assignment10

# assignment10 - E X 3 t 1 6 Let X t and Y t be independent...

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Assignment 10 1. Let Y ( t ) = X ( t + d ) - X ( t ), 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 ) and Y ( t + s ). (d) Show that Y ( t ) is a Gaussian random process. 2. 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 ). 3. Let Y ( t ) = X 2 ( t ), where X ( t ) is the Wiener process. (a) Find the pdf of Y ( t ). (b) Find the conditional pdf of Y ( t 2 ) given Y ( t 1 ). 4. Let Z ( t ) = X ( t ) - aX ( t - s ), where X ( t ) is the Wiener process. (a) Find the pdf of Z ( t ). (b) Find m Z ( t ) and C Z ( t 1 ,t 2 ). 5. Let X ( t ) be deﬁned by X ( t ) = A cos ωt + B sin ωt, Where A and B are iid random variables. (a) Under what conditions is X ( t ) wide-sense stationary? (b) Show that X ( t ) is not stationary. hint : Consider

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Unformatted text preview: E [ X 3 ( t )]. 1 6. Let X ( t ) and Y ( t ) be independent, wide-sense stationary random processes with zero means and the same covariance function C X ( τ ). Let Z ( t ) be deﬁned by Z ( t ) = 3 X ( t )-5 Y ( t ) (a) Determine whether Z ( t ) is also wide-sense stationary. (b) Determine the pdf of Z ( t ) if X ( t ) and Y ( t ) are also jointly Gaussian zero-mean random processes with C X ( τ ) = 4 e-| τ | . (c) Find the joint pdf of Z ( t 1 ) and Z ( t 2 ) in part b. (d) Find the cross-covariance between Z ( t ) and X ( t ). Are Z ( t ) and X ( t ) jointly stationary random processes? (e) Find the joint pdf of Z ( t 1 ) and X ( t 2 ) in part b. Hint : Use auxilliary variables. 2...
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assignment10 - E X 3 t 1 6 Let X t and Y t be independent...

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