On the Bivariate Nakagami-m Cumulative Distribution Function: Closed-Form Expression and Applications

Abstract

In this paper, we derive exact closed-form expressions for the bivariate Nakagami-m cumulative distribution function (CDF) with positive integer fading severity index m in terms of a class of hypergeometric functions. Particularly, we show that the bivariate Nakagami-m CDF can be expressed as a finite sum of elementary functions and bivariate confluent hypergeometric Φ 3 functions. Direct applications which arise from the proposed closed-form expression are the outage probability (OP) analysis of a dual-branch selection combiner in correlated Nakagami-m fading, or the calculation of the level crossing rate (LCR) and average fade duration (AFD) of a sampled Nakagami-m fading envelope. Index TermsAverage fade duration, Bivariate Nakagami-m, Correlation, Cumulative distribution function, Diversity reception, Level crossing rate, Nakagami-m fading. arXiv:1206.3965v1 [cs.IT] 18 Jun 2012While the PDF and the CDF of a Nakagami-m variate follow very simple and tractable expressions, the statistical characterization of the multivariate Nakagami-m distribution is rather involved and remains as an open problem in the literature. Although a closed-form expression for the joint PDF of two correlated Nakagami-m variates was given in the original manuscript by Nakagami [2], a closed-form expression for the bivariate Nakagami-m CDF has not been obtained yet to the best of our knowledge. Existing results provide expressions for this bivariate CDF in terms of infinite summations [4][5][6], and more recently [7] in terms of a single integral involving special functions. The result in [7] is particularly relevant, since it allows to express the PDF and the CDF of a number of multivariate distributions in terms of a single integral, whilst previous results were only available in terms on nested infinite summations [8,9].As the Rayleigh distribution is coincident with the Nakagami-m distribution when the fading parameter m = 1, it may result expectable that the existing closed-form expression for the bivariate Rayleigh CDF [10] in terms of the first order Marcum-Q function is a particular case of the bivariate Nakagami-m CDF. This aspect was discussed by Simon and Alouini in [1, p. 174], who pointed out that "One might anticipate that the bivariate Nakagami-m CDF could be expressed in a form analogous to (the bivariate Rayleigh CDF) depending instead on the m thorder Marcum Q-function ... ... Unfortunately, to the authors knowledge an expression analogous to (the bivariate Rayleigh CDF) has not been reported in the literature, and the authors have themselves been unable to arrive at one.".In this paper, we demonstrate that the bivariate Nakagami-m cumulative distribution function can be expressed in closed-form as a finite sum of elementary functions and Φ 3 functions, for a positive integer fading severity index m. The confluent hypergeometric function of two variables Φ 3 , which is well studied in classical books of integrals and special functions [11, 9.261.3] and Laplace transforms [12], is one of the bivariate forms of Kummer'...

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