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

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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2008 Lecture 19 Today: (1) Discussion of AA 5, (2) Several Example Random Processes Assign HW 9, due Tuesday Nov 25 at 5pm in HW locker. Application Assignment 5 due at 5pm today. (a) 5 10 15 0.5 1 1.5 2 2.5 Time Index n X n (b) 5 10 15 0.5 1 1.5 2 2.5 Time Index n Y n Figure 1: Realization of the (a) Bernoulli and (b) filtered Bernoulli process example covered in previous lecture. 1 Random Telegraph Wave Figure 2: 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 ECE 5510 Fall 2008 2 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 , (the number of arrivals in a Poisson process at time t ). X (0) is independent of N ( t ) for any time t . (Draw a graph.) 1. What is E X ( t ) [ X ( t )]? 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) 2. What is R X ( t, )? R X ( t, ) = E X bracketleftBig X (0)( 1) N ( t ) X (0)( 1) N ( t + ) bracketrightBig = E X bracketleftBig ( X (0)) 2 ( 1) N ( t )+ N ( t + ) bracketrightBig = E N bracketleftBig ( 1) N ( t )+ N ( t + ) bracketrightBig = E N bracketleftBig ( 1) N ( t )+ N ( t )+( N ( t + )- N ( t )) bracketrightBig = E N bracketleftBig ( 1) 2 N ( t ) ( 1) ( N ( t + )- N ( t )) bracketrightBig = E N bracketleftBig ( 1) ( N ( t + )- N ( t )) bracketrightBig Assume...
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lecture19 - ECE 5510: Random Processes Lecture Notes Fall...

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