Study of the optimal and simplified Kalman filters for echo cancellation

In this paper, we study the time-domain Kalman filter in the context of echo cancellation. We explain the fundamental differences between the Kalman filter and the recursive least-squares (RLS) algorithm. Also, we show that the normalized least-mean-square (NLMS) algorithm has a clear relationship with the Kalman filter. Furthermore, a simplified Kalman filter is derived and by a judicious choice of its parameters, this algorithm behaves like a variable step-size adaptive filter. Simulation results indicate the good performance of the optimal and simplified Kalman filtering algorithms.