Contents QR Decomposition: An Annotated Bibliography Introduction to Adaptive Filters

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1 Contents 1 QR Decomposition: An Annotated Bibliography... 1 Marcello L. R. de Campos and Gilbert Strang 1.1 Preamble EigenvaluesandEigenvectors Iterative Methods for the Solution of the Eigenproblem TheLRalgorithm TheQRalgorithm QR Decomposition for Orthogonalization The classical Gram Schmidt orthogonalization method The modified Gram Schmidt orthogonalization method Triangularization via Householder reflections Triangularization via Givens plane rotations QR Decomposition for Linear Least Squares Problems QR Decomposition by systolic arrays QR Decomposition for Recursive Least Squares Adaptive Filters Fast QR decomposition RLS adaptation algorithms Conclusion References Introduction to Adaptive Filters José A. Apolinário Jr. and Sergio L. Netto 2.1 Basic Concepts ErrorMeasurements The mean-square error The instantaneous square error The weighted least-squares Adaptation Algorithms LMS and normalized-lms algorithms xi

2 xii Contents Data-reusingLMSalgorithms RLS-typealgorithms ComputerSimulations Example1:MisadjustmentoftheLMSalgorithm Example 2: Convergence trajectories Example 3: Tracking performance Example 4: Algorithm stability Conclusion References Conventional and Inverse QRD-RLS Algorithms José A. Apolinário Jr. and Maria D. Miranda 3.1 The Least-Squares Problem and the QR Decomposition TheGivensRotationMethod The Conventional QRD-RLS Algorithm Initialization of the Triangularization Procedure On the Q θ (k) Matrix The backward prediction problem Theforwardpredictionproblem Interpreting the elements of Q θ (k) for a lower triangular Cholesky factor Interpreting the elements of Q θ (k) for an upper triangular Cholesky factor TheInverseQRD-RLSAlgorithm Conclusion Appendix Appendix Appendix References Fast QRD-RLS Algorithms José A. Apolinário Jr. and Paulo S. R. Diniz 4.1 Introduction Upper Triangularization Algorithms (UpdatingForwardPredictionErrors) The FQR POS Falgorithm The FQR PRI Falgorithm Lower Triangularization Algorithms (Updating Backward Prediction Errors) The FQR POS Balgorithm The FQR PRI Balgorithm The Order Recursive Versions of the Fast QRD Algorithms Conclusion Appendix

3 Contents xiii Appendix Appendix References QRD Least-Squares Lattice Algorithms Jenq-Tay Yuan 5.1 Fundamentals of QRD-LSL Algorithms LSLInterpolatorandLSLPredictor LSLinterpolator Orthogonal bases for LSL interpolator LSLpredictor SRF Givens Rotation with Feedback Mechanism SRFQRD-LSLAlgorithms QRDbasedoninterpolation SRFQRD-LSLinterpolationalgorithm SRF QRD-LSL prediction algorithm and SRF joint process estimation SRF(QRD-LSL)-BasedRLSAlgorithm Simulations Conclusion References Multichannel Fast QRD-RLS Algorithms António L. L. Ramos and Stefan Werner 6.1 Introduction ProblemFormulation Redefining the input vector Input vector for sequential-type multichannel algorithms Input vector for block-type multichannel algorithms Sequential-Type MC-FQRD-RLS Algorithms Triangularization of the information matrix AprioriandAposterioriversions Alternativeimplementations Block-Type MC-FQRD-RLS Algorithms The backward and forward prediction problems AprioriandAposterioriversions Alternativeimplementations Order-Recursive MC-FQRD-RLS Algorithms Application Example and Computational Complexity Issues Applicationexample Computational complexity issues Conclusion References

4 xiv Contents 7 Householder-Based RLS Algorithms Athanasios A. Rontogiannis and Sergios Theodoridis 7.1 Householder Transforms Hyperbolic Householder transforms Row Householder transforms The Householder RLS (HRLS) Algorithm Applications The Householder Block Exact QRD-RLS Algorithm The Householder Block Exact Inverse QRD-RLS Algorithm Sliding Window (SW) Householder Block Implementation Conclusion References Numerical Stability Properties Phillip Regalia and Richard Le Borne 8.1 Introduction Preliminaries Conditioning, forward stability, and backward stability The Conditioning of the Least-Squares Problem The conditioning of the least-squares problem Consistency, stability, and convergence The Recursive QR Least-Squares Methods Full QR decomposition adaptive algorithm FastQRAlgorithms Past input reconstruction Reachable states in fast least-squares algorithms QR decomposition lattice algorithm Conclusion References Finite and Infinite-Precision Properties of QRD-RLS Algorithms Paulo S. R. Diniz and Marcio G. Siqueira 9.1 Introduction Precision Analysis of the QR-Decomposition RLS Algorithm Infinite-precision analysis Stability analysis Error propagation analysis in steady-state Simulationresults Precision Analysis of the Fast QRD-Lattice Algorithm Infinite-precision analysis Finite-precision analysis Simulationresults Conclusion References

5 Contents xv 10 On Pipelined Implementations of QRD-RLS Adaptive Filters Jun Ma and Keshab K. Parhi 10.1 QRD-RLSSystolicArchitecture The Annihilation-Reording Look-Ahead Technique Look-ahead through block processing Look-ahead through iteration Relationship with multiply add look-ahead Parallelism in annihilation-reording look-ahead Pipelinedandblockprocessingimplementations Invariance of bounded input and bounded output Pipelined CORDIC-Based RLS Adaptive Filters Pipelined QRD-RLS with implicit weight extraction Stability analysis Pipelined QRD-RLS with explicit weight extraction Conclusion Appendix References Weight Extraction of Fast QRD-RLS Algorithms Stefan Werner and Mohammed Mobien 11.1 FQRD-RLSPreliminaries QR decomposition algorithms FQR POS Balgorithm SystemIdentificationwithFQRD-RLS WeightextractionintheFQRD-RLSalgorithm Example Burst-trainedEqualizerwithFQRD-RLS Problemdescription Equivalent-outputfiltering Equivalent-output filtering with explicit weight extraction Example ActiveNoiseControlandFQRD-RLS Filtered-x RLS Modifiedfiltered-xFQRD-RLS Example Multichannel and Lattice Implementations Conclusion References On Linearly Constrained QRD-Based Algorithms Shiunn-Jang Chern 12.1 Introduction Optimal Linearly Constrained QRD-LS Filter The Adaptive LC-IQRD-RLS Filtering Algorithm The Adaptive GSC-IQRD-RLS Algorithm

6 xvi Contents 12.5 Applications Application 1: Adaptive LCMV filtering for spectrum estimation Application 2: Adaptive LCMV antenna array beamformer Conclusion References Index...347

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