A Variable Step Size LMS Algorithm -raymond

A Variable Step Size LMS Algorithm -raymond - IEEE...

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IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL 40, NO 7. JULY 1992 1633 A Variable Step Size LMS Algorithm Raymond H. Kwong, Member, IEEE, and Edward W. Johnston Abstract-A new LMS-type adaptive filter with a variable step size is introduced. The step size increases or decreases as the mean-square error increases or decreases, allowing the adap- tive filter to track changes in the system as well as produce a small steady state error. The convergence and steady state be- havior of the algorithm are analyzed. These results reduce to well-known ones when specialized to the constant step size case. Simulation results are presented to support the analysis and to compare the performance of the new algorithm with the usual LMS algorithm and another variable step algorithm. They show that the performance of the new algorithm compares favorably with these existing algorithms. I. INTRODUCTION NE of the most popular algorithms in adaptive signal 0 processing is the least mean square (LMS) algorithm of Widrow and Hoff [ 11. It has been extensively analyzed in the literature, and a large number of results on its steady state misadjustment and its tracking performance has been obtained [2]-[8]. The majority of these papers examine the LMS algorithm with a constant step size. The choice of the step size reflects a tradeoff between misadjustment and the speed of adaptation. In [l], approximate expres- sions were derived which showed that a small step size gives small misadjustment but also a longer convergence time constant. Subsequent works have discussed the issue of optimization of the step size or methods of varying the step size to improve performance [9], [lo]. It seems to us, however, that there is as yet no detailed analysis of a variable step size algorithm that is simple to implement and is capable of giving both fast tracking as well as small misadjustment . In this paper, we propose a variable step size LMS al- gorithm where the step size adjustment is controlled by the square of the prediction error. The motivation is that a large prediction error will cause the step size to increase to provide faster tracking while a small prediction error will result in a decrease in the step size to yield smaller misadjustment. The adjustment equation is simple to im- plement, and its form is such that a detailed analysis of the algorithm is possible under the standard independence assumptions commonly made in the literature [l] to sim- plify the analysis of LMS algorithms. Manuscript received June 27, 1989; revised February 5, 1991. This work was supported by the Natural Sciences and Engineering Research Council of Canada under Grant A0875. R. H. Kwong is with the Department of Electrical Engineering, Univer- slty of Toronto, Toronto, Ontario M5S 1A4, Canada.
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A Variable Step Size LMS Algorithm -raymond - IEEE...

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