Spectral Analysis

This book deals with these parametric methods, first discussing those based on time series models, Capon s method and its variants, and then estimators based on the notions of sub-spaces. However, the book also deals with the traditional analog methods, now called non-parametric methods, which are still the most widely used in practical spectral analysis.

Auteur
Francis Castanié is the Director of the Research Laboratory Telecommunications for Space and Aeronautics (TeSA). He joined the CNRS Institut de Recherche en Informatique de Toulouse (IRIT) in 2002, where he heads the Signal and Communication Group.

Contenu

Preface 9

Specific Notations 13

PART I. Tools and Spectral Analysis 15

Chapter 1. Fundamentals 17
Francis CASTANIÉ

1.1. Classes of signals 17

1.1.1. Deterministic signals 17

1.1.2. Random signals 20

1.2. Representations of signals 23

1.2.1. Representations of deterministic signals 23

1.2.1.1. Complete representations 23

1.2.1.2. Partial representations 25

1.2.2. Representations of random signals 27

1.2.2.1. General approach 27

1.2.2.2. 2nd order representations 28

1.2.2.3. Higher order representations 32

1.3. Spectral analysis: position of the problem 33

1.4. Bibliography 35

Chapter 2. Digital Signal Processing 37
Éric LE CARPENTIER

2.1. Introduction 37

2.2. Transform properties 38

2.2.1. Some useful functions and series 38

2.2.2. Fourier transform 43

2.2.3. Fundamental properties 47

2.2.4. Convolution sum 48

2.2.5. Energy conservation (Parseval's theorem) 50

2.2.6. Other properties 51

2.2.7. Examples 53

2.2.8. Sampling 55

2.2.9. Practical calculation, FFT 59

2.3. Windows 62

2.4. Examples of application 71

2.4.1. LTI systems identification 71

2.4.2. Monitoring spectral lines 75

2.4.3. Spectral analysis of the coefficient of tide fluctuation 76

2.5. Bibliography 78

Chapter 3. Estimation in Spectral Analysis 79
Olivier BESSON and André FERRARI

3.1. Introduction to estimation 79

3.1.1. Formalization of the problem 79

3.1.2. Cramér-Rao bounds 81

3.1.3. Sequence of estimators 86

3.1.4. Maximum likelihood estimation 89

3.2. Estimation of 1st and 2nd order moments 92

3.3. Periodogram analysis 97

3.4. Analysis of estimators based on cxx 􀀋m􀀌?n101

3.4.1. Estimation of parameters of an AR model 103

3.4.2. Estimation of a noisy cisoid by MUSIC 106

3.5. Conclusion 108

3.6. Bibliography 108

Chapter 4. Time-Series Models 111
Francis CASTANIÉ

4.1. Introduction 111

4.2. Linear models 113

4.2.1. Stationary linear models 113

4.2.2. Properties 116

4.2.2.1. Stationarity 116

4.2.2.2. Moments and spectra 117

4.2.2.3. Relation with Wold's decomposition 119

4.2.3. Non-stationary linear models 120

4.3. Exponential models 123

4.3.1. Deterministic model 123

4.3.2. Noisy deterministic model 124

4.3.3. Models of random stationary signals 125

4.4. Non-linear models 126

4.5. Bibliography 126

PART II. Non-Parametric Methods 129

Chapter 5. Non-Parametric Methods 131
Éric LE CARPENTIER

5.1. Introduction 131

5.2. Estimation of the power spectral density 136

5.2.1. Filter bank method 136

5.2.2. Periodogram method 139

5.2.3. Periodogram variants 142

5.3. Generalization to higher order spectra 146

5.4. Bibliography 148

PART III. Parametric Methods 149

Chapter 6. Spectral Analysis by Stationary Time Series Modeling 151
Corinne MAILHES and Francis CASTANIÉ

6.1. Parametric models 151

6.2. Estimation of model parameters 153

6.2.1. Estimation of AR parameters 153

6.2.2. Estimation of ARMA parameters 160

6.2.3. Estimation of Prony parameters 161

6.2.4. Order selection criteria 164

6.3. Properties of spectral estimators produced 167

6.4. Bibliography 172 Chapter 7. Minimum Variance 175&lt...