QR Decomposition: An Annotated Bibliography | p. 1 |
Preamble | p. 1 |
Eigenvalues and Eigenvectors | p. 2 |
Iterative Methods for the Solution of the Eigenproblem | p. 3 |
The LR Algorithm | p. 3 |
The QR algorithm | p. 4 |
QR Decomposition for Orthogonalization | p. 5 |
The classical Gram-Schmidt orthogonalization method | p. 6 |
The modified Gram-Schmidt orthogonalization method | p. 8 |
Triangularization via Householder reflections | p. 9 |
Triangularization via Givens plane rotations | p. 10 |
QR Decomposition for Linear Least Squares Problems | p. 12 |
QR Decomposition by systolic arrays | p. 14 |
QR Decomposition for Recursive Least Squares Adaptive Filters | p. 14 |
Fast QR Decomposition RLS adaptation algorithms | p. 16 |
Conclusion | p. 17 |
References | p. 18 |
Introduction to Adaptive Filters | p. 23 |
Basic Concepts | p. 23 |
Error Measurements | p. 28 |
The mean-square error | p. 28 |
The instantaneous square error | p. 29 |
The weighted least-squares | p. 29 |
Adaptation Algorithms | p. 30 |
LMS and normalized-LMS algorithms | p. 31 |
Data-reusing LMS algorithms | p. 34 |
RLS-type algorithms | p. 40 |
Computer Simulations | p. 42 |
Example 1: Misadjustment of the LMS algorithm | p. 42 |
Example 2: Convergence trajectories | p. 43 |
Example 3: Tracking performance | p. 43 |
Example 4: Algorithm stability | p. 46 |
Conclusion | p. 47 |
References | p. 48 |
Conventional and Inverse QRD-RLS Algorithms | p. 51 |
The Least-Squares Problem and the QR Decomposition | p. 51 |
The Givens Rotation Method | p. 57 |
The Conventional QRD-RLS Algorithm | p. 60 |
Initialization of the Triangularization Procedure | p. 64 |
On the Q_{¿}(k) Matrix | p. 66 |
The backward prediction problem | p. 69 |
The forward prediction problem | p. 71 |
Interpreting the elements of Q_{¿}(k) for a lower triangular Cholesky factor | p. 74 |
Interpreting the elements of Q_{¿}(k) for an upper triangular Cholesky factor | p. 75 |
The Inverse QRD-RLS Algorithm | p. 76 |
Conclusion | p. 77 |
| p. 79 |
| p. 80 |
| p. 81 |
References | p. 84 |
Fast QRD-RLS Algorithms | p. 87 |
Introduction | p. 87 |
Upper Triangualarization Algorithms (Updating Forward Prediction Errors) | p. 89 |
The FQR_POS_F algorithm | p. 90 |
The FQR_PRI_F algorithm | p. 92 |
Lower Triangularization Algorithms (Updating Backward Prediction Errors) | p. 93 |
The FQR_POS_B algorithm | p. 95 |
The FQR_PRI_B algorithm | p. 98 |
The Order Recursive Versions of the Fast QRD Algorithms | p. 100 |
Conclusion | p. 104 |
| p. 105 |
| p. 107 |
| p. 111 |
References | p. 113 |
QRD Least-Squares Lattice Algorithms | p. 115 |
Fundamentals of QRD-LSL Algorithms | p. 116 |
LSL Interpolator and LSL Predictor | p. 118 |
LSL interpolator | p. 119 |
Orthogonal bases for LSL interpolator | p. 121 |
LSL predictor | p. 122 |
SRF Givens Rotation with Feedback Mechanism | p. 123 |
SRF QRD-LSL Algorithms | p. 125 |
QRD based on interpolation | p. 126 |
SRF QRD-LSL interpolation algorithm | p. 129 |
SRF QRD-LSL prediction algorithm and SRF joint process estimation | p. 136 |
SRF (QRD-LSL)-Based RLS Algorithm | p. 139 |
Simulations | p. 140 |
Conclusion | p. 142 |
References | p. 143 |
Multichannel Fast QRD-RLS Algorithms | p. 147 |
Introduction | p. 147 |
Problem Formulation | p. 149 |
Redefining the input vector | p. 151 |
Input vector for sequential-type multichannel algorithms | p. 152 |
Input vector for block-type multichannel algorithms | p. 153 |
Sequential-Type MC-FQRD-RLS Algorithms | p. 153 |
Triangularization of the information matrix | p. 154 |
A priori and A posteriori versions | p. 157 |
Alternative implementations | p. 159 |
Block-Type MC-FQRD-RLS Algorithms | p. 162 |
The backward and forward prediction problems | p. 162 |
A priori and A posteriori versions | p. 166 |
Alternative implementations | p. 169 |
Order-Recursive MC-FQRD-RLS Algorithms | p. 171 |
Application Example and Computational Complexity Issues | p. 176 |
Application example | p. 176 |
Computational complexity issues | p. 178 |
Conclusion | p. 179 |
References | p. 179 |
Householder-Based RLS Algorithms | p. 181 |
Householder Transforms | p. 181 |
Hyperbolic Householder transforms | p. 184 |
Row Householder transforms | p. 184 |
The Householder RLS (HRLS) Algorithm | p. 186 |
Applications | p. 190 |
The Householder Block Exact QRD-RLS Algorithm | p. 192 |
The Householder Block Exact Inverse QRD-RLS Algorithm | p. 196 |
Sliding Window (SW) Householder Block Implementation | p. 199 |
Conclusion | p. 202 |
References | p. 202 |
Numerical Stability Properties | p. 205 |
Introduction | p. 205 |
Preliminaries | p. 206 |
Conditioning, forward stability, and backward stability | p. 208 |
The Conditioning of the Least-Squares Problem | p. 210 |
The conditioning of the least-squares problem | p. 211 |
Consistency, stability, and convergence | p. 212 |
The Recursive QR Least-Squares Methods | p. 214 |
Full QR decomposition adaptive algorithm | p. 214 |
Fast QR Algorithms | p. 220 |
Past input reconstruction | p. 223 |
Reachable states in fast least-squares algorithms | p. 227 |
QR decomposition lattice algorithm | p. 229 |
Conclusion | p. 231 |
References | p. 232 |
Finite and Infinite-Precision Properties of QRD-RLS Algorithms | p. 235 |
Introduction | p. 235 |
Precision Analysis of the QR-Decomposition RLS Algorithm | p. 236 |
Infinite-precision analysis | p. 237 |
Stability analysis | p. 242 |
Error propagation analysis in steady-state | p. 244 |
Simulation results | p. 255 |
Precision Analysis of the Fast QRD-Lattice Algorithm | p. 256 |
Infinite-precision analysis | p. 258 |
Finite-precision analysis | p. 261 |
Simulation results | p. 265 |
Conclusion | p. 266 |
References | p. 266 |
On Pipelined Implementations of QRD-RLS Adaptive Filters | p. 269 |
QRD-RLS Systolic Architecture | p. 270 |
The Annihilation-Reording Look-Ahead Technique | p. 273 |
Look-ahead through bloack processing | p. 274 |
Look-ahead through iteration | p. 276 |
Relationship with multiply-and look-ahead | p. 277 |
Parallelism in annihilation-recording look-ahead | p. 279 |
Pipelined and block processing implementations | p. 280 |
Invariance of bounded input and bounded output | p. 283 |
Pipelined CORDIC-Based RLS Adaptive Filters | p. 283 |
Pipelined QRD-RLS with implicit weight extraction | p. 284 |
Stability analysis | p. 286 |
Pipelined QRD-RLS with explicit weight extraction | p. 288 |
Conclusion | p. 291 |
Appendix | p. 294 |
References | p. 296 |
Weight Extraction of Fast QRD-RLS Algorithms | p. 299 |
FQRD-RLS Preliminaries | p. 300 |
QR decomposition algorithms | p. 300 |
FQR_POS_B algorithm | p. 301 |
System Identification with FQRD-RLS | p. 303 |
Weight extraction in the FQRD-RLS algorithm | p. 304 |
Example | p. 306 |
Burst-trained Equalizer with FQRD-RLS | p. 308 |
Problem description | p. 309 |
Equvalent-output filtering | p. 309 |
Equivalent-output filtering with explicit weight extraction | p. 311 |
Example | p. 313 |
Active Noise Control and FQRD-RLS | p. 314 |
Filtered-s RLS | p. 315 |
Modified filtered-x FQRD-RLS | p. 316 |
Example | p. 319 |
Multichannel and Lattice Implementations | p. 320 |
Conclusion | p. 320 |
References | p. 321 |
On Linearly Constrained QRD-Based Algorithms | p. 323 |
Introduction | p. 323 |
Optimal Linearly Constrained QRD-LS Filter | p. 325 |
The Adaptive LC-IQRD-RLS Filtering Algorithm | p. 327 |
The Adaptive GSC-IQRD-RLS Algorithm | p. 331 |
Applications | p. 335 |
Application 1: Adaptive LCMV filtering for spectrum estimation | p. 335 |
Application 2: Adaptive LCMV antenna array beamformer | p. 338 |
Conclusion | p. 343 |
References | p. 343 |
Index | p. 347 |
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