Least squares approximation to lognormal sum distribution via piecewise linear functions

In this paper, the least squares approximation via a piecewise linear function approach is applied to solve the approximation problem of a sum of lognormal random variables. A number of linear basis functions are applied and the corresponding coefficients are specified to form a piecewise linear approximation to the sum lognormal cumulative distribution function. By using the proposed approach, arbitrary precision of the approximation can be achieved by increasing the number of basis functions. The computation complexity shows no significant increase with the increase of the number of basis functions. Simulation results exhibit a desirable performance by using the proposed approximation method through piecewise linear basis functions.

I. Introduction

The sum of lognormal random variables (RVs) has found important applications in communication problems and has attracted lots of research attention recently (see for example, [1] – [9] and the references cited in). Early work [10] shows that the received power (or signal-to-interference ratio) of a wireless link suffering from shadowing and fading follows lognormal distribution after the power control. Therefore, when we try to evaluate the overall interference of the desired wireless communication link, the problem falls down to find the distribution of a sum of lognormal random variables.