IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 8, AUGUST 2008 1241 Bounds on the Distribution of a Sum of Correlated Lognormal Random Variables and Their Application C. Tellambura, Senior Member, IEEE Abstract—The cumulative distribution function (cdf) of a sum of correlated or even independent lognormal random variables (RVs), which is of wide interest in wireless communications, remains unsolved despite long standing efforts. Several cdf approximations are thus widely used. This letter derives bounds for the cdf of a sum of 2 or 3 arbitrarily correlated lognormal RVs and of a sum of any number of equally-correlated lognormal RVs. The bounds are single-fold integrals of readily computable functions and extend previously known bounds for independent lognormal summands. An improved set of bounds are also derived which are expressed as 2-fold integrals. For correlated lognormal fading channels, new expressions are derived for the moments of the output SNR and amount of fading for maximal ratio combining (MRC), selection combining (SC) and equal gain combining (EGC) and outage probability expressions for SC. Index Terms—Amount of fading, cochannel interference, log- normal distribution, diversity combining. I. I NTRODUCTION A LOGNORMAL power sum of type I = n  i=1 e Xi (1) where each X i is a Gaussian random variable (RV) appears pervasively in wireless communications. Applications include modeling and assessing cochannel interference, evaluating coverage for cellular mobile networks and modeling fading and shadowing [1], [2], and the book by Aitchison and Brown [3] lists over 100 applications. Nevertheless, no exact closed- form formula for the distribution of I is known. Even the characteristic function (chf) of a lognormal RV is not known in closed-form. Thus, several approximations have been de- veloped such as moment matching [4], cumulant matching [5] and recursive methods [6] - for comparisons see [2] [7]. For more recent results on lognormal sums the reader is referred to [8]–[17]. Although independent lognormal sums have been widely studied, applications where correlation among the summands in (1) exists occur just as frequently. Therefore, the distribu- tion of correlated lognormal sums has applications such as macro-diversity systems [18], soft handoff algorithms, single frequency networks and others. Moment matching or cumulant matching approximations that have been developed for inde- pendent lognormal sums can also be extended for correlated lognormal sums [19]–[21]. Paper approved by F. Santucci, the Editor for Wireless System Performance of the IEEE Communications Society. Manuscript received December 15, 2003; revised September 16, 2006. This work has been supported in part by NSERC and iCORE. C. Tellambura is with the Department of Electrical and Computer Engi- neering, University of Alberta, Edmonton, AB, Canada T6G 2V4 (e-mail: chintha@ece.ualberta.ca). Digital Object Identifier 10.1109/TCOMM.2008.030947. In this paper, we derive bounds for the cdf of a sum of two or three correlated lognormal RVs with arbitrary correlation and of a sum of any number of equally-correlated lognormal RVs. Several previously derived bounds are shown to be special cases of these. A further set of improved bounds (2-fold integrals) are derived. The simulation results confirm the tightness of these bounds. Although diversity combiners such as maximal ratio combining (MRC), selection combining (SC) and equal gain combining (EGC) have been widely studied over various channel models, their performance over correlated lognormal channels is not available in detail. For correlated lognormal fading channels, we also derive new expressions for the moments of the output SNR and amount of fading for MRC, SC and EGC and outage probability expres- sions for SC. For this development, we utilize S-function and T-function [22], [23], which are generalizations of the well- known Q-function and are not known widely in the wireless community. These results facilitate the rapid evaluation of the performance of diversity schemes over correlated lognormal channels, where no results have been published for diversity order more than two. Our bounds generalize previously published bounds for independent lognormal sums by Slimane [24] and Farley’s lower bound (or approximation) for independent lognormal sums [1] [2]. Abu-Dayya & Beaulieu [19] provide a detailed analysis of outage probabilities in the presence of multiple correlated lognormal interferers, using Wilkinson’s approach, Schwartz and Yeh’s approach and cumulant matching. Outage probability estimation in the presence of multiple lognormal components has been discussed by Ligeti [21]. Pratesi et al [20] treat outage analysis in mobile radio with generically correlated lognormal interferers. They extend several approx- imations for the independent case to those for the generically correlated case. Berggren and Slimane [25], by applying the arithmetic-geometric mean inequality, give a lower bound expressed by a single Q-function. Our work is different from all these in that we develop bounds for the correlated case and analyze diversity schemes. A detailed performance analysis of dual-branch diversity schemes over correlated lognormal fading has been developed by Alouini & Simon [26]. Our results generalize some of their results to multibranch (n> 2) diversity systems. This paper is organized as follows. Section II lists several relevant results for the multinormal distribution. Section III develops bounds for sums of correlated lognormal variables. Section IV derives the performance of SC, MRC and EGC receivers over correlated lognormal channels and Section V concludes the paper. 0090-6778/08$25.00 c  2008 IEEE Authorized licensed use limited to: UNIVERSITY OF ALBERTA. Downloaded on December 17, 2009 at 15:11 from IEEE Xplore. Restrictions apply. "©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE."