Simple Approximations of the Error Function Q(x) for Communications Applications

Börjesson, Per Ola; Sundberg, C.-E.W.

doi:10.1109/tcom.1979.1094433

Cited by 207 publications

(119 citation statements)

References 3 publications

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“…It should be noted here that the approximation results of the Mills' ratio r(x) are commonly derived through empirical approaches. We start with the approximations to Q(·) developed by Borjesson and Sundberg [11] as…”

Section: The Mills'-ratio-based Formmentioning

confidence: 99%

“…The study in [10] represented the result of (1) with dissimilar A i over Nakagami-m fading, but N is limited up to 3, i.e., N ≤ 3. Hence, generally speaking, the need of approximating or bounding the Q-function is actually necessary and so far, this problem has been still a hot topic and got high attention [11][12][13][14][15][16][17][18][19][20][21]. This paper aims to give an overview about Qfunction approximation with emphasis on its applications.…”

Section: Introductionmentioning

confidence: 99%

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A Survey on Approximations of One-Dimensional Gaussian Q-Function

Bao

Tuyen

Tue

2016

REV-JEC

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Abstract-Predicting the digital communication system performance plays a very important role in the process of system design. This performance is usually quantified by symbol error probability or bit error probability. Computing these probabilities in presence of Additive White Gaussian Noise requires to work with integrals involving the Gaussian Qfunction, which cannot be expressed in closed-form in terms of elementary functions. As a result, approximating the Gaussian Q-function in closed-form expressions with high accuracy becomes a necessity. In this paper, we give an overview about the Gaussian Q-function approximations and via some illustrating examples, we discuss their accuracy, tractability as well as their computational complexity.

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Section: The Mills'-ratio-based Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Survey on Approximations of One-Dimensional Gaussian Q-Function

Bao

Tuyen

Tue

2016

REV-JEC

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“…A number of authors have looked for free-form approximations, with no constraints on the form. Borjesson and Sundberg [2] derived a class PBCS of approximations to the complementary error function (closely related to the Gaussian Q-function). Rewriting in terms of Q, it has the form:…”

Section: Q-function Approximationmentioning

confidence: 99%

“…The Q-function's definition in the form of an improper integral makes it hard to conduct exact analyses for communication systems [2,12]. Thus it would be highly desirable to obtain a closed (analytical) form using elementary functions.…”

Section: Introductionmentioning

confidence: 99%

Evolving the best known approximation to the Q function

Phong

Hoài

McKay

et al. 2012

Proceedings of the 14th Annual Conference on Genetic and Evolutionary Computation

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The Gaussian Q-function is the integral of the tail of the Gaussian distribution; as such, it is important across a vast range of fields requiring stochastic analysis. No elementary closed form is possible, so a number of approximations have been proposed. We use a Genetic Programming (GP) system, Tree Adjoining Grammar Guided GP (TAG3P) with local search operators to evolve approximations of the Qfunction in the form given by Benitez [1]. We found more accurate approximations than any previously published. This confirms the practical importance of local search in TAG3P.

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“…This mathematical function plays a vital role in the analysis and design of digital communications since the conditional error probability (CEP) of a broad class of coherent modulation schemes can be expressed either in terms of Q(x) alone or as a weighted sum of its integer powers (e.g., see E Q x from its canonical integral representation of (1) (owing to the presence of the argument of the function in the lower limit of the integral) have led to the development of alternative exponential-type integral representations for the Q-function and its integer powers [1,Eqs. (4.2), (4.9), (4.31) and (4.32)], analytically simple and tight closed-form bounds and approximations for Q(x) [3]- [16], characteristic function method [21] and the asymptotic analysis approach [17]- [19]. Table 1.…”

Section: Introductionmentioning

confidence: 99%