Recursive Least-Squares Adaptive Filters Array Algorithms Elementary Circular Rotations • Elementary Hyperbolic Rotations • Square-Root-Free and Householder Transformations • A Numerical Example Geometric Interpretation • Statistical Interpretation Geometric Interpretation • Statistical Interpretation Reducing to the Regularized Form • Time Updates Estimation Errors and the Conversion Factor • Update of the Minimum Cost Motivation • A Very Useful Lemma • The Inverse QR Algorithm • The QR Algorithm The Prewindowed Case • Low-Rank Property • A Fast Array Algorithm • The Fast Transversal Filter Joint Process Estimation • The Backward Prediction Error Vectors • The Forward Prediction Error Vectors • A Nonunity. | Ali H. Sayed et. Al. Recursive Least-Squares Adaptive Filters. 2000 CRC Press LLC. http . Recursive Least-Squares Adaptive Filters Ali H. Sayed University of California Los Angeles Thomas Kailath Stanford University Array Algorithms Elementary Circular Rotations Elementary Hyperbolic Rotations Square-Root-Free and Householder Transformations A Numerical Example The Least-Squares Problem Geometric Interpretation Statistical Interpretation The Regularized Least-Squares Problem Geometric Interpretation Statistical Interpretation The Recursive Least-Squares Problem Reducing to the Regularized Form Time Updates The RLS Algorithm Estimation Errors and the Conversion Factor Update of the Minimum Cost RLS Algorithms in Array Forms Motivation A Very Useful Lemma The Inverse QR Algorithm The QR Algorithm Fast Transversal Algorithms The Prewindowed Case Low-Rank Property A Fast Array Algorithm The Fast Transversal Filter Order-Recursive Filters Joint Process Estimation The Backward Prediction Error Vectors The Forward Prediction Error Vectors A Nonunity Forgetting Factor The QRD Least-Squares Lattice Filter The Filtering or Joint Process Array Concluding Remarks References The central problem in estimation is to recover to good accuracy a set of unobservable parameters from corrupted data. Several optimization criteria have been used for estimation purposes over the years but the most important at least in the sense of having had the most applications are criteria that are based on quadratic cost functions. The most striking among these is the linear least-squares criterion which was perhaps first developed by Gauss ca. 1795 in his work on celestial mechanics. Since then it has enjoyed widespread popularity in many diverse areas as a result of its attractive computational and statistical properties. Among these attractive properties the most notable are the facts that least-squares solutions can be explicitly .
