Maximum and minimum tracking performances of adaptive filteringalgorithms over target weight cross correlations

摘要

This paper is concerned with studying the dependence of thentracking performance of the LMS, RLS, sign, signed regressor, andnsign-sign algorithms on the cross correlations among the fluctuations ofnindividual target weights. In practical applications, these crossncorrelations are usually unknown. Therefore, it is useful for designnpurposes to find the extreme values of the performance measures over allnpossible cross correlations. The paper derives, for each one of thenabove five algorithms, the conditions of target weight crossncorrelations that maximize and the ones that minimize the steady-statenexcess mean-square error Ξ and the steady-state mean-square weightnmisalignment Η. The relationship between the step sizesnΜΞ and ΜΗ that minimize Ξ andnΗ, respectively, for given target weight cross correlations isnstudied. Maxima and minima of ΜΞ and ΜΗn over all target weight cross correlations are found. Thennecessary and sufficient condition of equality of ΜΞnand ΜΗ for all target weight cross correlations isnderived. A rule that maps the tracking performance measures of the LMSnalgorithm to the ones of the RLS algorithm is found. The necessary andnsufficient condition of equality of the tracking capabilities of the RLSnand LMS algorithms for all target weight cross correlations is derived.nA measure of the degree of ambiguity of the tracking performance, due tonignorance of target weight cross correlations, is defined. It is foundnthat all of the above algorithms share the same degree of ambiguity, andnthat this degree increases with the eigenvalue spread of the inputncovariance matrix