Convergence analysis of the sign algorithm without the independence and gaussian assumptions

Abstract

The paper is concerned with rigorous convergence analysis of the sign algorithm (SA) in the context of adaptive plant identification. Asymptotic time-averaged convergence for the mean absolute weight misalignment is proved for all values of the algorithm step size and initial weight vector. The paper has three main contributions with respect to available convergence results of the SA. The first is the deletion of the Gaussian assumption, which is important when covering the case of discrete valued data. No assumption about the distribution of the regressor sequence is used, except for the usual assumption of positive definite covariance matrix. The assumptions used about the noise allow nonexistence, unboundedness, and vanishing of the noise probability density function for arguments strictly different from zero. The second contribution is the deletion of the assumption of independent successive regressors. This deletion is important since, in applications, two successive regressors usually share all their components except two. Hence, they are strongly dependent, even for white plant input. The case of colored noise is also analyzed. Finally, the third contribution is the extension of the above results to the nonstationary case. The used assumptions allow nonstationarity of the plant input, plant noise, and plant parameters.