Discrete Stochastic Processes and Optimal Filtering

Jean-Claude Bertein has a Master's and PhD degree from the University of Paris VI. He was formerly a research engineer at Alcatel and today is Professor in the Department of Mathematics and Physics at the Graduate School of Electrical and Electronic Engineering (ESIEE) Paris. Roger Ceschi is an ENSEA engineer and holds a Master's and PhD degree from the University of Paris XI. He was formerly Director of the ENSEA and today he is Director General of the ESIEE Amiens. He is also the author of a theorem on analytic signals and is Visiting Professor at the BIPT and BIT Universities in China.

• Preface ixIntroduction xiChapter 1. Random Vectors 11.1. Definitions and general properties. 11.2. Spaces L1 (dP) and L2 (dP) 201.3. Mathematical expectation and applications 231.4. Second order random variables and vectors. 391.5. Linear independence of vectors of L2 (dP) 461.6. Conditional expectation (concerning random vectors with density function) 511.7. Exercises for Chapter 1 56Chapter 2. Gaussian Vectors 632.1. Some reminders regarding random Gaussian vectors 632.2. Definition and characterization of Gaussian vectors 662.3. Results relative to independence 682.4. Affine transformation of a Gaussian vector 722.5. The existence of Gaussian vectors. 742.6. Exercises for Chapter 2 84Chapter 3. Introduction to Discrete Time Processes 933.1. Definition 933.2. WSS processes and spectral measure 1053.3. Spectral representation of a WSS process 1093.4. Introduction to digital filtering 1143.5. Important example: autoregressive process 1273.6. Exercises for Chapter 3 132Chapter 4. Estimation 1394.1. Position of the problem 1394.2. Linear estimation 1424.3. Best estimate – conditional expectation 1544.4. Example: prediction of an autoregressive process AR (1) 1624.5. Multivariate processes 1634.6. Exercises for Chapter 4 172Chapter 5. The Wiener Filter 1775.1. Introduction 1775.2. Resolution and calculation of the FIR filter 1795.3. Evaluation of the least error 1815.4. Resolution and calculation of the IIR filter 1835.5. Evaluation of least mean square error 1875.6. Exercises for Chapter 5 188Chapter 6. Adaptive Filtering: Algorithm of the Gradient and the LMS 1956.1. Introduction 1956.2. Position of problem 1986.3. Data representation 2006.4. Minimization of the cost function 2026.5. Gradient algorithm 2096.6. Geometric interpretation 2126.7. Stability and convergence 2166.8. Estimation of gradient and LMS algorithm 2216.9. Example of the application of the LMS algorithm 2246.10. Exercises for Chapter 6 233Chapter 7. The Kalman Filter 2357.1. Position of problem 2357.2. Approach to estimation 2397.3. Kalman filtering 2437.4. Exercises for Chapter 7 2617.5. Appendices 2677.6. Examples treated using Matlab software 273Table of Symbols and Notations 281Bibliography 283Index 285