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A Signal Theoretic Introduction to Random Processes

Book Description

A fresh introduction to random processes utilizing signal theory

By incorporating a signal theory basis, A Signal Theoretic Introduction to Random Processes presents a unique introduction to random processes with an emphasis on the important random phenomena encountered in the electronic and communications engineering field. The strong mathematical and signal theory basis provides clarity and precision in the statement of results. The book also features:

  • A coherent account of the mathematical fundamentals and signal theory that underpin the presented material

  • Unique, in-depth coverage of material not typically found in introductory books

  • Emphasis on modeling and notation that facilitates development of random process theory

  • Coverage of the prototypical random phenomena encountered in electrical engineering

  • Detailed proofs of results

  • A related website with solutions to the problems found at the end of each chapter

  • A Signal Theoretic Introduction to Random Processes is a useful textbook for upper-undergraduate and graduate-level courses in applied mathematics as well as electrical and communications engineering departments. The book is also an excellent reference for research engineers and scientists who need to characterize random phenomena in their research.

    Table of Contents

    1. COVER
    2. TITLE PAGE
    3. ABOUT THE AUTHOR
    4. PREFACE
    5. 1 A SIGNAL THEORETIC INTRODUCTION TO RANDOM PROCESSES
      1. 1.1 INTRODUCTION
      2. 1.2 MOTIVATION
      3. 1.3 BOOK OVERVIEW
    6. 2 BACKGROUND: MATHEMATICS
      1. 2.1 INTRODUCTION
      2. 2.2 SET THEORY
      3. 2.3 FUNCTION THEORY
      4. 2.4 MEASURE THEORY
      5. 2.5 MEASURABLE FUNCTIONS
      6. 2.6 LEBESGUE INTEGRATION
      7. 2.7 CONVERGENCE
      8. 2.8 LEBESGUE–STIELTJES MEASURE
      9. 2.9 LEBESGUE–STIELTJES INTEGRATION
      10. 2.10 MISCELLANEOUS RESULTS
      11. 2.11 PROBLEMS
      12. APPENDIX 2.A PROOF OF THEOREM 2.1
      13. APPENDIX 2.B PROOF OF THEOREM 2.2
      14. APPENDIX 2.C PROOF OF THEOREM 2.7
      15. APPENDIX 2.D PROOF OF THEOREM 2.8
      16. APPENDIX 2.E PROOF OF THEOREM 2.10
    7. 3 BACKGROUND
      1. 3.1 INTRODUCTION
      2. 3.2 SIGNAL ORTHOGONALITY
      3. 3.3 THEORY FOR DIRICHLET POINTS
      4. 3.4 DIRAC DELTA
      5. 3.5 FOURIER THEORY
      6. 3.6 SIGNAL POWER
      7. 3.7 THE POWER SPECTRAL DENSITY
      8. 3.8 THE AUTOCORRELATION FUNCTION
      9. 3.9 POWER SPECTRAL DENSITY–AUTOCORRELATION FUNCTION
      10. 3.10 RESULTS FOR THE INFINITE INTERVAL
      11. 3.11 CONVERGENCE OF FOURIER COEFFICIENTS
      12. 3.12 CRAMER’S REPRESENTATION AND TRANSFORM
      13. 3.13 PROBLEMS
      14. APPENDIX 3.A PROOF OF THEOREM 3.5
      15. APPENDIX 3.B PROOF OF THEOREM 3.8
      16. APPENDIX 3.C FOURIER TRANSFORM AND PSD OF A SINUSOID
      17. APPENDIX 3.D PROOF OF Theorem 3.14
      18. APPENDIX 3.E PROOF OF Theorem 3.19
      19. APPENDIX 3.F PROOF OF Theorem 3.23
      20. APPENDIX 3.G PROOF OF THEOREM 3.24
      21. APPENDIX 3.H PROOF OF THEOREM 3.25
      22. APPENDIX 3.I PROOF OF THEOREM 3.26
      23. APPENDIX 3.J CRAMER TRANSFORM OF UNIT STEP FUNCTION
      24. APPENDIX 3.K CRAMER TRANSFORM FOR SINUSOIDAL SIGNALS
      25. APPENDIX 3.L PROOF OF THEOREM 3.30
      26. APPENDIX 3.M PROOF OF THEOREM 3.31
      27. APPENDIX 3.N PROOF OF THEOREM 3.32
      28. APPENDIX 3.O PROOF OF THEOREM 3.33
    8. 4 BACKGROUND: PROBABILITY AND RANDOM VARIABLE THEORY
      1. 4.1 INTRODUCTION
      2. 4.2 BASIC CONCEPTS: EXPERIMENTS-PROBABILITY THEORY
      3. 4.3 THE RANDOM VARIABLE
      4. 4.4 DISCRETE AND CONTINUOUS RANDOM VARIABLES
      5. 4.5 STANDARD RANDOM VARIABLES
      6. 4.6 FUNCTIONS OF A RANDOM VARIABLE
      7. 4.7 EXPECTATION
      8. 4.8 GENERATION OF DATA CONSISTENT WITH DEFINED PDF
      9. 4.9 VECTOR RANDOM VARIABLES
      10. 4.10 PAIRS OF RANDOM VARIABLES
      11. 4.11 COVARIANCE AND CORRELATION
      12. 4.12 SUMS OF RANDOM VARIABLES
      13. 4.13 JOINTLY GAUSSIAN RANDOM VARIABLES
      14. 4.14 STIRLING’S FORMULA AND APPROXIMATIONS TO BINOMIAL
      15. 4.15 PROBLEMS
      16. APPENDIX 4.A PROOF OF THEOREM 4.6
      17. APPENDIX 4.B PROOF OF THEOREM 4.8
      18. APPENDIX 4.C PROOF OF THEOREM 4.9
      19. APPENDIX 4.D PROOF OF THEOREM 4.21
      20. APPENDIX 4.E PROOF OF STIRLING’S FORMULA
      21. APPENDIX 4.F PROOF OF THEOREM 4.27
      22. APPENDIX 4.G PROOF OF THEOREM 4.29
    9. 5 INTRODUCTION TO RANDOM PROCESSES
      1. 5.1 RANDOM PROCESSES
      2. 5.2 DEFINITION OF A RANDOM PROCESS
      3. 5.3 EXAMPLES OF RANDOM PROCESSES
      4. 5.4 EXPERIMENTS AND EXPERIMENTAL OUTCOMES
      5. 5.5 PROTOTYPICAL EXPERIMENTS
      6. 5.6 RANDOM VARIABLES DEFINED BY A RANDOM PROCESS
      7. 5.7 CLASSIFICATION OF RANDOM PROCESSES
      8. 5.8 CLASSIFICATION: ONE-DIMENSIONAL RPs
      9. 5.9 SUMS OF RANDOM PROCESSES
      10. 5.10 PROBLEMS
    10. 6 PROTOTYPICAL RANDOM PROCESSES
      1. 6.1 INTRODUCTION
      2. 6.2 BERNOULLI RANDOM PROCESSES
      3. 6.3 POISSON RANDOM PROCESSES
      4. 6.4 CLUSTERED RANDOM PROCESSES
      5. 6.5 SIGNALLING RANDOM PROCESSES
      6. 6.6 JITTER
      7. 6.7 WHITE NOISE
      8. 6.8 1/<i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ibooks="http://vocabulary.itunes.apple.com/rdf/ibooks/vocabulary-extensions-1.0">f</i> NOISE NOISE
      9. 6.9 BIRTH–DEATH RANDOM PROCESSES
      10. 6.10 ORTHOGONAL INCREMENT RANDOM PROCESSES
      11. 6.11 LINEAR FILTERING OF RANDOM PROCESSES
      12. 6.12 SUMMARY OF RANDOM PROCESSES
      13. 6.13 PROBLEMS
      14. APPENDIX 6.A PROOF OF THEOREM 6.4
    11. 7 CHARACTERIZING RANDOM PROCESSES
      1. 7.1 INTRODUCTION
      2. 7.2 TIME EVOLUTION OF PMF OR PDF
      3. 7.3 FIRST-, SECOND-, AND HIGHER-ORDER CHARACTERIZATION
      4. 7.4 AUTOCORRELATION AND POWER SPECTRAL DENSITY
      5. 7.5 CORRELATION
      6. 7.6 NOTES ON AVERAGE POWER AND AVERAGE ENERGY
      7. 7.7 CLASSIFICATION: STATIONARITY VS NON-STATIONARITY
      8. 7.8 CRAMER’S REPRESENTATION
      9. 7.9 STATE SPACE CHARACTERIZATION of Random Processes
      10. 7.10 TIME SERIES CHARACTERIZATION
      11. 7.11 PROBLEMS
      12. APPENDIX 7.A PROOF OF THEOREM 7.2
      13. APPENDIX 7.B PROOF OF THEOREMS 7.3 AND 7.4
      14. APPENDIX 7.C PROOF OF THEOREM 7.5
      15. APPENDIX 7.D PROOF OF THEOREM 7.6
      16. APPENDIX 7.E PROOF OF THEOREM 7.11
      17. APPENDIX 7.F PROOF OF THEOREM 7.12
      18. APPENDIX 7.G PROOF OF THEOREM 7.16
      19. APPENDIX 7.H PROOF OF THEOREM 7.17
      20. APPENDIX 7.I PROOF OF THEOREM 7.18
      21. APPENDIX 7.J PROOF OF THEOREM 7.20
      22. APPENDIX 7.K PROOF OF THEOREM 7.21
      23. APPENDIX 7.L PROOF OF THEOREM 7.23
      24. APPENDIX 7.M PROOF OF THEOREM 7.24
    12. 8 PMF AND PDF EVOLUTION
      1. 8.1 INTRODUCTION
      2. 8.2 PROBABILITY MASS/DENSITY FUNCTION ESTIMATION
      3. 8.3 NON/SEMI-PARAMETRIC PDF ESTIMATION
      4. 8.4 PMF/PDF EVOLUTION: SIGNAL PLUS NOISE
      5. 8.5 PMF EVOLUTION OF A RANDOM WALK
      6. 8.6 PDF EVOLUTION: BROWNIAN MOTION
      7. 8.7 PDF EVOLUTION: SIGNALLING RANDOM PROCESS
      8. 8.8 PDF EVOLUTION: GENERALIZED SHOT NOISE
      9. 8.9 PDF EVOLUTION: SWITCHING IN A CMOS INVERTER
      10. 8.10 PDF EVOLUTION: GENERAL CASE
      11. 8.11 PROBLEMS
      12. APPENDIX 8.A PROOF OF THEOREM 8.1
      13. APPENDIX 8.B PROOF OF THEOREM 8.5
      14. APPENDIX 8.C PROOF OF THEOREM 8.11
      15. APPENDIX 8.D PROOF OF THEOREM 8.12
    13. 9 THE AUTOCORRELATION FUNCTION
      1. 9.1 INTRODUCTION
      2. 9.2 NOTATION AND DEFINITIONS
      3. 9.3 BASIC RESULTS AND INDEPENDENCE INFORMATION
      4. 9.4 SINUSOID WITH RANDOM AMPLITUDE AND PHASE
      5. 9.5 RANDOM TELEGRAPH SIGNAL
      6. 9.6 GENERALIZED SHOT NOISE
      7. 9.7 SIGNALLING RANDOM PROCESS-FIXED PULSE CASE
      8. 9.8 GENERALIZED SIGNALLING RANDOM PROCESS
      9. 9.9 AUTOCORRELATION: JITTERED RANDOM PROCESSES
      10. 9.10 RANDOM WALK
      11. 9.11 PROBLEMS
      12. APPENDIX 9.A PROOF OF THEOREM 9.6
      13. APPENDIX 9.B PROOF OF THEOREM 9.7
      14. APPENDIX 9.C PROOF OF THEOREMS 9.8 AND 9.9
      15. APPENDIX 9.D PROOF OF THEOREM 9.12
      16. APPENDIX 9.E PROOF OF THEOREM 9.16
      17. APPENDIX 9.F PROOF OF THEOREM 9.17
      18. APPENDIX 9.G PROOF OF THEOREM 9.19
      19. APPENDIX 9.H PROOF OF THEOREM 9.20
    14. 10 POWER SPECTRAL DENSITY THEORY
      1. 10.1 INTRODUCTION
      2. 10.2 POWER SPECTRAL DENSITY THEORY
      3. 10.3 POWER SPECTRAL DENSITY OF A PERIODIC PULSE TRAIN
      4. 10.4 PSD OF A SIGNALLING RANDOM PROCESS
      5. 10.5 DIGITAL TO ANALOGUE CONVERSION
      6. 10.6 PSD OF SHOT NOISE RANDOM PROCESSES
      7. 10.7 WHITE NOISE
      8. 10.8 1/<i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ibooks="http://vocabulary.itunes.apple.com/rdf/ibooks/vocabulary-extensions-1.0">f</i> NOISE NOISE
      9. 10.9 PSD OF A JITTERED BINARY RANDOM PROCESS
      10. 10.10 PSD OF A JITTERED PULSE TRAIN
      11. 10.11 PROBLEMS
      12. APPENDIX 10.A PROOF OF THEOREM 10.1
      13. APPENDIX 10.B PROOF OF THEOREM 10.2
      14. APPENDIX 10.C PROOF OF THEOREM 10.3
      15. APPENDIX 10.D PROOF OF THEOREM 10.5
      16. APPENDIX 10.E PROOF OF THEOREM 10.6
      17. APPENDIX 10.F PROOF OF THEOREM 10.7
      18. APPENDIX 10.G PROOF OF THEOREM 10.8
      19. APPENDIX 10.H PROOF OF THEOREM 10.10
      20. APPENDIX 10.I PROOF OF THEOREM 10.13
      21. APPENDIX 10.J PROOF OF THEOREM 10.15
    15. 11 ORDER STATISTICS
      1. 11.1 INTRODUCTION
      2. 11.2 ORDERED RANDOM VARIABLE THEORY
      3. 11.3 IDENTICAL RVs WITH UNIFORM DISTRIBUTION
      4. 11.4 UNIFORM DISTRIBUTION AND INFINITE INTERVAL
      5. 11.5 PROBLEMS
      6. APPENDIX 11.A PROOF OF THEOREM 11.1
      7. APPENDIX 11.B PROOF OF THEOREM 11.4
      8. APPENDIX 11.C PROOF OF THEOREM 11.5
      9. APPENDIX 11.D PROOF OF THEOREM 11.6
      10. APPENDIX 11.E PROOF OF THEOREMS 11.8 AND 11.9
      11. Appendix 11.F PROOF OF THEOREM 11.10
      12. Appendix 11.G PROOF: MARGINAL PDF FROM JOINT PDF
      13. APPENDIX 11.H PROOF OF THEOREM 11.12
      14. APPENDIX 11.I PROOF OF THEOREM 11.13
      15. APPENDIX 11.J PROOF OF THEOREM 11.15
      16. APPENDIX 11.K PROOF OF THEOREM 11.16
      17. APPENDIX 11.L PROOF OF THEOREM 11.18
      18. APPENDIX 11.M PROOF OF THEOREM 11.20
      19. APPENDIX 11.N PROOF OF THEOREM 11.24
      20. APPENDIX 11.O PROOF OF THEOREM 11.25
      21. APPENDIX 11.P PROOF OF THEOREM 11.26
      22. APPENDIX 11.Q PROOF OF THEOREM 11.25
      23. APPENDIX 11.R PROOF OF THEOREM 11.28
      24. APPENDIX 11.S PROOF OF THEOREM 11.29
      25. APPENDIX 11.T PROOF OF THEOREM 11.31
      26. APPENDIX 11.U PROOF OF THEOREM 11.36
    16. 12 POISSON POINT RANDOM PROCESSES
      1. 12.1 INTRODUCTION
      2. 12.2 CHARACTERIZING POISSON RANDOM PROCESSES
      3. 12.3 PMF: NUMBER OF POINTS IN A SUBSET OF AN INTERVAL
      4. 12.4 RESULTS FROM ORDER STATISTICS
      5. 12.5 ALTERNATIVE CHARACTERIZATION FOR INFINITE INTERVAL
      6. 12.6 MODELLING WITH UNORDERED OR ORDERED TIMES
      7. 12.7 ZERO CROSSING TIMES OF RANDOM TELEGRAPH SIGNAL
      8. 12.8 POINT PROCESSES: THE GENERAL CASE
      9. 12.9 PROBLEMS
      10. APPENDIX 12.A PROOF OF THEOREM 12.5
      11. APPENDIX 12.B PROOF OF THEOREM 12.6
      12. APPENDIX 12.C PROOF OF THEOREM 12.9
      13. APPENDIX 12.D PROOF OF THEOREM 12.12
      14. APPENDIX 12.E PROOF OF THEOREM 12.16
      15. APPENDIX 12.F PROOF OF THEOREM 12.18
      16. APPENDIX 12.G PROOF OF THEOREM 12.19
      17. APPENDIX 12.H EQUIVALENCE: ORDERED–UNORDERED TIMES
    17. 13 BIRTH–DEATH RANDOM PROCESSES
      1. 13.1 INTRODUCTION
      2. 13.2 DEFINING AND CHARACTERIZING BIRTH–DEATH PROCESSES
      3. 13.3 CONSTANT BIRTH RATE, ZERO DEATH RATE PROCESS
      4. 13.4 STATE DEPENDENT BIRTH RATE - ZERO DEATH RATE
      5. 13.5 CONSTANT DEATH RATE, ZERO BIRTH RATE, PROCESS
      6. 13.6 CONSTANT BIRTH AND CONSTANT DEATH RATE PROCESS
      7. 13.7 PROBLEMS
      8. APPENDIX 13.A PROOF OF THEOREM 13.1
      9. APPENDIX 13.B PROOF OF THEOREM 13.2
      10. APPENDIX 13.C PROOF OF THEOREM 13.5
      11. APPENDIX 13.D PROOF OF THEOREM 13.8
      12. APPENDIX 13.E PROOF OF THEOREM 13.9
    18. 14 THE FIRST PASSAGE TIME
      1. 14.1 INTRODUCTION
      2. 14.2 FIRST PASSAGE TIME
      3. 14.3 APPROACHES: ESTABLISHING THE FIRST PASSAGE TIME
      4. 14.4 MAXIMUM LEVEL AND THE FIRST PASSAGE TIME
      5. 14.5 SOLUTIONS FOR THE First Passage TIME PDF
      6. 14.6 PROBLEMS
      7. APPENDIX 14.A PROOF OF THEOREM 14.3
      8. APPENDIX 14.B PROOF OF THEOREM 14.4
      9. APPENDIX 14.C PROOF OF THEOREM 14.5
      10. APPENDIX 14.D PROOF OF THEOREM 14.7
      11. APPENDIX 14.E PROOF OF THEOREM 14.8
      12. APPENDIX 14.F PROOF OF THEOREM 14.9
    19. REFERENCE MATERIAL
      1. 1. FOURIER TRANSFORM RELATIONSHIPS
      2. 2. COMMON RANDOM VARIABLE
      3. 3. NOTATION
      4. 4. ACRONYMS
    20. REFERENCES
    21. INDEX
    22. END USER LICENSE AGREEMENT