# Probability and Random Processes

## 2nd Edition

### With Applications to Signal Processing and Communications

**Authors:**Scott Miller

**Hardcover ISBN:**9780123869814

**eBook ISBN:**9780123870131

**Imprint:**Academic Press

**Published Date:**11th January 2012

**Page Count:**522

## Description

*Probability and Random Processes, Second Edition* presents pertinent applications to signal processing and communications, two areas of key interest to students and professionals in today's booming communications industry. The book includes unique chapters on narrowband random processes and simulation techniques. It also describes applications in digital communications, information theory, coding theory, image processing, speech analysis, synthesis and recognition, and others.

Exceptional exposition and numerous worked out problems make this book extremely readable and accessible. The authors connect the applications discussed in class to the textbook. The new edition contains more real world signal processing and communications applications. It introduces the reader to the basics of probability theory and explores topics ranging from random variables, distributions and density functions to operations on a single random variable. There are also discussions on pairs of random variables; multiple random variables; random sequences and series; random processes in linear systems; Markov processes; and power spectral density.

This book is intended for practicing engineers and students in graduate-level courses in the topic.

## Key Features

- Exceptional exposition and numerous worked out problems make the book extremely readable and accessible
- The authors connect the applications discussed in class to the textbook
- The new edition contains more real world signal processing and communications applications
- Includes an entire chapter devoted to simulation techniques

## Readership

Graduate-level courses in the topic with secondary interest to professionals

## Table of Contents

Preface

CHAPTER 1. Introduction

1.1 A Speech Recognition System

1.2 A Radar System

1.3 A Communication Network

CHAPTER 2. Introduction to Probability Theory

2.1 Experiments, Sample Spaces, and Events

2.2 Axioms of Probability

2.3 Assigning Probabilities

2.4 Joint and Conditional Probabilities

2.5 Basic Combinatorics

2.6 Bayes’s Theorem

2.7 Independence

2.8 Discrete Random Variables

2.9 Engineering Application—An Optical Communication System

CHAPTER 3. Random Variables, Distributions, and Density Functions

3.1 The Cumulative Distribution Function

3.2 The Probability Density Function

3.3 The Gaussian Random Variable

3.4 Other Important Random Variables

3.5 Conditional Distribution and Density Functions

3.6 Engineering Application: Reliability and Failure Rates

CHAPTER 4. Operations on a Single Random Variable

4.1 Expected Value of a Random Variable

4.2 Expected Values of Functions of Random Variables

4.3 Moments

4.4 Central Moments

4.5 Conditional Expected Values

4.6 Transformations of Random Variables

4.7 Characteristic Functions

4.8 Probability-Generating Functions

4.9 Moment-Generating Functions

4.10 Evaluating Tail Probabilities

4.11 Engineering Application—Scalar Quantization

4.12 Engineering Application—Entropy and Source Coding

CHAPTER 5. Pairs of Random Variables

5.1 Joint Cumulative Distribution Functions

5.2 Joint Probability Density Functions

5.3 Joint Probability Mass Functions

5.4 Conditional Distribution, Density, and Mass Functions

5.5 Expected Values Involving Pairs of Random Variables

5.6 Independent Random Variables

5.7 Jointly Gaussian Random Variables

5.8 Joint Characteristic and Related Functions

5.9 Transformations of Pairs of Random Variables

5.10 Complex Random Variables

5.11 Engineering Application: Mutual Information, Channel Capacity, and Channel Coding

CHAPTER 6. Multiple Random Variables

6.1 Joint and Conditional PMFs, CDFs, and PDFs

6.2 Expectations Involving Multiple Random Variables

6.3 Gaussian Random Variables in Multiple Dimensions

6.4 Transformations Involving Multiple Random Variables

6.5 Estimation and Detection

6.6 Engineering Application: Linear Prediction of Speech

CHAPTER 7. Random Sums and Sequences

7.1 Independent and Identically Distributed Random Variables

7.2 Convergence Modes of Random Sequences

7.3 The Law of Large Numbers

7.4 The Central Limit Theorem

7.5 Confidence Intervals

7.6 Random Sums of Random Variables

7.7 Engineering Application: A Radar System

CHAPTER 8. Random Processes

8.1 Definition and Classification of Processes

8.2 Mathematical Tools for Studying Random Processes

8.3 Stationary and Ergodic Random Processes

8.4 Properties of the Autocorrelation Function

8.5 Gaussian Random Processes

8.6 Poisson Processes

8.7 Engineering Application—Shot Noise in a p–n Junction Diode

CHAPTER 9. Markov Processes

9.1 Definition and Examples of Markov Processes

9.2 Calculating Transition and State Probabilities in Markov Chains

9.3 Characterization of Markov Chains

9.4 Continuous Time Markov Processes

9.5 Engineering Application: A Computer Communication Network

9.6 Engineering Application: A Telephone Exchange

CHAPTER 10. Power Spectral Density

10.1 Definition of PSD

10.2 The Wiener–Khintchine–Einstein Theorem

10.3 Bandwidth of a Random Process

10.4 Spectral Estimation

10.5 Thermal Noise

10.6 Engineering Application: PSDs of Digital Modulation Formats

CHAPTER 11. Random Processes in Linear Systems

11.1 Continuous Time Linear Systems

11.2 Discrete-Time Linear Systems

11.3 Noise Equivalent Bandwidth

11.4 Signal-to-Noise Ratios

11.5 The Matched Filter

11.6 The Wiener Filter

11.7 Bandlimited and Narrowband Random Processes

11.8 Complex Envelopes

11.9 Engineering Application: An Analog Communication System

CHAPTER 12. Simulation Techniques

12.1 Computer Generation of Random Variables

12.2 Generation of Random Processes

12.3 Simulation of Rare Events

12.4 Engineering Application: Simulation of a Coded Digital Communication System

APPENDIX A. Review of Set Theory

APPENDIX B. Review of Linear Algebra

APPENDIX C. Review of Signals and Systems

APPENDIX D. Summary of Common Random Variables

Continuous Random Variables

Discrete Random Variables

APPENDIX E. Mathematical Tables

A. Trigonometric Identities

B. Series Expansions

C. Some Common Indefinite Integrals

D. Some Common Definite Integrals

E. Definitions of Some Common Continuous Time Signals

F. Fourier Transforms

G. z-Transforms

H. Laplace Transforms

I. Table of the Q-function

APPENDIX F. Numerical Methods for Evaluating the Q-Function

Index

## Details

- No. of pages:
- 522

- Language:
- English

- Copyright:
- © Academic Press 2012

- Published:
- 11th January 2012

- Imprint:
- Academic Press

- eBook ISBN:
- 9780123870131

- Hardcover ISBN:
- 9780123869814

## About the Author

### Scott Miller

### Affiliations and Expertise

Texas A & M University

## Reviews

"...a utilitarian toolkit, to help the reader learn how to solve problems, while skirting technical issues such as measure theory…fills a particular niche in the literature, and is certainly recommended by me." **--MathSciNet**

"...primarily focused toward undergraduate students in areas of electrical and computer engineering...the book is very well written and wasy to read and follow." **--Ali Esmaili, in TECHNOMETRICS, VOL. 47, 2005**

"...very well written...I think this is a highly valuable textbook that is very recommendable for students, researchers as well as practitioners interested in signal processing and communications." **--Stefan Reh, Carnegie Mellon University**

"...it is well written, providing the intended readership with tools and methods to study and solve problems concerning random signals and systems." **--Evelyn Buckwar, Zentralblatt MATH Berlin**

"Electrical and computer engineers Miller (Texas A&M U.) and Childers (emeritus, U. of Florida) present a textbook for an upper-division undergraduate course in probability, or an introductory graduate course in random processes within an electrical engineering curriculum. Students are assumed to have the background appropriate to those levels. The area is primarily mathematical, but they treat the mathematics as a tool for engineers rather than a rigorous or elegant entity in its own right. They seek a balance between explaining elementary concepts clearly and providing enough depth that students can study modern communications systems, control systems, signal processing techniques, and other applications." **--Reference and Research Book News, Inc.**