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

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ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 17 Today: Random Processes: (1) Autocorrelation & Autocovari- ance, (2) Wide Sense Stationarity, (3) PSD As a brief overview of the rest of the semester. We’re going to talk about autocovariance (and autocorrela- tion, a similar topic), that is, the covariance of a random signal with itself at a later time. The autocovariance tells us something about our ability to predict future values ( k in ad- vance) of Y k . The higher C Y [ k ] is, the more the two values separated by k can be predicted. Autocorrelation is critical to the next topic: What does the random signal look like in the frequency domain? More specif- ically, what will it look like in a spectrum analyzer (set to av- erage). This is important when there are specific limits on the bandwidth of the signal (imposed, for example, by the FCC) and you must design the process in order to meet those limits. We can analyze what happens to the spectrum of a random process when we pass it through a filter. Filters are every- where in ECE, so this is an important tool. We’ll also discuss new random processes, including Gaussian random processes, and Markov chains. Markov chains are particularly useful in the analysis of many discrete-time engi- neered systems, for example, computer programs, networking protocols, and networks like the Internet. 1 Expectation of Random Processes 1.1 Expected Value and Correlation Def’n: Expected Value of a Random Process The expected value of continuous time random process X ( t ) is the deterministic function μ X ( t ) = E X ( t ) [ X ( t )] for the discrete-time random process X n , μ X [ n ] = E X n [ X n ]
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ECE 5510 Fall 2009 2 Example: What is the expected value of a Poisson process? Let Poisson process X ( t ) have arrival rate λ . We know that μ X ( t ) = E X ( t ) [ X ( t )] = summationdisplay x =0 x ( λt ) x x ! e - λt = e - λt summationdisplay x =0 x ( λt ) x x ! = e - λt summationdisplay x =1 ( λt ) x ( x - 1)! = ( λt ) e - λt summationdisplay x =1 ( λt ) x - 1 ( x - 1)! = ( λt ) e - λt summationdisplay y =0 ( λt ) y ( y )! = ( λt ) e - λt e λt = λt (1) This is how we intuitively started deriving the Poisson process - we said that it is the process in which on average we have λ arrivals per unit time. Thus we’d certainly expect to see λt
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