Optimal Filtering

This graduate-level text augments and extends studies of signal processing, particularly in regard to communication systems and digital filtering theory. Topics include filtering, linear systems, and estimation; the discrete-time Kalman filter; time-invariant filters; properties of Kalman filters; computational aspects; smoothing of discrete-time signals; and more. 24 figures. 1979 edition.

Preface

ix

Introduction

1

(8)

Filtering

1

(1)

History of Signal Filtering

2

(2)

Subject Matter of this Book

4

(2)

Outline of the Book

6

(3)

References

7

(2)

Filtering, Linear Systems, and Estimation

9

(27)

Systems, Noise, Filtering, Smoothing, and Prediction

9

(3)

The Gauss-Markov Discrete-time Model

12

(11)

Estimation Criteria

23

(13)

References

34

(2)

The Discrete-Time Kalman Filter

36

(26)

The Kalman Filter

36

(10)

Best Linear Estimator Property of the Kalman Filter

46

(4)

Identification as a Kalman Filtering Problem

50

(3)

Application of Kalman Filters

53

(9)

References

59

(3)

Time-Invariant Filters

62

(28)

Background to Time Invariance of the Filter

62

(1)

Stability Properties of Linear, Discrete-time Systems

63

(5)

Stationary Behaviour of Linear Systems

68

(8)

Time Invariance and Asymptotic Stability of the Filter

76

(9)

Frequency Domain Formulas

85

(5)

References

88

(2)

Kalman Filter Properties

90

(39)

Introduction

90

(2)

Minimum Variance and Linear Minimum Variance Estimation; Orthogonality and Projection

92

(8)

The Innovations Sequence

100

(5)

The Kalman Kilter

105

(10)

True Filtered Estimates and the Signal-to-Noise Ratio Improvement Property

115

(7)

Inverse Problems: When is a Filter Optimal?

122

(7)

References

127

(2)

Computational Aspects

129

(36)

Signal Model Errors, Filter Divergence, and Data Saturation

129

(6)

Exponential Data Weighting---A Filter with Prescribed Degree of Stability

135

(3)

The Matrix Inversion Lemma and the Information Filter

138

(4)

Sequential Processing

142

(5)

Square Root Filtering

147

(6)

The High Measurement Noise Case

153

(2)

Chandrasekhar-Type, Doubling, and Nonrecursive Algorithms

155

(10)

References

162

(3)

Smoothing of Discrete-Time Signals

165

(28)

Introduction to Smoothing

165

(5)

Fixed-point Smoothing

170

(6)

Fixed-lag Smoothing

176

(11)

Fixed-interval Smoothing

187

(6)

References

190

(3)

Applications in Nonlinear Filtering

193

(30)

Nonlinear Filtering

193

(2)

The Extended Kalman Filter

195

(10)

A Bound Optimal Filter

205

(6)

Gaussian Sum Estimators

211

(12)

References

221

(2)

Innovations Representations, Spectral Factorization, Wiener and Levinson Filtering

223

(44)

Introduction

223

(4)

Kalman Filter Design from Covariance Data

227

(3)

Innovations Representations with Finite Initial Time

230

(8)

Stationary Innovations Representations and Spectral Factorization

238

(16)

Wiener Filtering

254

(4)

Levinson Filters

258

(9)

References

264

(3)

Parameter Identification and Adaptive Estimation

267

(21)

Adaptive Estimation via Parallel Processing

267

(12)

Adaptive Estimation via Extended Least Squares

279

(9)

References

286

(2)

Colored Noise and Suboptimal Reduced Order Filters

288

(63)

General Approaches to Dealing with Colored Noise

288

(2)

Filter Design with Markov Output Noise

290

(2)

Filter Design with Singular or Near-singular Output Noise

292

(4)

Suboptimal Design Given Colored Input or Measurement Noise

296

(5)

Suboptimal Filter Design by Model Order Reduction

301

(6)

References

304

(3)

APPENDIXES

A Brief Review of Results of Probability Theory

307

(17)

A.1 Pure Probability Theory

308

(8)

A.2 Stochastic Processes

316

(4)

A.3 Gaussian Random Variables, Vectors, and Processes

320

(3)

References

323

(1)

B Brief Review of Some Results of Matrix Theory

324

(16)

References

339

(1)

C Brief Review of Several Major Results of Linear System Theory

340

(7)

References

346

(1)

D Lyapunov Stability

347

(4)

References

349

(2)

Author Index

351

(3)

Subject Index

354

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