A Clarification of the Proper-Integral Form for the Gaussian Q-Function and Some New Results Involving the F-Function

We discuss the origin of the well-known integral form for the Gaussian Q-function. Although this form is originally attributed to Craig in his 1991 paper, it actually was inferred in previous communications theory references in the literature by Weinstein, and explicitly stated by Pawula, Rice and R...

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Bibliographic Details
Published inIEEE communications letters Vol. 18; no. 9; pp. 1495 - 1498
Main Authors Lopez-Martinez, F. Javier, Pawula, R. F., Martos-Naya, Eduardo, Paris, Jose F.
Format Journal Article
LanguageEnglish
Published IEEE 01.09.2014
Subjects
Error analysis
Error probability
Fading
Frequency modulation
Gaussian noise
Integral equations
Vectors
hypergeometric functions
gaussian Q-function
F-function
Communication theory
special functions
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Summary:We discuss the origin of the well-known integral form for the Gaussian Q-function. Although this form is originally attributed to Craig in his 1991 paper, it actually was inferred in previous communications theory references in the literature by Weinstein, and explicitly stated by Pawula, Rice and Roberts. Specifically, this form can be seen as a particular case of the general F-function, that is related to the distribution of the phase angle between two vectors perturbed by Gaussian noise. We show that the F-function includes as particular cases the one- and two-dimensional Gaussian Q-functions, and is connected with a class of confluent hypergeometric functions through the multivariate Φ 1 (n) function. This connection generalizes the existing relation between the Gaussian Q-functions and this family of hypergeometric functions.
ISSN:1089-7798
1558-2558
DOI:10.1109/LCOMM.2014.2331056