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Approximating a Sum of Random Variables

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 7, JULY 2007
Approximating a Sum of Random Variables with a Lognormal
Neelesh B. Mehta, Senior Member, IEEE, Jingxian Wu, Member, IEEE, Andreas F. Molisch, Fellow, IEEE, Jin Zhang, Senior Member, IEEE
Abstract— A simple, novel, and general method is presented in this paper for approximating the sum of independent or arbitrarily correlated lognormal random variables (RV) by a single lognormal RV. The method is also shown to be applicable for approximating the sum of lognormal-Rice and Suzuki RVs by a single lognormal RV. A sum consisting of a mixture of the above distributions can also be easily handled. The method uses the moment generating function (MGF) as a tool in the approximation and does so without the extremely precise numerical computations at a large number of points that were required by the previously proposed methods in the literature. Unlike popular approximation methods such as the Fenton-Wilkinson method and the Schwartz-Yeh method, which have their own respective short-comings, the proposed method provides the parametric flexibility to accurately approximate different portions of the lognormal sum distribution. The accuracy of the method is measured both visually, as has been done in the literature, as well as quantitatively, using curve-fitting metrics. An upper bound on the sensitivity of the method is also provided. Index Terms— lognormal distribution, correlation, Suzuki distribution, lognormal-Rice distribution, moment methods, characteristic function, moment generating function, approximation methods, co-channel interference.
TR2007-067
August 2007
Abstract A simple, novel, and general method is presented in this paper for approximating the sum of independent or arbitrarily correlated lognormal random variables (RV) by a single lognormal RV. The method is also shown to be applicable for approximating the sum of lognormal-Rice and Suzuki RVs by a single lognormal RV. A sum consisting of a mixture of the above distributions can also be easily handled. The method uses the moment generating function (MGF) as a tool in the approximation and does so without the extremely precise numerical computations at a large number of points that were required by the previously proposed methods in the literature. Unlike popular approximation methods such as the Fenton-Wilkinson method and the Schwartz-Yeh method, which have their own respective short-comings, the proposed method provides the parametric flexibility to accurately approximate different portions of the lognormal sum distribution. The accuracy of the method is measured both visually, as has been done in the literature, as well as quantitatively, using curve-fitting metrics. An upper bound on the sensitivity of the method is also provided. IEEE Transactions on Wireless Communications
This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Copyright c Mitsubishi Electric Research Laboratories, Inc., 2007 201 Broadway, Cambridge, Massachusetts 02139
MITSUBISHI ELECTRIC RESEARCH LABORATORIES
Approximating a Sum of Random Variables with a Lognormal
Neelesh Mehta, Jingxian Wu, Andreas Molisch, Jin Zhang
Manuscript received December 20, 2005; revised December 13, 2006; accepted December 13, 2006. The associate editor coordinating the review of this paper and approving it for publication was P. Jung. A part of this work was presented at Globecom 2005 and ICC 2006. N. B. Mehta, A. F. Molisch, and J. Zhang are with Mitsubishi Electric Research Labs (MERL), 201 Broadway, Cambridge, MA 02139, USA (email: {mehta, molisch, jzhang}@). A. F. Molisch is also at the Department of Electroscience, Lund University, Sweden. J. Wu is with the Department of Engineering Science, Sonoma State University, Rohnert Park, CA 94928, USA (email: jingxian.wu@). J. Wu was at MERL during the course of this work. Digital Object Identifier 10.1109/TWC.2007.051000.

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N −1 x =0 =0 ux N
F (u ) = ∑ f ( x)e 1 f ( x) = N
N −1
− j 2π
u=0,1,…,N-1
∑ F (u )e
u =0
j 2π
ux N
x=0,1,…,N-1
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Use Euler Formula
F(u) can be described as follows:
N −1 x=0 ux N N −1 x=0
F (u) = ∑ f ( x)e
− j 2π
= ∑ f ( x)[cos(2πux / N ) − j sin(2πux / N )]
− jφ ( u )
F (u ) = R(u ) + jI (u ) = F (u ) e
polar coordinates
R (F) + jI (F)
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F(u,v) is a complex number, with real and imaginary parts. We can thus define the magnitude and phase of F(u,v):
∑∑
x =0 y =0
ux vy f ( x, y ) sin 2π + M N
– Real part
1 Fr (u , v) = MN
M −1 N −1
∑∑
x =0 y =0
ux vy f ( x, y ) cos 2π + M N
33
•Inverse Fourier Transfo ∞
− ∞− ∞ ∞ ∞
∫∫
f ( x, y )e − j 2π (ux + vy ) dxdy =
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Cumulative distribution function of random variable
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