Skip to main content
Springer Nature Link
Log in

Geometric algebra and cosine-function based variable step-size adaptive filtering algorithms

  • Original Paper
  • Published:
Signal, Image and Video Processing Aims and scope Submit manuscript

Abstract

Adaptive filtering algorithms are currently successfully employed in a number of fields.But, a disadvantage of traditional real-valued fixed step-size adaptive filtering algorithm is that it is unable to meet both requirement of convergence rate and the steady-state error.In this article, we presented cosine function and geometric algebra(GA) based variable step-size technique for adaptive filtering.Initially, a multi-dimensional signal is represented as a GA multi-vector for the vectorization process in the proposed approach of adaptive filtering with variable step-size based on GA.Then, by establishing a non-linear function relationship between error signal e(n) and the step-size factor \(\mu \), given method of adaptive filtering resolves the contradiction among steady-state error and the convergence rate.Finally, simulation results illustrate that in comparison with other existing adaptive filtering algorithms, the defined approach performs better in terms of convergence rate, steady-state error, robustness against impulsive noise and the computational complexity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Data availability

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

References

  1. Widrow, B., McCool, J., Larimore, M.G. and Johnson, C.R.: Stationary and nonstationary learning characteristics of the LMS adaptive filter. In Aspects of Signal Processing: With Emphasis on Underwater Acoustics Part 1 Proceedings of the NATO Advanced Study Institute held at Portovenere, La Spezia, Italy 30 August 11 September 1976 (pp. 355-393). Springer Netherlands (1977)

  2. Chen, Y., Tian, J. and Liu, Y.: Variable step size LMS algorithm based on modified Sigmoid function. In 2014 International Conference on Audio, Language and Image Processing (pp. 627-630). IEEE (2014) July

  3. Liu, F.C., Zhang, Y.X., Wang, Y.J.: A variable step size LMS adaptive filtering algorithm based on the number of computing mechanisms. In 2009 International Conference on Machine Learning and Cybernetics (Vol. 4, pp. 1904-1908). IEEE (2009) July

  4. Ao, W., Xiang, W.Q., Zhang, Y.P., Wang, L., Lv, C.Y., Wang, Z.H.: A new variable step size LMS adaptive filtering algorithm. In 2012 International Conference on Computer Science and Electronics Engineering (Vol. 2, pp. 265-268). IEEE (2012) March

  5. Gao, Y., Xie, S.L.: A variable step size LMS adaptive filtering algorithm and its analysis. Acta Electronica Sinica 29(8), 1094–1097 (2001)

    Google Scholar 

  6. LU, B., Feng, C.Q., Long, G.N.: A new variable step-size LMS algorithm based on sine function’. Journal of Air Force Engineering University 44(2), 47–50 (2013)

    Google Scholar 

  7. Deng, J., Hou, X., Wu, Z.: Variable step adaptive filtering LMS algorithm based on tongue-like curve. Journal of Data Acquisition, pp.282-285 (2004)

  8. Youran, H., Yunyao, Z.: A Variable Step Size Adaptive Filtering Algorithm with Feedback Mechanism. In Proceedings of the 2019 2nd International Conference on Electronics, Communications and Control Engineering (pp. 3-6) (2019) April

  9. Zhang, L.Y., Wang, B.M., LIU, S.: A novel variable step-size adaptive interference cancellation algorithm. Acta Electronica Sinica 45(2), 321–327 (2017)

    Google Scholar 

  10. Yang, Y., Jing, X., Zhang, Z., Chen, Q.: Adaptive interference cancellation for coherent and uncorrelated interference signals coexistence. In 2013 22nd Wireless and Optical Communication Conference (pp. 1-5). IEEE (2013) May

  11. Zhang, J.: Variable Step Size LMS Algorithm Based on Cosine Function. In 2018 8th International Conference on Manufacturing Science and Engineering (ICMSE 2018) (pp. 693-697). Atlantis Press (2018) May

  12. Took, C.C., Mandic, D.P.: A quaternion widely linear adaptive filter. IEEE Transactions on Signal Processing 58(8), 4427–4431 (2010)

    Article  MathSciNet  Google Scholar 

  13. Jahanchahi, C., Took, C.C., Mandic, D.P.: The widely linear quaternion recursive least squares filter. In 2010 2nd International Workshop on Cognitive Information Processing (pp. 87-92). IEEE (2010) June

  14. Hubscher, P.I., Bermudez, J.C.M., Nascimento, V.H.: A mean-square stability analysis of the least mean fourth adaptive algorithm. IEEE Transactions on Signal Processing 55(8), 4018–4028 (2007)

    Article  MathSciNet  Google Scholar 

  15. Zou, Y., Chan, S.C., Ng, T.S.: Least mean M-estimate algorithms for robust adaptive filtering in impulse noise. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing 47(12), 1564–1569 (2000)

    Google Scholar 

  16. Chambers, J.A., Tanrikulu, O., Constantinides, A.G.: Least mean mixed-norm adaptive filtering. Electronics letters 30(19), 1574–1575 (1994)

    Article  Google Scholar 

  17. Tanrikulu, O., Constantinides, A.G.: Least-mean kurtosis: a novel higher-order statistics based adaptive filtering algorithm. Electronics letters 30(3), 189–190 (1994)

    Article  Google Scholar 

  18. H bscher, P.I., Bermudez, J.C.: A model for the behavior of the least mean kurtosis (LMK) adaptive algorithm with Gaussian inputs. In International Telecommunications Symposium-ITS (2002)

  19. Bershad, N.J., Bermudez, J.C.: Stochastic analysis of the least mean kurtosis algorithm for Gaussian inputs. Digital Signal Processing 54, 35–45 (2016)

  20. Lu, L., Zhao, H.: Improved filtered-x least mean kurtosis algorithm for active noise control. Circuits, Systems, and Signal Processing 36, 1586–1603 (2017)

    Article  Google Scholar 

  21. Meng, E.C., Acir, N.: An augmented complex-valued least-mean kurtosis algorithm for the filtering of noncircular signals. IEEE Transactions on Signal Processing 66(2), 438–448 (2017)

    Article  MathSciNet  Google Scholar 

  22. Meng, E.C., Acir, N.: Kurtosis-based CRTRL algorithms for fully connected recurrent neural networks. IEEE Transactions on Neural Networks and Learning Systems 29(12), 6123–6131 (2018)

    Article  Google Scholar 

  23. Meng, E.C., Acir, N., Mandic, D.P.: Widely linear quaternion-valued least-mean kurtosis algorithm. IEEE Transactions on Signal Processing 68, 5914–5922 (2020)

    Article  MathSciNet  Google Scholar 

  24. Wang, R., Shi, Y., Cao, W.: GA-SURF: A new speeded-up robust feature extraction algorithm for multispectral images based on geometric algebra. Pattern Recognition Letters 127, 11–17 (2019)

    Article  Google Scholar 

  25. Wang, R., Shen, M., Wang, T., Cao, W.: L1-norm minimization for multi-dimensional signals based on geometric algebra. Advances in Applied Clifford Algebras 29, 1–18 (2019)

    Article  Google Scholar 

  26. Wang, R., Shen, M., Cao, W.: Multivector sparse representation for multispectral images using geometric algebra. IEEE access 7, 12755–12767 (2019)

    Article  Google Scholar 

  27. Shen, M., Wang, R., Cao, W.: Joint sparse representation model for multi-channel image based on reduced geometric algebra. Ieee Access 6, 24213–24223 (2018)

    Article  Google Scholar 

  28. Wang, R., Wang, K., Cao, W., Wang, X.: Geometric algebra in signal and image processing: A survey. IEEE Access 7, 156315–156325 (2019)

    Article  Google Scholar 

  29. Su, H., Bo, Z.: Conformal geometric algebra based band selection and classification for hyperspectral imagery. In 2016 8th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS) (pp. 1-4). IEEE (2016) August

  30. Pham, M.T., Yoshikawa, T., Furuhashi, T., Tachibana, K.: Robust feature extractions from geometric data using geometric algebra. In 2009 IEEE International Conference on Systems, Man and Cybernetics (pp. 529-533). IEEE (2009) October

  31. Lopes, W.B., Al-Nuaimi, A., Lopes, C.G.: Geometric-algebra LMS adaptive filter and its application to rotation estimation. IEEE signal processing letters 23(6), 858–862 (2016)

    Article  Google Scholar 

  32. Al-Nuaimi, A., Steinbach, E., Lopes, W.B., Lopes, C.G.: 6DOF point cloud alignment using geometric algebra-based adaptive filtering. In 2016 IEEE Winter Conference on Applications of Computer Vision (WACV) (pp. 1-9). IEEE (2016) March

  33. Wang, R., He, Y., Huang, C., Wang, X., Cao, W.: A novel least-mean kurtosis adaptive filtering algorithm based on geometric algebra. IEEE access 7, 78298–78310 (2019)

    Article  Google Scholar 

  34. He, Y., Wang, R., Wang, X., Zhou, J., Yan, Y.: Novel adaptive filtering algorithms based on higher-order statistics and geometric algebra. IEEE Access 8, 73767–73779 (2020)

    Article  Google Scholar 

  35. Wang, H., He, Y., Li, Y., Wang, R.: An approach to adaptive filtering with variable step size based on geometric algebra. IET Communications 16(10), 1094–1105 (2022)

    Article  Google Scholar 

  36. Baylis, W.E.: Clifford (Geometric) Algebras: with applications to physics, mathematics, and engineering. Springer Science and Business Media (2012)

    Google Scholar 

  37. Xie, W., Cao, W., Meng, S.: Coverage analysis for sensor networks based on Clifford algebra. Science in China Series F: Information Sciences 51, 460–475 (2008)

    Google Scholar 

  38. Bayro-Corrochano, E.J., Arana-Daniel, N.: Clifford support vector machines for classification, regression, and recurrence. IEEE Transactions on Neural Networks 21(11), 1731–1746 (2010)

    Article  Google Scholar 

  39. Hestenes, D.: New foundations for classical mechanics (Vol. 15). Springer Science and Business Media(2012)

  40. Crowe, M.J.: A history of vector analysis: The evolution of the idea of a vectorial system. Courier Corporation (1994)

    Google Scholar 

  41. Vaz, J., Jr., da Rocha Jr, R.: An introduction to Clifford algebras and spinors. Oxford University Press (2016)

    Book  Google Scholar 

  42. Grassmann, H.: LM Society, Ausdehnungslehre (History of Mathematics). American Mathematical Society, Providence, RI, USA (2000)

    Google Scholar 

  43. Hestenes, D., Sobczyk, G.: Clifford algebra to geometric calculus: a unified language for mathematics and physics (Vol. 5). Springer Science and Business Media (2012)

  44. Strogatz, S.H.: Nonlinear dynamics and chaos with student solutions manual: With applications to physics, biology, chemistry, and engineering. CRC Press (2018)

  45. Lopes, W.B., Lopes, C.G.: Geometric-algebra adaptive filters. IEEE Transactions on Signal Processing 67(14), 3649–3662 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

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

    Khurram Shahzad, Rui Wang, Yichen Feng & Kaiwen Zhou

Authors
  1. Khurram Shahzad
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Rui Wang
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Yichen Feng
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Kaiwen Zhou
    View author publications

    You can also search for this author inPubMed Google Scholar

Contributions

KS involved in methodology, conceptualization, simulation, and writing original draft of manuscript. WR took part in validation, project administration, supervision, review and editing the manuscript, and funding acquisition. FY and KZ Involved in review, editing and formal analysis.

Corresponding authors

Correspondence to Khurram Shahzad or Rui Wang.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.All the authors listed have approved the manuscript that is enclosed.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shahzad, K., Wang, R., Feng, Y. et al. Geometric algebra and cosine-function based variable step-size adaptive filtering algorithms. SIViP 18, 7641–7654 (2024). https://doi.org/10.1007/s11760-024-03417-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11760-024-03417-5

Keywords