New Very Simply Explicitly Invertible Approximation of the Gaussian Q-Function

The approximations for the Gaussian Q-function found in the literature have been often developed with the objective of obtaining high estimation accuracies, to derive the error probability for digital modulation schemes. Unfortunately, the obtained mathematical expressions are often too complex, and, most of all, not explicitly invertible. The simple explicit invertibility may be required when, given the error probability needed by a communication system, one may be interested in deriving the operating signal-to-noise ratio needed by the considered application. This paper proposes a simply explicitly invertible mathematical approximation of the Gaussian Q- function on the basis of the one presented in Soranzo et al.'s 2014 paper, whose objective was that of finding a good invertible approximation for the Φ-function. As explained in the paper, the approximation for the Φ-function in Soranzo et al.'s 2014 paper presents a very good relative error with respect to the Φ-function, but can be certainly improved as far as the relative error with respect to the $Q$-function, simply derivable from the Φ-function, is concerned.