A 4Term ExponentialQuadratic Approximation for Gaussian Q or Error Functions Accurate to $1.65\times 10^{4}$
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C. Hastings, Jr.
Approximations for Digital Computers, Princeton, NJ: Princeton University Press, pp. 167169, 1955.
C. W. Clenshaw.
Chebyshev series for mathematical functions, National Physical Laboratory Mathematical Tables, vol. 5, Her Majesty's Stationery Office, London, 1962.
W. J. Cody.
``Rational Chebyshev approximations for the error function'', Mathematics of Computation, Vol. 23, No. 107, pp. 631637, July 1969.
J. L. Schonfelder.
``Chebyshev expansions for the error and related functions'', Mathematics of Computation. vol. 32, no. 144, pp. 12321240, 1978.
M. Abramowitz and I. A. Stegun.
(eds.) Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C., 1964.
S. Chevillard.
The functions erf and erfc computed with arbitrary precision and explicit error bounds. Information and Computation, 216:7295, 2012.
Marvin K. Simon.
Some new twists to problems involving the gaussian probability integral. IEEE Trans. on Communications, 46(2):200210, 1998.
M. K. Simon.
``Single integral representations of certain integer powers of the Gaussian Qfunction and their application'', IEEE Commun. Letters, vol. 6, no. 12, pp. 532534, 2002.
M. K. Simon.
``A simpler form of the Craig representation for the twodimensional joint Gaussian Qfunction'', IEEE Commun. Letters, vol. 6, no. 2, pp. 4951, 2002.
R. M. Radaydeh and M. M. Matalgah.
Results for integrals involving mth power of the Gaussian Qfunction over rayleigh fading channels with applications. Int. Conf. on Communications, pages 59105914, 2007.
I. S. Gradshteyn and I. M. Ryzhik.
Table of Integrals, Series and Products. Academic Press, 7th edition, 2007.
H. A. Fayed and A. F. Atiya.
``An evaluation of the integral of the product of the error function and the normal probability density with application to the bivariate normal integral'', Mathematics of Computation, vol. 83, no. 285, pp. 235250, 2013.
C.R. Selvakumar.
``Approximations to complementary error function by method of least squares'', Proceedings of the IEEE, vol. 70, no. 4, pp. , 1982.
S. A. Dyer and J. S. Dyer.
``Approximations to Error Functions'', IEEE Instrumentation & Measurement Magazine, vol. 10, no. 6, pp. 4548, 2007.
S. Aggarwal.
``A SurveycumTutorial on Approximations to Gaussian Q Function for Symbol Error Probability Analysis Over Nakagami m Fading Channels'', IEEE Communications Surveys & Tutorials, vol. 21, no. 3, pp. 21952223, 2019.
V. N. Q. Bao, L. P. Tuyen, and H. H. Tue.
A Survey on Approximations of OneDimensional Gaussian QFunction. REV Journal on Electronics and Communications, 5(12):114, 2015.
P. Borjesson and C. E. Sundberg.
``Simple Approximations of the Error Function Q(x) for Communications Applications'', IEEE Trans. on Communications, vol. 27, no. 3, pp. 639643, 1979.
W. M. Jang.
``A simple upper bound of the Gaussian Qfunction with closedform error bound'', IEEE Communications Letters, vol. 14, no. 2, p. 157, Feb. 2011.
G. K. Karagiannidis and A. S. Lioumpas.
``An improved approximation for the Gaussian Qfunction'', IEEE Commun. Lett., vol. 11, no. 8, pp.644646, Aug. 2007.
P. Loskot and N. C. Beaulieu.
``Prony and polynomial approximations for evaluation of the average probability of error over slowfading channels'', IEEE Trans. Vehicular Technol., vol. 58, no. 3, pp. 12691280, Mar. 2009.
L. M. Benitez and F. Casadevall.
``Versatile, accurate, and analytically tractable approximation for the gaussian Q function'', IEEE Trans. on Communications, vol. 59, no. 4, pp. 917922, Apr. 2011.
Marco Chiani, Davide Dardari, and Marvin K. Simon.
New exponential bounds and approximations for the computation of error probability in fading channels. IEEE Trans. on Wireless Communications, 2(4):840845, 2003.
P. C. Sofotasios and S. Freear.
``Novel expressions for the Marcum and one dimensional Qfunctions'', Proc. Int. Conf. on Wireless Info. Techn. & Systems (ICWITS'10), pp. 736740, 2010.
A. Annamalai, E. Adebola and O. Olabiyi.
``Simple ClosedForm Approximations for the ASER of Digital Modulations over Fading Channels'', International Conference on Wireless Networks, pp. 17, 2012.
P. Dao Ngoc, U. Nguyen Quang, H. Nguyen Xuan, and R. McKay.
``Evolving approximations for the gaussian Qfunction by genetic programming with semantic based crossover'', Congress on Evolutionary Computation, pp. 16, 2012.
P. Van Halen.
``Accurate analytical approximations for error function and its integral'', Electronics Letters, vol. 25, no. 9, pp. 561563, 1989.
R. M. Howard.
``Arbitrarily Accurate Analytical Approximations for the Error Function'', Technical Report, School of Elec. Eng. Comp. & Math. Sci., Curtin University, Perth, Australia, pp.144, Dec. 2020.
I. M. Tanash and T. Riihonen.
``Global minimax approximations and bounds for the Gaussian Qfunction by sums of exponentials'', IEEE Trans. on Communications, vol. 68 no. 10, pp. 65146524, 2020.
C. de Boor.
A Practical Guide to Splines, SpringerVerlag, 1978.
E. W. Ng and M. Geller.
A table of integrals of the error functions. J. Research of the National Bureau of Standards, 73B(1):120, 1968.
A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev.
Integrals and Series, Volume 2  Special Functions. Gordon & Breach Science Publishers, New York, 1986.
N. E. Korotkov and A. N. Korotkov.
Integrals Related to the Error Function. Chapman and Hall, 1st edition, 2020.
DOI: http://dx.doi.org/10.21553/revjec.276
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