A Survey-cum-Tutorial on Approximations to Gaussian Q Function for Symbol Error Probability Analysis Over Nakagami-m Fading Channels

This paper is a survey-cum-tutorial on the Gaussian Q function and its approximations; and their significance in computing the error probabilities over Nakagami-m fading. The goal is to focus on benefits of approximate computing over perfect solutions. Undoubtedly their exist closed-form solutions to some integrals involved in the error probabilities, but these either involve complex mathematical computations or are undefined for complete range of parameter variations, i.e., are valid only for certain specific parameter values. On the other hand approximations result in general solutions which reduce the computation complexity and are also valid for the entire range of parameter variations; though this comes at a cost, as approximations tend to introduce slight errors in computations. Through this paper, we analyze approximations for their simplicity, analytical tractability, and accuracy to understand how it can benefit research in domains where perfect solutions are either lacking or complex. We focus on Nakagami-m as it spans wide range of fading and covers other Nakagami distributions as special cases. For various digital modulation schemes the error probabilities are first expressed in closed-form using approximations, which are then compared for their accuracy over wide range of signal-to-noise ratio (SNR) [-10, 60] dB, along with variations in the fading parameter m.