$N{\ast}$Nakagami: A Novel Stochastic Model for Cascaded Fading Channels

Abstract

A generic and novel distribution, referred to as Nakagami, constructed as the product of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> statistically independent, but not necessarily identically distributed, Nakagami- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> random variables (RVs), is introduced and analyzed. The proposed distribution turns out to be a very convenient tool for modelling cascaded Nakagami- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> fading channels and analyzing the performance of digital communications systems operating over such channels. The moments-generating, probability density, cumulative distribution, and moments functions of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sub> Nakagami distribution are developed in closed form using the Meijer's G -function. Using these formulas, generic closed-form expressions for the outage probability, amount of fading, and average error probabilities for several binary and multilevel modulation signals of digital communication systems operating over the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sub> Nakagami fading and the additive white Gaussian noise channel are presented. Complementary numerical and computer simulation performance evaluation results verify the correctness of the proposed formulation. The suitability of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sub> Nakagami fading distribution to approximate the lognormal distribution is also being investigated. Using Kolmogorov--Smirnov tests, the rate of convergence of the central limit theorem as pertaining to the multiplication of Nakagami- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> RVs is quantified.