Nonlinear Signal Processing: A Statistical Approach focuses on unifying the study of a broad and important class of nonlinear signal processing algorithms which emerge from statistical estimation principles, and where the underlying signals are non-Gaussian, rather than Gaussian, processes. Notably, by concentrating on just two non-Gaussian models, a large set of tools is developed that encompass a large portion of the nonlinear signal processing tools proposed in the literature over the past several decades. Key features include: * Numerous problems at the end of each chapter to aid development and understanding * Examples and case studies provided throughout the book in a wide range of applications bring the text to life and place the theory into context * A set of 60+ MATLAB software m-files allowing the reader to quickly design and apply any of the nonlinear signal processing algorithms described in the book to an application of interest is available on the accompanying FTP site.

GONZALO R. ARCE received a PhD degree in electrical engineering from Purdue University in 1982. Since 1982, he has been with the faculty of the Department of Electrical and Computer Engineering at the University of Delaware where he is currently Charles Black Evans Distinguished Professor and Chairman. He has held visiting professor appointments at the Unisys Corporate Research Center and at the International Center for Signal and Image Processing, Tampere University of Technology, in Tampere, Finland. He holds seven U.S. patents, and his research has been funded by DoD, NSF, and numerous industrial organizations. He is an IEEE Fellow for his contributions to the theory and applications of nonlinear signal processing.

Preface

vii

Acknowledgments

xi

Acronyms

xix

1 Introduction

1

(16)

1.1 NonGaussian Random Processes

7

(3)

1.1.1 Generalized Gaussian Distributions and Weighted Medians

9

(1)

1.1.2 Stable Distributions and Weighted Myriads

10

(1)

1.2 Statistical Foundations

10

(2)

1.3 The Filtering Problem

12

(5)

1.3.1 Moment Theory

13

(4)

Part I Statistical Foundations

2 NonGaussian Models

17

(26)

2.1 Generalized Gaussian Distributions

18

(1)

2.2 Stable Distributions

19

(11)

2.2.1 Definitions

22

(1)

2.2.2 Symmetric Stable Distributions

23

(5)

2.2.3 Generalized Central Limit Theorem

28

(1)

2.2.4 Simulation of Stable Sequences

29

(1)

2.3 Lower-Order Moments

30

(11)

2.3.1 Fractional Lower-Order Moments

30

(3)

2.3.2 Zero-Order Statistics

33

(3)

2.3.3 Parameter Estimation of Stable Distributions

36

(5)

Problems

41

(2)

3 Order Statistics

43

(18)

3.1 Distributions Of Order Statistics

44

(5)

3.2 Moments Of Order Statistics

49

(5)

3.2.1 Order Statistics From Uniform Distributions

50

(2)

3.2.2 Recurrence Relations

52

(2)

3.3 Order Statistics Containing Outliers

54

(2)

3.4 Joint Statistics Of Ordered And NonOrdered Samples

56

(2)

Problems

58

(3)

4 Statistical Foundations of Filtering

61

(20)

4.1 Properties of Estimators

62

(2)

4.2 Maximum Likelihood Estimation

64

(8)

4.3 Robust Estimation

72

(3)

Problems

75

(6)

Part II Signal Processing with Order Statistics

5 Median and Weighted Median Smoothers

81

(58)

5.1 Running Median Smoothers

81

(13)

5.1.1 Statistical Properties

83

(5)

5.1.2 Root Signals (Fixed Points)

88

(6)

5.2 Weighted Median Smoothers

94

(17)

5.2.1 The Center-Weighted Median Smoother

102

(5)

5.2.2 Permutation-Weighted Median Smoothers

107

(4)

5.3 Threshold Decomposition Representation

111

(13)

5.3.1 Stack Smoothers

114

(10)

5.4 Weighted Medians in Least Absolute Deviation (LAD) Regression

124

(12)

5.4.1 Foundation and Cost Functions

126

(5)

5.4.2 LAD Regression with Weighted Medians

131

(3)

5.4.3 Simulation

134

(2)

Problems

136

(3)

6 Weighted Median Filters

139

(112)

6.1 Weighted Median Filters With Real-Valued Weights

139

(17)

6.1.1 Permutation-Weighted Median Filters

154

(2)

6.2 Spectral Design of Weighted Median Filters

156

(13)

6.2.1 Median Smoothers and Sample Selection Probabilities

158

(1)

6.2.2 SSPs for Weighted Median Smoothers

159

(3)

6.2.3 Synthesis of WM Smoothers

162

(3)

6.2.4 General Iterative Solution

165

(2)

6.2.5 Spectral Design of Weighted Median Filters Admitting Real-Valued Weights

167

(2)

6.3 The Optimal Weighted Median Filtering Problem

169

(16)

6.3.1 Threshold Decomposition For Real-Valued Signals

170

(6)

6.3.2 The Least Mean Absolute (LMA) Algorithm

176

(9)

6.4 Recursive Weighted Median Filters

185

(17)

6.4.1 Threshold Decomposition Representation of Recursive WM Filters

188

(2)

6.4.2 Optimal Recursive Weighted Median Filtering

190

(12)

6.5 Mirrored Threshold Decomposition and Stack Filters

202

(8)

6.5.1 Stack Filters

203

(4)

6.5.2 Stack Filter Representation of Recursive WM Filters

207

(3)

6.6 Complex-Valued Weighted Median Filters

210

(21)

6.6.1 Phase-Coupled Complex WM Filter

214

(1)

6.6.2 Marginal Phase-Coupled Complex WM Filter

214

(1)

6.6.3 Complex threshold decomposition

215

(1)

6.6.4 Optimal Marginal Phase-Coupled Complex WM

216

(10)

6.6.5 Spectral Design of Complex-Valued Weighted Medians

226

(5)

6.7 Weighted Median Filters for Multichannel Signals

231

(18)

6.7.1 Marginal WM filter

232

(1)

6.7.2 Vector WM filter

233

(2)

6.7.3 Weighted Multichannel Median Filtering Structures

235

(3)

6.7.4 Filter Optimization

238

(11)

Problems

249

(2)

7 Linear Combination of Order Statistics

251

(52)

7.1 L-Estimates of Location

252

(6)

7.2 L-Smoothers

258

(4)

7.3 Ll-Filters

262

(8)

7.3.1 Design and Optimization of Ll-filters

265

(5)

7.4 Ljl Permutation Filters

270

(5)

7.5 Hybrid Median/Linear FIR Filters

275

(11)

7.5.1 Median and FIR Affinity Trimming

275

(11)

7.6 Linear Combination of Weighted Medians

286

(11)

7.6.1 LCWM Filters

289

(2)

7.6.2 Design of LCWM filters

291

(2)

7.6.3 Symmetric LCWM Filters

293

(4)

Problems

297

(6)

Part III Signal Processing with the Stable Model

8 Myriad Smoothers

303

(44)

8.1 FLOM Smoothers

304

(2)

8.2 Running Myriad Smoothers

306

(16)

8.3 Optimality of the Sample Myriad

322

(3)

8.4 Weighted Myriad Smoothers

325

(7)

8.5 Fast Weighted Myriad Computation

332

(4)

8.6 Weighted Myriad Smoother Design

336

(10)

8.6.1 Center-Weighted Myriads for Image Denoising

336

(2)

8.6.2 Myriadization

338

(8)

Problems

346

(1)

9 Weighted Myriad Filters

347

(18)

9.1 Weighted Myriad Filters With Real-Valued Weights

347

(3)

9.2 Fast Real-valued Weighted Myriad Computation

350

(1)

9.3 Weighted Myriad Filter Design

351

(11)

9.3.1 Myriadization

351

(2)

9.3.2 Optimization

353

(9)

Problems

362

(3)

References

365

(16)

Appendix A Software Guide

381

(74)

Index

455

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