# hw7 - Virginia Tech The Bradley Department of Electrical...

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Virginia Tech The Bradley Department of Electrical and Computer Engineering ECE 5605 – Stochastic Signals and Systems – Fall 08 Problem Set 7 Problem 1. Let θ be a random variable uniformly distributed on [0 , 2 π ). Consider the process X ( t ) = cos 2 ( ωt + θ ) . Find the mean value of X ( t ). Problem 2. Let Y ( t ) = X ( t ) - aX ( t + d ), where X ( t ) is a Gaussian random process. a. Find the mean and autocovariance of Y ( t ). b. Find the pdf of Y ( t ). c. Find the joint pdf of Y ( t 1 ) and Y ( t 2 ). Problem 3. Let X ( t ) be a zero-mean Gaussian random process with autocovariance function given by C X ( t 1 , t 2 ). If X ( t ) is the input to a “square law detector,” then the output is Y ( t ) = X ( t ) 2 . Find the mean and autocovariance of the output Y ( t ). Hint : If X 1 , X 2 , X 3 , X 4 are zero-mean jointly Gaussian random variables, E [ X 1 X 2 X 3 X 4 ] = E [ X 1 X 2 ] E [ X 3 X 4 ] + E [ X 1 X 3 ] E [ X 2 X 4 ] + E [ X 1 X 4 ] E [ X 2 X 3 ] . Problem 4. Consider a linear combination of two sinusoids: X ( t ) = A 1 cos ( ω 0 t + θ 1 ) + A 2 cos 2 ω 0 t + θ 2 , where θ 1 and θ 2

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