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Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials

Research output: Contribution to journalArticleScientificpeer-review


Original languageEnglish
Article number 20036044
Pages (from-to) 6514 - 6524
Number of pages11
JournalIEEE Transactions on Communications
Early online date2020
Publication statusPublished - 3 Jul 2020
Publication typeA1 Journal article-refereed


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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