lecture19 - ECE 5510 Random Processes Lecture Notes Fall...

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ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 19 Today: (1) Discussion of AA 5, (2) Several Example Random Processes (Y&G 10.12) HW 8 due today at 5pm in HW locker. Application Assign- ment 5 due Tue, Nov. 24. HW 9 is assigned this coming Tue., due Thu. after Thanks- giving (Dec. 3). For Tuesday, move on to Chapter 11 of Y&G (11.1, 11.5, 11.8). 1 Random Telegraph Wave Figure 1: The telegraph wave process is generated by switching between +1 and -1 at every arrival of a Poisson process. This was originally used to model the signal sent over telegraph lines. Today it is still useful in digital communications, and digital control systems. We model each flip as an arrival in a Poisson process. It is a model for a binary time-varying signal: X ( t ) = X (0)( 1) N ( t ) Where X (0) is -1 with prob. 1/2, and 1 with prob. 1/2, and N ( t ) is a Poisson counting process with rate λ , (the number of arrivals in a Poisson process at time t ). X (0) is independent of N ( t ) for any time t . See Figure 1. 1. What is E X ( t ) [ X ( t )]? μ X ( t ) = E X ( t ) bracketleftBig X (0)( 1) N ( t ) bracketrightBig = E X [ X (0)] E N bracketleftBig ( 1) N ( t ) bracketrightBig = 0 · E N bracketleftBig ( 1) N ( t ) bracketrightBig = 0 (1)
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ECE 5510 Fall 2009 2 2. What is R X ( t, δ )? (Assume τ 0.) R X ( t, τ ) = E X bracketleftBig X (0)( 1) N ( t ) X (0)( 1) N ( t + τ ) bracketrightBig = E X bracketleftBig ( X (0)) 2 ( 1) N ( t )+ N ( t + τ ) bracketrightBig = E N bracketleftBig ( 1) N ( t )+ N ( t + τ ) bracketrightBig = E N bracketleftBig ( 1) N ( t )+ N ( t )+( N ( t + τ ) - N ( t )) bracketrightBig = E N bracketleftBig ( 1) 2 N ( t ) ( 1) ( N ( t + τ ) - N ( t )) bracketrightBig = E N bracketleftBig ( 1) 2 N ( t ) bracketrightBig E N bracketleftBig ( 1) ( N ( t + τ ) - N ( t )) bracketrightBig = E N bracketleftBig ( 1) ( N ( t + τ ) - N ( t )) bracketrightBig Remember the trick you see inbetween lines 3 and 4? N ( t ) and N ( t + τ ) represent the number of arrivals in overlap- ping intervals . Thus ( 1) N ( t ) and ( 1) N ( t + τ ) are NOT in- dependent. But N ( t ) and N ( t + τ ) N ( t ) DO represent the number of arrivals in non-overlapping intervals, so we can pro- ceed to simplify the expected value of the product (in line 6) to the product of the expected values (in line 7). This dif- ference is just the number of arrivals in a period τ , call it K = N ( t + τ )
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