Remez Exchange Algorithm for Approximating Powers of the Q-Function by Exponential Sums

Abstract

In this paper, we present simple and tight approximations for the integer powers of the Gaussian Q-function, in the form of exponential sums. They are based on optimizing the corresponding coefficients in the minimax sense using the Remez exchange algorithm. In particular, the best exponential approximation is characterized by the alternation of its absolute error function, which results in extrema that alternate in sign and have the same magnitude of error. The extrema are described by a system of nonlinear equations that are solved using Newton– Raphson method in every iteration of the Remez algorithm, which eventually leads to a uniform error function. This approximation can be employed in the evaluation of average symbol error probability (ASEP) under additive white Gaussian noise and various fading models. Especially, we present several application examples on evaluating ASEP in closed forms with Nakagami-m, Fisher–Snedecor $\mathcal{F}$, η − µ, and κ − µ channels. The numerical results show that our approximations outperform the existing ones with the same form in terms of the global error. In addition, they achieve high accuracy for the whole range of the argument with and without fading, and it can even be improved further by increasing the number of exponential terms.