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Adaptive Filtering, Normalized LMS Algorithm, Tracking
The paper analyzes the performance of the normalized LMS (NLMS) adaptive filter in the rarely-studied situation of a non-stationary input. The analysis is done in the context of tracking a Markov plant. An upper bound of the asymptotic time-averaged mean square normalized excess estimation error is derived. The bound is equal to a weighted sum of the asymptotic time-averaged mean square plant noise to input ratio and the asymptotic time-averaged mean square plant parameter increments. The bound holds for all values of the algorithm step size between 0 and 2, all degrees of nonstationarity of the plant parameters, and all weight initializations. The derived boundedness result implies that the algorithm is non-divergent as long as the asymptotic long term averages of both the mean squared plant noise to input ratio and the mean squared plant parameter increments are finite. An expression for the optimum step-size is derived. The analytical results are supported by simulations.
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