LECTURES IN TIME SERIES AND FILTERING
0. THE METHODS OF TIME-SERIES ANALYSIS
1. THE ALGEBRA FOR TIME-SERIES ANALYSIS
Rational Functions and the z-transform
The Expansion of a Rational Function
Representations via Toeplitz Matrices
Representations via Circulant Matrices
The Spectral Factorisation of a Circulant Matrix
Linear and Circular Convolutions
2. PROPERTIES OF TRANSFER FUNCTIONS
The Impulse Response function
Stability of a Transfer Function
Response to a Sinusoidal Input
Spectral Representation of a Stationary Stochastic Process
The Frequency Response
3. THE DISCRETE FOURIER TRANSFORM
Complex Numbers
Trigonometrical Identities
Trigonometrical Orthogonality Conditions
The Fourier Decomposition of a Time Series
The Periodogram and the Spectral Analysis of Variance
The Periodogram and the Empirical Autocovariances
Complex Exponential Forms
4. ALTERNATIVE REPRESENTATIONS OF THE DFT
The Roots of Unity
Circulant Matrices and the Discrete Fourier Transform
The Matrix Discrete Fourier Transform
Circular Autocovariances
5. THE CLASSES OF FOURIER TRANSFORMS
The Discrete Fourier Transform
The Classical Fourier Series
The Discrete-Time Fourier Transform
The Fourier Integral Transform
Sampling and Sinc Function Interpolation
6. STATIONARY LINEAR STOCHASTIC MODELS
Autoregressive Moving-average Processes
Autocovariances of a Moving-Average Process
Autocovariance Generating Function
The Spectral Density Function
The White-Noise Spectrum
7. WIENER-KOLMOGOROV FILTER THEORY
The Classical Wiener--Kolmogorov Theory
The Butterworth Filter
The Hordick--Prescott (Leser) Filter
8. FILTERING SHORT NONSTSATIONARY SEQUENCES
Wiener-Kolmogorov Filtering of Short Stationary Sequences
Filtering via Fourier Methods
Dealing with Trended Data
Recovering the Trend Component
Wiener--Kolmogorov Estimates from Trended Data