Tight Approximations for the Two Dimensional Gaussian $Q-$function

TL;DR

This paper derives two highly accurate, simple closed-form approximations for the two-dimensional Gaussian Q-function, facilitating analytical and numerical evaluations in digital communications where this function is not readily available.

Contribution

The work introduces novel, simple, and accurate closed-form approximations for the two-dimensional Gaussian Q-function, aiding performance analysis in digital communications.

Findings

Two new approximations are highly accurate.

Approximations are expressed in closed-form for ease of use.

Useful in digital communication performance evaluations.

Abstract

The aim of this work is the derivation of two approximated expressions for the two dimensional Gaussian Q-function, $Q(x,y;\rho)$. These expressions are highly accurate and are expressed in closed-form. Furthermore, their algebraic representation is relatively simple and therefore, convenient to handle both analytically and numerically. This feature is particularly useful for two reasons: firstly because it renders the derived expressions useful mathematical tools that can be utilized in numerous analytic performance evaluation studies in digital communications under fading; secondly because the two dimensional Gaussian Q-function is neither tabulated nor a built-in function in popular mathematical software packages such as $Maple$, $Mathematica$ and $Matlab$.