A Clarification of the Proper-Integral Form for the Gaussian $Q$-Function and Some New Results Involving the $F$-Function

摘要

We discuss the origin of the well-known integral form for the Gaussian $Q$-function. Although this form is originally attributed to Craig in his 1991 paper, it actually was inferred in previous communications theory references in the literature by Weinstein, and explicitly stated by Pawula, Rice and Roberts. Specifically, this form can be seen as a particular case of the general $F$-function, that is related to the distribution of the phase angle between two vectors perturbed by Gaussian noise. We show that the $F$-function includes as particular cases the one- and two-dimensional Gaussian $Q$-functions, and is connected with a class of confluent hypergeometric functions through the multivariate $Phi_{1}^{(n)}$ function. This connection generalizes the existing relation between the Gaussian $Q$-functions and this family of hypergeometric functions.