17recit - EECS 501 RECITATION Nov. 19-21 Fall 2001 Given:...

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Unformatted text preview: EECS 501 RECITATION Nov. 19-21 Fall 2001 Given: Random process x ( t ) = A cos( t + ) where is a known constant. Random: A and are independent random variables . f () = 1 2 , < < 2 . Goal: Compute the mean E [ x ( t )] and covariance K x ( t, s ) functions of x ( t ). Mean: E [ x ( t )] = E [ A ] E [cos( t + )] = E [ A ] R 2 1 2 cos( t + ) d = 0. Covar- K x ( t, s ) = R x ( t, s ) = E [ x ( t ) x ( s )] = E [ A 2 ] E [cos( t + ) cos( s + )] iance: = E [ A 2 ] 1 2 E [cos( ( t + s ) + 2 ) + cos( ( t- s ))] = 1 2 E [ A 2 ] cos( ( t- s )). Note: (1) x ( t ) WSS; (2) Need only E [ A 2 ], not f a ( A ); (3) Can have E [ A ] 6 = 0. Now: Let A > 0 be a known constant. Compute f x ( t ) ( X ) and f x ( t ) | x ( s ) ( X t | X s ). Note: Sample functions are sinusoids with frequency and amplitude A . Different sample points different phases different sample functions....
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