Time Series with Mixed Spectra

Introduction
Periodicity and Sinusoidal Functions
Sampling and Aliasing
Time Series with Mixed Spectra
Complex Time Series with Mixed Spectra

Basic Concepts
Parameterization of Sinusoids
Spectral Analysis of Stationary Processes
Gaussian Processes and White Noise
Linear Prediction Theory .
Asymptotic Statistical Theory

Cramér-Rao Lower Bound
Cramér-Rao Inequality
CRLB for Sinusoids in Gaussian Noise
Asymptotic CRLB for Sinusoids in Gaussian Noise
CRLB for Sinusoids in NonGaussian White Noise

Autocovariance Function
Autocovariances and Autocorrelation Coefficients
Consistency and Asymptotic Unbiasedness
Covariances and Asymptotic Normality
Autocovariances of Filtered Time Series

Linear Regression Analysis
Least Squares Estimation
Sensitivity to Frequency Offset
Frequency Identification
Frequency Selection
Least Absolute Deviations Estimation

Fourier Analysis Approach
Periodogram Analysis
Detection of Hidden Sinusoids
Extension of the Periodogram
Continuous Periodogram
Time-Frequency Analysis

Estimation of Noise Spectrum
Estimation in the Absence of Sinusoids
Estimation in the Presence of Sinusoids
Detection of Hidden Sinusoids in Colored Noise

Maximum Likelihood Approach
Maximum Likelihood Estimation
Maximum Likelihood under Gaussian White Noise
The Case of Laplace White Noise
The Case of Gaussian Colored Noise
Determining the Number of Sinusoids

Autoregressive Approach
Linear Prediction Method
Autoregressive Reparameterization
Extended Yule-Walker Method
Iterative Filtering Method
Iterative Quasi Gaussian Maximum Likelihood Method

Covariance Analysis Approach
Eigenvalue Decomposition of Covariance Matrix
Principal Component Analysis Method
Subspace Projection Method
Subspace Rotation Method
Estimating the Number of Sinusoids
Sensitivity to Colored Noise

Further Topics
Single Complex Sinusoid
Tracking Time-Varying Frequencies
Periodic Functions in Noise
Beyond Single Time Series
Quantile Periodogram

Appendix
Trigonometric Series
Probability Theory
Numerical Analysis
Matrix Theory
Asymptotic Theory

Bibliography