ece531

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

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 241

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online