Piece-wise approximation for Gaussian ${\text{Q}}$-function and its applications

Abstract

The Gaussian ${\text{Q}}$-function and its integer powers are quite versatile in various fields of science. In this paper, we propose a novel exponential-based approximation for the Gaussian ${\text{Q}}$-function and compare the tightness of the approximation against existing methods. The proposed approximation is a piece-wise function, which is derived using a numerical method of integration. With the help of numerical results, we show that the proposed approximation yields accurate results for a wide range of ${\text{Q}}$-function arguments and at the same time outperforms existing approximations. Using the results of the proposed approximation, we evaluate the symbol error probability (SEP) of triangular quadrature amplitude modulation (TQAM) schemes for different constellations over additive white Gaussian noise (AWGN) and various fading channels including Nakagami-m, $\alpha - \mu$, $\kappa - \mu$ and $\alpha - \kappa - \mu$ fading channels. We facilitate the closed-form solution of the intractable integrals used in SEP calculation over the Nakagami-m fading channel. To further validate the results, we perform Monte Carlo simulations for SEP calculation of TQAM schemes over various fading channels and indicate a close match with the exact SEP calculation.