Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

Contents
1 Introduction 1
1.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Motivation and Goals of the Thesis . . . . . . . . . . . . . . . . . . 5
1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Linear Systems and Prediction 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Linear Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Linear Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Forward and backward prediction . . . . . . . . . . . . . . . 12
2.3.2 Estimating AR coefficients via Yule-Walker Equations . . . . 12
2.3.3 Estimating AR coefficients via Burg’s algorithm . . . . . . . 14
2.3.4 Computing the AR coefficients without limiting the aperture 16
2.4 1-D Synthetic and Real Data Examples . . . . . . . . . . . . . . . . 19
2.5 Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Nonlinear Prediction 26
3.1 Nonlinear Processes via the Volterra Series . . . . . . . . . . . . . . 26
3.1.1 Time domain representation . . . . . . . . . . . . . . . . . . 27
3.1.2 Frequency domain representation of Volterra kernels . . . . . 28
3.1.3 Symmetry property of Volterra kernels . . . . . . . . . . . . 30
3.2 Nonlinear Modeling of Time Series . . . . . . . . . . . . . . . . . . 32
3.3 1-D Synthetic and Real Data Examples . . . . . . . . . . . . . . . . 40
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