Convergence of an adaptive filter with signed filtered error

The need for high-speed adaptive filters has prompted the search for alternatives to the popular LMS algorithm. One modification is the replacement of the prediction error term in the LMS update kernel by its signum function. At the same time, in noise and echo cancellation problems, reduced residual noise variance often requires error filtering. In the paper, the authors consider the situation where the signum function of such a filtered error appears in the update kernel. Without the signum function, the error model of the filtered algorithms is similar to those in output error identifiers, and a strictly positive real (SPR) error filter suffices for convergence. The authors analyze the signed filtered error algorithm to show that because of the sign operator a SPR error filter no longer guarantees convergence. A new concept in systems theory, namely, that of strictly dominant passivity (SDP) is introduced instead. A system is SDP if the average product of its output and the signum of the input is positive. The authors conduct a systems theoretic investigation of the SDP condition and prove that the algorithm under study is convergent if the inverse of the error filter is SDP