Table of Contents

• Preface ... x
• Conventions and Notation ... xiv
• Data and Software ... xxii
  
  
• Chapter 1: Introduction to Spectral Analysis ... 1
  • 1.0 Introduction ... 1
  • 1.1 Some Aspects of Time Series Analysis ... 1
  • 1.2 Spectral Analysis for a Simple Time Series Model ... 9
  • 1.3 Nonparametric Estimation of the Spectrum from Data ... 17
  • 1.4 Parametric Estimation of the Spectrum from Data ... 22
  • 1.5 Uses of Spectral Analysis ... 23
  • 1.6 Exercises ... 25
  
  
• Chapter 2: Stationary Stochastic Processes ... 30
  • 2.0 Introduction ... 30
  • 2.1 Stochastic Processes ... 30
  • 2.2 Notation ... 32
  • 2.3 Basic Theory for Stochastic Processes ... 33
  • 2.4 Real-Valued Stationary Processes ... 35
  • 2.5 Complex-Valued Stationary Processes ... 39
  • 2.6 Examples of Discrete Parameter Stationary Processes ... 41
  • 2.7 Comments on Continuous Parameter Processes ... 49}
  • 2.8 Use of Stationary Processes as Models for Data ... 49
  • 2.9 Exercises ... 53
  
  
• Chapter 3: Deterministic Spectral Analysis ... 56
  • 3.0 Introduction ... 56
  • 3.1 Fourier Theory -- Continuous Time/Discrete Frequency ... 57
  • 3.2 Fourier Theory -- Continuous Time and Frequency ... 64
  • 3.3 Band-Limited and Time-Limited Functions ... 69
  • 3.4 Continuous/Continuous Reciprocity Relationships ... 70
  • 3.5 Concentration Problem -- Continuous/Continuous Case ... 75
  • 3.6 Convolution Theorem -- Continuous Time and Frequency ... 81
  • 3.7 Fourier Theory -- Discrete Time/Continuous Frequency ... 87
  • 3.8 Aliasing Problem -- Discrete Time/Continuous Frequency ... 97
  • 3.9 Concentration Problem -- Discrete/Continuous Case ... 101
  • 3.10 Fourier Theory -- Discrete Time and Frequency ... 108
  • 3.11 Summary of Fourier Theory ... 116
  • 3.12 Exercises ... 120
  
  
• Chapter 4: Foundations for Stochastic Spectral Analysis ... 126
  • 4.0 Introduction ... 126
  • 4.1 Spectral Representation of Stationary Processes ... 127
  • 4.2 Alternative Definitions for the Spectral Density Function ... 136
  • 4.3 Basic Properties of the Spectrum ... 138
  • 4.4 Classification of Spectra ... 141
  • 4.5 Sampling and Aliasing ... 144
  • 4.6 Comparison of SDFs and ACVSs as Characterizations ... 147
  • 4.7 Exercises ... 149
  
  
• Chapter 5: Linear Time-Invariant Filters ... 153
  • 5.0 Introduction ... 153
  • 5.1 Basic Theory of LTI Analog Filters ... 154
  • 5.2 Basic Theory of LTI Digital Filters ... 164
  • 5.3 Convolution as an LTI filter ... 165
  • 5.4 Determination of SDFs by LTI Digital Filtering ... 167
  • 5.5 Some Filter Terminology ... 169
  • 5.6 Interpretation of Spectrum via Band-Pass Filtering ... 170
  • 5.7 An Example of LTI Digital Filtering ... 171
  • 5.8 Least Squares Filter Design ... 176
  • 5.9 Use of DPSSs in Low-Pass Filter Design ... 180
  • 5.10 Exercises ... 183
  
  
• Chapter 6: Nonparametric Spectral Estimation ... 187
  • 6.0 Introduction ... 187
  • 6.1 Estimation of the Mean ... 188
  • 6.2 Estimation of the Autocovariance Sequence ... 190
  • 6.3 A Naive Spectral Estimator -- the Periodogram ... 196
  • 6.4 Bias Reduction -- Tapering ... 206
  • 6.5 Bias Reduction -- Prewhitening ... 218
  • 6.6 Statistical Properties of Direct Spectral Estimators ... 220
  • 6.7 Smoothing Direct Spectral Estimators ... 235
  • 6.8 First Moment Properties of Lag Window Estimators ... 243
  • 6.9 Second Moment Properties of Lag Window Estimators ... 246
  • 6.10 Asymptotic Distribution of Lag Window Estimators ... 254
  • 6.11 Examples of Lag Windows ... 258
  • 6.12 Choice of Lag Window ... 271
  • 6.13 Choice of Lag Window Parameter ... 274
  • 6.14 Estimation of Spectral Bandwidth ... 277
  • 6.15 Automatic Smoothing of Log Spectral Estimators ... 280
  • 6.16 Computational Details ... 284
  • 6.17 Welch's Overlapped Segment Averaging ... 289
  • 6.18 Examples of Nonparametric Spectral Analysis ... 295
  • 6.19 Comments on Complex-Valued Time Series ... 317
  • 6.20 Summary of Nonparametric Spectral Estimation ... 318
  • 6.21 Exercises ... 322
  
  
• Chapter 7: Multitaper Spectral Estimation ... 331
  • 7.0 Introduction ... 331
  • 7.1 A Motivating Example ... 333
  • 7.2 Multitapering of Gaussian White Noise ... 347
  • 7.3 Quadratic Spectral Estimators and Multitapering ... 351
  • 7.4 Regularization and Multitapering ... 361
  • 7.5 Multitaper Spectral Analysis of Ocean Wave Data ... 371
  • 7.6 Summary of Multitaper Spectral Estimation ... 374
  • 7.7 Exercises ... 375
  
  
• Chapter 8: Calculation of Discrete Prolate Spheroidal Sequences ... 378
  • 8.0 Introduction ... 378
  • 8.1 Calculating DPSSs from Defining Equation ... 379
  • 8.2 Calculating DPSSs from Numerical Integration ... 384
  • 8.3 Tridiagonal Formulation ... 386
  • 8.4 Substitutes for the DPSSs ... 387
  • 8.5 Exercises ... 390
  
  
• Chapter 9: Parametric Spectral Estimation ... 391
  • 9.0 Introduction ... 391
  • 9.1 Notation ... 391
  • 9.2 The Autoregressive Model ... 392
  • 9.3 The Yule--Walker Equations ... 393
  • 9.4 The Levinson--Durbin Recursions ... 397
  • 9.5 Burg's Algorithm ... 414
  • 9.6 The Maximum Entropy Argument ... 420
  • 9.7 Least Squares Estimation ... 426
  • 9.8 Maximum Likelihood Estimation ... 429
  • 9.9 Order Selection for AR(p) Processes ... 434
  • 9.10 Prewhitened Spectral Estimators and An Example ... 437
  • 9.11 Use of Other Models for Parametric SDF Estimation ... 442
  • 9.12 Confidence Intervals Using AR Spectral Estimators ... 445
  • 9.13 Comments on Complex-Valued Time Series ... 450
  • 9.14 Summary of Parametric Spectral Analysis ... 451
  • 9.15 Exercises ... 454
  
  
• Chapter 10: Harmonic Analysis ... 456
  • 10.0 Introduction ... 456
  • 10.1 Harmonic Processes with Purely Discrete Spectra ... 457
  • 10.2 Harmonic Processes with Discrete Spectra ... 460
  • 10.3 Spectral Representation of Discrete and Mixed Spectra ... 467
  • 10.4 An Example from Tidal Analysis ... 468
  • 10.5 A Special Case of Unknown Frequencies ... 473
  • 10.6 General Case of Unknown Frequencies ... 475
  • 10.7 An Artificial Example from Kay and Marple ... 480
  • 10.8 Tapering and the Identification of Frequencies ... 486
  • 10.9 Tests for Periodicity -- White Noise Case ... 489
  • 10.10 Periodicity Tests with Ocean Noise Data ... 495
  • 10.11 Tests for Periodicity -- Colored Noise Case ... 496
  • 10.12 Completing a Harmonic Analysis ... 501
  • 10.13 Harmonic Analysis of River Flow Data ... 504
  • 10.14 A Parametric Approach to Harmonic Analysis ... 514
  • 10.15 Parametric Approach with River Flow Data ... 520
  • 10.16 Problems with the Parametric Approach ... 525
  • 10.17 Singular Value Decomposition Approach ... 531
  • 10.18 SVD Approach with River Flow Data ... 537
  • 10.19 Summary of Harmonic Analysis ... 539
  • 10.20 Exercises ... 544
  
  
• References ... 546
• Author Index ... 562
• Subject Index ... 566