Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials

TL;DR

This paper introduces a systematic method to derive tight, simple approximations and bounds for the Gaussian Q-function using sums of exponentials, optimizing for minimal maximum error and extending to polynomial functions of the Q-function.

Contribution

The authors develop a novel numerical approach to generate globally optimal exponential sum approximations and bounds for the Q-function and its polynomial transformations, improving accuracy and efficiency.

Findings

Achieves tighter bounds with fewer exponential terms than existing methods.

Provides a flexible framework to control absolute and relative errors.

Demonstrates superior accuracy in numerical comparisons.

Abstract

This paper presents a novel systematic methodology to obtain new simple and tight approximations, lower bounds, and upper bounds for the Gaussian Q-function, and functions thereof, in the form of a weighted sum of exponential functions. They are based on minimizing the maximum absolute or relative error, resulting in globally uniform error functions with equalized extrema. In particular, we construct sets of equations that describe the behaviour of the targeted error functions and solve them numerically in order to find the optimized sets of coefficients for the sum of exponentials. This also allows for establishing a trade-off between absolute and relative error by controlling weights assigned to the error functions' extrema. We further extend the proposed procedure to derive approximations and bounds for any polynomial of the Q-function, which in turn allows approximating and bounding many functions of the Q-function that meet the Taylor series conditions, and consider the integer powers of the Q-function as a special case. In the numerical results, other known approximations of the same and different forms as well as those obtained directly from quadrature rules are compared with the proposed approximations and bounds to demonstrate that they achieve increasingly better accuracy in terms of the global error, thus requiring significantly lower number of sum terms to achieve the same level of accuracy than any reference approach of the same form.