An Oracle Normalized Least Mean Square (NLMS) Algorithm and a Simple Bayesian Detection NLMS Algorithm Robust to Impulse Noise

This paper suggests an oracle normalized least-mean-square (NLMS) algorithm and a simple Bayesian detection NLMS impulse noise detection algorithm as an effective adaptive algorithm against impulsive noises. Initially, to have a fast algorithm, an optimization problem is introduced and then an oracle NLMS algorithm is devised. It has the largest reduction in misalignment error at each iteration with respect to the previous iteration. Oracle NLMS algorithm needs the values of impulse noises and hence is not practical. To have a practical variant of oracle NLMS algorithm, a simple Bayesian impulse noise detection NLMS algorithm is proposed. It is based on a MAP detection criterion and the impulse noise detection rule is proved to be a comparison of the absolute value of the error of the adaptive filter with a threshold and hence is very simple. Also, by assuming the sparsity of the impulse noises, the value of threshold is obtained via a simple statistical estimation. The simulation results in both dispersive and sparse system cases, show the effectiveness of the suggested algorithm in terms of convergence rate and complexity.