On the robustness of set-membership adaptive filtering algorithms

In this paper, we address the robustness, in the sense of l2-stability, of the set-membership normalized least-mean-square (SM-NLMS) and the set-membership affine projection (SM-AP) algorithms. For the SM-NLMS algorithm, we demonstrate that it is robust regardless of the choice of its parameters and that the SM-NLMS enhances the parameter estimation in most of the iterations in which an update occurs, two advantages over the classical NLMS algorithm. Moreover, we also prove that if the noise bound is known, then we can set the SM-NLMS so that it never degrades the estimate. As for the SM-AP algorithm, we demonstrate that its robustness depends on a judicious choice of one of its parameters: the constraint vector (CV). We prove the existence of CVs satisfying the robustness condition, but practical choices remain unknown. We also demonstrate that both the SM-AP and SM-NLMS algorithms do not diverge, even when their parameters are selected naively, provided the additional noise is bounded. Numerical results that corroborate our analyses are presented.

[1]  Y. F. Huang,et al.  On the value of information in system identification - Bounded noise case , 1982, Autom..

[2]  Wei-Lung Mao,et al.  Robust Set-Membership Filtering Techniques on GPS Sensor Jamming Mitigation , 2017, IEEE Sensors Journal.

[3]  Paulo S. R. Diniz,et al.  Steady-state analysis of the set-membership affine projection algorithm , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

[4]  Yih-Fang Huang,et al.  BEACON: an adaptive set-membership filtering technique with sparse updates , 1999, IEEE Trans. Signal Process..

[5]  Giovanni L. Sicuranza,et al.  Analysis of a Multichannel Filtered-x Set-Membership Affine Projection Algorithm , 2006, 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings.

[6]  John G. Proakis,et al.  Digital Communications , 1983 .

[7]  Juraci Ferreira Galdino,et al.  A set-membership NLMS algorithm with time-varying error bound , 2006, 2006 IEEE International Symposium on Circuits and Systems.

[8]  Paulo S. R. Diniz,et al.  On the robustness of the set-membership NLMS algorithm , 2016, 2016 IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM).

[9]  Paulo S. R. Diniz,et al.  Convergence Performance of the Simplified Set-Membership Affine Projection Algorithm , 2011, Circuits Syst. Signal Process..

[10]  Paulo S. R. Diniz,et al.  Steady-State MSE Performance of the Set-Membership Affine Projection Algorithm , 2013, Circuits, Systems, and Signal Processing.

[11]  José Antonio Apolinário,et al.  Set-Membership Proportionate Affine Projection Algorithms , 2007, EURASIP J. Audio Speech Music. Process..

[12]  Paulo S. R. Diniz,et al.  Optimal constraint vectors for set-membership affine projection algorithms , 2017, Signal Process..

[13]  Shirish Nagaraj,et al.  Set-membership filtering and a set-membership normalized LMS algorithm with an adaptive step size , 1998, IEEE Signal Processing Letters.

[14]  Paulo S. R. Diniz,et al.  On the steady-state MSE performance of the set-membership NLMS algorithm , 2010, 2010 7th International Symposium on Wireless Communication Systems.

[15]  Ali H. Sayed,et al.  Adaptive Filters , 2008 .

[16]  P. L. Combettes.  The foundations of set theoretic estimation , 1993 .

[17]  Markus Rupp,et al.  Pseudo Affine Projection Algorithms Revisited: Robustness and Stability Analysis , 2011, IEEE Transactions on Signal Processing.

[18]  P. L. Combettes,et al.  Foundation of set theoretic estimation , 1993 .

[19]  Yih-Fang Huang,et al.  Frequency-Domain Set-Membership Filtering and Its Applications , 2007, IEEE Transactions on Signal Processing.

[20]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[21]  Isao Yamada,et al.  Steady-State Mean-Square Performance Analysis of a Relaxed Set-Membership NLMS Algorithm by the Energy Conservation Argument , 2009, IEEE Transactions on Signal Processing.

[22]  Simon Haykin,et al.  Adaptive Filter Theory 4th Edition , 2002 .

[23]  Kazuhiko Ozeki,et al.  An adaptive filtering algorithm using an orthogonal projection to an affine subspace and its properties , 1984 .

[24]  Sheng Zhang,et al.  Set-Membership NLMS Algorithm With Robust Error Bound , 2014, IEEE Transactions on Circuits and Systems II: Express Briefs.

[25]  Andreas Antoniou,et al.  Robust Set-Membership Affine-Projection Adaptive-Filtering Algorithm , 2012, IEEE Transactions on Signal Processing.

[26]  Paulo S. R. Diniz,et al.  Fast learning set theoretic estimation , 2013, 21st European Signal Processing Conference (EUSIPCO 2013).

[27]  J. Deller.  Set membership identification in digital signal processing , 1989, IEEE ASSP Magazine.

[28]  Kutluyil Dogançay,et al.  Tracking performance analysis of the set-membership NLMS Adaptive Filtering Algorithm , 2012, Proceedings of The 2012 Asia Pacific Signal and Information Processing Association Annual Summit and Conference.

[29]  Paulo S. R. Diniz,et al.  Adaptive Filtering: Algorithms and Practical Implementation , 1997 .

[30]  Yih-Fang Huang,et al.  Set-membership adaptive equalization and an updator-shared implementation for multiple channel communications systems , 1998, IEEE Trans. Signal Process..

[31]  W.A. Martins,et al.  Semi-blind data-selective equalizers for QAM , 2008, 2008 IEEE 9th Workshop on Signal Processing Advances in Wireless Communications.

[32]  S. Haykin,et al.  Adaptive Filter Theory , 1986 .

[33]  P. Diniz,et al.  Set-membership affine projection algorithm , 2001, IEEE Signal Processing Letters.

[34]  Paulo S. R. Diniz,et al.  New Trinion and Quaternion Set-Membership Affine Projection Algorithms , 2017, IEEE Transactions on Circuits and Systems II: Express Briefs.