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ece531 - Statistical Signal Processing Don H Johnson Rice...

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Unformatted text preview: Statistical Signal Processing Don H. Johnson Rice University c circlecopyrt 2010 Contents 1 Introduction 1 2 Probability and Stochastic Processes 3 2.1 Foundations of Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.2 Random Variables and Probability Density Functions . . . . . . . . . . . . . . . . . 4 2.1.3 Function of a Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.4 Expected Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.5 Jointly Distributed Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.6 Random Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.7 Single function of a random vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.8 Several functions of a random vector . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.9 The Gaussian Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.10 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.2 The Gaussian Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.3 Sampling and Random Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.4 The Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Linear Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.2 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.3 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.4 Separable Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.5 The Vector Space L 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.6 A Hilbert Space for Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.7 Karhunen-Lo`eve Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 Optimization Theory 45 3.1 Unconstrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.1 Equality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.2 Inequality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
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