Very Simple Tight Bounds on the Q-Function

Abstract

We present new lower and upper bounds on the Gaussian Q-function, unified in a single and simple algebraic expression which contains only two exponential terms with a constant and a rational coefficient, respectively. Lower- and upper-bounding properties are obtained from such unified expression by selecting the coefficients accordingly. Despite the remarkable simplicity, the bounds are found to be as tight as multi-term alternatives obtained e.g. from the Exponential [2] and Jensen-Cotes [3] families of bounds. A corollary result is that the n-th integer power of Q(x) can also be tightly bounded both below and above with only n+1 algebraic terms. In addition to offering remarkable accuracy and mathematical tractability combined, the new bounds are very consistent, in which both lower and upper counterparts are similarly tight over the entire domain.