Novel approximation for the Gaussian Q-function and related applications

Using the semi-infinite Gauss-Hermite quadrature rule defined in (0,∞), we present an accurate and efficient approximation to the Gaussian Q-function, which is expressed as a finite sum of exponential functions. Based on this approximation, we address a product of Gaussian Q-functions averaged over Nakagami-m fading, ending up with a closed-form solution applicable for any real m ≥ 0.5. We further consider more general situations, in which the Gaussian Q-function is involved in more complicated ways. Numerical examples show that the proposed method with very few terms can give error probabilities (in closed form) that are virtually indistinguishable from the exact results obtained by numerical integration.