Bayesian adaptive filtering at linear cost

Standard adaptive filtering algorithms, including the popular LMS and RLS algorithms, possess only one parameter (step-size, forgetting factor) to adjust the tracking speed in a non-stationary environment. Furthermore, existing techniques for the automatic adjustment of this parameter are not totally satisfactory and are rarely used. In this paper we pursue the concept of Bayesian adaptive filtering (BAF) that we introduced earlier, based on modeling the optimal adaptive filter coefficients as a stationary vector process, in particular a diagonal AR(1) model. Optimal adaptive filtering with such a state model becomes Kalman filtering. The AR(1) model parameters are determined with an adaptive version of the EM algorithm, which leads to linear prediction on reconstructed optimal filter correlations, and hence a meaningful approximation/estimation compromise. The resulting algorithm, of complexity O(N²), is shown by simulations to have performance close to that of the Kalman filter with true model parameters. In this paper, we apply a component-wise EM approach to further reduce the complexity to being linear in the number of adaptive filtering coefficients. The good performance of the resulting algorithm is illustrated in simulations. The AR(1) state model can be further approximated by a random walk, leading to further simplified adaptive filter that can be interpreted an LMS algorithm with a variable step-size per filter tap