Analysis of the Error Signal of the LMS Algorithm

An analysis of the error signal of the least-mean-square (LMS) algorithm is conducted from the robust control theory viewpoint. The difference equation that relates the input of the LMS algorithm and the error signal is presented. This equation is used to build the matrix S that maps the input vector to the error vector. It is shown that S has at least one singular value greater than 1. Therefore, the system may amplify noise at high frequencies. Nevertheless, the tap-weight vector may be chosen to prevent that noise amplification and improve the disturbance rejection performance of the LMS algorithm.