Nonlinear Dynamics

, Volume 92, Issue 3, pp 1243–1259 | Cite as

Fractional-order algorithms for tracking Rayleigh fading channels

  • Syed Muslim Shah
  • Raza Samar
  • Muhammad Asif Zahoor Raja
Original Paper


This paper presents the tracking behavior of fractional-order (FO) variants of the normalized least mean square (NLMS) algorithm in a nonstationary environment modeled as time-varying Rayleigh fading sequence. The celebrated recursive least squares (RLS) or its variant extended RLS (E-RLS) algorithms fail in such situations although they exhibit faster convergence but with the undesired feature of higher computational complexity. The FO algorithms are based on the Riemann–Liouville differintegral operator which is used in the gradient calculation; such schemes provide two step sizes and an FO to control the rate of convergence. In evaluation, we consider a high-speed mobile environment with a Rayleigh channel which results in different Doppler frequency shifts depending upon the transmission frequency, relative velocity of the transmitter and receiver. The proposed algorithms are compared with the NLMS, RLS and E-RLS schemes, and numerical experiments show the superiority of the FO variants over these schemes in terms of stability and model accuracy in the steady state. A hybrid scheme is also shown where the weights of an FO variant are initially trained with RLS and then performs self-adaptation; the FO scheme is confirmed to have better performance than all traditional counterparts.


Fractional derivatives Least mean square Mean square error Rayleigh fading channels Recursive least squares 


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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Syed Muslim Shah
    • 1
  • Raza Samar
    • 1
  • Muhammad Asif Zahoor Raja
    • 2
  1. 1.Department of Electrical EngineeringCapital University of Science and TechnologyIslamabadPakistan
  2. 2.Department of Electrical EngineeringCIIT-AttockAttockPakistan

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