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Unformatted text preview: EEM3066: Random Processes & Queueing Theory Chapter 2: Random Processes Page 1 of 30 Chapter 2: Random Processes Objective: To introduce the concept of random process and related terms Syllabus: • Basic concepts and definition. • Classification of random processes. • Stationary process and independence property. • Autocorrelation and correlation functions. • Ergodicity. At the completion of the chapter, students should be able to: Understand the concepts of random processes. Contents: 2.1 Introduction 2.2 Basic Concepts of Random Process 2.3. Description of a random process 2.4 Probabilistic Descriptions of Random Processes 2.5 Statistical Descriptions of Random Processes 2.5.1 Ensemble average (or statistical average), { }{ } ⋅ ⋅ , E 2.5.2 TimeAverage, ⋅ 2.6 Autocorrelation function 2.6.1 Concept of Correlation 2.6.2 Autocorrelation 2.7 Stationary Random Process 2.7.1 Strictsense Stationary (SSS) Random Process 2.7.2 Strictsense Stationary (SSS) Random Sequence EEM3066: Random Processes & Queueing Theory Chapter 2: Random Processes Page 2 of 30 2.7.3 Stationary of order k . 2.7.4 Widesense Stationary (WSS) Random Process 2.8 TimeAverage 2.9 Ergodic Processes 2.10 Autocorrelation Function Properties 2.11 CrossCorrelation and Covariance Functions 2.11.1 Joint Random Processes 2.11.2 CrossCorrelation Function 2.11.3 (Cross) Covariance Function 2.11.4 Autocovariance 2.11.5 Correlation coefficient 2.12 Summary References Appendix A: Fourier Transform Pair EEM3066: Random Processes & Queueing Theory Chapter 2: Random Processes Page 3 of 30 2.1 Introduction A random variable (RV) is a number , ( 29 x ζ or ( 29 X ζ or simply X , assigned to every outcome ζ of an experiment. In other words, a random variable is defined as a function that maps each point in a sample space to a point on the real line. This allows us to describe outcome of experiment in term of numerical number rather than the event. The definition of random process involves generalisation of the concept of random variable (function with parameter ζ ) to include time as additional parameter. 2.2 Basic Concepts of Random Process If we take into account the time factor, we will obtain a function of time t and the experiment outcome ζ , ( 29 ζ , t X or simply X ( t ). This function X ( t ) is referred to as a random (or stochastic) process. We may view a stochastic process as a family of random variables indexed by the parameter t , { } I t t X ), , ( ε ζ . This family is called a random process . Figure 2.1: Random process (H Stark pp373, Fig8.11) In general, instead of time dependence, we may have space dependence, etc. The function is still called a random process....
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This note was uploaded on 08/21/2009 for the course FET 44 taught by Professor ;im during the Spring '09 term at Multimedia University, Cyberjaya.
 Spring '09
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