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Unformatted text preview: Example: Oscillator with Random Phase Consider the output of a sinusoidal oscillator that has a random phase and amplitude of the form: X ( t ) = cos(Ω c t + Θ) , where Θ ∼ U ([0 , 2 π ]). Writing out the explicit dependence on the underlying sample space S the oscillator output can be written as x ( t, Θ) = cos(Ω c t + Θ) . (31) This random signal falls in the continuoustime, continuous parameter, and continuous am plitude category and is useful in modeling propagation phenomena such as multipath fading. The first order distribution of this process can be found by looking at the distribution of the R.V X t (Θ) = cos(Θ + θ o ) , where Ω c t = θ o is a non random quantity. This can easily be shown via the derivative method shown in class to be of the form: f X ( x ) = 1 π √ 1 x 2 ,  x  < 1 . (32) Note that this distribution is dependent only on the set of values that the process takes and is independent of the particular sampling instant t and the constant phase offset θ o . If the secondorder distribution is needed then we use the conditional distribution of x ( t 2 ) as in : f x ( t 1 ) ,x ( t 2 ) ( x 1 ,x 2 ) = f x ( t 2 ) ( x 2 ) f x ( t 1 )  x ( t 2 ) ( x 1  x 2 ) (33) If the value of x ( t 2 ) is to be equal to x 2 then we require cos(Θ + Ω c t 2 ) = x 2 . This can happen only when : Θ = cos 1 ( x 2 ) Ω c t 2 or Θ = 2 π cos 1 ( x 2 ) Ω c t 2 , (34) where 0 ≤ cos 1 ( x 2 ) ≤ π . All other possible solutions lie outside the desired interval [0....
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This note was uploaded on 10/26/2011 for the course MATH 180C taught by Professor Eggers during the Winter '09 term at Aarhus Universitet.
 Winter '09
 Eggers

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