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Unformatted text preview: Midterm Exam of Stochastic Processes Nov. 16, 2004 1. Let ( ) j t t ae = ω X be a random process, where a is a real constant and ω is a random variable with pdf ( ) f ϖ ω . Let ( ) ( ) ju u f e d ϖ ϖ ϖ Φ = ω ω be the characteristic function of ω . (a) (10 points) Express the mean ( ) t η and autocorrelation 1 2 ( , ) R t t of ( ) t X in terms of ( ) Φ ω . (b) (4 points) Determine whether ( ) t X is a WSS process. 2. (15 points) Let ( ) t X be a WSS random process with autocorrelation ( ) R τ XX and ( ) t Y be the output of an linear timeinvariant (LTI) system with input ( ) t X . Let the LTI system can be expressed by 3 ( ) 2 ( ) ( ) ( ) 2 ( ) t t t t t t + + = + Y Y Y X X (a) Find the system function ( ) H s of the LTI system. (b) Find the power spectral density ( ) S ϖ YY of ( ) t Y . (c) Find 2 { ( ) } E t Y . 3. (10 points) Assume that the random points i t , which represent the times that earthquakes ( o ) occur, are Poisson points. Let λ be the average number of...
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This note was uploaded on 07/21/2009 for the course CM EM5102 taught by Professor Sinhorngchen during the Fall '08 term at National Chiao Tung University.
 Fall '08
 SinHorngChen

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