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

This paper is concerned with studying the dependence of the tracking performance of the LMS, RLS, sign, signed regressor, and sign-sign algorithms on the cross correlations among the fluctuations of individual target weights. In practical applications, these cross correlations are usually unknown. Therefore, it is useful for design purposes to find the extreme values of the performance measures over all possible cross correlations. The paper derives, for each one of the above five algorithms, the conditions of target weight cross correlations that maximize and the ones that minimize the steady-state excess mean-square error ξ and the steady-state mean-square weight misalignment η. The relationship between the step sizes μ_ξ and μ_η that minimize ξ and η, respectively, for given target weight cross correlations is studied. Maxima and minima of μ_ξ and μ_η over all target weight cross correlations are found. The necessary and sufficient condition of equality of μ_ξ and μ_η for all target weight cross correlations is derived. A rule that maps the tracking performance measures of the LMS algorithm to the ones of the RLS algorithm is found. The necessary and sufficient condition of equality of the tracking capabilities of the RLS and LMS algorithms for all target weight cross correlations is derived. A measure of the degree of ambiguity of the tracking performance, due to ignorance of target weight cross correlations, is defined. It is found that all of the above algorithms share the same degree of ambiguity, and that this degree increases with the eigenvalue spread of the input covariance matrix