Time Series with Mixed Spectra

Time Series With Mixed Spectra......Page 1
Dedication......Page 3
Contents......Page 4
Preface......Page 8
1.1 Periodicity and Sinusoidal Functions......Page 10
1.2 Sampling and Aliasing......Page 12
1.3 Time Series with Mixed Spectra......Page 13
1.4 Complex Time Series with Mixed Spectra......Page 16
2.1 Parameterization of Sinusoids......Page 21
2.2 Spectral Analysis of Stationary Processes......Page 26
2.3 Gaussian Processes and White Noise......Page 31
2.4 Linear Prediction Theory......Page 36
2.5 Asymptotic Statistical Theory......Page 39
3.1 Cramér- Rao Inequality......Page 44
3.2 CRLB for Sinusoids in Gaussian Noise......Page 47
3.3 Asymptotic CRLB for Sinusoids in Gaussian Noise......Page 54
3.4 CRLB for Sinusoids in NonGaussian White Noise......Page 65
3.5 Proof of Theorems......Page 71
4.1 Autocovariances and Autocorrelation Coefficients......Page 82
4.2 Consistency and Asymptotic Unbiasedness......Page 84
4.3 Covariances and Asymptotic Normality......Page 87
4.4 Autocovariances of Filtered Time Series......Page 97
4.5 Proof of Theorems......Page 98
5.1 Least Squares Estimation......Page 118
5.2 Sensitivity to Frequency Offset......Page 127
5.3 Frequency Identification......Page 129
5.4 Frequency Selection......Page 133
5.5 Least Absolute Deviations Estimation......Page 143
5.6 Proof of Theorems......Page 154
6.1 Periodogram Analysis......Page 173
6.2 Detection of Hidden Sinusoids......Page 190
6.3.1 Sinusoids with NonFourier Frequencies......Page 200
6.3.2 Refined Periodogram......Page 203
6.3.3 Secondary Analysis......Page 207
6.3.4 Interpolation Estimators......Page 210
6.4.1 Statistical Properties......Page 214
6.4.2 Periodogram Maximization......Page 216
6.4.3 Resolution Limit......Page 229
6.5 Time- Frequency Analysis......Page 231
6.6 Proof of Theorems......Page 235
7.1 Estimation in the Absence of Sinusoids......Page 259
7.1.1 Periodogram Smoother......Page 260
7.1.2 Lag- Window Spectral Estimator......Page 264
7.1.3 Autoregressive Spectral Estimator......Page 267
7.1.4 Numerical Examples......Page 277
7.2 Estimation in the Presence of Sinusoids......Page 282
7.2.1 Modified Periodogram Smoother......Page 283
7.2.2 M Spectral Estimator......Page 285
7.2.3 Modified Autoregressive Spectral Estimator......Page 289
7.2.4 A Comparative Example......Page 293
7.3 Detection of Hidden Sinusoids in Colored Noise......Page 296
7.4 Proof of Theorems......Page 302
8.1 Maximum Likelihood Estimation......Page 317
8.2 Maximum Likelihood under Gaussian White Noise......Page 319
8.2.1 Multivariate Periodogram......Page 320
8.2.2 Statistical Properties......Page 328
8.3 Maximum Likelihood under Laplace White Noise......Page 337
8.3.1 Multivariate Laplace Periodogram......Page 338
8.3.2 Statistical Properties......Page 343
8.4 The Case of Gaussian Colored Noise......Page 347
8.5 Determining the Number of Sinusoids......Page 357
8.6 Proof of Theorems......Page 361
9.1 Linear Prediction Method......Page 381
9.1.1 Linear Prediction Estimators......Page 382
9.1.2 Statistical Properties......Page 391
9.2 Autoregressive Reparameterization......Page 398
9.3 Extended Yule- Walker Method......Page 402
9.3.1 Extended Yule- Walker Estimators......Page 403
9.3.2 Statistical Properties......Page 405
9.4 Iterative Filtering Method......Page 410
9.4.1 Iterative Filtering Estimator: Complex Case......Page 411
9.4.2 Iterative Filtering Estimator: Real Case......Page 419
9.5 Iterative Quasi Gaussian Maximum Likelihood Method......Page 430
9.5.1 Iterative Generalized Least- Squares Algorithm......Page 432
9.5.2 Iterative Least- Eigenvalue Algorithm......Page 442
9.5.3 Self Initialization......Page 448
9.6 Proof of Theorems......Page 450
10.1 Eigenvalue Decomposition of Covariance Matrix......Page 460
10.2 Principal Component Analysis Method......Page 463
10.2.1 Reduced Rank Autoregressive Estimators......Page 464
10.2.2 Statistical Properties......Page 477
10.3.1 MUSIC and Minimum- Norm Estimators......Page 487
10.3.2 Statistical Properties......Page 494
10.4.1 Matrix- Pencil and ESPRIT Estimators......Page 497
10.4.2 Statistical Properties......Page 505
10.5 Estimating the Number of Sinusoids......Page 508
10.6 Sensitivity to Colored Noise......Page 515
10.7 Proof of Theorems......Page 518
11.1 Single Complex Sinusoid......Page 533
11.2 Tracking Time- Varying Frequencies......Page 538
11.3 Periodic Functions in Noise......Page 544
11.4 Beyond Single Time Series......Page 549
11.5 Quantile Periodogram......Page 556
12.1 Trigonometric Series......Page 571
12.2 Probability Theory......Page 577
12.3 Numerical Analysis......Page 579
12.4 Matrix Theory......Page 581
12.5 Asymptotic Theory......Page 589
Bibliography......Page 615
Index......Page 641