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Design and Analysis of the Fractional-Order Complex Least Mean Square (FoCLMS) Algorithm

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Abstract

In this work, a new class of stochastic gradient algorithm is developed based on fractional calculus. Unlike the existing algorithms, the concept of complex fractional gradient is introduced by employing Caputo’s fractional derivative which results in a fractional steepest descent algorithm and a fractional-order complex LMS (FoCLMS) algorithm. We demonstrate that with the Caputo’s fractional gradient definition, the Weiner solution remains invariant. Convergence analysis of the proposed FoCLMS algorithm is presented for both transient and steady state scenarios. Consequently, expressions for the learning curves and steady state EMSE are derived. Our theoretical developments are validated by simulation experiments. Extensive simulations are presented to investigate all possible scenarios: channel with negative weights and real input data, channel with positive weights and complex input data, and channel with complex weights and complex input data.

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Authors and Affiliations

  1. Department of Electrical Engineering, Usman Institute of Technology, ST-13, Block-7, Gulshan-e-Iqbal, Abu-Hasan Isphahani Road, Karachi, 75300, Pakistan

    Jawwad Ahmad

  2. Faculty of Engineering, Sciences and Technology, IQRA University, North Karachi Campus, Sector 7b North Karachi Twp, Karachi, 75850, Pakistan

    Muhammad Zubair

  3. Faculty of Computer Science, Shaheed Zulfikar Ali Bhutto Institute of Science and Technology, SZABIST, Block 5 Clifton, Karachi, 75600, Pakistan

    Syed Sajjad Hussain Rizvi

  4. Electrical and Computer Engineering Department, King Abdulaziz University, Al Ehtifalat St, Jeddah, 21589, Saudi Arabia

    Muhammad Shafique Shaikh

  5. Center of Excellence in Intelligent Engineering Systems (CEIES), King Abdulaziz University, Al Ehtifalat St, Jeddah, 21589, Saudi Arabia

    Muhammad Shafique Shaikh

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  1. Jawwad Ahmad
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Ahmad, J., Zubair, M., Rizvi, S.S.H. et al. Design and Analysis of the Fractional-Order Complex Least Mean Square (FoCLMS) Algorithm. Circuits Syst Signal Process 40, 5152–5181 (2021). https://doi.org/10.1007/s00034-021-01715-8

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