Open Access

Step Size Bound of the Sequential Partial Update LMS Algorithm with Periodic Input Signals

  • Pedro Ramos1Email author,
  • Roberto Torrubia2,
  • Ana López1,
  • Ana Salinas1 and
  • Enrique Masgrau2
EURASIP Journal on Audio, Speech, and Music Processing20062007:010231

DOI: 10.1155/2007/10231

Received: 9 June 2006

Accepted: 5 October 2006

Published: 14 December 2006

Abstract

This paper derives an upper bound for the step size of the sequential partial update (PU) LMS adaptive algorithm when the input signal is a periodic reference consisting of several harmonics. The maximum step size is expressed in terms of the gain in step size of the PU algorithm, defined as the ratio between the upper bounds that ensure convergence in the following two cases: firstly, when only a subset of the weights of the filter is updated during every iteration; and secondly, when the whole filter is updated at every cycle. Thus, this gain in step-size determines the factor by which the step size parameter can be increased in order to compensate the inherently slower convergence rate of the sequential PU adaptive algorithm. The theoretical analysis of the strategy developed in this paper excludes the use of certain frequencies corresponding to notches that appear in the gain in step size. This strategy has been successfully applied in the active control of periodic disturbances consisting of several harmonics, so as to reduce the computational complexity of the control system without either slowing down the convergence rate or increasing the residual error. Simulated and experimental results confirm the expected behavior.

[123456789101112131415161718]

Authors’ Affiliations

(1)
Communication Technologies Group, Aragón Institute for Engineering Research (I3A), EUPT, University of Zaragoza
(2)
Communication Technologies Group, Aragón Institute for Engineering Research (I3A), CPS Ada Byron, University of Zaragoza

References

  1. Lueg P: Process of silencing sound oscillations. U.S. Patent no. 2.043.416, 1936Google Scholar
  2. Morgan DR: Analysis of multiple correlation cancellation loops with a filter in the auxiliary path. IEEE Transactions on Acoustics, Speech, and Signal Processing 1980,28(4):454-467. 10.1109/TASSP.1980.1163430View ArticleGoogle Scholar
  3. Widrow B, Shur D, Shaffer S: On adaptive inverse control. Proceedings of the15th Asilomar Conference on Circuits, Systems, and Computers, November 1981, Pacific Grove, Calif, USA 185-195.Google Scholar
  4. Burgess JC: Active adaptive sound control in a duct: a computer simulation. Journal of the Acoustical Society of America 1981,70(3):715-726. 10.1121/1.386908View ArticleGoogle Scholar
  5. Ramos P, Torrubia R, López A, Salinas A, Masgrau E: Computationally efficient implementation of an active noise control system based on partial updates. Proceedings of the International Symposium on Active Control of Sound and Vibration (ACTIVE '04), September 2004, Williamsburg, Va, USA paper 003Google Scholar
  6. Douglas SC: Adaptive filters employing partial updates. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing 1997,44(3):209-216. 10.1109/82.558455View ArticleGoogle Scholar
  7. Aboulnasr T, Mayyas K: Selective coefficient update of gradient-based adaptive algorithms. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '97), April 1997, Munich, Germany 3: 1929-1932.Google Scholar
  8. Doǧançay K, Tanrikulu O: Adaptive filtering algorithms with selective partial updates. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing 2001,48(8):762-769. 10.1109/82.959866View ArticleMATHGoogle Scholar
  9. Sanubari J: Fast convergence LMS adaptive filters employing fuzzy partial updates. Proceedings of IEEE Conference on Convergent Technologies for Asia-Pacific Region (TENCON '03), October 2003, Bangalore, India 4: 1334-1337.View ArticleGoogle Scholar
  10. Naylor PA, Cui J, Brookes M: Adaptive algorithms for sparse echo cancellation. Signal Processing 2006,86(6):1182-1192. 10.1016/j.sigpro.2005.09.015View ArticleMATHGoogle Scholar
  11. Kuo SM, Tahernezhadi M, Hao W: Convergence analysis of narrow-band active noise control system. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing 1999,46(2):220-223. 10.1109/82.752958View ArticleGoogle Scholar
  12. Bjarnason E: Analysis of the filtered-X LMS algorithm. IEEE Transactions on Speech and Audio Processing 1995,3(6):504-514. 10.1109/89.482218View ArticleGoogle Scholar
  13. Kuo SM, Morgan DR: Active Noise Control Systems: Algorithms and DSP Implementations. John Wiley & Sons, New York, NY, USA; 1996.Google Scholar
  14. Vicente L, Masgrau E: Novel FxLMS convergence condition with deterministic reference. IEEE Transactions on Signal Processing 2006,54(10):3768-3774.View ArticleGoogle Scholar
  15. Elliott SJ: Signal Processing for Active Control. Academic Press, London, UK; 2001.Google Scholar
  16. Widrow B, Stearns SD: Adaptive Signal Processing. Prentice Hall, Englewood Cliffs, NJ, USA; 1985.MATHGoogle Scholar
  17. Haykin S: Adaptive Filter Theory. Prentice Hall, Upper Saddle River, NJ, USA; 2002.MATHGoogle Scholar
  18. Texas Instruments Digital Signal Processing Products : TMS320C6000 CPU and Instruction Set Reference Guide. 1999Google Scholar

Copyright

© Pedro Ramos et al. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.