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Highly Accurate Analytic Approximation to the Gaussian Q-function Based on the Use of Nonlinear Least Squares Optimization Algorithm

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Abstract

In this paper, as an extension of a previous study, an improved approximation for the Gaussian Q-function is presented. The nonlinear least squares algorithm is employed to optimize the coefficients of the proposed approximation. The accuracy of the presented approximation is evaluated using extensive computer simulations. Results show that the proposed approximation has superior accuracy in high arguments’ region when compared to the performance of other approaches introduced in the literature.

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References

  1. Simon, M.K.: Probability Distributions Involving Gaussian Random Variables: A Handbook for Engineers and Scientists. Springer, Berlin (2006)

    Google Scholar 

  2. Beaulieu, N.C.: A simple series for personal computer computation of the error function Q(). IEEE Trans. Commun. 37, 989–991 (1989)

    Article  Google Scholar 

  3. Tellambura, C., Annamalai, A.: Efficient computation of \(Q\left(x\right)\) for large arguments. IEEE Trans. Commun. 48, 529–532 (2000)

    Article  MATH  Google Scholar 

  4. Benitez, M.L., Casadevall, F.: Versatile, accurate, and analytically tractable approximation for the Gaussian Q-function. IEEE Trans. Commun. 59, 917–922 (2011)

    Article  Google Scholar 

  5. Börjesson, P.O., Sundberg, C.E.W.: Simple approximations of the error function \(Q\left(x\right)\) for communications applications. IEEE Trans. Commun. COM-27, 639–643 (1979)

    Article  Google Scholar 

  6. Chen, Y., Beaulieu, N.C.: A simple polynomial approximation to the Gaussian Q-function and its application. IEEE Commun. Lett. 13, 124–126 (2009)

    Article  Google Scholar 

  7. Chiani, M., Dardari, D., Simon, M.K.: New exponential bounds and approximations for the computation of error probability in fading channels. IEEE Trans. Wirel. Commun. 2, 840–845 (2003)

    Article  Google Scholar 

  8. Dyer, J.S., Dyer, S.A.: Corrections to, and comments on, “An improved approximation for the Gaussian Q-function”. IEEE Commun. Lett. 12, 231 (2008)

    Article  Google Scholar 

  9. Isukapalli, Y., Rao, B.D.: An analytically tractable approximation for the Gaussian Q-function. IEEE Commun. Lett. 12, 669–671 (2008)

    Article  Google Scholar 

  10. Karagiannidis, G.K., Lioumpas, A.S.: An improved approximation for the Gaussian Q-function. IEEE Commun. Lett. 11, 644–646 (2007)

    Article  Google Scholar 

  11. Loskot, P., Beaulieu, N.C.: Prony and polynomial approximations for evaluation of the average probability of error over slow-fading channels. IEEE Trans. Veh. Technol. 58, 1269–1280 (2009)

    Article  Google Scholar 

  12. Develi, I., Akdagli, A.: High-order exponential approximations for the Gaussian Q-function obtained by genetic algorithm. Int. J. Electron. doi:10.1080/00207217.2012.713024

  13. Develi, I.: A new approximation based on the differential evolution algorithm for the Gaussian Q-function. Int. J. Innov. Comput. Inf. Control 8, 7095–7102 (2012)

    Google Scholar 

  14. Levenberg, K.: A method for the solution of certain nonlinear problems in least-squares. Q. Appl. Math. II, 164–168 (1944)

    MathSciNet  Google Scholar 

  15. Marquardt, D.W.: An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11, 431–441 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dennis, J.E.: In: Nonlinear Least-Squares: State of the Art in Numerical Analysis, pp. 269–312. Academic Press, New York (1977)

    Google Scholar 

  17. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Dept. of Electrical and Electronics Engineering, Erciyes University, Kayseri, 38039, Turkey

    I. Develi

  2. Dept. of Computer Engineering, Erciyes University, Kayseri, 38039, Turkey

    A. Basturk

Authors
  1. I. Develi
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  2. A. Basturk
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Correspondence to I. Develi.

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Develi, I., Basturk, A. Highly Accurate Analytic Approximation to the Gaussian Q-function Based on the Use of Nonlinear Least Squares Optimization Algorithm. J Optim Theory Appl 159, 183–191 (2013). https://doi.org/10.1007/s10957-012-0217-0

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  • DOI: https://doi.org/10.1007/s10957-012-0217-0

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