STOCHASTIC METHODS IN SIGNAL TRANSMISSION

CONTENT OF THE COURSE
(taken from the book published)

Preface , 15

1. PROBABILITY SPACES. RANDOM VARIABLES
1.1 Brief information upon Theory of Probability
1.2 Concept about a random Variable, 19
1.3 Generalizations   22
1.4 Convergence   22

2. STOCHASTIC PROCESSES
2.1 Stochastic Processes as Models of Signals and Noises, 27
2.2 Random Functions   30
2.3 Finite-Dimensional Distribution Function of a Random Process   37
2.4 Other Classes of Random Processes   45
2.5 Problems   60

3. SECOND ORDER RANDOM PROCESSES
3.1 Covariance   63
3.2 Mean-Square Analysis   64
3.3 Examples and Applications   74
3.4 Problems   83

4. MORE ABOUT MARKOV PROCESSES AND WIENER PROCESSES
ITO PROCESSES
4.1 Markov Processes   85
4.2 Some Peculiarities of Wiener Process. White Noise   91
4.3 Ito Processes   101
4.4 Problems   108

5. STATIONARY RANDOM PROCESSES
5.1 Concept about Stationary Process   110
5.2 Correlation Function   115
5.3 Ergodic Processes   122
5.4 Examples and Applications   131
5.5 Problems   139

6. HARMONIC ANALYSIS OF STATIONARY RANDOM PROCESSES
6.1 Processes that can be Analyzed   141
6.2 Wiener-Khintchine Theorem   145
6.3 Problems 155

7. SYSTEMS FOR TRANSMISSION OF SIGNALS
7.1 Random Processes and Linear Systems   157
7.2 Random Processes and Non-linear Systems   162
7.3 Problems   167

8. METHODS OF STATISTICAL SYNTHESIS AND ANLYSIS OF INFORMATION SYSTEMS
8.1 Problems of the Synthesis with a Complete Prior Information   169
8.2 Problems of the Synthesis in Conditions of Prior Uncertainty   180

9. STATISTICAL HYPOTHESES TESTING
9.1 One Step Algorithms for Testing of a Simple Hypothesis Against a Simple Alternative, 189
9.2 Hypotheses Testing in Conditions of Parametric Prior Uncertainty, 196
9.3 Hypotheses Testing for the Mean Value of a Gaussian Random Variable, 197
9.4 Testing of a Simple Hypothesis for the Mean Value Vector of Multidimensional Gaussian Distribution Against a Simple Alternative   204
9.5 Hypotheses Testing in Conditions of Non-parametric Prior Uncertainty, 205
9.6 Sequential (Multi-steps) Algorithms for Hypotheses Testing, 209
9.7 Problems   213

10. ESTIMATION OF UNKNOWN CHARACTERISTICS
10.1 Estimation in Conditions of Parametric Prior Uncertainty, 215
10.2 Estimation in Conditions of Non-parametric Prior Uncertainty, 227
10.3 Problems, 231

11. OPTIMAL FILTERING OF RANDOM PROCESSES
11.1 General Information, 233
11.2 Matched Filter, 234
11.3 Linear Filtering of Stationary Random Processes (Theory of Wiener), 239
11.4 A Physically Realizable Discrete Filter. Kalman Filtering, 249
11.5 Non-linear Filtering, 251
11.6 Innovation Approach,p 257
11.7 Problems, 261

12. ASYMPTOTIC OPTIMAL ALGORITHMS FOR DETECTING AND IDENTIFICATIONS OF SIGNALS ON A BACKGROUND OF NOISE
12.1 Structure of AO Algorithms for Signal Detection According to Independent Observations, 263
12.2 AO Tests for Detection of Signals According to Independent Quantized Observations, 269
12.3 Structure of AO Algorithms for Detection of Signals in m-Dependent Markov Noise, 275
12.4 Identification of Signals in m-Dependent Markov Noise, 282

13. ASYMPTOTIC OPTIMAL ALGORITHMS FOR DETECTION OF SIGNALS IN DEPENDENT VALUE’S INTERFERENCE AND WHITE NOISE 285
13.1 Asymptotic Optimal Algorithms with Continuous Time, 285
13.2 Asymptotic Optimal Algorithms with Discrete Time, 286

14. PARAMETRIC METHODS IN SPECTRAL ANALYSIS OF GAUSSIAN TIME SERIES
14.1 Parametric Statistical Inferences for Time Series, 294
14.2 Locally Asymptotic Normality for Parametric Models of Gaussian Time Series, 298
14.3 Regular Stationary Processes and Their Approximation by ARMA Models, 305
14.4 Detection of Signals in Noise According to Dependent Gaussian Observations, 309
14.5 Detection of Disorders of ARMA Processes, 315

15. AREAS OF PRACTICAL APPLICATIONS
15.1 Transmission of Signals, 317
15.2 Radar, 322
15.3 Statistical Modeling, 326

BIBLIOGRAPHY, 329



STATISTICS OF DISCRETE RANDOM SIGNALS



CONTENTS

Preface 1

1. INTRODUCTION
1.1 Discrete Random Signals, 2
1.2 Statistical Signal Processing, 3
1.3 Application of Statistical Signal Processing, 4

2. ESTIMATION, 14
2.1 Estimation of Parameters, 15
2.2 Estimation of First and Second Moments for a Random Process, 36
2.3 Bayes Estimation of Random Variables, 46
2.4 Linear Mean-Square Estimation, 54
2.5 Problems, 63

3. OPTIMAL FILTERING, 70
3.1 The Orthogonality Principle, 71
3.2 Linear Predictive Filtering, 74
3.3 General Optimal Filtering – the FIR Case, 80
3.4 General Optimal Filtering – the IIR Case, 89
3.5 Recursive Filtering, 112
3.6 Wold Decomposition, 123
3.7 Problems 131

4. LINEAR PREDICTION, 139
4.1 Another Look at Linear Prediction,140
4.2 The Autoregressive (AR) Model, 141
4.3 Linear Prediction for AR Processes, 145
4.4 Backward Linear Prediction and the Anticausal AR Model,148
4.5 The Levinson Recursion, 152
4.6 Partial Correlation Interpretation of the Reflection Coefficients,160
4.7 Minimum-Phase Property of the Prediction Error,165
4.8 The Schur Algorithm,168
4.9 Problems, 182

5. LINEAR MODELS,187
5.1 Linear Modeling of Random Processes, 188
5.2 Estimation of Model Parameters from Data, 198
5.3 Principles of Least-Squares, 204
5.4 AR Modeling via Linear Prediction, 221
5.5 ARMA Modeling: A Deterministic Approach,236
5.6 Least Squares Methods and the Yule-Walker Equations,250
5.7 Problems,254

6. SPECTRUM ESTIMATION, 260
6.1 Classical Spectrum Estimation, 261
6.2 Spectrum Estimation Based on Linear Models, 271
6.3 “Maximum Likelihood” Spectrum Estimation, 282
6.4 Subspace Methods: Estimating the Discrete Components, 289
6.5 Problems, 333

BIBLIOGRAPHY, 337