HW4 27 - Yarmouk University Hijjawi Faculty of Engineering...

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Yarmouk University Hijjawi Faculty of Engineering and Technology Department of Communication Engineering Random Processes CME 610 First Semester 2009/2010 Problem Set # 4 Instructor: Dr. Khaled Gharaibeh Dec. 30, 2009 P r o b l e m 1 Given the complex process () ()  tt j t zx y where ()and () x y are zero mean independent processes with  RA a n d RB  xy If z( t ) is applied to a linear filter L whose impulse response is ct ht e Ut such that () [ () ] tL t vz a) Find the autocorrelation function of () t v b) Find the power spectral density of () t v c) Find the power of v (t) d) Find the rms bandwidth of () t v Problem 2 Given the random process: 0 cos ( ) Xt A t  where A and 0 are constants and is a random variable uniformly distributed on the interval ] 2 , 0 [ . (a) Find the autocovariance functions of X ( t ). (b) Determine if X ( t ) is a wide-sense stationary process. (c) Define new random variables ) ( ) ( ) ( ) ( ) ( 3 2 1 t X t X t Y t X t Y where is a constant (This is called Gram-Schmidt Orthogonalization). Find such that the two processes ) ( ), ( 2 1 t Y t Y are orthogonal. Problem 3 A real random process is defined by x ( t ) = A cos ( ω o t ) + w ( t ) where A is a Gaussian random variable with mean zero and variance σ A 2 and w(t) is a white noise process independent of A with variance σ w 2 (a) What is the correlation function of x (t)? (b) Can the power spectral density of x (t) be defined? If so, what is the power spectral density function?
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HW4 27 - Yarmouk University Hijjawi Faculty of Engineering...

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