Extension of LMS stability condition over a wide set of signals
SummaryA sufficient condition for least mean squares (LMS) algorithm stability with a small set of assumptions is derived in this paper. The derivation is not, contrary to the majority of currently known conditions, based on the independence assumption or other statistic properties of the input sign...
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| Published in | International journal of adaptive control and signal processing Vol. 29; no. 5; pp. 653 - 670 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Bognor Regis
Blackwell Publishing Ltd
01.05.2015
Wiley Subscription Services, Inc |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0890-6327 1099-1115 |
| DOI | 10.1002/acs.2500 |
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| Abstract | SummaryA sufficient condition for least mean squares (LMS) algorithm stability with a small set of assumptions is derived in this paper. The derivation is not, contrary to the majority of currently known conditions, based on the independence assumption or other statistic properties of the input signals. Moreover, it does not make use of the small‐step‐size assumption, neither does it assume the input signals are stationary. Instead, it uses a theory of discrete systems and properties of a discrete state‐space matrix. Therefore, the result can be applied to a wide set of signals, including deterministic and nonstationary signals. The location of all eigenvalues of the matrix responsible for the LMS algorithm stability has been calculated. Simulation experiments, where the step size reaches a couple of hundreds without loss of stability, are shown to support the theory. On the other hand, simulation where the calculations based on the small‐step‐size theory provide a too large estimation of the upper bound for the step size, while the new condition gives a proper solution, is also presented. Therefore, the new condition may be used in cases where fast adaptation is necessary and when the independence theory or the small‐step‐size assumptions do not hold. Copyright © 2014 John Wiley & Sons, Ltd. |
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| AbstractList | A sufficient condition for least mean squares (LMS) algorithm stability with a small set of assumptions is derived in this paper. The derivation is not, contrary to the majority of currently known conditions, based on the independence assumption or other statistic properties of the input signals. Moreover, it does not make use of the small‐step‐size assumption, neither does it assume the input signals are stationary. Instead, it uses a theory of discrete systems and properties of a discrete state‐space matrix. Therefore, the result can be applied to a wide set of signals, including deterministic and nonstationary signals. The location of all eigenvalues of the matrix responsible for the LMS algorithm stability has been calculated. Simulation experiments, where the step size reaches a couple of hundreds without loss of stability, are shown to support the theory. On the other hand, simulation where the calculations based on the small‐step‐size theory provide a too large estimation of the upper bound for the step size, while the new condition gives a proper solution, is also presented. Therefore, the new condition may be used in cases where fast adaptation is necessary and when the independence theory or the small‐step‐size assumptions do not hold. Copyright © 2014 John Wiley & Sons, Ltd. A sufficient condition for least mean squares (LMS) algorithm stability with a small set of assumptions is derived in this paper. The derivation is not, contrary to the majority of currently known conditions, based on the independence assumption or other statistic properties of the input signals. Moreover, it does not make use of the small-step-size assumption, neither does it assume the input signals are stationary. Instead, it uses a theory of discrete systems and properties of a discrete state-space matrix. Therefore, the result can be applied to a wide set of signals, including deterministic and nonstationary signals. The location of all eigenvalues of the matrix responsible for the LMS algorithm stability has been calculated. Simulation experiments, where the step size reaches a couple of hundreds without loss of stability, are shown to support the theory. On the other hand, simulation where the calculations based on the small-step-size theory provide a too large estimation of the upper bound for the step size, while the new condition gives a proper solution, is also presented. Therefore, the new condition may be used in cases where fast adaptation is necessary and when the independence theory or the small-step-size assumptions do not hold. Copyright copyright 2014 John Wiley & Sons, Ltd. Summary A sufficient condition for least mean squares (LMS) algorithm stability with a small set of assumptions is derived in this paper. The derivation is not, contrary to the majority of currently known conditions, based on the independence assumption or other statistic properties of the input signals. Moreover, it does not make use of the small-step-size assumption, neither does it assume the input signals are stationary. Instead, it uses a theory of discrete systems and properties of a discrete state-space matrix. Therefore, the result can be applied to a wide set of signals, including deterministic and nonstationary signals. The location of all eigenvalues of the matrix responsible for the LMS algorithm stability has been calculated. Simulation experiments, where the step size reaches a couple of hundreds without loss of stability, are shown to support the theory. On the other hand, simulation where the calculations based on the small-step-size theory provide a too large estimation of the upper bound for the step size, while the new condition gives a proper solution, is also presented. Therefore, the new condition may be used in cases where fast adaptation is necessary and when the independence theory or the small-step-size assumptions do not hold. Copyright © 2014 John Wiley & Sons, Ltd. SummaryA sufficient condition for least mean squares (LMS) algorithm stability with a small set of assumptions is derived in this paper. The derivation is not, contrary to the majority of currently known conditions, based on the independence assumption or other statistic properties of the input signals. Moreover, it does not make use of the small‐step‐size assumption, neither does it assume the input signals are stationary. Instead, it uses a theory of discrete systems and properties of a discrete state‐space matrix. Therefore, the result can be applied to a wide set of signals, including deterministic and nonstationary signals. The location of all eigenvalues of the matrix responsible for the LMS algorithm stability has been calculated. Simulation experiments, where the step size reaches a couple of hundreds without loss of stability, are shown to support the theory. On the other hand, simulation where the calculations based on the small‐step‐size theory provide a too large estimation of the upper bound for the step size, while the new condition gives a proper solution, is also presented. Therefore, the new condition may be used in cases where fast adaptation is necessary and when the independence theory or the small‐step‐size assumptions do not hold. Copyright © 2014 John Wiley & Sons, Ltd. |
| Author | Bismor, Dariusz |
| Author_xml | – sequence: 1 givenname: Dariusz surname: Bismor fullname: Bismor, Dariusz email: Correspondence to: Dariusz Bismor, Institute of Automatic Control, Silesian University of Technology, ul. Akademicka 16, 44-100 Gliwice, Poland., Dariusz.Bismor@polsl.pl organization: Institute of Automatic Control, Silesian University of Technology, 44-100 Gliwice, Poland |
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| Cites_doi | 10.1109/TAC.1968.1098860 10.1016/0165-1684(84)90013-6 10.1109/5.60921 10.1109/ICASSP.1995.480504 10.1109/78.236504 10.1109/TASSP.1985.1164493 10.1109/TASSP.1986.1164993 10.1109/LSP.2009.2032996 10.1109/TSP.2008.2007618 10.1109/MSP.2009.932168 10.1016/j.sigpro.2009.05.002 10.1155/1993/438416 10.1109/PROC.1976.10286 10.1016/S0003-682X(02)00027-0 10.1016/0165-1684(87)90016-8 10.1109/TAES.1986.310809 10.1017/CBO9780511810817 10.1109/TAC.1964.1105698 10.1109/TIT.1984.1056892 10.1109/TAC.1967.1098599 10.1016/j.sigpro.2010.05.015 10.1109/81.928156 10.1016/j.sigpro.2010.07.015 10.1109/78.205720 10.2478/v10168-010-0019-z |
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| References | Widrow B, McCool J, Larimore M, Johnson J. CR. Stationary and nonstationary learning characteristics of the LMS adaptive filter. Proceedings of the IEEE August 1976; 64(8): 1151-1162. Givens L. Enhanced-convergence normalized LMS algorithm. IEEE Signal Processing Magazine 2009; 26(3): 81-95. Kaczorek T. Control and System Theory, Wydawnictwo Naukowe PWN: Warsaw, 1993. Nitzberg R. Normalized LMS algorithm degradation due to estimation noise. IEEE Transactions on Aerospace and Electronic Systems November 1986; AES-22(6): 740-750. Bubnicki Z. On the stability condition of nonlinear sampled-data systems. IEEE Transactions on Automatic Control July 1964; 9(3): 280-281. Butterweck HJ. Steady-state analysis of the long LMS adaptive filter. Signal Processing 2011; 91(4): 690-701. Pawelczyk M. Analogue active noise control. Applied Acoustics 2002; 63(11): 1193-1213. Gardner WA. Learning characteristics of stochastic-gradient-descent algorithms: a general study, analysis and critique. Signal Processing 1984; 6: 113-133. Shi K, Shi P. Convergence analysis of sparse LMS algorithms with l1-norm penalty based on white input signal. Signal Processing 2010; 90(12): 3289-3293. Latos M, Paweczyk M. Adaptive algorithms for enhancement of speech subject to a high-level noise. Archives of Acoustics 2010; 35(2): 203-212. McLernon D, Lara M, Orozco-Lugo A. On the convergence of the LMS algorithm with a rank-deficient input autocorrelation matrix. Signal Processing 2009; 89(11): 2244-2250. Bubnicki Z. Modern Control Theory, Springer-Verlag: Berlin, 2005. Bendat J, Piersol A. Random Data. Analysis and Measurement Procedures, John Wiley & Sons: New York, 1986. Florian S, Feuer A. Performance analysis of the LMS algorithm with a tapped delay line (two-dimensional case). IEEE Transactions on Acoustics, Speech and Signal Processing December 1986; 34(6): 1542-1549. Rupp M. The behavior of LMS and NLMS algorithms in the presence of spherically invariant processes. IEEE Transactions on Signal Processing March 1993; 41(3): 1149-1160. Pawelczyk M. Active noise control-a review of control-related problems. Archives of Acoustics 2008; 33(4): 509-520. Nagumo J, Noda A. A learning method for system identification. IEEE Transactions on Automatic Control June 1967; 12(3): 282-287. Butterweck H. A wave theory of long adaptive filters. IEEE Transactions on Circuits and Systems 2001; 48(6): 739-747. Bendat J, Piersol A. Engineering Applications of Correlation and Spectral Analysis, John Wiley & Sons: New York, 1993. Aihara S, Bagchi A. Adaptive filtering for stochastic risk premia in bond market. International Journal of Innovative Computing, Information and Control 2012; 8(3 B): 2203-2214. Kaczorek T. Vectors and Matrices in Automatic Control and Electrotechnics, Wydawnictwo Naukowo-Techniczne: Warsaw, 1998. Boland FM, Foley JB. Stochastic convergence of the LMS algorithm in adaptive systems. Signal Processing 1987; 13(4): 339-352. Sayed AH. Fundamentals of Adaptive Filtering, John Wiley & Sons: New York, 2003. Zeidler J. Performance analysis of LMS adaptive prediction filters. Proceedings of the IEEE December 1990; 78(12): 1781-1806. Shi K, Ma X. A frequency domain step-size control method for LMS algorithms. IEEE Signal Processing Letters 2010; 17(2): 125-128. Slock DTM. On the convergence behavior of the LMS and the normalized LMS algorithms. IEEE Transactions on Signal Processing 1993; 41(9): 2811-2825. Bubnicki Z. On the linear conjecture in the deterministic and stochastic stability of discrete systems. IEEE Transactions on Automatic Control April 1968; 13(2): 199-200. Haykin S. Adaptive Filter Theory (4th edn). Prentice Hall: New York, 2002. Feuer A, Weinstein E. Convergence analysis of LMS filters with uncorrelated Gaussian data. IEEE Transactions on Acoustics, Speech and Signal Processing February 1985; 33(1): 222-230. Widrow B, Walach E. On the statistical efficiency of the LMS algorithm with nonstationary inputs. IEEE Transactions on Information Theory March 1984; 30(2): 211-221. Zhao S, Man Z, Khoo S, Wu HR. Stability and convergence analysis of transform-domain LMS adaptive filters with second-order autoregressive process. IEEE Transactions on Signal Processing 2009; 57(1): 119-130. Söderström TPS. System Identification, Prentice Hall International, Inc.: New York, 1989. Zhij Y, Chen B, Hu J. Adaptive filtering with adaptive p-power error criterion. International Journal of Innovative Computing, Information and Control 2011; 7(4): 1725-1737. Horn RA, Johnson CR. Matrix Analysis, Cambridge University Press: New York, 1985. 1987; 13 2009; 89 1976; 64 1986; AES‐22 1990; 78 2010; 35 1964; 9 2011 2010; 17 1986; 34 1993; 41 1998 2008 1995 2001; 48 2005 2008; 33 1993 2003 2002 2011; 7 2009; 26 1984; 30 2009; 57 1968; 13 2002; 63 2011; 91 1967; 12 1984; 6 1986 1985 1985; 33 2010; 90 2012; 8 1989 e_1_2_9_30_1 e_1_2_9_31_1 Kaczorek T (e_1_2_9_38_1) 1998 Kaczorek T (e_1_2_9_37_1) 1993 e_1_2_9_11_1 Bendat J (e_1_2_9_29_1) 1986 e_1_2_9_34_1 Haykin S (e_1_2_9_2_1) 2002 e_1_2_9_10_1 e_1_2_9_35_1 e_1_2_9_13_1 e_1_2_9_12_1 Aihara S (e_1_2_9_15_1) 2012; 8 e_1_2_9_39_1 e_1_2_9_17_1 e_1_2_9_36_1 Zhij Y (e_1_2_9_16_1) 2011; 7 e_1_2_9_19_1 e_1_2_9_18_1 e_1_2_9_20_1 e_1_2_9_22_1 e_1_2_9_21_1 Söderström TPS (e_1_2_9_33_1) 1989 e_1_2_9_23_1 e_1_2_9_8_1 e_1_2_9_7_1 e_1_2_9_6_1 e_1_2_9_5_1 e_1_2_9_4_1 Pawelczyk M (e_1_2_9_24_1) 2008; 33 e_1_2_9_3_1 Bubnicki Z (e_1_2_9_28_1) 2005 Sayed AH (e_1_2_9_14_1) 2003 e_1_2_9_9_1 e_1_2_9_26_1 e_1_2_9_25_1 Bismor D (e_1_2_9_32_1) 2011 e_1_2_9_27_1 |
| References_xml | – reference: Zhij Y, Chen B, Hu J. Adaptive filtering with adaptive p-power error criterion. International Journal of Innovative Computing, Information and Control 2011; 7(4): 1725-1737. – reference: Boland FM, Foley JB. Stochastic convergence of the LMS algorithm in adaptive systems. Signal Processing 1987; 13(4): 339-352. – reference: Shi K, Ma X. A frequency domain step-size control method for LMS algorithms. IEEE Signal Processing Letters 2010; 17(2): 125-128. – reference: Kaczorek T. Control and System Theory, Wydawnictwo Naukowe PWN: Warsaw, 1993. – reference: Horn RA, Johnson CR. Matrix Analysis, Cambridge University Press: New York, 1985. – reference: Widrow B, McCool J, Larimore M, Johnson J. CR. Stationary and nonstationary learning characteristics of the LMS adaptive filter. Proceedings of the IEEE August 1976; 64(8): 1151-1162. – reference: Sayed AH. Fundamentals of Adaptive Filtering, John Wiley & Sons: New York, 2003. – reference: Bendat J, Piersol A. Engineering Applications of Correlation and Spectral Analysis, John Wiley & Sons: New York, 1993. – reference: Aihara S, Bagchi A. Adaptive filtering for stochastic risk premia in bond market. International Journal of Innovative Computing, Information and Control 2012; 8(3 B): 2203-2214. – reference: Gardner WA. Learning characteristics of stochastic-gradient-descent algorithms: a general study, analysis and critique. Signal Processing 1984; 6: 113-133. – reference: Shi K, Shi P. Convergence analysis of sparse LMS algorithms with l1-norm penalty based on white input signal. Signal Processing 2010; 90(12): 3289-3293. – reference: Givens L. Enhanced-convergence normalized LMS algorithm. IEEE Signal Processing Magazine 2009; 26(3): 81-95. – reference: Bubnicki Z. On the stability condition of nonlinear sampled-data systems. IEEE Transactions on Automatic Control July 1964; 9(3): 280-281. – reference: Florian S, Feuer A. Performance analysis of the LMS algorithm with a tapped delay line (two-dimensional case). IEEE Transactions on Acoustics, Speech and Signal Processing December 1986; 34(6): 1542-1549. – reference: McLernon D, Lara M, Orozco-Lugo A. On the convergence of the LMS algorithm with a rank-deficient input autocorrelation matrix. Signal Processing 2009; 89(11): 2244-2250. – reference: Rupp M. The behavior of LMS and NLMS algorithms in the presence of spherically invariant processes. IEEE Transactions on Signal Processing March 1993; 41(3): 1149-1160. – reference: Butterweck H. A wave theory of long adaptive filters. IEEE Transactions on Circuits and Systems 2001; 48(6): 739-747. – reference: Zhao S, Man Z, Khoo S, Wu HR. Stability and convergence analysis of transform-domain LMS adaptive filters with second-order autoregressive process. IEEE Transactions on Signal Processing 2009; 57(1): 119-130. – reference: Söderström TPS. System Identification, Prentice Hall International, Inc.: New York, 1989. – reference: Bendat J, Piersol A. Random Data. Analysis and Measurement Procedures, John Wiley & Sons: New York, 1986. – reference: Widrow B, Walach E. On the statistical efficiency of the LMS algorithm with nonstationary inputs. IEEE Transactions on Information Theory March 1984; 30(2): 211-221. – reference: Pawelczyk M. Active noise control-a review of control-related problems. Archives of Acoustics 2008; 33(4): 509-520. – reference: Latos M, Paweczyk M. Adaptive algorithms for enhancement of speech subject to a high-level noise. Archives of Acoustics 2010; 35(2): 203-212. – reference: Slock DTM. On the convergence behavior of the LMS and the normalized LMS algorithms. IEEE Transactions on Signal Processing 1993; 41(9): 2811-2825. – reference: Haykin S. Adaptive Filter Theory (4th edn). Prentice Hall: New York, 2002. – reference: Kaczorek T. Vectors and Matrices in Automatic Control and Electrotechnics, Wydawnictwo Naukowo-Techniczne: Warsaw, 1998. – reference: Feuer A, Weinstein E. Convergence analysis of LMS filters with uncorrelated Gaussian data. IEEE Transactions on Acoustics, Speech and Signal Processing February 1985; 33(1): 222-230. – reference: Bubnicki Z. Modern Control Theory, Springer-Verlag: Berlin, 2005. – reference: Pawelczyk M. Analogue active noise control. Applied Acoustics 2002; 63(11): 1193-1213. – reference: Zeidler J. Performance analysis of LMS adaptive prediction filters. Proceedings of the IEEE December 1990; 78(12): 1781-1806. – reference: Butterweck HJ. Steady-state analysis of the long LMS adaptive filter. Signal Processing 2011; 91(4): 690-701. – reference: Bubnicki Z. On the linear conjecture in the deterministic and stochastic stability of discrete systems. IEEE Transactions on Automatic Control April 1968; 13(2): 199-200. – reference: Nagumo J, Noda A. A learning method for system identification. IEEE Transactions on Automatic Control June 1967; 12(3): 282-287. – reference: Nitzberg R. Normalized LMS algorithm degradation due to estimation noise. IEEE Transactions on Aerospace and Electronic Systems November 1986; AES-22(6): 740-750. – volume: 91 start-page: 690 issue: 4 year: 2011 end-page: 701 article-title: Steady‐state analysis of the long LMS adaptive filter publication-title: Signal Processing – year: 2011 – year: 1985 – volume: 12 start-page: 282 issue: 3 year: 1967 end-page: 287 article-title: A learning method for system identification publication-title: IEEE Transactions on Automatic Control – volume: 7 start-page: 1725 issue: 4 year: 2011 end-page: 1737 article-title: Adaptive filtering with adaptive p‐power error criterion publication-title: International Journal of Innovative Computing, Information and Control – volume: AES‐22 start-page: 740 issue: 6 year: 1986 end-page: 750 article-title: Normalized LMS algorithm degradation due to estimation noise publication-title: IEEE Transactions on Aerospace and Electronic Systems – year: 2005 – volume: 63 start-page: 1193 issue: 11 year: 2002 end-page: 1213 article-title: 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| Snippet | SummaryA sufficient condition for least mean squares (LMS) algorithm stability with a small set of assumptions is derived in this paper. The derivation is not,... A sufficient condition for least mean squares (LMS) algorithm stability with a small set of assumptions is derived in this paper. The derivation is not,... Summary A sufficient condition for least mean squares (LMS) algorithm stability with a small set of assumptions is derived in this paper. The derivation is... |
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| SubjectTerms | adaptive filtering Algorithms Computer simulation Derivation Discrete systems discrete-time systems Eigenvalues least mean squares (LMS) method Mathematical analysis Stability Statistics |
| Title | Extension of LMS stability condition over a wide set of signals |
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