Adaptive total least squares filtering

A new performance criterion known as Total Least Squares (TLS) has recently been applied to problems in adaptive filtering. The criterion has been proposed as a method to remove the parameter bias in the coefficient vector estimate when both the input data sequence and the output data sequence are corrupted by noise interference. This method is applicable for both Finite Impulse Response (FIR) filtering as well as Infinite Impulse Response (IIR) filtering. A related technique, known as mixed Least Squares-Total Least Squares (LS-TLS), is applicable for removing the parameter bias in output noise only scenarios for the Equation Error (EE) IIR framework.; The solution to a TLS or mixed LS-TLS problem involves computing the Singular Value Decomposition (SVD) of the data matrix. Due to the large computational burden associated with iteratively computing the SVD of a growing matrix in on-line applications, alternate fast and efficient algorithms are desired. To that end, we propose two classes of algorithms for computing the solution. The first class consists of fast, O(N2) complexity algorithms based on a QR-decomposition of the data covariance matrix. These fast algorithms exploit QR-decomposition such that the desired solution can be computed without the need of matrix inversion. The second class of algorithms are of O(N) complexity and are designed to be comparable to the ubiquitous Least Mean Squares (LMS) algorithm. Our O(N) algorithms are based upon the steepest descent gradient of a cost function whose minimum is known to lead to the TLS or mixed LS-TLS solution, depending on a suitably chosen weighting matrix. We show, for the first time, a complete convergence stability proof that shows that these algorithms converge to the desired solution.; Performance benefits gained with the TLS or mixed LS-TLS performance criterion as compared to the LS criterion are given for several classical filtering problems. We also describe a new application for mixed LS-TLS to remove the coefficient bias in an otherwise well-studied non-linear filtering problem.