Preconditioner structures for the CLMS adaptive filtering algorithm

LMS filtering can be viewed as solving the Wiener-Hopf equation iteratively using the Richardson´s iteration with an identity matrix for a preconditioner. The ideal preconditioner in this situation is the inverse of the autocorrelation matrix of the input signal. This is why LMS is the optimal adaptive filter for white input signals. In situations where the input signal is not white one can improve the convergence of the adaptive filter by specifying a fixed preconditioning matrix other than the identity matrix by using approximate a priori knowledge about the input signal´s autocorrelation. This is the main idea behind the CLMS algorithm. We develop methods to obtain such preconditioning matrices with different structures that also make the algorithm computationally efficient and test these matrices for convergence rate on AR-1 signals