A new nonstationary LMS algorithm for tracking Markovian time varying systems

We propose in this paper a new adaptive algorithm, designed to track system impulse responses, characterized by stochastic Markovian time variations. The proposed nonstationary least mean square (NSLMS) algorithm is designed so that it explicitly takes into account the structure of the nonstationarity. Hence, unlike the classical LMS algorithm, the NSLMS algorithm is not blind with respect to the time variations of the system impulse response to identify.The proposed algorithm structure is based on a coupling between the estimation of the Markovian parameter that characterizes the nonstationarity and the estimation of the adaptive filter that identifies the system impulse response. The adaptive identification of the Markovian parameter is performed by an LMS algorithm, based on the minimization of the mean square of the system identification error.A theoretical analysis of the transient and the steady-state behaviors of the NSLMS adaptive filter is carried out. In particular, an analytical expression of the step size that guarantees the stability of the latter is established. The theoretical misadjustment that measures the tracking ability of the NSLMS algorithm is computed for an i.i.d. input.We prove that in the steady-state, the NSLMS algorithm exhibits better performance than the classical LMS algorithm, and goes beyond the limitation of the LMS algorithm to track severe filter time variations.The experimental results reported here are in perfect agreement with the theory. They display the good properties of the NSLMS algorithm and demonstrate its ability to yield good performances in a hard Markovian time varying environment.