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# HW2 - 1 EE 7615 Problem Set 2 Due 9,19,11 1 Please attempt...

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1 EE 7615 Problem Set # 2 Due: 9,19,11 1) Please attempt each problem on a new page. 2) Show all your work clearly. 1) A system has input X which is random with mean E ( X ) = m . The output is Y . Suppose the conditional density of Y given X = x is Gaussian with mean x , i.e. f Y | X ( y | x ) = 1 2 πσ exp[ - ( y - x ) 2 2 σ 2 ] Find E ( Y ) . 2) { X ( t ) } is a WSS random process with m X = 0 and σ 2 X = 5 . Can any of the following functions be a candidate for the autocorrelation function of { X ( t ) } ? Justify your answers. a) R X ( τ ) = 6 u ( τ ) e - 3 τ where u ( τ ) = 1 for τ 0 and 0 otherwise. b) R X ( τ ) = 5 sin(5 τ ) . c) R X ( τ ) = 5 cos(5 τ ) . d) R X ( τ ) = 5(1 + 2 τ 2 ) - 1 + 5 sin( τ ) . 3) A linear time invariant filter has transfer function H ( f ) where H ( f ) = ( 1 1 ≤ | f | ≤ 2 0 otherwise. The input to this filter is the Gaussian WSS random process { X ( t ) } with mean zero and autocorrelation function R X ( τ ) = e -| τ | . a) Find and sketch the power spectral density of the output process { Y ( t ) } . b) Find the pdf of the random variable Y t .

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