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Mean-Square Performance of the Modified Filtered-x Affine Projection Algorithm

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Abstract

The modified filtered-x affine projection (MFxAP) algorithm is effective for active noise control owing to its good convergence behavior and medium computational burden. The transient and steady-state performances of the MFxAP algorithm have been analyzed in previous studies, which presented a relatively good agreement between the theory and measured results. However, the correlation between the weight-error vector and the past noise vectors is disregarded in the existing methods. Hence, a more accurate theoretical analysis for the MFxAP algorithm is presented herein, in which the effect of the past noise vector on the weight-error vector is considered comprehensively. Simulation results indicate that the proposed theoretical results match the experimental results more precisely than the previous studies, in particular, at the steady state.

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References

  1. F. Albu, Efficient multichannel filtered-x affine projection algorithm for active noise control. Electron. Lett. 42(7), 421–423 (2006)

    Article  Google Scholar 

  2. F. Albu, M. Bouchard, Y. Zakharov, Pseudo-affine projection algorithms for multichannel active noise control. IEEE Trans. Audio Speech Lang. Process. 15(3), 1044–1052 (2007)

    Article  Google Scholar 

  3. F. Albu, Y. Zakharov, C. Paleologu, Modified filtered-x dichotomous coordinate descent recursive affine projection algorithm, in 2009 IEEE International Conference on Acoustics, Speech and Signal Processing (2009), pp. 257–260

  4. N.J. Bershad, D. Linebarger, S. McLaughlin, A stochastic analysis of affine projection algorithm for gaussian autoregressive inputs, in IEEE International Conference on Acoustics Speech and Signal Processing (2001), pp. 3837–3840

  5. E. Bjarnason, Analysis of the filtered-X LMS algorithm, in IEEE International Conference on Acoustics Speech and Signal Processing Proceedings, vol. 3 (1993), pp. 511–514

  6. M. Bouchard, Multichannel affine and fast affine projection algorithms for active noise control and acoustic equalization systems. IEEE Trans. Speech Audio Process. 16(4), 13–41 (1999)

    Google Scholar 

  7. A. Carini, G.L. Sicuranza, Analysis of a multichannel filtered-x set membership affine projection algorithm, in IEEE International Conference on Acoustics Speech and Signal Processing (2006a), pp. 2644–2647

  8. A. Carini, G.L. Sicuranza, Transient and steady-state analysis of filtered-x affine projection algorithm. IEEE Trans. Signal Process. 54(2), 665–678 (2006b)

    Article  Google Scholar 

  9. A. Carini, G.L. Sicuranza, Analysis of transient and steady-state behavior of a multichannel filtered-x partial-error affine projection algorithm. EURASIP J. Audio Speech Music Process. 1, 9–9 (2007)

    Google Scholar 

  10. P.S.R. Diniz, Convergence performance of the simplified set-membership affine projection algorithm. Circuits Syst. Signal Process. 30(2), 439–462 (2011)

    Article  MathSciNet  Google Scholar 

  11. S.C. Douglas, The fast affine projection algorithm for active noise control. Asilomar Conf. Signals 2, 1245–1249 (1995)

    Google Scholar 

  12. S.J. Elliott, P.A. Nelson, Active noise control. IEEE Signal Process. Mag. 10(4), 12–35 (1993)

    Article  Google Scholar 

  13. M. Ferrer, A. Gonzalez, M. de Diego, G. Piñero, Fast affine projection algorithms for filtered-x multichannel active noise control. IEEE Trans. Audio Speech Lang. Process. 16(8), 1396–1408 (2008)

    Article  Google Scholar 

  14. M. Ferrer, A. Gonzalez, M. de Diego, G. Piñero, Transient analysis of the conventional filtered-x affine projection algorithm for active noise control. IEEE Trans. Audio Speech Lang. Process. 19(3), 652–657 (2011)

    Article  Google Scholar 

  15. M. Ferrer, A. Gonzalez, M. de Diego, G. Piñero, Steady-state mean square performance of the multichannel filtered-x affine projection algorithm. IEEE Trans. Signal Process. 60(6), 2771–2785 (2012)

    Article  MathSciNet  Google Scholar 

  16. A. González, M. Ferrer, M. de Diego, L. Fuster, Efficient implementation of matrix recursions in the multichannel affine projection algorithm for multichannel sound, in IEEE Workshop on Multimedia Signal Processing (2004), pp. 215–218

  17. Y. Kajikawa, W.S. Gan, S.M. Kuo, Recent advances on active noise control: open issues and innovative applications. APSIPA Trans. Signal Inf. Process. 1(2), e3 (2012)

    Article  Google Scholar 

  18. S.E. Kim, J.W. Lee, W.J. Song, A theory on the convergence behavior of the affine projection algorithm. IEEE Trans. Signal Process. 59(12), 6233–6239 (2011)

    Article  MathSciNet  Google Scholar 

  19. S.M. Kuo, D.R. Morgan, Active Noise Control Systems: Algorithms and DSP Implementations (Wiley, New York, 1996)

    Google Scholar 

  20. M.V. Lima, P.S. Diniz, Steady-state mse performance of the set-membership affine projection algorithm. Circuits Syst. Signal Process. 32(4), 1811–1837 (2013)

    Article  Google Scholar 

  21. T.K. Moon, W.C. Stirling, Mathematical Methods and Algorithms for Signal Processing (Prentice-Hall, New York, 2000)

    Google Scholar 

  22. T.K. Paul, T. Ogunfunmi, On the convergence behavior of the affine projection algorithm for adaptive filters. IEEE Trans. Circuits Syst. I Regul. Pap. 58(8), 1813–1826 (2011)

    Article  MathSciNet  Google Scholar 

  23. S.G. Sankaran, A.A.L. Beex, Convergence behavior of affine projection algorithms. IEEE Trans. Signal Process. 48, 1086–1096 (2000)

    Article  MathSciNet  Google Scholar 

  24. H.C. Shin, A.H. Sayed, Mean-square performance of a family of affine projection algorithms. IEEE Trans. Signal Process. 52(1), 90–102 (2004)

    Article  MathSciNet  Google Scholar 

  25. D.T.M. Slock, On the convergence behavior of the LMS and normalized LMS algorithms. IEEE Trans. Signal Process. 41(9), 2811–2825 (1993)

    Article  MathSciNet  Google Scholar 

  26. F. Yang, J. Yang, A comparative survey of fast affine projection algorithms. Digit. Signal Process. 83, 297–322 (2018)

    Article  Google Scholar 

  27. F. Yang, M. Wu, J. Yang, Z. Kuang, A fast exact filtering approach to a family of affine projection-type algorithms. Signal Process. 101, 1–10 (2014)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by Youth Innovation Promotion Association of Chinese Academy of Sciences under Grant 2018027, the Strategic Priority Research Program of Chinese Academy of Sciences under Grant XDC02020400, IACAS Young Elite Researcher Projects QNYC201812 and QNYC201722, National Key R&D Program of China under Grant 2017YFC0804900, and National Natural Science Foundation of China under Grants 61501449, 11674348, and 11804368.

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Authors and Affiliations

  1. Key Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of Sciences, Beijing, 100190, China

    Jianfeng Guo, Feiran Yang & Jun Yang

  2. State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing, 100190, China

    Feiran Yang & Jun Yang

  3. School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing, 100049, China

    Jianfeng Guo, Feiran Yang & Jun Yang

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  1. Jianfeng Guo
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  2. Feiran Yang
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  3. Jun Yang
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Appendix A

Appendix A

From (17), we expand the term \(\prod \nolimits _{l = 0}^{k - 1} {\left( {{\mathbf{{I}}_L} - \mu \mathbf{{P}}(n - l)} \right) }\) as follows:

$$\begin{aligned} \begin{aligned}&\prod \limits _{l = 0}^{k - 1} {\left( {{\mathbf{{I}}_L} - \mu \mathbf{{P}}(n - l)} \right) } \\&\quad = \left( {{\mathbf{{I}}_L} - \mu \mathbf{{P}}(n)} \right) \left( {{\mathbf{{I}}_L} - \mu \mathbf{{P}}(n - 1)} \right) \cdots \left( {{\mathbf{{I}}_L} - \mu \mathbf{{P}}(n - k + 1)} \right) \\&\quad = {\mathbf{{I}}_L} - \mu \left( {\mathbf{{P}}(n) + \mathbf{{P}}(n - 1) + \cdots \mathbf{{P}}(n - k + 1)} \right) \\&\qquad + {\mu ^2}\left( {\mathbf{{P}}(n)\mathbf{{P}}(n - 1) + \mathbf{{P}}(n)\mathbf{{P}}(n - 2) + \cdots \mathbf{{P}}(n - k)\mathbf{{P}}(n - k + 1)} \right) \\&\qquad + \cdots + {( - \mu )^k}\left( {\mathbf{{P}}(n)\mathbf{{P}}(n - 1) \cdots \mathbf{{P}}(n - k + 1)} \right) \\&\quad = {\mathbf{{I}}_L} - \mu \sum \limits _{{n_0} = 0}^{k - 1} {\mathbf{{P}}(n - {n_0})} + {\mu ^2}\sum \limits _{{n_0} = 0}^{k - 2} {\sum \limits _{{n_1} = {n_0} + 1}^{k - 1} {\mathbf{{P}}(n - {n_0})\mathbf{{P}}(n - {n_1})} } + \cdots \\&\qquad + {( - \mu )^k}\sum \limits _{{n_0} = 0}^0 {\sum \limits _{{n_1} = {n_0} + 1}^1 { \cdots \sum \limits _{{n_{l - 1}} = {n_{l - 2}} + 1}^{k - 1} {\mathbf{{P}}(n - {n_0})\mathbf{{P}}(n - {n_1}) \cdots \mathbf{{P}}(n - {n_{l - 1}})} } }\\&\quad = \sum \limits _{l = 0}^k {{{( - \mu )}^l}{} \mathbf{{G}}_k^l\left( {\mathbf{{P}}(n)} \right) }, \end{aligned} \end{aligned}$$
(35)

where \(\mathbf{{G}}_k^l\left( {\mathbf{{P}}(n)} \right) \) is the coefficient related to \({( - \mu )^l}\) which can be written as

$$\begin{aligned} \mathbf{{G}}_k^l\left( {\mathbf{{P}}(n)} \right) = \sum \limits _{{n_0} = 0}^{k - l} {\sum \limits _{{n_1} = {n_0} + 1}^{k - l + 1} { \cdots \sum \limits _{{n_{l - 1}} = {n_{l - 2}} + 1}^{k - 1} {\mathbf{{P}}(n - {n_0})\mathbf{{P}}(n - {n_1}) \cdots \mathbf{{P}}(n - {n_{l - 1}})} } }.\nonumber \\ \end{aligned}$$
(36)

In particular, \(\mathbf{{G}}_k^0\left( {\mathbf{{P}}(n)} \right) = {\mathbf{{I}}_L}\).

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Guo, J., Yang, F. & Yang, J. Mean-Square Performance of the Modified Filtered-x Affine Projection Algorithm. Circuits Syst Signal Process 39, 4243–4257 (2020). https://doi.org/10.1007/s00034-020-01365-2

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