Abstract
In the field of echo cancellation, the normalized least mean squares (NLMS) algorithm is the most popular adaptive algorithm due to its simplicity and ease of implementation. However, this category of algorithms presents a conflict between several performance criteria: the initial convergence speed, the tracking ability and the root mean square error of filtering (MSE) in the steady state. Variable-step algorithms (VSS) address the trade-off between convergence speed and low final MSE. Nevertheless, due to a fairly small adaptive step-size in the steady-state regime, they fail to adequately track variations of the unknown system and they are all implemented with the original NLMS algorithm. In this contribution, a new improved variable adaptation step algorithm capable to track time variations of the unknown system even after good convergence in the steady state is suggested. It is based on the use of the fast-normalized adaptive algorithm (FNLMS) for system identification and acoustic echo cancellation context. The purposes of using the FNLMS algorithm in the field of VSS are on the one hand to improve its final MSE and, on the other hand, to obtain a VSS algorithm with better convergence and tracking compared to the VSS NLMS algorithms. Simulation results show that the proposed VSS-Fast NLMS algorithm outperforms the original FNLMS algorithm in terms of steady-state error reduction and minimization after the initial transition phase while maintaining similar convergence speed and tracking capacity. Furthermore, it achieves visible improvements in terms of two objective criteria, i.e., a faster initial convergence rate and a better tracking ability than the ones of the non-parametric VSS-NLMS (NPVSS-NLMS) and traditional NLMS algorithms.
Similar content being viewed by others
References
Benesty, J. Gänsler, T. Morgan, D. R. Sondhi, M. M. and Gay, S. L.: Advances in network and acoustic echo cancellation. Springer, Berlin, Heidelberg. 2001. https://doi.org/10.1007/978-3-662-04437-7
Puder, H. Dreiseitel, P.: Implementation of a hands-free car phone with echo cancellation and noise-dependent loss control. In: 2000 Proceedings of the IEEE international conference on acoustics, speech and signal processing, vol 6, pp. 3622–3625. 2000. https://doi.org/10.1109/ICASSP.2000.860186
Kellermann, W.: “Echo Cancellation”. In: Havelock, D., Kuwano, S., Vorländer, M. (eds) Handbook of Signal Processing in Acoustics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30441-0_46
Sondhi, M. Kellermann, W.: Adaptive echo cancellation for speech signal. In: Furui S, Sondhi M (eds) Chapter 11 in Advances in speech signal processing. Dekker, New York. 1992.
Benesty, J. Paleologu, C. Gänsler, T. Ciochina, S. A.: Perspective on Stereophonic Acoustic Echo Cancellation. Springer-Verlag. vol.4. 2011.
Diniz, P.S.R.: Adaptive Filtering: Algorithms and Practical Implementations. Springer, Boston, MA, Fifth Edition (2020)
Simon, O. H.: Adaptive Filter Theory. 5th Edition. Pearson. 2013
Qin, L., Bellanger, M. G.: Convergence Analysis of a Variable Step-Size Normalized LMS Adaptive Filter Algorithm. In: 8th European Signal Processing Conference (EUSIPCO 1996), IEEE, Trieste, Italy, pp. 1–4. September 1996
Sulyman, A. I., Zerguine, A.: Echo Cancellation Using a Variable Step-Size NLMS Algorithm. In: 12th European Signal Processing Conference, (EUSIPCO 2004), IEEE, pp. 401–404. Septembre 2004
Shin, H.C., Sayed, A.H., Song, W.J.: Variable step-size NLMS and affine projection algorithms. IEEE Signal Process. Lett. 11(2), 132–135 (2004)
Benesty, J., Rey, H., Vega, L.R., Tressens, S.: A non-parametric VSS-NLMS algorithm. IEEE Signal Process. Lett. 13(10), 581–584 (2006). https://doi.org/10.1109/LSP.2006.876323
Iqbal, M. A., Grant, S. L.: Novel variable step size NLMS algorithms for echo cancellation. In: IEEE International Conference on Acoustics, Speech and Signal Processing, IEEE ICASSP, pp. 241–244. Mar 2008. https://doi.org/10.1109/ICASSP.2008.4517591
Paleologu, C., Benesty, J., Grant, S. L., Osterwise, C.: Variable Step-Size NLMS algorithms designed for echo cancellation. In: conference Record of the forty-third Asilomar Conference on Signals, Systems and Computers, IEEE, pp. 633–637. 2009
Casco-Sánchez, F.M., Medina-Ramírez, R.C., López-Guerrero, M.A.: A new variable step-size NLMS algorithm and its performance evaluation in echo cancelling applications. J. Appl. Res. Technol. 9(3), 302–313 (2011). https://doi.org/10.22201/icat.16656423.2011.9.03.425
Huang, H.C., Lee, J.: A new variable step-size NLMS Algorithm and its performance analysis. IEEE Trans. Signal Process. 60(4), 2055–2060 (2012). https://doi.org/10.1109/TSP.2011.2181505
Zhu, Y.G., Li, Y.G., Guan, S.Y., Chen, Q.S.: A novel variable step-size NLMS algorithm and its analysis. Proc. Eng. 29, 1181–1185 (2012)
Kanadi, M., Akhtar, M. T., Mitsuhashi, W.: A variable step-size-based ICA method for a fast and robust acoustic echo cancellation system without requiring double-talk detector. In: IEEE China Summit and International Conference on Signal and Information Processing. IEEE, pp.118–121. Jul 2013.https://doi.org/10.1109/ChinaSIP.2013.6625310
Hamidia, M., Amrouche, A.: Improved variable step-size NLMS adaptive filtering algorithm for acoustic echo cancellation. Dig. Signal Process. 49, 44–55 (2016). https://doi.org/10.1016/j.dsp.2015.10.015
Casco‐Sanchez, F., Lopez‐Guerrero, M., et al.: A variable‐step size NLMS algorithm based on the cross‐correlation between the squared output error and the near‐end input signal. Trans. Electr. Electron. Eng. 14(8), 1197–1202 (2019). https://doi.org/10.1002/tee.22918
Wei, Z., Long, Z.: Improved variable step size NLMS algorithm. In: 4th International Conference on Automatic Control and Mechatronic Engineering (ACME). 2019. https://doi.org/10.23977/amce.2019.008
Bershad, N.J., Bermudez, C.M.: A switched variable step size NLMS adaptive filter. Dig. Signal Process. 101, 102730 (2020). https://doi.org/10.1016/j.dsp.2020.102730
Rusu, A.G., Paleologu, C., Benesty, J., Ciochină, S.: A variable step size normalized least-mean-square algorithm based on data reuse. Algorithms 15(4), 111 (2022). https://doi.org/10.3390/a15040111
Gilloire, A., Moulines, E., Slock, D., Duhamel, P.: State of the art in acoustic echo cancellation. In: Figueiras-Vidal, A.R. (eds) Digital Signal Processing in Telecommunications. Springer, London, pp. 45–91. 1996. https://doi.org/10.1007/978-1-4471-1019-4_2
Benallal, A., Arezki, M.: A fast convergence normalized least-mean-square type algorithm for adaptive filtering. Int. J. Adapt. Control Signal Process. 28(10), 1073–1080 (2013). https://doi.org/10.1002/acs.2423
Mader, A., Puder, H., Schmidt, G.U.: Step-size control for acoustic echo cancellation filters – an overview. Signal Process. 80(9), 1698–1719 (2000). https://doi.org/10.1016/S0165-165-1684(00)00082-7
Gueraini, I., Benallal, A. et al.: The NP-VSS NLMS Algorithm with Noise Power Estimation Methods For Acoustic Echo Cancellation. In: 2022 2nd International Conference on Advanced Electrical Engineering (ICAEE), IEEE, pp. 1–6. Oct 2022. https://doi.org/10.1109/ICAEE53772.2022.9962001
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there is no conflict of interest.
Authors' contributions
IG is the major author who led all stages of the study. AB contributed to the theoretical aspect of adaptive filtering. AT contributed to the implementation part of the algorithms as well as to the writing part. All authors read and approved the final manuscript.
Funding
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
Data availability
The data used to support the findings of this study are available from the corresponding author upon request.
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.
About this article
Cite this article
Gueraini, I., Benallal, A. & Tedjani, A. New variable step-size fast NLMS algorithm for non-stationary systems. SIViP 17, 3099–3107 (2023). https://doi.org/10.1007/s11760-023-02531-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11760-023-02531-0