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Geometric algebra based least-mean absolute third and least-mean mixed third-fourth adaptive filtering algorithms

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Abstract

With regards to the problem of multidimensional signal processing in the field of adaptive filtering, geometric algebra based higher-order statistics algorithms were proposed. For instance, to express a multidimensional signal as a multi-vector, the adaptive filtering algorithms described in this work leverage all of the benefits of GA theory in multidimensional signal processing. GA space is employed to extend the traditional least-mean absolute third (LMAT) and newly deduced least-mean mixed third-fourth (LMMTF) adaptive filtering methods for multi-dimensional signal processing. The objective of the presented GA-based least-mean absolute third (GA-LMAT) and GA-based least-mean mixed third-fourth (GA-LMMTF) algorithms is to minimize the cost functions by using higher-order statistics of the error signal e(n) in GA space. The simulation’s results revealed that at significantly smaller step size, the given GA-LMAT algorithm is better than the others in terms of steady-state error and convergence rate. Besides, the defined GA-LMMTF algorithm mitigates for the instability of GA-LMAT as the step size increases and illustrates an improved performance relative to mean absolute error and convergence rate.

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Data availibility statement

The data analyzed in this study is available upon reasonable request.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (NSFC) under Grant 61771299.

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Authors and Affiliations

  1. School of Communication and Information Engineering, Shanghai University, Shanghai, 200444, Shanghai, People’s Republic of China

    Khurram Shahzad, Yichen Feng & Rui Wang

Authors
  1. Khurram Shahzad
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  2. Yichen Feng
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  3. Rui Wang
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Contributions

All authors contributed to the study conception and design. Khurram shahzad and Yichen Feng analyzed the data, deduced new algorithm, performed experiments and wrote manuscript. Prof. Rui wang provided administrative support and supervised this project.

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Correspondence to Khurram Shahzad or Rui Wang.

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Shahzad, K., Feng, Y. & Wang, R. Geometric algebra based least-mean absolute third and least-mean mixed third-fourth adaptive filtering algorithms. SIViP 18, 5253–5267 (2024). https://doi.org/10.1007/s11760-024-03230-0

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