Least Mean Fourth Algorithm Research Articles

Variable Weight Mixed-Norm Adaptive Algorithm

In this work, the convergence analysis of the variable weight mixed-norm least-mean squares (LMS)-least mean-fourth (LMF) adaptive algorithm is derived. The proposed algorithm minimizes an objective function defined as a weighted sum of the LMS and LMF cost functions, where the weighting factor is time varying and adapts itself to allow the algorithm to keep track of the variations in the environment. As a by-product of this novel approach, new necessary and sufficient conditions for the LMF algorithm have been derived. Furthermore, a more general expression for the excess steady-state error for the LMF algorithm has been derived.

MVSE adaptive filtering subject to a constraint on MSE

An adaptive filtering algorithm that minimizes the variance on the squared error subject to a constraint on the mean-squared error (MSE) is proposed. For the adaptive-linear-combiner signal-processing configuration, this algorithm, called the LVCMS algorithm (for least variance subject to a constraint on mean squared error) has a gradient that is a weighted linear combination of the least-mean-square (LMS) and least-mean-fourth (LMF) algorithm gradients. Theoretical expressions are derived for the LVCMS algorithm convergence factor and misadjustment, and comparisons are made with the LMS and LMF adaptive rules for Gaussian, Laplacian, and uniform plant noise nd driving term distributions. Performance comparisons of the LVCMS, LMS, and LMF algorithms based on Monte Carlo simulation studies indicate that the LVCMS adaptation rule not only can yield a small variance of the squared error but it also produces favorable values of the mean-squared error and the mean-squared coefficient error. >

The least mean fourth (LMF) adaptive algorithm and its family

New steepest descent algorithms for adaptive filtering and have been devised which allow error minimization in the mean fourth and mean sixth, etc., sense. During adaptation, the weights undergo exponential relaxation toward their optimal solutions. Time constants have been derived, and surprisingly they turn out to be proportional to the time constants that would have been obtained if the steepest descent least mean square (LMS) algorithm of Widrow and Hoff had been used. The new gradient algorithms are insignificantly more complicated to program and to compute than the LMS algorithm. Their general form is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W_{j+1} = W_{j} + 2 \mu K \epsilon_{j}^{2K-1}X_{j},</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W_{j}</tex> is the present weight vector, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W_{j+1}</tex> is the next weight vector, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon_{j}</tex> is the present error, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X_{j}</tex> is the present input vector, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\mu</tex> is a constant controlling stability and rate of convergence, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2K</tex> is the exponent of the error being minimized. Conditions have been derived for weight-vector convergence of the mean and of the variance for the new gradient algorithms. The behavior of the least mean fourth (LMF) algorithm is of special interest. In comparing this algorithm to the LMS algorithm, when both are set to have exactly the same time constants for the weight relaxation process, the LMF algorithm, under some circumstances, will have a substantially lower weight noise than the LMS algorithm. It is possible, therefore, that a minimum mean fourth error algorithm can do a better job of least squares estimation than a mean square error algorithm. This intriguing concept has implications for all forms of adaptive algorithms, whether they are based on steepest descent or otherwise.