On the Distribution of the Sum of Nakagami-$ m$ Random Variables

In this paper, we investigate the classical problem of finding the probability density function (pdf) of the sum of Nakagami-
m
random variables. Exact infinite series representations are derived for the sum of three and four identically and independently distributed (i.i.d.) Nakagami-
m
random variables, and subsequently, it is extended to the sum of general number of random variables. A useful pattern emerged as a result of these extensions, and a new Fourier transform pair that involves parabolic cylinder function and Gauss hypergeometric function is obtained. Bounds on the error resulting from truncation of the infinite series of pdfs are also presented. These pdfs are used to analyze the performance of dual, triple, and quadruple branch predetection equal gain combining (EGC) receivers over Nakagami-
m
fading environment. Furthermore, a hypergeometric relation is derived as a result of those derivations. Subsequently, the analysis is extended to the case of L branch EGC receiver performance as well. Selected simulation plots are provided to compare the results with the approximate results available in the literature and to illustrate the validity of the formulation.