Digital Signal and Image Processing Using MATLAB: Advances and Applications - The Stochastic Case: V

Volume 3 of the second edition of the fully revised and updated Digital Signal and Image Processing using MATLAB, after first two volumes on the “Fundamentals” and “Advances and Applications: The Deterministic Case”, focuses on the stochastic case. It will be of particular benefit to readers who already possess a good knowledge of MATLAB, a command of the fundamental elements of digital signal processing and who are familiar with both the fundamentals of continuous-spectrum spectral analysis and who have a certain mathematical knowledge concerning Hilbert spaces.

This volume is focused on applications, but it also provides a good presentation of the principles. A number of elements closer in nature to statistics than to signal processing itself are widely discussed. This choice comes from a current tendency of signal processing to use techniques from this field.

More than 200 programs and functions are provided in the MATLAB language, with useful comments and guidance, to enable numerical experiments to be carried out, thus allowing readers to develop a deeper understanding of both the theoretical and practical aspects of this subject.

Foreword ix

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