Bivariate Fisher–Snedecor \( {\user2{\mathcal{F}}} \) Distribution with Arbitrary Fading Parameters

  • Weijun ChengEmail author
  • Xianmeng Xu
  • Xiaoting Wang
  • Xiaohan Liu
Conference paper
Part of the Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering book series (LNICST, volume 309)


A bivariate Fisher–Snedecor \( {\mathcal{F}} \) composite distribution with arbitrary fading parameters (not necessary identical) is presented in this paper. We derive novel theoretical formulations of the statistical characteristics for the correlated \( {\mathcal{F}} \) composite fading model, which include the joint probability density function, the joint cumulative distribution function, the joint moments and the power correlation coefficient. Capitalizing on the joint cumulative distribution function, the bit error rate for binary digital modulation systems and the outage probability of a correlated dual-branch selection diversity system, and the level crossing rate and the average fade duration of a sampled Fisher-Snedecor \( {\mathcal{F}} \) composited fading envelope are obtained, respectively. Finally, we employ numerical and simulation results to demonstrate the validity of the theoretical analysis under various correlated fading and shadowing scenarios.


Fisher–Snedecor \( {\mathcal{F}} \) distribution Correlated composite fading Selection diversity Second-order statistics 


  1. 1.
    Yoo, S.K., Cotton, S.L., Sofotasios, P.C., Matthaiou, M., Valkama, M., Karagiannidis, G.K.: The Fisher-Snedecor \( {\mathcal{F}} \) distribution: a simple and accurate composite fading model. IEEE Commun. Lett. 21(7), 1661–1664 (2017)CrossRefGoogle Scholar
  2. 2.
    Badarneh, O.S., da Costa, D.B., Sofotasios, P.C., Muhaidat, S., Cotton, S.L.: On the sum of Fisher-Snedecor \( {\mathcal{F}} \) variates and its application to maximal-ratio combining. IEEE Wirel. Commun. Lett. 7(6), 966–969(2018)Google Scholar
  3. 3.
    Kong, L., Kaddoum, G.: On physical layer security over the Fisher-Snedecor \( {\mathcal{F}} \) wiretap fading channels. IEEE Access 6, 39466–39472 (2018)CrossRefGoogle Scholar
  4. 4.
    Yoo, S.K, Sofotasios, P.C., Cotton, S.L., et al.: A comprehensive analysis of the achievable channel capacity in \( {\mathcal{F}} \) composite fading channels. IEEE Access 7, 34078–34094 (2019)CrossRefGoogle Scholar
  5. 5.
    Yoo, S.K., Sofotasios, P.C., Cotton, S.L., Muhaidat, S., Badarneh, O.S., Karagiannidis, G.K.: Entropy and energy detection-based spectrum sensing over \( {\mathcal{F}} \)-composite fading channels. IEEE Trans. Commun. 67(7), 4641–4653 (2019)CrossRefGoogle Scholar
  6. 6.
    Al-Hmood, H., Al-Raweshidy, H.S.: Selection combining scheme over non-identically distributed Fisher-Snedecor \( {\mathcal{F}} \) fading channels. arXiv:1905.05595 (2019)
  7. 7.
    Zhao, H., Yang, L., Salem, A.S., Alouini, M.: Ergodic capacity under power adaption over Fisher-Snedecor \( {\mathcal{F}} \) fading channels. IEEE Commun. Lett. 23(3), 546–549 (2019)Google Scholar
  8. 8.
    Chen, S., Zhang, J., Karagiannidis, G.K., Ai, B.: Effective rate of MISO systems over Fisher-Snedecor \( {\mathcal{F}} \) fading channels. IEEE Commun. Lett. 22(12), 2619–2622 (2018)Google Scholar
  9. 9.
    Aldalgamouni, T., Ilter, M.C., Badarneh, O.S., Yanikomeroglu, H.: Performance analysis of Fisher-Snedecor \( {\mathcal{F}} \) composite fading channels. In: 2018 IEEE Middle East and North Africa Communications Conference (MENACOMM), pp. 1–5, IEEE, Jounieh (2018)Google Scholar
  10. 10.
    Reig, J., Rubio, L., Cardona, N.: Bivariate Nakagami-m distribution with arbitrary fading parameters. Electron. Lett. 38(25), 1715–1717 (2002)CrossRefGoogle Scholar
  11. 11.
    Zhang, R., Wei, J., Michelson, D.G., Leung, V.C.M.: Outage probability of MRC diversity over correlated shadowed fading channels. IEEE Wirel. Commun. Lett. 1(5), 516–519 (2012)CrossRefGoogle Scholar
  12. 12.
    Bithas, P.S., Sagias, N.C., Mathiopoulos, P.T., Kotsopoulos, S.A., Maras, A.M.: On the correlated K-distribution with arbitrary fading parameters. IEEE Sig. Process Lett. 15, 541–544 (2008)CrossRefGoogle Scholar
  13. 13.
    Bithas, P.S., Sagias, N.C., Mathiopoulos, P.T.: The bivariate generalized-K (KG) distribution and its application to diversity receivers. IEEE Trans. Commun. 57(9), 2655–2662 (2009)CrossRefGoogle Scholar
  14. 14.
    Reddy, T., Subadar, R., Sahu, P.R.: Outage probability of selection combiner over exponentially correlated Weibull-gamma fading channels for arbitrary number of branches. In: 2010 National Conference on Communications, pp. 1–5, IEEE, Madras (2010)Google Scholar
  15. 15.
    Ni, Z., Zhang, X., Liu, X., Yang, D.: Bivariate Weibull-gamma composite distribution with arbitrary fading parameters. Electron. Lett. 48(18), 1165–1167 (2012)CrossRefGoogle Scholar
  16. 16.
    Trigui, I., Laourine, A., Affes, S., Stéphenne, A.: Bivariate \( {\mathcal{G}} \) distribution with arbitrary fading parameters. In: 2009 3rd International Conference on Signals, Circuits and Systems (SCS), pp. 1–5, IEEE, Medenine (2009)Google Scholar
  17. 17.
    Reig, J., Rubio, L., Rodrigo-Peñarrocha, V.M.: On the bivariate Nakagami-lognormal distribution and its correlation properties. Int. J. Antennas Propag. 2014, 1–8 (2014)Google Scholar
  18. 18.
    Gradshteyn, I., Ryzhik, I.: Table of Integrals, Series, and Products, 8th edn. Academic Press, London (2007)zbMATHGoogle Scholar
  19. 19.
    Chun, Y.J., Cotton, S.L., Dhillon, H.S., Lopez-Martinez, F.J., Paris, J.F., Yoo, S.K.: A comprehensive analysis of 5G heterogeneous cellular systems operating over \( \kappa -\mu \) shadowed fading channels. IEEE Trans. Wirel. Commun. 16(11), 6995–7010 (2017)Google Scholar
  20. 20.
    Simon, M.K., Alouni, M.-S.: Digital Communication over Fading Channels, 2nd edn. Wiley, New York (2005)Google Scholar
  21. 21.
    Ansari, I.S., Al-Ahmadi, S., Yilmaz, F., Alouini, M., Yanikomeroglu, H.: A new formula for the BER of binary modulations with dual-branch selection over generalized-K composite fading channels. IEEE Trans. Commun. 59(10), 2654–2658 (2011)CrossRefGoogle Scholar
  22. 22.
    Wojnar, A.H.: Unknown bounds on performance in Nakagami channels. IEEE Trans. Commun. 34(1), 22–24 (1986)CrossRefGoogle Scholar
  23. 23.
    García-Corrales, C., Canete, F.J., Paris, J.F.: Capacity of \( \kappa -\mu \) shadowed fading channels. Int. J. Antennas Propag. 2014, 1–8 (2014)Google Scholar
  24. 24.
    López-Martínez, F.J., Martos-Naya, E., Paris, J.F., Fernández-Plazaola, U.: Higher order statistics of sampled fading channels with applications. IEEE Trans. Veh. Technol. 61(7), 3342–3346 (2012)CrossRefGoogle Scholar
  25. 25.
    López-Fernández, J., Paris, J.F., Martos-Naya, E.: Bivariate Rician shadowed fading model. IEEE Trans. Veh. Technol. 67(1), 378–384 (2018)CrossRefGoogle Scholar
  26. 26.
    Reig, J., Martinez-Amoraga, M.A., Rubio, L.: Generation of bivariate Nakagami-m fading envelopes with arbitrary not necessary identical fading parameters. Wirel. Commun. Mob. Comput. 2007(7), 531–537 (2007)CrossRefGoogle Scholar

Copyright information

© ICST Institute for Computer Sciences, Social Informatics and Telecommunications Engineering 2020

Authors and Affiliations

  • Weijun Cheng
    • 1
    Email author
  • Xianmeng Xu
    • 1
  • Xiaoting Wang
    • 1
  • Xiaohan Liu
    • 1
  1. 1.School of Information EngineeringMinzu University of ChinaBeijingPeople’s Republic of China

Personalised recommendations