A Stable Normalized Least Mean Fourth Algorithm With Improved Transient and Tracking Behaviors

The stability problems of the least mean fourth (LMF) algorithm put a limitation on its tracking capability. The paper investigates the possibility of solving this problem via stabilization of the algorithm. The analysis is done for a Markov plant. It is found that the available stable normalized LMF (NLMF) algorithm has a tracking limitation for high signal-to-noise ratio. The paper presents a new stable NLMF algorithm that is free of this limitation. Mean-square stability of the algorithm is proved. Expressions are derived for the minimum steady-state mean square deviation (MSD) and the corresponding convergence time. The new algorithm outperforms the available stable NLMF algorithm in both the transient and steady states. The new algorithm is also compared with the NLMS algorithm when the adaptation parameter of each algorithm is set to the value that minimizes its steady-state MSD. For large initial MSD, the algorithm outperforms the NLMS algorithm, even for Gaussian noise. For small initial MSD, the algorithm outperforms the NLMS algorithm for sub-Gaussian noise, while the situation is opposite for Gaussian noise. Analytical results are supported by simulations.

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

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

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

[4]  Michel Loève,et al.  Probability Theory I , 1977 .

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

[6]  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.

[7]  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.

[8]  Ali H. Sayed,et al.  Fundamentals Of Adaptive Filtering , 2003 .

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

[10]  Odile Macchi,et al.  Adaptive Processing: The Least Mean Squares Approach with Applications in Transmission , 1995 .

[11]  A. Constantinides,et al.  Least mean mixed-norm adaptive filtering , 1994 .

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

[13]  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).

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

[15]  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.

[16]  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).

[17]  Eweda Eweda,et al.  Mean-Square Stability Analysis of a Normalized Least Mean Fourth Algorithm for a Markov Plant , 2014, IEEE Transactions on Signal Processing.

[18]  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.

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

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

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

[23]  Abubakr Gafar Abdalla,et al.  Probability Theory , 2017, Encyclopedia of GIS.

[24]  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).

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