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

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EECS 501 RECITATION Nov. 19-21 Fall 2001 Given: Random process x ( t ) = A cos( ωt + θ ) where ω is a known constant. Random: A and θ are independent random variables . f θ (Θ) = 1 2 π , 0 < Θ < 2 π . 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( ωt + θ )] = E [ A ] R 2 π 0 1 2 π cos( ωt + Θ) d Θ = 0. Covar- K x ( t, s ) = R x ( t, s ) = E [ x ( t ) x ( s )] = E [ A 2 ] E [cos( ωt + θ ) cos( ωs + θ )] iance: = E [ A 2 ] 1 2 E [cos( ω ( t + s ) + 2 θ ) + cos( ω ( t - s ))] = 1 2 E [ A 2 ] cos( ω ( 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 ω and amplitude A . Different sample points different phases different sample functions. EX: x ( t ) is ideal oscillator with known amp. and freq. but random phase. f x ( t ) ( X ): t is fixed this is a derived distribution problem from θ to x ( θ ). Jaco-: f x ( X ) = 2 1 (1 / | dx | ) f θ i ) | Θ i = x - 1 ( X ) , where dx = d A cos( ωt + θ ) bian: = - A sin( ωt + θ ) = ± A p 1 - cos 2 ( ωt + θ ) = ± A 2 - X 2 . f x ( X ) = 1 A 2 - X 2 1 2 π + 1 A 2 - X 2 1 2 π = 1 / ( π A 2 - X 2 ) , | X | < A . Note: Integrates to one; diverges at X = ± A (most likely values of x ( t )). Note: 2 solns Θ 1 , Θ 2 to x ( θ ) = X θ
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