Spectral Analysis

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.

• Preface 9Specific Notations 13PART I. Tools and Spectral Analysis 15Chapter 1. Fundamentals 17Francis CASTANIÉ1.1. Classes of signals 171.1.1. Deterministic signals 171.1.2. Random signals 201.2. Representations of signals 231.2.1. Representations of deterministic signals 231.2.1.1. Complete representations 231.2.1.2. Partial representations 251.2.2. Representations of random signals 271.2.2.1. General approach 271.2.2.2. 2nd order representations 281.2.2.3. Higher order representations 321.3. Spectral analysis: position of the problem 331.4. Bibliography 35Chapter 2. Digital Signal Processing 37Éric LE CARPENTIER2.1. Introduction 372.2. Transform properties 382.2.1. Some useful functions and series 382.2.2. Fourier transform 432.2.3. Fundamental properties 472.2.4. Convolution sum 482.2.5. Energy conservation (Parseval’s theorem) 502.2.6. Other properties 512.2.7. Examples 532.2.8. Sampling 552.2.9. Practical calculation, FFT 592.3. Windows 622.4. Examples of application 712.4.1. LTI systems identification 712.4.2. Monitoring spectral lines 752.4.3. Spectral analysis of the coefficient of tide fluctuation 762.5. Bibliography 78Chapter 3. Estimation in Spectral Analysis 79Olivier BESSON and André FERRARI3.1. Introduction to estimation 793.1.1. Formalization of the problem 793.1.2. Cramér-Rao bounds 813.1.3. Sequence of estimators 863.1.4. Maximum likelihood estimation 893.2. Estimation of 1st and 2nd order moments 923.3. Periodogram analysis 973.4. Analysis of estimators based on cˆxx m?n1013.4.1. Estimation of parameters of an AR model 1033.4.2. Estimation of a noisy cisoid by MUSIC 1063.5. Conclusion 1083.6. Bibliography 108Chapter 4. Time-Series Models 111Francis CASTANIÉ4.1. Introduction 1114.2. Linear models 1134.2.1. Stationary linear models 1134.2.2. Properties 1164.2.2.1. Stationarity 1164.2.2.2. Moments and spectra 1174.2.2.3. Relation with Wold’s decomposition 1194.2.3. Non-stationary linear models 1204.3. Exponential models 1234.3.1. Deterministic model 1234.3.2. Noisy deterministic model 1244.3.3. Models of random stationary signals 1254.4. Non-linear models 1264.5. Bibliography 126PART II. Non-Parametric Methods 129Chapter 5. Non-Parametric Methods 131Éric LE CARPENTIER5.1. Introduction 1315.2. Estimation of the power spectral density 1365.2.1. Filter bank method 1365.2.2. Periodogram method 1395.2.3. Periodogram variants 1425.3. Generalization to higher order spectra 1465.4. Bibliography 148PART III. Parametric Methods 149Chapter 6. Spectral Analysis by Stationary Time Series Modeling 151Corinne MAILHES and Francis CASTANIÉ6.1. Parametric models 1516.2. Estimation of model parameters 1536.2.1. Estimation of AR parameters 1536.2.2. Estimation of ARMA parameters 1606.2.3. Estimation of Prony parameters 1616.2.4. Order selection criteria 1646.3. Properties of spectral estimators produced 1676.4. Bibliography 172Chapter 7. Minimum Variance 175Nadine MARTIN7.1. Principle of the MV method 1797.2. Properties of the MV estimator 1827.2.1. Expressions of the MV filter 1827.2.2. Probability density of the MV estimator 1867.2.3. Frequency resolution of the MV estimator 1927.3. Link with the Fourier estimators 1937.4. Link with a maximum likelihood estimator 1967.5. Lagunas methods: normalized and generalized MV 1987.5.1. Principle of normalized MV 1987.5.2. Spectral refinement of the NMV estimator 2007.5.3. Convergence of the NMV estimator 2027.5.4. Generalized MV estimator 2047.6. The CAPNORM estimator 2067.7. Bibliography 209Chapter 8. Subspace-based Estimators 213Sylvie MARCOS8.1. Model, concept of subspace, definition of high resolution 2138.1.1. Model of signals 2138.1.2. Concept of subspaces 2148.1.3. Definition of high-resolution 2168.1.4. Link with spatial analysis or array processing 2178.2. MUSIC 2178.2.1. Pseudo-spectral version of MUSIC 2208.2.2. Polynomial version of MUSIC 2218.3. Determination criteria of the number of complex sine waves 2238.4. The MinNorm method 2248.5. “Linear” subspace methods 2268.5.1. The linear methods 2268.5.2. The propagator method 2268.5.2.1. Propagator estimation using least squares technique 2288.5.2.2. Determination of the propagator in the presence of a white noise 2298.6. The ESPRIT method 2328.7. Illustration of subspace-based methods performance 2358.8. Adaptive research of subspaces 2368.9. Bibliography 242Chapter 9. Introduction to Spectral Analysis of Non-Stationary Random Signals 245Corinne MAILHES and Francis CASTANIÉ9.1. Evolutive spectra 2469.1.1. Definition of the “evolutive spectrum” 2469.1.2. Evolutive spectrum properties 2479.2. Non-parametric spectral estimation 2489.3. Parametric spectral estimation 2499.3.1. Local stationary postulate 2509.3.2. Elimination of a stationary condition 2519.3.3. Application to spectral analysis 2549.4. Bibliography 255List of Authors 259Index 261