Stochastic analysis of the Least Mean Kurtosis algorithm for Gaussian inputs

The Least Mean Kurtosis (LMK) algorithm was initially proposed as an adaptive algorithm that is robust to the observation noise distribution. Good performances of this algorithm have been shown for non-Gaussian additive measurement noise. However, the complexity of the algorithm imposes difficulties for the development of a reasonably complete theoretical stochastic model for its behavior. The purpose of this paper is to contribute to the development of such a model. We study the stochastic behavior of Least Mean Kurtosis (LMK) algorithm for Gaussian inputs and for additive noises with even probability density functions. Deterministic recursions are derived for the adaptive weight error covariance matrix in a very novel manner, leading to a recursive model for the excess mean square error (EMSE) behavior that is shown to be accurate for Gaussian, uniform and binary noise distributions. The analysis results are then used to compare the performances of LMK with the least mean squares (LMS) and least mean fourth (LMF) algorithms under different circumstances.

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

[2]  Chien-Cheng Tseng,et al.  Least mean p-power error criterion for adaptive FIR filter , 1994, IEEE J. Sel. Areas Commun..

[3]  Anthony G. Constantinides,et al.  Least-mean kurtosis: a novel higher-order statistics based adaptive filtering algorithm , 1994 .

[4]  Junibakti Sanubari.  The least mean kurtosis: Its steady state performance with gaussian and non-gaussian noise , 2005, SIP.

[5]  Hong Zhao,et al.  Self-similar traffic prediction using least mean kurtosis , 2003, Proceedings ITCC 2003. International Conference on Information Technology: Coding and Computing.

[6]  PooGyeon Park,et al.  An Improved Least Mean Kurtosis (LMK) Algorithm for Sparse System Identification , 2012 .

[7]  Anthony G. Constantinides,et al.  The LMK algorithm with time-varying forgetting factor for adaptive system identification in additive output-noise , 1996, 1996 IEEE International Conference on Acoustics, Speech, and Signal Processing Conference Proceedings.

[8]  Jonathan M. Nichols,et al.  An Isserlis’ Theorem for Mixed Gaussian Variables: Application to the Auto-Bispectral Density , 2009 .

[9]  Y. Bar-Ness,et al.  Blind adaptive asynchronous CDMA multiuser detector using prediction least mean kurtosis algorithm , 2000, Proceedings of the Tenth IEEE Workshop on Statistical Signal and Array Processing (Cat. No.00TH8496).

[10]  Jeng-Kuang Hwang,et al.  A Gradient-Based Variable Step-Size Scheme for Kurtosis of Estimated Error , 2010, IEEE Signal Processing Letters.

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

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

[13]  B. Farhang-Boroujeny,et al.  Adaptive Filters: Theory and Applications , 1999 .

[14]  Andrew H. Jazwinski,et al.  Adaptive filtering , 1969, Autom..

[15]  J.S. Awitri.  The performance of the fixed-point least mean kurtosis and noisy inputs , 2005, 48th Midwest Symposium on Circuits and Systems, 2005..

[16]  Pierre-Olivier Amblard,et al.  Adaptive estimation of the fourth-order cumulant of a white stochastic process , 1995, Signal Process..

[17]  Nanning Zheng,et al.  Steady-State Mean-Square Error Analysis for Adaptive Filtering under the Maximum Correntropy Criterion , 2014, IEEE Signal Processing Letters.

[18]  Tung-Sang Ng,et al.  Fast least mean M-estimate algorithms for robust adaptive filtering in impulse noise , 2000, 2000 10th European Signal Processing Conference.

[19]  O. Tanrikulu.  Adaptive signal processing algorithms with accelerated convergence and noise immunity , 1995 .

[20]  Dimitris G. Manolakis,et al.  Statistical and Adaptive Signal Processing , 2000 .

[21]  Junibakti Sanubari.  Analysis of steady-state excess mean-square-error of the least mean kurtosis adaptive algorithm , 2006, 2006 14th European Signal Processing Conference.

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

[23]  José Carlos M. Bermudez,et al.  Properties of the kurtosis performance surface in linear estimation: application to adaptive filtering , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

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

[25]  Anthony G. Constantinides,et al.  A novel kurtosis driven variable step-size adaptive algorithm , 1999, IEEE Trans. Signal Process..