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Second-Order Adaptive Algorithms

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Fundamentals of Adaptive Signal Processing

Part of the book series: Signals and Communication Technology ((SCT))

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Abstract

This chapter introduces the second-order algorithms for the solution of the Yule–Walker normal equations with online recursive methods, such as error sequential regression (ESR) algorithm [1, 2].

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Notes

  1. 1.

    Recall that, for the symmetrical nature of the matrix P n , it holds that [P n−1 x n ]H = x H n P n−1.

  2. 2.

    In this context \( \widehat{\mathbf{v}} \) indicates an RV that represents an estimate of a deterministic vector v.

  3. 3.

    We remind the reader that in Riemannian geometry, for two vectors w and w+δ w the metric distance d w(.,.), which by definition depends on the space point in which it is located w, is defined as \( d\left(\mathbf{w},\mathbf{w}+\delta \mathbf{w}\right)=\sqrt{{\displaystyle \sum_{i=0}^{M-1}{\displaystyle \sum_{j=0}^{M-1}\delta {w}_i\delta {w}_j{g}_{ij}\left(\mathbf{w}\right)}}}=\sqrt{\delta {\mathbf{w}}^T\mathbf{G}\left(\mathbf{w}\right)\delta \mathbf{w}} \) where G(w) ∈ ℝM×M is a positive definite matrix representing the Riemann metric tensor. The G(w) characterizes the curvature of the particular manifold of the M-dimensional space. Namely, G(w) represents a “correction” of Euclidean distance defined for G(w) = I.

References

  1. Widrow B, Stearns SD (1985) Adaptive signal processing. Prentice Hall, Englewood Cliffs, NJ

    MATH  Google Scholar 

  2. Ahmed N, Soldan DL, Hummels DR, Parikh DD (1977) Sequential regression considerations of adaptive filter. IEE Electron Lett 13(15):446–447

    Article  Google Scholar 

  3. Ahmed N, Hummels DR, Uhl M, Soldan DL (1979) A short term sequential regression algorithm. IEEE Trans Acoust Speech Signal Process ASSP-27:453

    Article  Google Scholar 

  4. Shin HC, Sayed AH (2004) Mean-square performance of a family of affine projection algorithms. IEEE Trans Signal Process 52(1):90–102

    Article  MathSciNet  Google Scholar 

  5. Sankaran SG, (Louis) Beex AA (2000) Convergence behavior of affine projection algorithms. IEEE Trans Signal Process 48:1086–1096

    Article  MathSciNet  MATH  Google Scholar 

  6. Manolakis DG, Ingle VK, Kogon SM (2000) Statistical and adaptive signal processing. McGraw-Hill, New York, NY

    Google Scholar 

  7. Sayed AH (2003) Fundamentals of adaptive filtering. Wiley, New York, NY

    Google Scholar 

  8. Haykin S (2001) Kalman filter. In: Haykin S (ed) Kalman filtering and neural networks. Wiley. ISBN 0-471-36998-5

    Google Scholar 

  9. Kalman RE (1960) A new approach to linear filtering and prediction problems. J Basic Eng 82:34–45

    Google Scholar 

  10. Roweis S, Ghahramani Z (1999) A unifying review of linear Gaussian models. Neural Comput 11(2):305–345

    Article  Google Scholar 

  11. Welch G, Bishop G (2006) An introduction to the Kalman filter. TR 95-041, Department of Computer Science, University of North Carolina at Chapel Hill (NC 27599-3175), July

    Google Scholar 

  12. Macchi O (1996) The theory of adaptive filtering in a random time-varying environment. In: Figueiras-Vidal AR (ed) Digital signal processing in telecommunications. Springer, London

    Google Scholar 

  13. Huang Y, Benesty J, Chen J (2006) Acoustic MIMO signal processing. Springer series on signal and communication technology. ISBN 10 3-540-37630-5

    Google Scholar 

  14. Benesty J, Gänsler T, Eneroth P (2000) Multi-channel sound, acoustic MIMO echo cancellation, and multi-channel time-domain adaptive filtering. In: Acoustic signal processing for telecommunication. Kluwer. ISBN 0-7923-7814-8

    Google Scholar 

  15. Martin RK, Sethares WA, Williamson RC, Johnson CR Jr (2002) Exploiting sparsity in adaptive filters. IEEE Trans Signal Process 50(8):1883–1894

    Article  Google Scholar 

  16. Benesty J, Gänsler T, Huang Y, Rupp M (2004) Adaptive algorithms for MIMO acoustic echo cancellation. In: Audio signal processing for next-generation multimedia communication systems. Kluwer. ISBN 1-4020-7768-8

    Google Scholar 

  17. Vega LR, Rey H, J. Benesty J, Tressens S (2009) A family of robust algorithms exploiting sparsity in adaptive filters. IEEE Trans Audio Speech Lang Process 17(4):572–581

    Article  Google Scholar 

  18. Amari S (1998) Natural gradient works efficently in learning. Neural Comput 10:251–276

    Article  Google Scholar 

  19. Kivinen J, Warmuth MK (1997) Exponentiated gradient versus gradient descent for linear predictors. Inform Comput 132:1–64

    Article  MathSciNet  MATH  Google Scholar 

  20. Benesty J, Gänsler T, Gay L, Sondhi MM (2000) A robust proportionate affine projection algorithm for network echo cancellation. In: Proceedings of IEEE international conference on acoustics, speech, and signal processing, ICASSP ‘00, pp II-793–II-796

    Google Scholar 

  21. Haykin S (1996) Adaptive filter theory, 3rd edn. Prentice Hall, Upper Saddle River, NJ

    Google Scholar 

  22. Golub GH, Van Loan CF (1989) Matrix computation. John Hopkins University Press, Baltimore, MD. ISBN 0-80183772-3

    Google Scholar 

  23. Sherman J, Morrison WJ (1950) Adjustment of an inverse matrix corresponding to a change in one element of a given matrix. Ann Math Stat 21(1):124–127

    Article  MathSciNet  MATH  Google Scholar 

  24. Ozeki K, Umeda T (1984) An adaptive filtering algorithm using an orthogonal projection to an affine subspace and its properties. Electron Commun Jpn J67-A(5):126–132

    MathSciNet  Google Scholar 

  25. Rupp M, Sayed AH (1996) A time-domain feedback analysis of filtered error adaptive gradient algorithms. IEEE Trans Signal Process 44(6):1428–1439

    Article  Google Scholar 

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Authors and Affiliations

  1. DIET, Sapienza University of Rome, Rome, Italy

    Aurelio Uncini

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  1. Aurelio Uncini
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Uncini, A. (2015). Second-Order Adaptive Algorithms. In: Fundamentals of Adaptive Signal Processing. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-02807-1_6

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  • DOI: https://doi.org/10.1007/978-3-319-02807-1_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-02806-4

  • Online ISBN: 978-3-319-02807-1

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