A Signal Theoretic Introduction to Random Processes E-Book

Roy M. Howard

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 704

Veröffentlichungsjahr: 2015

Beschreibung

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.

Leseprobe

CONTENTS

COVER

TITLE PAGE

ABOUT THE AUTHOR

PREFACE

1 A SIGNAL THEORETIC INTRODUCTION TO RANDOM PROCESSES

1.1 INTRODUCTION

1.2 MOTIVATION

1.3 BOOK OVERVIEW

2 BACKGROUND: MATHEMATICS

2.1 INTRODUCTION

2.2 SET THEORY

2.3 FUNCTION THEORY

2.4 MEASURE THEORY

2.5 MEASURABLE FUNCTIONS

2.6 LEBESGUE INTEGRATION

2.7 CONVERGENCE

2.8 LEBESGUE–STIELTJES MEASURE

2.9 LEBESGUE–STIELTJES INTEGRATION

2.10 MISCELLANEOUS RESULTS

2.11 PROBLEMS

APPENDIX 2.A PROOF OF THEOREM 2.1

APPENDIX 2.B PROOF OF THEOREM 2.2

APPENDIX 2.C PROOF OF THEOREM 2.7

APPENDIX 2.D PROOF OF THEOREM 2.8

APPENDIX 2.E PROOF OF THEOREM 2.10

3 BACKGROUND

3.1 INTRODUCTION

3.2 SIGNAL ORTHOGONALITY

3.3 THEORY FOR DIRICHLET POINTS

3.4 DIRAC DELTA

3.5 FOURIER THEORY

3.6 SIGNAL POWER

3.7 THE POWER SPECTRAL DENSITY

3.8 THE AUTOCORRELATION FUNCTION

3.9 POWER SPECTRAL DENSITY–AUTOCORRELATION FUNCTION

3.10 RESULTS FOR THE INFINITE INTERVAL

3.11 CONVERGENCE OF FOURIER COEFFICIENTS

3.12 CRAMER’S REPRESENTATION AND TRANSFORM

3.13 PROBLEMS

APPENDIX 3.A PROOF OF THEOREM 3.5

APPENDIX 3.B PROOF OF THEOREM 3.8

APPENDIX 3.C FOURIER TRANSFORM AND PSD OF A SINUSOID

APPENDIX 3.D PROOF OF Theorem 3.14

APPENDIX 3.E PROOF OF Theorem 3.19

APPENDIX 3.F PROOF OF Theorem 3.23

APPENDIX 3.G PROOF OF THEOREM 3.24

APPENDIX 3.H PROOF OF THEOREM 3.25

APPENDIX 3.I PROOF OF THEOREM 3.26

APPENDIX 3.J CRAMER TRANSFORM OF UNIT STEP FUNCTION

APPENDIX 3.K CRAMER TRANSFORM FOR SINUSOIDAL SIGNALS

APPENDIX 3.L PROOF OF THEOREM 3.30

APPENDIX 3.M PROOF OF THEOREM 3.31

APPENDIX 3.N PROOF OF THEOREM 3.32

APPENDIX 3.O PROOF OF THEOREM 3.33

4 BACKGROUND: PROBABILITY AND RANDOM VARIABLE THEORY

4.1 INTRODUCTION

4.2 BASIC CONCEPTS: EXPERIMENTS-PROBABILITY THEORY

4.3 THE RANDOM VARIABLE

4.4 DISCRETE AND CONTINUOUS RANDOM VARIABLES

4.5 STANDARD RANDOM VARIABLES

4.6 FUNCTIONS OF A RANDOM VARIABLE

4.7 EXPECTATION

4.8 GENERATION OF DATA CONSISTENT WITH DEFINED PDF

4.9 VECTOR RANDOM VARIABLES

4.10 PAIRS OF RANDOM VARIABLES

4.11 COVARIANCE AND CORRELATION

4.12 SUMS OF RANDOM VARIABLES

4.13 JOINTLY GAUSSIAN RANDOM VARIABLES

4.14 STIRLING’S FORMULA AND APPROXIMATIONS TO BINOMIAL

4.15 PROBLEMS

APPENDIX 4.A PROOF OF THEOREM 4.6

APPENDIX 4.B PROOF OF THEOREM 4.8

APPENDIX 4.C PROOF OF THEOREM 4.9

APPENDIX 4.D PROOF OF THEOREM 4.21

APPENDIX 4.E PROOF OF STIRLING’S FORMULA

APPENDIX 4.F PROOF OF THEOREM 4.27

APPENDIX 4.G PROOF OF THEOREM 4.29

5 INTRODUCTION TO RANDOM PROCESSES

5.1 RANDOM PROCESSES

5.2 DEFINITION OF A RANDOM PROCESS

5.3 EXAMPLES OF RANDOM PROCESSES

5.4 EXPERIMENTS AND EXPERIMENTAL OUTCOMES

5.5 PROTOTYPICAL EXPERIMENTS

5.6 RANDOM VARIABLES DEFINED BY A RANDOM PROCESS

5.7 CLASSIFICATION OF RANDOM PROCESSES

5.8 CLASSIFICATION: ONE-DIMENSIONAL RPs

5.9 SUMS OF RANDOM PROCESSES

5.10 PROBLEMS

6 PROTOTYPICAL RANDOM PROCESSES

6.1 INTRODUCTION

6.2 BERNOULLI RANDOM PROCESSES

6.3 POISSON RANDOM PROCESSES

6.4 CLUSTERED RANDOM PROCESSES

6.5 SIGNALLING RANDOM PROCESSES

6.6 JITTER

6.7 WHITE NOISE

6.8 1/

f

NOISE

6.9 BIRTH–DEATH RANDOM PROCESSES

6.10 ORTHOGONAL INCREMENT RANDOM PROCESSES

6.11 LINEAR FILTERING OF RANDOM PROCESSES

6.12 SUMMARY OF RANDOM PROCESSES

6.13 PROBLEMS

APPENDIX 6.A PROOF OF THEOREM 6.4

7 CHARACTERIZING RANDOM PROCESSES

7.1 INTRODUCTION

7.2 TIME EVOLUTION OF PMF OR PDF

7.3 FIRST-, SECOND-, AND HIGHER-ORDER CHARACTERIZATION

7.4 AUTOCORRELATION AND POWER SPECTRAL DENSITY

7.5 CORRELATION

7.6 NOTES ON AVERAGE POWER AND AVERAGE ENERGY

7.7 CLASSIFICATION: STATIONARITY VS NON-STATIONARITY

7.8 CRAMER’S REPRESENTATION

7.9 STATE SPACE CHARACTERIZATION of Random Processes

7.10 TIME SERIES CHARACTERIZATION

7.11 PROBLEMS

APPENDIX 7.A PROOF OF THEOREM 7.2

APPENDIX 7.B PROOF OF THEOREMS 7.3 AND 7.4

APPENDIX 7.C PROOF OF THEOREM 7.5

APPENDIX 7.D PROOF OF THEOREM 7.6

APPENDIX 7.E PROOF OF THEOREM 7.11

APPENDIX 7.F PROOF OF THEOREM 7.12

APPENDIX 7.G PROOF OF THEOREM 7.16

APPENDIX 7.H PROOF OF THEOREM 7.17

APPENDIX 7.I PROOF OF THEOREM 7.18

APPENDIX 7.J PROOF OF THEOREM 7.20

APPENDIX 7.K PROOF OF THEOREM 7.21

APPENDIX 7.L PROOF OF THEOREM 7.23

APPENDIX 7.M PROOF OF THEOREM 7.24

8 PMF AND PDF EVOLUTION

8.1 INTRODUCTION

8.2 PROBABILITY MASS/DENSITY FUNCTION ESTIMATION

8.3 NON/SEMI-PARAMETRIC PDF ESTIMATION

8.4 PMF/PDF EVOLUTION: SIGNAL PLUS NOISE

8.5 PMF EVOLUTION OF A RANDOM WALK

8.6 PDF EVOLUTION: BROWNIAN MOTION

8.7 PDF EVOLUTION: SIGNALLING RANDOM PROCESS

8.8 PDF EVOLUTION: GENERALIZED SHOT NOISE

8.9 PDF EVOLUTION: SWITCHING IN A CMOS INVERTER

8.10 PDF EVOLUTION: GENERAL CASE

8.11 PROBLEMS

APPENDIX 8.A PROOF OF THEOREM 8.1

APPENDIX 8.B PROOF OF THEOREM 8.5

APPENDIX 8.C PROOF OF THEOREM 8.11

APPENDIX 8.D PROOF OF THEOREM 8.12

9 THE AUTOCORRELATION FUNCTION

9.1 INTRODUCTION

9.2 NOTATION AND DEFINITIONS

9.3 BASIC RESULTS AND INDEPENDENCE INFORMATION

9.4 SINUSOID WITH RANDOM AMPLITUDE AND PHASE

9.5 RANDOM TELEGRAPH SIGNAL

9.6 GENERALIZED SHOT NOISE

9.7 SIGNALLING RANDOM PROCESS-FIXED PULSE CASE

9.8 GENERALIZED SIGNALLING RANDOM PROCESS

9.9 AUTOCORRELATION: JITTERED RANDOM PROCESSES

9.10 RANDOM WALK

9.11 PROBLEMS

APPENDIX 9.A PROOF OF THEOREM 9.6

APPENDIX 9.B PROOF OF THEOREM 9.7

APPENDIX 9.C PROOF OF THEOREMS 9.8 AND 9.9

APPENDIX 9.D PROOF OF THEOREM 9.12

APPENDIX 9.E PROOF OF THEOREM 9.16

APPENDIX 9.F PROOF OF THEOREM 9.17

APPENDIX 9.G PROOF OF THEOREM 9.19

APPENDIX 9.H PROOF OF THEOREM 9.20

10 POWER SPECTRAL DENSITY THEORY

10.1 INTRODUCTION

10.2 POWER SPECTRAL DENSITY THEORY

10.3 POWER SPECTRAL DENSITY OF A PERIODIC PULSE TRAIN

10.4 PSD OF A SIGNALLING RANDOM PROCESS

10.5 DIGITAL TO ANALOGUE CONVERSION

10.6 PSD OF SHOT NOISE RANDOM PROCESSES

10.7 WHITE NOISE

10.8 1/

f

NOISE

10.9 PSD OF A JITTERED BINARY RANDOM PROCESS

10.10 PSD OF A JITTERED PULSE TRAIN

10.11 PROBLEMS

APPENDIX 10.A PROOF OF THEOREM 10.1

APPENDIX 10.B PROOF OF THEOREM 10.2

APPENDIX 10.C PROOF OF THEOREM 10.3

APPENDIX 10.D PROOF OF THEOREM 10.5

APPENDIX 10.E PROOF OF THEOREM 10.6

APPENDIX 10.F PROOF OF THEOREM 10.7

APPENDIX 10.G PROOF OF THEOREM 10.8

APPENDIX 10.H PROOF OF THEOREM 10.10

APPENDIX 10.I PROOF OF THEOREM 10.13

APPENDIX 10.J PROOF OF THEOREM 10.15

11 ORDER STATISTICS

11.1 INTRODUCTION

11.2 ORDERED RANDOM VARIABLE THEORY

11.3 IDENTICAL RVs WITH UNIFORM DISTRIBUTION

11.4 UNIFORM DISTRIBUTION AND INFINITE INTERVAL

11.5 PROBLEMS

APPENDIX 11.A PROOF OF THEOREM 11.1

APPENDIX 11.B PROOF OF THEOREM 11.4

APPENDIX 11.C PROOF OF THEOREM 11.5

APPENDIX 11.D PROOF OF THEOREM 11.6

APPENDIX 11.E PROOF OF THEOREMS 11.8 AND 11.9

Appendix 11.F PROOF OF THEOREM 11.10

Appendix 11.G PROOF: MARGINAL PDF FROM JOINT PDF

APPENDIX 11.H PROOF OF THEOREM 11.12

APPENDIX 11.I PROOF OF THEOREM 11.13

APPENDIX 11.J PROOF OF THEOREM 11.15

APPENDIX 11.K PROOF OF THEOREM 11.16

APPENDIX 11.L PROOF OF THEOREM 11.18

APPENDIX 11.M PROOF OF THEOREM 11.20

APPENDIX 11.N PROOF OF THEOREM 11.24

APPENDIX 11.O PROOF OF THEOREM 11.25

APPENDIX 11.P PROOF OF THEOREM 11.26

APPENDIX 11.Q PROOF OF THEOREM 11.25

APPENDIX 11.R PROOF OF THEOREM 11.28

APPENDIX 11.S PROOF OF THEOREM 11.29

APPENDIX 11.T PROOF OF THEOREM 11.31

APPENDIX 11.U PROOF OF THEOREM 11.36

12 POISSON POINT RANDOM PROCESSES

12.1 INTRODUCTION

12.2 CHARACTERIZING POISSON RANDOM PROCESSES

12.3 PMF: NUMBER OF POINTS IN A SUBSET OF AN INTERVAL

12.4 RESULTS FROM ORDER STATISTICS

12.5 ALTERNATIVE CHARACTERIZATION FOR INFINITE INTERVAL

12.6 MODELLING WITH UNORDERED OR ORDERED TIMES

12.7 ZERO CROSSING TIMES OF RANDOM TELEGRAPH SIGNAL

12.8 POINT PROCESSES: THE GENERAL CASE

12.9 PROBLEMS

APPENDIX 12.A PROOF OF THEOREM 12.5

APPENDIX 12.B PROOF OF THEOREM 12.6

APPENDIX 12.C PROOF OF THEOREM 12.9

APPENDIX 12.D PROOF OF THEOREM 12.12

APPENDIX 12.E PROOF OF THEOREM 12.16

APPENDIX 12.F PROOF OF THEOREM 12.18

APPENDIX 12.G PROOF OF THEOREM 12.19

APPENDIX 12.H EQUIVALENCE: ORDERED–UNORDERED TIMES

13 BIRTH–DEATH RANDOM PROCESSES

13.1 INTRODUCTION

13.2 DEFINING AND CHARACTERIZING BIRTH–DEATH PROCESSES

13.3 CONSTANT BIRTH RATE, ZERO DEATH RATE PROCESS

13.4 STATE DEPENDENT BIRTH RATE - ZERO DEATH RATE

13.5 CONSTANT DEATH RATE, ZERO BIRTH RATE, PROCESS

13.6 CONSTANT BIRTH AND CONSTANT DEATH RATE PROCESS

13.7 PROBLEMS

APPENDIX 13.A PROOF OF THEOREM 13.1

APPENDIX 13.B PROOF OF THEOREM 13.2

APPENDIX 13.C PROOF OF THEOREM 13.5

APPENDIX 13.D PROOF OF THEOREM 13.8

APPENDIX 13.E PROOF OF THEOREM 13.9

14 THE FIRST PASSAGE TIME

14.1 INTRODUCTION

14.2 FIRST PASSAGE TIME

14.3 APPROACHES: ESTABLISHING THE FIRST PASSAGE TIME

14.4 MAXIMUM LEVEL AND THE FIRST PASSAGE TIME

14.5 SOLUTIONS FOR THE First Passage TIME PDF

14.6 PROBLEMS

APPENDIX 14.A PROOF OF THEOREM 14.3