Mean-Square Stability Analysis of a Normalized Least Mean Fourth Algorithm for a Markov Plant

Recently, it has been shown that the stability of the least mean fourth (LMF) algorithm depends on the nonstationarity of the plant. The present paper investigates the possibility of overcoming this problem by normalization of the weight vector update term. A rigorous mean-square stability analysis is provided for a recent normalized LMF algorithm, which is normalized by a term that is second order in the estimation error and fourth order in the regressor. The analysis is done for a Markov plant with a stationary white input with even probability density function and a stationary zero-mean white noise. It is proved that the mean-square deviation (MSD) of the algorithm is bounded for all finite values of the input variance, noise variance, initial MSD, and mean-square plant parameter increment. Analytical results are supported by simulations.

[1]  Eweda Eweda.  Comparison of least mean fourth and least mean square tracking , 2012, 2012 Conference Record of the Forty Sixth Asilomar Conference on Signals, Systems and Computers (ASILOMAR).

[2]  Paulo Sergio Ramirez,et al.  Fundamentals of Adaptive Filtering , 2002 .

[3]  Azzedine Zerguine,et al.  A normalized least mean fourth algorithm with improved stability , 2010, 2010 Conference Record of the Forty Fourth Asilomar Conference on Signals, Systems and Computers.

[4]  Tareq Y. Al-Naffouri,et al.  Transient analysis of adaptive filters , 2001, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221).

[5]  Azzedine Zerguine,et al.  New insights into the normalization of the least mean fourth algorithm , 2013, Signal Image Video Process..

[6]  Colin Cowan,et al.  Using a normalised LMF algorithm for channel equalisation with co-channel interference , 2002, 2002 11th European Signal Processing Conference.

[7]  Eweda Eweda,et al.  Global Stabilization of the Least Mean Fourth Algorithm , 2012, IEEE Transactions on Signal Processing.

[8]  Eweda Eweda,et al.  Stochastic analysis of the least mean fourth algorithm for non-stationary white Gaussian inputs , 2014, Signal Image Video Process..

[9]  Eweda Eweda,et al.  Dependence of the Stability of the Least Mean Fourth Algorithm on Target Weights Non-Stationarity , 2014, IEEE Transactions on Signal Processing.

[10]  Shin'ichi Koike.  Stability conditions for adaptive algorithms with non-quadratic error criteria , 2000, 2000 10th European Signal Processing Conference.

[11]  José Carlos M. Bermudez,et al.  Probability of divergence for the least-mean fourth algorithm , 2006, IEEE Transactions on Signal Processing.

[12]  Bernard Widrow,et al.  The least mean fourth (LMF) adaptive algorithm and its family , 1984, IEEE Trans. Inf. Theory.

[13]  José Carlos M. Bermudez,et al.  Mean-square stability of the Normalized Least-Mean Fourth algorithm for white Gaussian inputs , 2011, Digit. Signal Process..

[14]  Eweda Eweda,et al.  Stochastic Analysis of a Stable Normalized Least Mean Fourth Algorithm for Adaptive Noise Canceling With a White Gaussian Reference , 2012, IEEE Transactions on Signal Processing.

[15]  Tareq Y. Al-Naffouri,et al.  Convergence and tracking analysis of a variable normalised LMF (XE-NLMF) algorithm , 2009, Signal Process..

[16]  José Carlos M. Bermudez,et al.  An improved model for the Normalized LMS algorithm with Gaussian inputs and large number of coefficients , 2002, 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[17]  José Carlos M. Bermudez,et al.  A Mean-Square Stability Analysis of the Least Mean Fourth Adaptive Algorithm , 2007, IEEE Transactions on Signal Processing.

[18]  Eweda Eweda.  A new approach for analyzing the limiting behavior of the normalized LMS algorithm under weak assumptions , 2009, Signal Process..

[19]  Sung Ho Cho,et al.  Statistical convergence of the adaptive least mean fourth algorithm , 1996, Proceedings of Third International Conference on Signal Processing (ICSP'96).

[20]  H. Teicher,et al.  Probability theory: Independence, interchangeability, martingales , 1978 .

[21]  A. Zerguine.  Convergence behavior of the normalized least mean fourth algorithm , 2000, Conference Record of the Thirty-Fourth Asilomar Conference on Signals, Systems and Computers (Cat. No.00CH37154).

[22]  José Carlos M. Bermudez,et al.  An improved statistical analysis of the least mean fourth (LMF) adaptive algorithm , 2003, IEEE Trans. Signal Process..