Optimization of the Exponential Bounds on the Gaussian Q Function Using Interior Point Algorithm and Its Application in Communication Theory

Owing to various applications in the field of wireless communication systems, in this paper, using vector-based interior-point algorithm, we propose a generic methodology which optimizes simple exponential based approximations of the Gaussian $Q$ function (GQF) yielding extremely accurate optimized approximations. We optimize the relative error (RE) which is considered as one of the key metrics used to evaluate the performance of these approximations. Precisely, we target the points of $local~maximas$ where the RE is high, defining a new set of optimized coefficients yielding reduced RE at these points of concern. We also optimize the points of $local~minimas$ ; however at these points the percentage reduction in RE is not that significant. We then compute the harmonic mean of all these optimal coefficients which makes the originally proposed bounds much tighter, for the entire performance range of the GQF. We further illustrate the tightness of the optimized approximation by facilitating the accurate computation of the error performance metrics like symbol error probability of various coherent digital modulation schemes like square quadrature amplitude modulation (SQAM), rectangular-QAM, cross-QAM and hexagonal-QAM over the versatile $\kappa -\mu $ shadowed fading channel. The analysis is also validated with the help of Monte-Carlo simulations.