Preface |
|
v | |
|
|
1 | (30) |
|
|
1 | (4) |
|
Central Problem Statement |
|
|
5 | (7) |
|
A Brief Glimpse into Approximation Criteria |
|
|
12 | (2) |
|
|
14 | (5) |
|
|
19 | (12) |
|
|
19 | (4) |
|
Parametrizations and Variances |
|
|
23 | (4) |
|
Equation Error versus Output Error |
|
|
27 | (2) |
|
|
29 | (2) |
|
Recursive Filter Structures |
|
|
31 | (51) |
|
Review of Linear System Theory |
|
|
31 | (6) |
|
Controllability and observability Grammians |
|
|
32 | (2) |
|
Minimality and parametrization |
|
|
34 | (1) |
|
Balanced Forms and hankel Singular Values |
|
|
35 | (2) |
|
|
37 | (4) |
|
Parallel and Cascade Forms |
|
|
41 | (2) |
|
Tapped State Lattice Form |
|
|
43 | (39) |
|
|
48 | (7) |
|
|
55 | (2) |
|
|
57 | (1) |
|
Szego Polynomials and Orthonormal Basis Functions |
|
|
58 | (4) |
|
Relations with Direct Form Filter |
|
|
62 | (2) |
|
|
64 | (14) |
|
|
78 | (4) |
|
The Beurling-Lax Theorem, Hankel Forms, and Classical Identification |
|
|
82 | (63) |
|
|
83 | (14) |
|
Shift-Invariant Subspaces |
|
|
84 | (3) |
|
Orthogonal Filters and All-Pass Completions |
|
|
87 | (8) |
|
|
95 | (2) |
|
|
97 | (5) |
|
Pade Approximations (Prony's Method) |
|
|
102 | (8) |
|
|
110 | (11) |
|
|
112 | (3) |
|
|
115 | (6) |
|
|
121 | (10) |
|
|
131 | (14) |
|
|
133 | (9) |
|
|
142 | (3) |
|
Rational Approximation in Hankel Norm |
|
|
145 | (35) |
|
|
146 | (1) |
|
|
147 | (3) |
|
|
150 | (4) |
|
|
154 | (5) |
|
Constructing the Hankel Norm Approximant |
|
|
159 | (6) |
|
Repeated hankel Singular Values |
|
|
165 | (3) |
|
Some Bounds for Other Criteria |
|
|
168 | (12) |
|
|
169 | (9) |
|
|
178 | (2) |
|
Rational H2 Approximation |
|
|
180 | (48) |
|
Normality of the Rational H2 Approximation Problem |
|
|
182 | (5) |
|
The Reduced Error Surface |
|
|
187 | (11) |
|
Invariance to Frequency Transformations |
|
|
198 | (7) |
|
Index of Stationary Points |
|
|
205 | (3) |
|
Relations to the Hankel Norm Problem |
|
|
208 | (20) |
|
|
215 | (11) |
|
|
226 | (2) |
|
Stability of Time-Varying Recursive Filters |
|
|
228 | (30) |
|
Time-Varying Recursive Filters |
|
|
228 | (6) |
|
BIBO and Exponential Stability |
|
|
234 | (4) |
|
|
238 | (4) |
|
|
242 | (16) |
|
|
249 | (8) |
|
|
257 | (1) |
|
Gradient Descent Algorithms |
|
|
258 | (116) |
|
The Mean-Square Cost Function |
|
|
260 | (7) |
|
|
267 | (6) |
|
An Introduction to the ODE Method |
|
|
273 | (10) |
|
Heuristics of the ODE Approach |
|
|
275 | (3) |
|
Stability of Differential Equations |
|
|
278 | (1) |
|
The Direct Approach of Lyapunov |
|
|
279 | (2) |
|
The Indirect Method of Lyapunov |
|
|
281 | (2) |
|
Lattice Gradient Descent Algorithm |
|
|
283 | (7) |
|
Simplified Gradient Calculation |
|
|
290 | (16) |
|
A Partial Gradient Algorithm |
|
|
306 | (18) |
|
ODE for the Partial Gradient Algorithm |
|
|
313 | (4) |
|
|
317 | (7) |
|
A Simplified Partial Gradient Algorithm |
|
|
324 | (10) |
|
Alternate Formulae for the Rotation Angles |
|
|
334 | (10) |
|
On Bounds for the Stepsize Constant μ |
|
|
344 | (7) |
|
A Priori and A Posteriori Errors |
|
|
345 | (3) |
|
|
348 | (1) |
|
Linearization About a Minimum Point |
|
|
349 | (2) |
|
|
351 | (23) |
|
|
363 | (8) |
|
|
371 | (3) |
|
The Steiglitz-McBride Family of Algorithms |
|
|
374 | (113) |
|
The Steiglitz-McBride Methodology |
|
|
376 | (2) |
|
Off-Line Direct-Form Algorithm |
|
|
378 | (8) |
|
Stationary Points of the Steiglitz-McBride Iteration |
|
|
386 | (11) |
|
Influence of the Disturbance Term |
|
|
397 | (5) |
|
Interpolation Constraints for the White Noise Input Case |
|
|
402 | (3) |
|
Adaptive Filtering Algorithm: Direct Form |
|
|
405 | (13) |
|
ODE for the Direct Form Algorithm |
|
|
410 | (2) |
|
Convergence in the Sufficient-Order Case |
|
|
412 | (6) |
|
A Lattice Version of the Steiglitz-McBride Iteration |
|
|
418 | (7) |
|
Stationary Points of the Lattice Steiglit-McBride Iteration |
|
|
425 | (17) |
|
Equivalence with Direct Form for General Inputs |
|
|
433 | (3) |
|
Equivalence for White Noise Input Case |
|
|
436 | (6) |
|
An A Priori Error Bound for White Noise Inputs |
|
|
442 | (12) |
|
Eigenvalue Bound for Disturbance-Induced Term |
|
|
447 | (2) |
|
Eigenvalue Bound for the Signal-Induced Term |
|
|
449 | (5) |
|
On-Line Lattice Algorithm |
|
|
454 | (13) |
|
Associated Differential Equation |
|
|
460 | (7) |
|
|
467 | (2) |
|
|
469 | (18) |
|
|
473 | (11) |
|
|
484 | (3) |
|
|
487 | (67) |
|
|
489 | (11) |
|
|
490 | (3) |
|
Passive Impedance Functions |
|
|
493 | (2) |
|
|
495 | (2) |
|
Proof of Hyperstability Theorem |
|
|
497 | (3) |
|
Hyperstability and Adaptive Filtering |
|
|
500 | (7) |
|
A Simplified Hyperstable Algorithm |
|
|
507 | (3) |
|
The Associated Differential Equation |
|
|
510 | (4) |
|
A Lattice Version of SHARF |
|
|
514 | (11) |
|
Relaxation of the SPR Condition |
|
|
525 | (3) |
|
|
528 | (26) |
|
Stationary Points for General Inputs |
|
|
531 | (3) |
|
|
534 | (11) |
|
|
545 | (6) |
|
|
551 | (3) |
|
|
554 | (46) |
|
|
554 | (2) |
|
|
556 | (6) |
|
Notch Filter Approximations |
|
|
562 | (10) |
|
|
562 | (4) |
|
|
566 | (6) |
|
Gradient Discent Algorithms |
|
|
572 | (2) |
|
A Simplified Lattice Algorithm |
|
|
574 | (12) |
|
Pseudo Least-Squares Algorithms |
|
|
586 | (2) |
|
|
588 | (12) |
|
Gradient Descent Algorithms |
|
|
589 | (2) |
|
Simplified Lattice Algorithm |
|
|
591 | (5) |
|
|
596 | (2) |
|
|
598 | (2) |
|
Perspectives and Open Problems |
|
|
600 | (50) |
|
Convergence in the Undermodelled Case |
|
|
602 | (4) |
|
|
606 | (6) |
|
Spectrally Weighted L2 Criterion |
|
|
612 | (4) |
|
Spectrally Weighted Balanced Systems |
|
|
616 | (5) |
|
|
621 | (12) |
|
Hankel-Toeplitz Equations |
|
|
621 | (3) |
|
Data-Driven Interpretation |
|
|
624 | (9) |
|
Spectral Extensions of the Shift Operator |
|
|
633 | (17) |
|
Spectrally Weighted Shift Operator |
|
|
634 | (8) |
|
Prefiltered Signal Interpretation |
|
|
642 | (6) |
|
|
648 | (2) |
Appendix A: Computations with Lattice Filters |
|
650 | (16) |
Appendix B: List of Notations |
|
666 | (9) |
Index |
|
675 | |