Discrete Stochastic Processes and Optimal Filtering

Table of Contents

Preface

Introduction

Chapter 1. Random Vectors

1.1. Definitions and general properties

1.2. Spaces L¹(dP) and L²(dP)

1.2.1. Definitions

1.2.2. Properties

1.3. Mathematical expectation and applications

1.3.1. Definitions

1.3.2. Characteristic functions of a random vector

1.4. Second order random variables and vectors

1.5. Linear independence of vectors of L²(dP)

1.6. Conditional expectation (concerning random vectors with density function)

1.7. Exercises for Chapter 1

Chapter 2. Gaussian Vectors

2.1. Some reminders regarding random Gaussian vectors

2.2. Definition and characterization of Gaussian vectors

2.3. Results relative to independence

2.4. Affine transformation of a Gaussian vector

2.5. The existence of Gaussian vectors

2.6. Exercises for Chapter 2

Chapter 3. Introduction to Discrete Time Processes

3.1. Definition

3.2. WSS processes and spectral measure

3.2.1. Spectral density

3.3. Spectral representation of a WSS process

3.3.1. Problem

3.3.2. Results

3.3.2.1. Process with orthogonal increments and associated measurements

3.3.2.2. Wiener stochastic integral

3.3.2.3. Spectral representation

3.4. Introduction to digital filtering

3.5. Important example: autoregressive process

3.6. Exercises for Chapter 3

Chapter 4. Estimation

4.1. Position of the problem

4.2. Linear estimation

4.3. Best estimate – conditional expectation

4.4. Example: prediction of an autoregressive process AR (1)

4.5. Multivariate processes

4.6. Exercises for Chapter 4

Chapter 5. The Wiener ...