handout_week10_c

handout_week10_c - Stochastic Signals and Systems Random...

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Unformatted text preview: Stochastic Signals and Systems Random Processes Virginia Tech Fall 2008 Gaussian Random Variables A random process X ( t ) is a Gaussian random process if the samples X ( t 1 ) , X ( t 2 ) ,..., X ( t k ) are jointly Gaussian random variables for all k , and all choices of t 1 , t 2 ,..., t k . Gaussian random processes therefore have the special property that their joint pdfs are completely specified by the vector of means and by the covariance matrix m = m X ( t 1 ) . . . m X ( t k ) C = C X ( t 1 , t 1 ) C X ( t 1 , t 2 ) C X ( t 1 , t k ) C X ( t 2 , t 1 ) C X ( t 2 , t 2 ) C X ( t 2 , t k ) . . . . . . . . . C X ( t k , t 1 ) C X ( t k , t 2 ) C X ( t k , t k ) Gaussian random processes also have the property that the linear operation on a Gaussian process results in another Gaussian random process . Problem Let X ( t ) and Y ( t ) be independent Gaussian random processes with zero-means and the same covariance function C ( t 1 , t 2 ) ....
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handout_week10_c - Stochastic Signals and Systems Random...

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