Tracking analysis of the sign algorithm in nonstationary environments

A tracking analysis of the adaptive filters equipped with the sign algorithm and operating in nonstationary environments is presented. Under the assumption that the nonstationary can be modeled using a random disturbance, it is shown that the long-term time average of the mean-absolute error is bounded, and that there exists an optimal choice of the convergence constant mu which minimizes this quality. Applying the commonly used independence assumption, and under the assumption that the nonstationarity is solely due to the time-varying behavior of the optimal coefficients, it is shown that the distributions of the successive coefficient misalignment vectors converge to a limiting distribution when the adaptive filter is used in the system identification mode. Finally, under the additional assumption that the signals involved are zero mean and Gaussian, a set of nonlinear difference equations is derived that characterizes the mean and mean-squared behavior of the filter coefficients and the mean-squared estimation error during adaptation and tracking. Results of several experiments that show very good correlation with the theoretical analyses are presented.>