Quadrature-Based Exponential-Type Approximations for the Gaussian Q-Function

In this paper, we present a comprehensive overview of (perhaps) all possible approximations resulting from applying the most common numerical integration techniques on the Gaussian Q-function. We also present a unified method to optimize the coefficients of the resulting exponential approximation for any number of exponentials and using any numerical quadrature rule to produce tighter approximations. Two new tight approximations are provided as examples by implementing the Legendre numerical rule with Quasi-Newton method for two and three exponential terms. The performance of the different numerical integration techniques is evaluated and compared, and the accuracy of the optimized ones is verified for the whole argument-range of interest and in terms of the chosen optimization criterion.

Automated summary

This paper provides a comprehensive overview of approximations resulting from applying numerical integration techniques on the Gaussian Q-function and presents a unified method for optimizing the coefficients of the resulting exponential approximation. Two new tight approximations are introduced as examples using the Legendre numerical rule with the Quasi-Newton method. The performance of different numerical integration techniques is evaluated and the accuracy of the optimized approximations is verified for the entire parameter range of interest.