Improved Coefficients for the Karagiannidis-Lioumpas Approximations and Bounds to the Gaussian Q-Function

TL;DR

This paper improves the Karagiannidis-Lioumpas approximation of the Gaussian Q-function by optimizing coefficients for various error measures and extending the method to bounds, resulting in more accurate and tighter approximations.

Contribution

It introduces optimized coefficients for the KL approximation and bounds, enhancing accuracy and extending the approximation framework.

Findings

Optimized coefficients reduce maximum absolute and total errors.

Adding an extra coefficient yields tighter absolute and total errors.

Extended bounds provide more accurate lower and upper estimates.

Abstract

We revisit the Karagiannidis-Lioumpas (KL) approximation of the Q-function by optimizing its coefficients in terms of absolute error, relative error and total error. For minimizing the maximum absolute/relative error, we describe the targeted uniform error functions by sets of nonlinear equations so that the optimized coefficients are the solutions thereof. The total error is minimized with numerical search. We also introduce an extra coefficient in the KL approximation to achieve significantly tighter absolute and total error at the expense of unbounded relative error. Furthermore, we extend the KL expression to lower and upper bounds with optimized coefficients that minimize the error measures in the same way as for the approximations.