lecture17 - ECE 5510: Random Processes Lecture Notes Fall...

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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2008 Lecture 17 Today: Random Processes: (1) Expectation (2) Correlation & Covariance (3) Wide Sense Stationarity 1 Expectation of Random Processes 1.1 Expected Value and Correlation Defn: Expected Value of a Random Process The expected value of continuous time random process X ( t ) is the deterministic function X ( t ) = E X ( t ) [ X ( t )] for the discrete-time random process X n , X [ n ] = E X n [ X n ] Example: What is the expected value of a Poisson process? Let Poisson process X ( t ) have arrival rate . We know that X ( t ) = E X ( t ) [ X ( t )] = summationdisplay x =0 x ( t ) x x ! e- t = e- t summationdisplay x =0 x ( t ) x x ! = e- t summationdisplay x =1 ( t ) x ( x- 1)! = ( t ) e- t summationdisplay x =1 ( t ) x- 1 ( x- 1)! = ( t ) e- t summationdisplay y =0 ( t ) y ( y )! = ( t ) e- t e t = t (1) This is how we intuitively started deriving the Poisson process - we said that it is the process in which on average we have arrivals per unit time. Thus wed certainly expect to see t arrivals after a time duration t . ECE 5510 Fall 2008 2 Example: What is the expected value of X n , the number of successes in a Bernoulli process after n trials? We know that X n is Binomial, with mean np . This is the mean function, if we consider it to be a function of n : X [ n ] = E X [ X n ] = np . 1.2 Autocovariance and Autocorrelation These next two definitions are the most critical concepts of the rest of the semester. Generally, for a time-varying signal, we often want to know two things: How to predict its future value. (It is not deterministic.) WeHow to predict its future value....
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This note was uploaded on 09/15/2011 for the course ECE 5510 taught by Professor Chen,r during the Fall '08 term at University of Utah.

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lecture17 - ECE 5510: Random Processes Lecture Notes Fall...

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