Generalized Karagiannidis–Lioumpas Approximations and Bounds to the Gaussian Q-Function With Optimized Coefficients

Abstract

We develop extremely tight novel approximations, lower bounds and upper bounds for the Gaussian Q-function and offer multiple alternatives for the coefficient sets thereof, which are optimized in terms of the four most relevant criteria: minimax absolute/relative error and total absolute/relative error. To minimize error maximum, we modify the classic Remez algorithm to comply with the challenging nonlinearity that pertains to the proposed expression for approximations and bounds. On the other hand, we minimize the total error numerically using the quasi-Newton algorithm. The proposed approximations and bounds are so well matching to the actual Q-function that they can be regarded as virtually exact in many applications since absolute and relative errors of 10 −9 and 10 −5 , respectively, are reached with only ten terms. The significant advance in accuracy is shown by numerical comparisons with key reference cases.

New upper bounds on the Gaussian Q‐function via Jensen's inequality and integration by parts, and applications in symbol error probability analysis

“…The coefficients of lower and upper bounds and approximations in the form of weighted sum of purely exponential functions are numerically optimized in [3]. The approximations and bounds are generalized with coefficients optimized in [7]. Using integration by parts on the canonical form of the Gaussian Q-function in (1), an upper bound is obtained in [1].…”