Sigmoid Approximation to the Gaussian Q-function and its Applications to Spectrum Sensing Analysis

Most of the existing approximations for the Gaussian Q-function have been developed bearing in mind applications that require high estimation accuracy for large argument values (e.g., derivation of the bit/symbol error rates of digital communication systems, which are typically in the order of 10(-6) to 10(-12)). Such values correspond to positive arguments of the function and consequently most of the existing approximations are valid for positive arguments only. However, other relevant problems where the Gaussian Q-function can appear do not require such a level of accuracy (e.g., derivation of the detection probability of a signal detector, where accuracies of two or three decimal figures are sufficient) and, more importantly, require the evaluation of the Q-function over the whole range of values (i.e., both positive and negative arguments). In this context, this paper analyses a sigmoid approximation to the Q-function that provides adequate levels of accuracy for any real argument. As an illustrative example, this approximation is employed to obtain new closed-form expressions for the probability of detection of an energy detector under Rayleigh and Nakagami-m fading channels.