Paper Info
Title
A closed-form upper-bound for the distribution of the weighted sum of rayleigh variates
Abstract
The problem of finding the distribution of the sum of more than two Rayleigh fading envelopes has never been solved in terms of tabulated functions. In this letter, we present a closed-form union upper-bound for the cumulative distribution function of the weighted sum of N independent Rayleigh fading envelopes. Computer simulation results verify the tightness of the proposed bound for several values of N. The proposed bound can be efficiently applied in various wireless applications, such as satellite communications, equal-gain receivers, and radars. Beaulieu (3), an infinite series approach for determining this CDF has been developed. This paper also lists all the related works on this topic up to that time. In two other papers, Helstrom has computed the distribution of the sum using saddle-point integration for uniformly weighted RVs (4), as well as for arbitrary weights (5). More recently, Karagiannidis and Kotsopoulos have presented a semi-analytical approach based on Hermite numerical integration, for the calculation of the CDF of the weighted sum of Nakagami-m and Ricean RVs (6). However, although the problem of finding the distribution of the sum of Rayleigh distributed RVs has been well-studied, all presented methods are approximative solutions in which the truncation error has to be taken into account. The use of bounds, as opposed to approximations, serves as a safe technique of addressing this problem in a computational efficient and easy way. In this letter, a closed-form solution for the distribution of the product of N independent Rayleigh distributed RVs is presented. Using the well-known inequality between arith- metic and geometric mean, also applied in (7) for the sum of Lognormal variates, an efficient closed-form union upper- bound for the CDF of the weighted sum of N independent Rayleigh distributed RVs is derived. The tightness of the proposed bound is verified by comparison with performance evaluation results of the exact CDF, obtained by means of computer simulations.
Year
DOI
Venue
2005
10.1109/LCOMM.2005.1461673
IEEE Communications Letters
Keywords
DocType
Volume
Rayleigh channels,Distribution functions,Computer simulation,Radar applications,Spaceborne radar,Diversity reception,Closed-form solution,Application software,Satellite communication,Probability
Journal
9
Issue
ISSN
Citations 
7
1089-7798
9
PageRank 
References 
Authors
0.97
6
3
Name
Order
Citations
PageRank
George K. Karagiannidis15216362.34
Theodoros A. Tsiftsis21406116.77
Nikos C. Sagias345339.91