Hermite series expansion of the PDF of sum of random variables and its application to analysis of equal gain combining diversity reception

This paper presents an investigation on the applicability of the Hermite series expansion to analytically determine the probability density function of the sum of independent random variables and its application to analysis of problems that arise in digital communications. The density function of the sum of random variables is expressed in the form of an infinite series using Hermite polynomials. The coefficients of the series are expressed in terms of the central moments of the individual random variables. The knowledge of the characteristic function is not required. The newly derived series is applied to analyze the symbol error rate of M-ary CPSK signals received over Nakagami-m fading channels with equal gain combining diversity reception. An expression for the symbol error rate is derived. Our investigation shows that the error probabilities can be numerically computed efficiently with the new series approximately up to 10
⁻⁵
. The Hermite series expansion, however, exhibits slow convergence and it is not particularly suitable for evaluating very small error probabilities though seldom one requires such low probabilities.