Numerical properties of adaptive recursive least-squares (RLS) algorithms with linear constraints.

Abstract

Adaptive filters have found applications in many signal processing problems. In some situations, linear constraints are imposed on the filter weights such that the filter is forced to exhibit a certain desired response. Several algorithms for linearly constrained least-squares adaptive filtering have been developed in the literature. When implemented with finite precision arithmetic, these algorithms are inevitably subjected to rounding errors. It is essential to understand how these algorithms react to rounding errors.In this thesis, the numerical properties of three linearly constrained least-squares adaptive filtering algorithms, namely, the linearly constrained fast least algorithm, the linear systolic array for MVDR beamforming and the linearly constrained QRD-RLS algorithm, are studied. It is shown that all these algorithms can be separated into a constrained part and an unconstrained part. The numerical properties of unconstrained least-squares algorithms (i.e., the unconstrained part of the linearly constrained algorithms under study) are reviewed from the perspectives of error propagation, error accumulation and numerical persistency. It is shown that persistent excitation and sufficient numerical resolution are needed to ensure the stability of the CRLS algorithm, while the QRD-RLS algorithm is unconditionally stable. The numerical properties of the constrained algorithms are then examined. Based on the technique of how the constraints are applied, these algorithms can be grouped into two categories. The first two algorithms admit a similar structure in that the unconstrained parts preceed the constrained parts. Error propagation analysis shows that this structure gives rise to unstable error propagation in the constrained part. In contrast, the constrained part of the third algorithm preceeds the unconstrained part. It is shown that this algorithm gives an exact solution to a linearly constrained least-squares adaptive filtering problem with perturbed constraints and perturbed input data. A minor modification to the constrained part of the linearly constrained QRD-RLS algorithm is proposed to avoid a potential numerical difficulty due to the Gaussian elimination operation employed in the algorithm.