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Efficient Time-Varying q-Parameter Design for q-Incremental Least Mean Square Algorithm with Noisy Links

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Abstract

The quantum calculus provides an extra degree of freedom to search the local and global minima by inducing a q-parameter. Motivated by this fact, a quantum calculus-based noisy links incremental least mean squares (NL-qILMS) algorithm is proposed. Moreover, for the proposed NL-qILMS, we also devised various time-varying techniques for the selection of the optimal q-parameter to improve the performance. Furthermore, the closed-form solutions for the steady-state mean square deviation, excess mean square deviation and mean square error are derived. The analytical results are validated through simulation. Finally, extensive simulations have been done to evaluate the performance of the proposed algorithm for various choices of optimal q-values.

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Notes

  1. For the sake of compactness, the proposed NL-qILMS algorithm with (12) is termed as qSD1-ILMS.

  2. The proposed NL-qILMS algorithm with (13) is termed as qSD2-ILMS.

  3. The proposed NL-qILMS algorithm with (14) is termed as qML-ILMS.

  4. The proposed NL-qILMS algorithm with (15) is termed as qSN-ILMS.

  5. The proposed NL-qILMS algorithm with (16) is termed as qP-ILMS.

  6. The matrix \(\hat{\mathbf{G}}_{k}(i)\) is decoupled as \(\mu _{k}{} \mathbf{G}_{k}(i)\) for the analysis.

  7. Here we dropped the time index for simplicity.

  8. Note that \(R_{\mathbf {x},k}=\lambda _{k} I\), \(S_{k} = \sigma _{r,k}^{2}I\), \(\bar{G}_{k}=g_{k}I\), additionally for small step-size \(\bar{F}_{k}\) is reduced to, \(\bar{F}_{k} \approx (1 - 2 \mu _{k} \lambda g_{k})I\) and \(\hat{g}_{k}(i)=\mu _{k,\mathrm{min}}\) as \(i\rightarrow \infty \).

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Acknowledgements

The authors acknowledge the support of the Karachi Institute of Economics and Technology, Pakistan, to make this work possible.

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Authors and Affiliations

  1. Electrical Engineering Department, Karachi Institute of Economics and Technology, Karachi, Pakistan

    Muhammad Arif, Syed Safi Uddin Qadri & Imran Naseem

  2. Department of Electrical and Electronic Engineering, Universiti Teknologi Petronas, Seri Iskandar, Malaysia

    Saba Samreen Khan

  3. School of Electrical, Electronic and Computer Engineering, University of Western Australia, Crawley, Australia

    Imran Naseem

  4. Love for Data, Karachi, Pakistan

    Imran Naseem

  5. Electrical and Computer Engineering Department, King Abdul Aziz University, Jeddah, 21859, Saudi Arabia

    Muhammad Moinuddin

  6. Center of Excellence in Intelligent Engineering Systems (CEIES), King Abdul Aziz University, Jeddah, 21859, Saudi Arabia

    Muhammad Moinuddin

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Arif, M., Khan, S.S., Qadri, S.S.U. et al. Efficient Time-Varying q-Parameter Design for q-Incremental Least Mean Square Algorithm with Noisy Links. Circuits Syst Signal Process 41, 5699–5718 (2022). https://doi.org/10.1007/s00034-022-02048-w

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