Skip to main content
SpringerLink
Log in

Product of two Gaussian Q-functions as sum of exponentials and its applications

  • Published:
Annals of Telecommunications Aims and scope Submit manuscript

Abstract

In this paper, product of two Gaussian-Q functions is represented as sum of exponentials. It is further used to evaluate error probabilities of modulation techniques in fading distributions. The knowledge of moment generating function (MGF) is sufficient enough to derive closed-form solution to integrals appearing in symbol error probability (SEP). Numerical results demonstrate accuracy improvement over other existing competing approximations. Furthermore, the proposed solutions are fairly simple as MGF of fading models comprises of fundamental mathematical functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. (2020) Taylor series expansion of Gaussian Q function. https://math.stackexchange.com/questions/3672516/taylor-series-approximation-of-gaussian-q-function-q-left-x-right-frac. [Online]

  2. Aggarwal S (2019) Closed-form solutions to SEP integrals over generalized fading distributions η − μ and κ − μ using approximate computing. Int J Commun Syst 32(18):e4162

    Article  Google Scholar 

  3. Aggarwal S (2019) A survey-cum-tutorial on approximations to Gaussian Q function for symbol error probability analysis over Nakagami- m fading channels. IEEE Communications Surveys Tutorials 21(3):2195–2223

    Article  Google Scholar 

  4. Aggarwal S (2020) Approximate solution to SEP integral over Fluctuating Beckmann fading. Wireless Pers Commun 111:1367–1376

    Article  Google Scholar 

  5. Aldalgamouni T, Ilter MC, Badarneh OS, Yanikomeroglu H (2018) Performance analysis of fisher-snedecor F composite fading channels. In: 2018 IEEE Middle East and North Africa Communications Conference (MENACOMM), pp 1–5

  6. Beaulieu NC (2006) A useful integral for wireless communication theory and its application to rectangular signaling constellation error rates. IEEE Trans Commun 54(5):802–805

    Article  Google Scholar 

  7. Bilim M (2019) QAM signaling over κ − μ shadowed fading channels. Physical Communication 34:261–271

    Article  Google Scholar 

  8. Bilim M (2019) Some new results for integrals involving Gaussian Q-function and their applications to α − μ and η − μ fading channels. Wireless Pers Commun 109:1463–1469

    Article  Google Scholar 

  9. Bilim M, Kapucu N (2019) Average symbol error rate analysis of qam schemes over millimeter wave fluctuating two-ray fading channels. IEEE Access 7:105746–105754

    Article  Google Scholar 

  10. Chauhan PS, Kumar S, Soni SK (2019) New approximate expressions of average symbol error probability, probability of detection and auc with mrc over generic and composite fading channels. AEU - Int J Electron Commun 99:119–129

    Article  Google Scholar 

  11. Chiani M, Dardari D, Simon MK (2003) New exponential bounds and approximations for the computation of error probability in fading channels. IEEE Trans Wireless Commun 2(4):840–845

    Article  Google Scholar 

  12. Ermolova NY (2009) Useful integrals for performance evaluation of communication systems in generalized η − μ and κ − μ fading channels. IEEE Comm 3(2):303–308

    MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  14. Kapucu N (2019) Error performance of digital modulations over Fisher-Snedecor \(\mathcal {F}\) fading channels. AEU - Int J Electron Commun 108:73–78

    Article  Google Scholar 

  15. Karagiannidis GK, Lioumpas AS (2007) An improved approximation for the Gaussian Q-function. IEEE Commun Lett 11(8):644–646

    Article  Google Scholar 

  16. Kaur M, Yadav RK (2020) Performance analysis of beaulieu-xie fading channel with mrc diversity reception. Trans Emerg Telecommun Technol, pp e3949. https://doi.org/10.1002/ett.3949

  17. Kumar S, Jain P (2017) Micro-diversity analysis of error probability and channel capacity over hoyt-gamma fading. Radioengineering 26:1096–1103. https://doi.org/10.13164/re.2017.1096

    Article  Google Scholar 

  18. Kumar S, Soni SK, Jain P (2018) Performance of mrc receiver over hoyt-lognormal composite fading channel. Int J Electron 105(9):1433–1450

    Article  Google Scholar 

  19. Li R, Kam PY (2007) Averages of the product of two Gaussian Q-functions over fading statistics and applications. IEEE Commun Lett 11(1):58–60

    Article  Google Scholar 

  20. Lopez-Benitez M (2015) Average of arbitrary powers of Gaussian Q-function over η-μ and κ-μ fading channels. Electron Lett 51(11):869–871

    Article  Google Scholar 

  21. Loskot P, Beaulieu NC (2009) Prony and polynomial approximations for evaluation of the average probability of error over slow-fading channels. IEEE Trans Veh Technol 58(3):1269–1280

    Article  Google Scholar 

  22. Mallik RK (2008) Average of product of two Gaussian Q-functions and its application to performance analysis in Nakagami fading. IEEE Trans Commun 56(8):1289–1299

    Article  Google Scholar 

  23. Olabiyi O, Annamalai A (2012) Invertible exponential-type approximations for the Gaussian probability integral Q(x) with applications. IEEE Wireless Commun Lett 1(5):544–547

    Article  Google Scholar 

  24. Paris JF (2014) Statistical characterization of κ − μ Shadowed fading. IEEE Trans Veh Technol 63(2):518–526

    Article  Google Scholar 

  25. Radaydeh RM (2007) Average SEP of rectangular QAM in Rayleigh fading with GSC. IEEE Comm Letters 11(6):492–494

    Article  Google Scholar 

  26. Ramirez-Espinosa P, Lopez-Martinez FJ, Paris JF, Yacoub MD, Martos-Naya E (2018) An extension of the κ-μ shadowed fading model: Statistical characterization and applications. IEEE Trans Veh Technol 67(5):3826–3837

    Article  Google Scholar 

  27. Rasethuntsa TR, Kumar S (2019) An integrated performance evaluation of ed-based spectrum sensing over α − κ − μ and α − κ − μ-extreme fading channels. Trans Emerg Telecommun Technol 30(5):e3569

    Article  Google Scholar 

  28. Rasethuntsa TR, Kumar S, Kaur M (2019) A comprehensive performance evaluation of a df-based multi-hop system over α − κ − μ and α − κ − μ-extreme fading channels. arXiv: Information Theory

  29. Rugini L (2016) Symbol error probability of hexagonal QAM. IEEE Commun Lett 20(8):1523–1526

    Article  Google Scholar 

  30. Sadhwani D, Yadav RN (2020) A comprehensible form of the product of two Gaussian Q functions and its usefulness in κ − μ shadowed fading distribution. J Franklin Inst 357(8):5110–5123

    Article  MathSciNet  Google Scholar 

  31. Sadhwani D, Yadav RN, Aggarwal S (2017) Tighter bounds on the Gaussian Q function and its application in Nakagami-m fading channel. IEEE Wireless Commun Lett 6(5):574–577

    Article  Google Scholar 

  32. Sadhwani D, Yadav RN, Aggarwal S, Raghuvanshi DK (2018) Simple and accurate SEP approximation of hexagonal-QAM in AWGN channel and its application in parametric α − μ, η − μ, κ − μ fading, and log-normal shadowing. IET Communications 12(5):1454–1459

    Article  Google Scholar 

  33. Shi Q, Karasawa Y (2011) An accurate and efficient approximation to the Gaussian Q-function and its applications in performance analysis in Nakagami-m fading. IEEE Commun Lett 15(5):479–481

    Article  Google Scholar 

  34. Simon MK, Alouini MS (2005) Digital Communication over Fading Channels, 2nd edn. Wiley, Hoboken

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electronics and Communication Engineering, MANIT, Bhopal, India

    Supriya Aggarwal

Authors
  1. Supriya Aggarwal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Supriya Aggarwal.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aggarwal, S. Product of two Gaussian Q-functions as sum of exponentials and its applications. Ann. Telecommun. 77, 151–161 (2022). https://doi.org/10.1007/s12243-021-00864-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12243-021-00864-w

Keywords

Search

Navigation