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Approximating the sum of Log-Normal random variables (RVs) is a long-standing open issue, in the old and recent literature, and many approaches have been proposed to deal with that problem. However, all previous contributions 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. In this paper, we extend that analysis by assuming that the weights may also take negative values. We also point out the importance that this scenario may have for the accurate analysis of ultra wide band (UWB) systems when intra-pulse interference due to multipath propagation cannot be neglected. In this perspective, (i) we show that the weighted linear combination of Log-Normal RVs can be well approximated by the Pearson Type IV distribution in the logarithmic domain, (ii) we use the obtained results for the accurate estimation of the average bit error probability (ABEP) of UWB receiver architectures.