Discrete Stochastic Processes and Optimal Filtering [Hardcover]

Optimal filtering applied to stationary and non-stationary signals provides the most efficient means of dealing with problems arising from the extraction of noise signals. Moreover, it is a fundamental feature in a range of applications, such as in navigation in aerospace and aeronautics, filter processing in the telecommunications industry, etc. This book provides a comprehensive overview of this area, discussing random and Gaussian vectors, outlining the results necessary for the creation of Wiener and adaptive filters used for stationary signals, as well as examining Kalman filters which are used in relation to non-stationary signals. Exercises with solutions feature in each chapter to demonstrate the practical application of these ideas using Matlab.

Preface xi

Introduction xiii

Chapter 1. Random Vectors 1

1.1. Definitions and general properties 1

1.2. Spaces L1(dP) and L2(dP) 20

1.2.1. Definitions 20

1.2.2. Properties 22

1.3. Mathematical expectation and applications 23

1.3.1. Definitions 23

1.3.2. Characteristic functions of a random vector 34

1.4. Second order random variables and vectors 39

1.5. Linear independence of vectors of L2(dP) 47

1.6. Conditional expectation (concerning random vectors with density function) 51

1.7. Exercises for Chapter 1 57

Chapter 2. Gaussian Vectors 63

2.1. Some reminders regarding random Gaussian vectors 63

2.2. Definition and characterization of Gaussian vectors 66

2.3. Results relative to independence 68

2.4. Affine transformation of a Gaussian vector 72

2.5. The existence of Gaussian vectol#P