Discrete Stochastic Processes and Optimal Filtering

Jean-Claude Bertein, Roger Ceschi

• Publisher: Wiley-ISTE
• Year: 2013
• Print ISBN: 9781905209743
• eBook ISBN: 9781118615492
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Electrical Engineering & Telecommunications
• Index: Technology & Engineering

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.

Chapter 1

Random Vectors

1.1. Definitions and general properties

If we remember that

ⁿ = {x = (x₁,…,x_n)| x_j

;j=1 to n}, the set of real n -tuples can be fitted to two laws:

and

making it a vector space of dimension n.

The basis implicitly considered on

ⁿ will be the canonical base ℓ₁ = (1, 0,…, 0),…, ℓ_n = (0,…, 0, 1) and x

ⁿ expressed in this base will be denoted:

Definition of a real random vector

Beginning with a basic definition, without concerning ourselves at the moment with its rigor: we can say simply that a real vector

linked to a physical or biological phenomenon is random if the value taken by this vector is unknown and the phenomenon is not completed.

For typographical reasons, the vector will instead be written X^T = (X₁,…, X_n) or even X = (X₁,…, X_n) when there is no risk of confusion.

In other words, given a random vector X and Β ⊂

ⁿ we do not know if the assertion (also called the event) (X

Β) is true or false:

However, we do usually know the “chance” that X

Β; this is denoted P (X

B) and is called the probability of the event (X

B).

After completion of the phenomenon, the result (also called the realization) will be denoted

when there is no risk of confusion.

An exact definition of a real random vector of dimension n will now be given. We take as...

Table of contents

• Cover
• Title Page
• Copyright
• Dedication
• Preface
• Introduction
• Chapter 1: Random Vectors
• Chapter 2: Gaussian Vectors
• Chapter 3: Introduction to Discrete Time Processes
• Chapter 4: Estimation
• Chapter 5: The Wiener Filter
• Chapter 6: Adaptive Filtering: Algorithm of the Gradient and the LMS
• Chapter 7: The Kalman Filter
• Table of Symbols and Notations
• Bibliography
• Index