Tracking analysis of the sign algorithm in nonstationary environments

A tracking analysis of adaptive filters equipped with the sign algorithm and operating in nonstationary environments is presented. By modeling the nonstationarity as a random disturbance, it is shown that the long-term time-average of the mean-absolute estimation error is bounded for all finite and positive values of the convergence parameter μ, and that there exists an optimal choice of μ such that this bound is minimized. By making use of the commonly used independence assumption and further assuming that the nonstationarity is solely due to the time-varying behavior of the optimum coefficients caused by a stationary random disturbance, it is shown that the probability distribution functions of the successive coefficient misalignment vectors converge to a stationary limiting distribution. Finally, by further assuming that the signals involved are zero-mean and Gaussian, a set of nonlinear difference equations that characterizes the mean and mean-squared behavior of the filter coefficients and the mean-squared estimation error during the adaptation and tracking operations is derived