Novel approximations for the Q-function with application in SQNR calculation

In this paper, an upper bound expression for the Q-function approximation is proposed. This expression defines the class of approximations that are upper bounds under the particular condition derived in the paper. The proposed upper bound approximation of the Q-function and the approximations derived from it are applied in signal to quantization noise ratio (SQNR) calculation of variance-matched Gaussian source scalar quantization. The proposed Q-function approximation having a simple analytical form and being a parametric one is optimized in terms of its parameter with respect to relative error (RE) of approximation for the particular problem observed. Specifically, three different optimization methods are proposed, so that different Q-function approximations are provided in the paper. Moreover, the manner for expansion of the obtained results is provided in order to make our proposal applicable for facilitating any mathematical analysis involving Q-function calculation. The results indicate that the proposed Q-function approximations not only outperform the numerous previously reported approximations in terms of accuracy, but also provide the derivation of the reasonably accurate closed-form formula for SQNR of variance-matched scalar quantization for the Gaussian source, which is not achievable with the application of the previously reported Q-function approximations of similar analytical form complexity. The results presented in this paper are applicable to many signal processing and communication problems requiring Q-function calculation.