Digital Signal Processing
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The performance of adaptive FIR filters governed by the recursive least-squares (RLS) algorithm, the least mean square (LMS) algorithm, and the sign algorithm (SA), are compared when the optimal filtering vector is randomly time-varying. The comparison is done in terms of the steady-state excess mean-square estimation error ξ and the steady-state mean-square weight deviation, η. It is shown that ξ does not depend on the spread of eigenvalues of the input covariance matrix, R, in the cases of the LMS algorithm and the SA, while it does in the case of the RLS algorithm. In the three algorithms, η is found to be increasing with the eigenvalue spread. The value of the adaptation parameter that minimizes ξ is different from the one that minimizes η. It is shown that the minimum values of ξ and η attained by the RLS algorithm are equal to the ones attained by the LMS algorithm in any one of the three following cases: (1) if R has equal eigenvalues, (2) if the fluctuations of the individual elements of the optimal vector are mutually uncorrelated and have the same mean-square value, or (3) if R is diagonal and the fluctuations of the individual elements of the optimal vector have the same mean-square value. Conditions that make the values of ξ and η of the LMS algorithm smaller (or greater) than the ones of the RLS algorithm are derived. For Gaussian input data, the minimum values of ξ and η attained by the SA are found to exceed the ones attained by the LMS algorithm by 1 dB independently of R and the mutual correlation between the elements of the optimal vector