Tracking analysis of the sign algorithm in nonstationary environments

Abstract-This paper presents a tracking analysis of the adaptive filters equipped with the sign algorithm and operating in nonstationary environments. Under the assumption that the nonstationarity 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 p which minimizes this quantity. Using the commonly employed independence assumption and under the assumption that the nonstationarity is solely due to the time-varying behavior of the optimal coefficients, we then show 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, we derive 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 adaptation and tracking. Results of several experiments that show very good correlation with the theoretical analyses are also presented in this paper.