Gérard Blanchet is Professor at Telecom ParisTech, France. In addition to his research, teaching and consulting activities, he is the author of several books on automatic control systems, digital signal processing and computer architecture. He also develops tools and methodologies to improve knowledge acquisition in various fields.

Maurice Charbit is Professor at Telecom ParisTech, France. He is a teacher in probability theory, signal processing, communication theory and statistics for data processing. With regard to research, his main areas of interest are: (i) the Bayesian approach for hidden Markov models, (ii) the 3D model-based approach for face tracking, and (iii) processing for multiple sensor arrays with applications to infrasonic systems.

Notations and Abbreviations xiii

1 Mathematical Concepts 1

1.1 Basic concepts on probability 1

1.2 Conditional expectation 9

1.3 Projection theorem 10

1.4 Gaussianity 13

1.5 Random variable transformation 18

1.6 Fundamental statistical theorems 21

1.7 Other important probability distributions 23

2 Statistical Inferences 25

2.1 Statistical model 25

2.2 Hypothesis tests 27

2.3 Statistical estimation 41

3 Monte-Carlo Simulation 85

3.1 Fundamental theorems 85

3.2 Stating the problem 86

3.3 Generating random variables 88

3.4 Variance reduction 99

4 Second Order Stationary Process 107

4.1 Statistics for empirical correlation 107

4.2 Linear prediction of WSS processes 111

4.3 Non-parametric spectral estimation of WSS processes 124

5 Inferences on HMM 139

5.1 Hidden Markov Models (HMM) 130

5.2 Inferences on HMM 142

5.3 Gaussian linear case: the Kalman filter 143

5.4 Discrete finite Markov case 152

6 Selected Topics 163

6.1 High resolution methods 163

6.2 Digital Communications 186

6.3 Linear equalization and the Viterbi algorithm 211

6.4 Compression 220

7 Hints and Solutions 235

H1 Mathematical concepts 235

H2 Statistical inferences 237

H3 Monte-Carlo simulation 269

H4 Second order stationary process 283

H5 Inferences on HMM 283

H6 Selected Topics 300

8 Appendices 317

A1 Miscellaneous functions 317

A2 Statistical functions 318

Bibliography 329

Index 333

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