This paper presents a novel systematic methodology to obtain new simple and tight approximations, lower bounds, and upper bounds for the Gaussian Q-function, and functions thereof, in the form of a weighted sum of exponential functions. They are based on minimizing the maximum absolute or relative error, resulting in globally uniform error functions with equalized extrema. In particular, we construct sets of equations that describe the behaviour of the targeted error functions and solve them numerically in order to find the optimized sets of coefficients for the sum of exponentials. This also allows for establishing a trade-off between absolute and relative error by controlling weights assigned to the error functions' extrema. We further extend the proposed procedure to derive approximations and bounds for any polynomial of the Q-function, which in turn allows approximating and bounding many functions of the Q-function that meet the Taylor series conditions, and consider the integer powers of the Q-function as a special case. In the numerical results, other known approximations of the same and different forms as well as those obtained directly from quadrature rules are compared with the proposed approximations and bounds to demonstrate that they achieve increasingly better accuracy in terms of the global error, thus requiring significantly lower number of sum terms to achieve the same level of accuracy than any reference approach of the same form.

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We revisit the Karagiannidis-Lioumpas (KL) approximation of the Q-function by optimizing its coefficients in terms of absolute error, relative error and total error. For minimizing the maximum absolute/relative error, we describe the targeted uniform error functions by sets of nonlinear equations so that the optimized coefficients are the solutions thereof. The total error is minimized with numerical search. We also introduce an extra coefficient in the KL approximation to achieve significantly tighter absolute and total error at the expense of unbounded relative error. Furthermore, we extend the KL expression to lower and upper bounds with optimized coefficients that minimize the error measures in the same way as for the approximations.

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.