probability density function and the cumulative distribution function of a sum of independent channel envelopes following generalized, and possibly mixed, fading models. In the proposed approach, the approximate sum distribution can be chosen out of a broad class of statistical models. More fundamentally, given any chosen model, the parameters of the approximate sum distribution are calibrated by matching its asymptotic behavior around zero to that of the exact sum. In a subsidiary fashion, one or more moments of the exact and approximate sums are also matched to one another if the approximate distribution has three or more parameters to be adjusted, respectively. As illustrated through many numerical examples, our approach outperforms existing ones that are solely based on moment matching, by yielding statistical approximations that are remarkably accurate at medium to high signal-to-noise ratio — a paramount operational regime for communications systems. Our results find applicability in several wireless applications where fading sums arise, and can be readily extended to accommodate sums of fading (power) gains and, in a broader context, generic sums of non-negative random variables.