A Weighted Gaussian Kernel Least Mean Square Algorithm

Muhammad Moinuddin*, Azzedine Zerguine, Muhammad Arif

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In this work, a novel weighted kernel least mean square (WKLMS) algorithm is proposed by introducing a weighted Gaussian kernel. The learning behavior of the WKLMS algorithm is studied. Mean square error (MSE) analysis shows that the WKLMS algorithm outperforms both the least mean square (LMS) and KLMS algorithms in terms of transient state as well as steady-state responses. We study the effect of the weighted Gaussian kernel on the associated kernel matrix, its eigenvalue spread and distribution, and show how these parameters affect the convergence behavior of the algorithm. Both of the transient and steady-state mean-square-error (MSE) behaviors of the WKLMS algorithm are studied, and a stability bound is derived. For a non-stationary environment, tracking analysis for a correlated random walk channel is presented. We also prove that the steady-state excess MSE (EMSE) of the WKLMS is Schur convex function of the weight elements in its kernel weight matrix and hence it follows the majorization of the kernel weight elements. This helps to decide which kernel weight matrix can provide better MSE performance. Simulations results are provided to contrast the performance of the proposed WKLMS with those of its counterparts KLMS and LMS algorithms. The derived analytical results of the proposed WKLMS algorithm are also validated via simulations for various step-size values.

Original languageEnglish
Pages (from-to)5267-5288
Number of pages22
JournalCircuits, Systems, and Signal Processing
Issue number9
StatePublished - Sep 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.


  • Gaussian kernel
  • Kernel adaptive filtering
  • Kernel methods
  • Least mean square
  • Reproducing kernel Hilbert space

ASJC Scopus subject areas

  • Signal Processing
  • Applied Mathematics


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