A General Formula for Log-MGF Computation: Application to the Approximation of Log-Normal Power Sum via Pearson Type IV Distribution

We have recently proposed a general approach for approximating the power sum of log-normal random variables (RVs) by using the Pearson system of distributions, and then used it for performance analysis of Ultra Wide Band (UWB) wireless systems. Therein, we have also highlighted the main advantages of using the Pearson approximation instead of the usual Log-Normal one, and compared the proposed method with other approaches available in the open technical literature. However, despite being very accurate, the proposed method may be, in some circumstances, computational demanding since a non-linear least-squares problem needs to be solved numerically. Motivated by the above considerations, the aim of this paper is to provide an alternative approach for computing the parameters of the approximating Pearson Type IV distribution. The proposed solution is based on the method of moments (MoMs) in the logarithmic domain. In particular, the specific contribution of this paper is to provide closed-form expressions for the log- moments of the log-normal power sum. By using some known properties of the Laplace transform, we will show that the MGF of the log-normal power sum in the logarithmic domain (i.e., the log-MGF) can be obtained from the Mellin transform of the MGF of the Log-Normal power sum in the linear domain. From the estimated log-MGF, we will then compute the desired log-moments required for Pearson Type IV approximation. Numerical results will be also shown in order to substantiate the accuracy of the proposed method.