Abstract
The purpose of this note is to discuss some aspects of recently proposed fractional-order variants of complex least mean square (CLMS) and normalized least mean square (NLMS) algorithms in Shah et al. (Nonlinear Dyn. 88(2):839–858, 2017). It is observed that these algorithms do not always converge, whereas they have apparently no advantage over the CLMS and NLMS algorithms whenever they converge. Our claims are based on analytical reasoning and are supported by numerical simulations.
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References
Haykin, S.: Adaptive Filter Theory, 5th edn. Pearson, London (2014)
Diniz, P.S.R.: Adaptive Filtering: Algorithms and Practical Implementation. Springer, New York (2013)
Widrow, B., McCool, J., Ball, M.: The complex LMS algorithm. Proc. IEEE 63, 719–720 (1975)
Khalili, A., Rastegarnia, A., Sanei, S.: Quantized augmented complex least-mean square algorithm: Derivation and performance analysis. Signal Process. 121, 54–59 (2016)
Chen, B., Zhao, S., Zhu, P., Príncipe, J.C.: Quantized kernel least mean square algorithm. IEEE Trans. Neural Netw. Learn. Syst. 23(1), 22–32 (2012)
Nagumo, J.I., Noda, A.: A learning method for system identification. IEEE Trans. Autom. Control 12(3), 282–287 (1967)
Al-Saggaf, U.M., Moinuddin, M., Arif, M., Zerguine, A.: The q-least mean squares algorithm. Signal Process. 111, 50–60 (2015)
Shah, S.M., Samar, R., Khan, N.M., Raja, M.A.Z.: Design of fractional-order variants of complex LMS and NLMS algorithms for adaptive channel equalization. Nonlinear Dyn. 88(2), 839–858 (2017)
Raja, M.A.Z., Qureshi, I.M.: A modified least mean square algorithm using fractional derivative and its application to system identification. Eur. J. Sci. Res. 35(1), 14–21 (2009)
Bershad, N.J., Wen, F., So, H.C.: Comments on “Fractional LMS algorithm”. Signal Process. 133, 219–226 (2017)
Wahab, A., Khan, S.: Comments on “Fractional extreme value adaptive training method: Fractional steepest descent approach”. IEEE Trans. Neural Netw. Learn. Syst. 31(3), 1066–1068 (2020)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2016)
Kreutz-Delgado, K.: The complex gradient operator and the CR-calculus. arXiv preprint (arXiv:0906.4835) (2009). Retrieved 15 Dec 2017
Tarasov, V.E.: On chain rule for fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 30, 1–4 (2016)
Khan, S., Wahab, A., Naseem, I., Moinudddin, M.: Simulation code for Comments on “Design of fractional-order variants of complex LMS and NLMS algorithms for adaptive channel equalization”. Nonlinear Dyn. (2020). https://doi.org/10.5281/zenodo.3723686
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Khan, S., Wahab, A., Naseem, I. et al. Comments on “Design of fractional-order variants of complex LMS and NLMS algorithms for adaptive channel equalization”. Nonlinear Dyn 101, 1053–1060 (2020). https://doi.org/10.1007/s11071-020-05850-w
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DOI: https://doi.org/10.1007/s11071-020-05850-w