Discrete Stochastic Processes and Optimal Filtering

Introduction

The object of this book is to present the bases of discrete optimal filtering in a progressive and rigorous manner. The optimal character can be understood in the sense that we always choose that criterion at the minimum of the norm −L² of error.

Chapter 1 tackles random vectors, their principal definitions and properties.

Chapter 2 covers the subject of Gaussian vectors. Given the practical importance of this notion, the definitions and results are accompanied by numerous commentaries and explanatory diagrams.

Chapter 3 is by its very nature more “physics” heavy than the preceding ones and can be considered as an introduction to digital filtering. Results that will be essential for what follows will be given.

Chapter 4 provides the pre-requisites essential for the construction of optimal filters. The results obtained on projections in Hilbert spaces constitute the cornerstone of future demonstrations.

Chapter 5 covers the Wiener filter, an electronic device that is well adapted to processing stationary signals of second order. Practical calculations of such filters, as an answer to finite or infinite pulses, will be developed.

Adaptive filtering, which is the subject of Chapter 6, can be considered as a relatively direct application of the determinist or stochastic gradient method. At the end of the process of adaptation or convergence, the Wiener filter is again encountered.

The book is completed with a study of Kalman filtering which allows stationary or non-stationary ...