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Approximating the sum of log-normal random variables (RVs) is a long-standing open issue and many studies in the old and recent literature resort to assume that the "power sum" can be well modeled via another log-normal RV. Such a log-normal approximation can be effectively used for accurate outage probability (Pout) analysis and modeling of signal-to-(noise+interference) ratio (SNIR) behaviors in the context of radio resource management in wireless systems; nevertheless, it is recognized that i) an adequate accuracy can only be obtained for a sub-range of values taken by the distribution; ii) the achievable accuracy depends on the parameters of the set of RVs; iii) log-normal approximation may not provide a satisfactory accuracy for estimating bit error probabilities. Moreover, all contributions in the open technical literature are referred to model the "power sum" distribution, i.e., the weighted linear combination of log-normal RVs with weights that may take only positive values. While this setup is consistent with typical scenarios where summands add incoherently, it has recently been shown that in some high frequency-selective multipath channels with log-normal distributed shadow- and fast-fading, e.g., in impulse radio ultra wide band (IR-UWB) systems, the signal- to-noise ratio (SNR) is more accurately modeled by the weighted linear combination of log-normal RVs with weights that may assume both positive and negative values. Motivated by the above considerations, and moving from recent results, which show that approximating the "power sum" of log-normal RVs can boil down to a model selection problem within the pearson system of distributions, the specific contribution of this paper can be summarized as follows, i) We show that the pearson type IV distribution can not only be used to accurately approximate the probability density function (PDF) of the "power sum" of log-normal RVs with positive weights, but it can also accurately model the scenario with both p- - ositive and negative weights, as well as the scenario with generically correlated RVs. ii) For the scenario with independent RVs, we propose a numerical technique, which is based on the moment generating function (MGF) matching principle and uses a least- squares criterion, to accurately estimate the four parameters of the pearson type IV distribution, iii) A byproduct of this analysis.
Date of Publication: February 2009