Non-Linear Transformations of Stochastic Processes (eBook)

Front Cover 1
Non-Linear Transformations of Stochastic Processes 4
Copyright Page 5
Table of Contents 6
FOREWORD TO ENGLISH EDITION 9
AUTHORS' FOREWORD 11
FOREWORD 14
CHAPTER I. Some Problems of the Theory of Stochastic Processes 18
1. THE TRANSMISSION OF CERTAIN RANDOM FUNCTIONS THROUGH LINEAR SYSTEMS 20
§ 1. CONVERSION FORMULAE FOR CORRELATION FUNCTIONS 20
§ 2. NORMALIZATION OF THE LINEAR SYSTEM OF PROBABILITY DISTRIBUTION LAWS 27
APPENDIX I. FORMULAE FOR CONVERTING CHARACTERISTIC FUNCTIONS IN THE LINEAR TRANSFORMATION OF RANDOM QUANTITIES 35
APPENDIX II. CONVERSION FORMULAE FOR CORRELATION FUNCTIONS (DISCONTINUOUS TIME) 38
APPENDIX III. CONVERSION FORMULAE FOR CORRELATI ON FUNCTIONS (CONTINUOUS TIME) 43
CONCLUSIONS 44
REFERENCES 44
2. THE TRANSMISSION OF RANDOM FUNCTIONS THROUGH NON-LINEAR SYSTEMS 46
§ 1. TRANSFORMATION OF MOMENT FUNCTIONS FOR THE SIMPLEST TYPE OF NON - LINEAR TRANSFORMATION 47
§2. TRANSFORMATION OF THE MOMENT FUNCTIONS FOR THE GENERAL NON-LINEAR TRANSFORMATION 49
§ 3. EXAMPLES 53
§4. ALTERNATIVE METHOD 63
APPENDIX I. APPROXIMATING THE CHARACTERISTICS OF NON - LINEAR ELEMENTS 70
APPENDIX II. THE RELATIONSHIP BETWEEN MOMENT FUNCTIONS AND CORRELATION FUNCTIONS 72
REFERENCES 75
3. QUASI-MOMENT FUNCTIONS IN THE THEORY OF RANDOM PROCESSES 76
REFERENCES 80
4. THE EFFECT OF NON-NORMAL FLUCTUATIONS ON LINEAR SYSTEMS 81
1. THE EXPANSION OF NON-NORMAL PROBABILITY DENSITY FUNCTIONS AS SERIES IN CHEBYSHEV-HERMITE POLYNOMIALS 82
2. THE PROBABILITY DENSITY FUNCTION FOR FLUCTUATIONS AT THE OUTPUT OF A LINEAR SYSTEM 85
3. EXCURSIONS OF NON-NORMAL FLUCTUATIONS 90
REFERENCES 93
5. CORRELATION FUNCTIONS IN THE THEORY OF THE BROWNIAN MOTION GENERALIZATION OF THE FOKKER-PLANCK EQUATION 94
1. THE FREE BROWNIAN PARTICLE 95
2. A PARTICLE IN INHOMOGENEOUS SPACE 108
3. PARTICULAR CASES 114
CONCLUSION 116
REFERENCES 117
6. A NOTE ON THE MATHEMATICAL THEORY OF CORRELATED RANDOM POINTS 118
REFERENCES 132
7. THE CORRELATION FUNCTIONS OF RANDOM SEQUENCES OF RECTANGULAR PULSES 133
1. INTRODUCTION 133
2. METHOD OF SOLUTION 136
3. PARTICULAR EXAMPLES 138
4. PULSE SEQUENCE WITH CONSTANT REPETITION PERIOD AND RANDOM DURATION 143
5. PULSE SEQUENCES WITH DURATIONS AND INTERVALS BOTH RANDOM 146
REFERENCES 150
CHAPTER II. The Effect of Noise on Certain Non-linear Elements 152
8. A METHOD OF DETERMINING THE ENVELOPE OF QUASI-HARMONIC FLUCTUATIONS 154
REFERENCES 159
9. THE EFFECT OF ELECTRICAL FLUCTUATIONS ON A DETECTOR (ENVELOPE METHOD) 160
REFERENCES 173
10. THE EFFECT OF FLUCTUATIONS ON A DETECTOR AND FILTER 174
REFERENCES 182
11. A NOTE ON THE PROBLEM OF MEASURING ELECTRICAL FLUCTUATIONS WITH THE AID OF THERMOELECTRIC DEVICES 184
REFERENCES 191
12. THE RESPONSE OF TYPICAL NON-LINEAR ELEMENTS TO NORMALLY FLUCTUATING INPUTS 192
REFERENCES 203
13. THE EFFECT OF SIGNAL AND NOISE ON NON-LINEAR ELEMENTS (THE DIRECT METHOD) 204
INTRODUCTION 204
1. THE EXPANSION IN SERIES OF DUAL PROBABILITY DENSITY 205
2. LINEAR DETECTOR WITH BIAS 206
3. THE SYMMETRICAL LIMITER 209
4. THE APPROXIMATION METHOD 210
5. AMPLITUDE MODULATION OF A HARMONIC SIGNAL BY NORMAL NOISE 212
6. INFLUENCE OF NOISE ON A FREQUENCY MULTIPLIER 214
APPENDIX: SOME PROPERTIES OF THE FUNCTIONS p(n)(.) AND Y(y)(.). 216
REFERENCES 220
14. APPROXIMATE METHOD OF CALCULATING THE CORRELATION FUNCTION OF STOCHASTIC SIGNALS 221
REFERENCES 225
15. THE EFFECT OF FLUCTUATIONS ON THE OPERATION OF AN AUTOMATIC RANGE FINDER 226
1. THE OPERATION OF AUTOMATIC RANGE FINDERS 226
2. THE EFFECT OF NOISE 229
3. STATISTICAL CHARACTERISTICS 232
4. SYSTEM STABILITY 235
CONCLUSION 236
REFERENCES 236
CHAPTER III. The Effect of Fluctuations on Oscillator Operation 238
16. THE EFFECT OF ELECTRICAL FLUCTUATIONS ON A VALVE OSCILLATOR 240
REFERENCES 245
17. THE EFFECT OF SMALL FLUCTUATIONS ON OSCILLATOR OPERATION 246
1. FLUCTUATIONS OF OSCILLATOR VALVES 247
2. METHOD OF ANALYSIS 251
3. THE SOLUTION OF THE EQUATION FOR SMALL FLUCTUATIONS 256
4. DISCUSSION OF THE RESULTS 259
REFERENCES 260
18. THE FLUCTUATING NATURE OF THE ESTABLISHMENT OF OSCILLATION AMPLITUDE IN AN OSCILLATOR 261
REFERENCES 267
19. THE EFFECT OF SLOW FLUCTUATIONS ON AN OSCILLATOR 269
REFERENCES 275
20. THE EFFECT OF NOISE ON AN OSCILLATOR WITH FIXED EXCITATION 276
1. THE FUNDAMENTAL EQUATIONS 277
2. ANALYSIS OF THE STATIONARY PROBABILITY 278
3. ANALYSIS OF THE PROBABILITY OF EXCITATION AND INTERRUPTION OF SELF-OSCILLATION 280
REFERENCES 285
21. OSCILLATOR SYNCHRONIZATION IN THE PRESENCE OF NOISE 286
INTRODUCTION 286
1. BASIC EQUATIONS 287
2. THE STATIONARY AMPLITUDE DISTRIBUTION 290
3. STATIONARY PHASE DISTRIBUTION AND MEAN FREQUENCY OF OSCILLATION 291
4. ANALYSIS OF SPECIAL CASES 293
5. DIFFUSION OF NUMBER OF OSCILLATIONS 296
APPENDIX 296
REFERENCES 299
22. ESTABLISHMENT OF THE SYNCHRONOUS PHASE IN A SELF-OSCILLATORY CIRCUIT IN THE PRESENCE OF NOISE 300
REFERENCES 306
23. ESTABLISHMENT OF AMPLITUDE IN A SYNCHRONIZED OSCILLATOR IN THE PRESENCE OF FLUCTUATION NOISE 307
REFERENCES 314
24. THE EFFECT OF NOISE ON THE OPERATION OF A PHASE AFC CIRCUIT 315
STATEMENT OF THE PROBLEM 316
SOLUTION OF THE FOKKER - PLANCK EQUATION 318
STATISTICAL CHARACTERISTICS 321
CONCLUSION 325
REFERENCES 325
25. OPERATION OF A PHASE AFC CIRCUIT IN THE PRESENCE OF NOISE 327
INTRODUCTION 327
THE TWO - DIMENSIONAL FOKKER - PLANCK EQUATION 328
APPROXIMATE SOLUTION OF THE FOKKER - PLANCK EQUATION 332
SPECIAL EXAMPLE 336
REFERENCES 338
26. PARAMETRIC EFFECT OF A RANDOM FORCE ON LINEAR AND NON-LINEAR OSCILLATORY SYSTEMS 339
EEFERENCES 343
27. THE SIMULTANEOUS PARAMETRIC EFFECT OF A HARMONIC AND RANDOM FORCE ON OSCILLATORY SYSTEMS 344
1. DERIVATION OF SIMPLIFIED EQUATIONS 344
2. DETERMINATION OF THE CONDITION OF PARAMETRIC EXCITATION 346
3. DETERMINATION OF THE LIMITS OF INSTABILITY AS A FUNCTION OF SYSTEM PARAMETERS 349
REFERENCES 355
CHAPTER IV. Random Function Excursions 356
28. ON THE DURATION OF EXCURSIONS OF RANDOM FUNCTIONS 358
REFERENCES 370
29. THE DISTRIBUTION OF THE DURATION OF EXCURSIONS OF NORMAL FLUCTUATIONS 371
REFERENCES 384
30. AN EXPERIMENTAL STUDY OF THE LENGTH DISTRIBUTION LAW OF FLUCTUATION EXCURSIONS 385
REFERENCES 391
31. FLUCTUATION EXCURSIONS AND THEIR CORRELATION 392
REFERENCES 398
32. PROBABILITY DENSITIES FOR THE DURATION OF FLUCTUATION EXCURSIONS 399
1. INTRODUCTION 399
2. APPROXIMATE METHODS OF CALCULATING THE PROBABILITY DENSITY (RICE METHOD) 400
3. METHOD OF NON-CORRELATED PULSES 401
3. EXPERIMENTAL RESULTS 406
4. DISCUSSION OF THE RESULTS 408
REFERENCES 412
33. THE EFFECT OF DIFFERENTIATING AND INTEGRATING ON THE AVERAGE NUMBER OF EXCURSIONS 413
REFERENCES 419
CHAPTER V. Optimum Filtration 420
34. THE CONDITIONAL DISTRIBUTION OF CORRELATED RANDOM POINTS AND THE USE OF CORRELATION FOR OPTIMUM FILTRATION OF A PULSE SIGNAL FROM NOISE 422
1. INTRODUCTION 422
2. THE CALCULATION OF CONDITIONAL PROBABILITIES 424
3. THE EVALUATION OF A POSTERIORI DISTRIBUTION FUNCTIONS 426
4. EXAMPLES 428
APPENDIX 1. NOTES ON THE THEORY OF CORRELATED RANDOM POINTS 431
APPENDIX 2. FORMULAE FOR CONDITIONAL DISTRIBUTION FUNCTIONS IN MORE GENERAL FORM 434
REFERENCES 438
35. A CONTRIBUTION TO THE THEORY OF OPTIMUM NON-LINEAR FILTRATION OF RANDOM FUNCTIONS 439
36. CONDITIONAL MARKOV PROCESSES 444
INTRODUCTION 444
1. CONDITIONAL MARKOV CHAINS AND CONDITIONAL PROBABILITIES 445
2. REVERSAL OF THE TIME SIGN 452
3. EXAMPLE: NOISE WITH INDEPENDENT VALUES 460
4. THE FIRST-TYP E EQUATION FOR A CONTINUOUS CONDITIONAL MARKOV PROCESS 465
REFERENCES 470
37. OPTIMUM NON-LINEAR SYSTEMS FOR ISOLATING A SIGNAL WITH CONSTANT PARAMETERS FROM NOISE 471
INTRODUCTION 471
1. THE PROBABILITY FUNCTIONAL OF MARKOV PROCESSES AND THE INVERSE PROBABILITY FORMULA 473
2. THE DIFFERENTIAL EQUATIONS OF OPTIMUM FILTRATION 478
3. EXAMPLE.DETERMINATION OF THE DISPLACEMENT IN NON - GAUSSIAN NOISE 481
REFERENCES 483
38. APPLICATION OF THE THEORY OF MARKOV PROCESSES TO THE OPTIMUM FILTRATION OF SIGNALS 484
INTRODUCTION 484
1. THE EQUATION FOR THE PROBABILITY OF PULSE COORDINATES FORMING A MARKOV CHAIN 487
2. DETERMINATION OF THE PHASE OF A NARROW BAND SIGNAL IN WHITE NOISE 490
3. THE DIFFERENTIAL EQUATION OF OPTIMUM FILTRATION 494
4. BREAKING THE CHAIN OF EQUATIONS 496
5. THE EQUATIONS OF OPTIMUM FILTRATION IN A GAUSSIAN APPROXIMATION 499
REFERENCES 501
39. SOME PROBLEMS IN CONDITIONAL PROBABILITY AND QUASI-MOMENT FUNCTIONS 502
REFERENCES 510
SUBJECT INDEX 512