New Results on the Sum of Gamma Random Variates With Application to the Performance of Wireless Communication Systems over Nakagami-m Fading Channels

TL;DR

This paper derives closed-form expressions for the sum of Gamma variates relevant to wireless communication over Nakagami-m channels, enabling improved performance analysis and error rate calculations.

Contribution

It provides new closed-form formulas using Meijer's G and Fox's H-bar functions for sums of Gamma variates with both integer and non-integer m, advancing statistical analysis in wireless systems.

Findings

Closed-form PDFs and CDFs for sums of Gamma variates.

Analytical expressions for bit error rate over Nakagami-m channels.

Complementary to previous infinite-sum based results.

Abstract

The probability density function (PDF) and cumulative distribution function of the sum of L independent but not necessarily identically distributed Gamma variates, applicable to the output statistics of maximal ratio combining (MRC) receiver operating over Nakagami-m fading channels or in other words to the statistical analysis of the scenario where the sum of squared Nakagami-m distributions are user-of-interest, is presented in closed-form in terms of well-known Meijer's G function and easily computable Fox's H-bar function for integer valued and non-integer valued m fading parameters. Further analysis, particularly on bit error rate via a PDF-based approach is also offered in closed form in terms of Meijer's G function and Fox's H-bar function for integer valued fading parameters, and extended Fox's H-bar function (H-hat) for non-integer valued fading parameters. Our proposed results complement previous known results that are either expressed in terms of infinite sums, nested sums, or higher order derivatives of the fading parameter m.