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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 continuous-time, continuous parameter, and continuous am- plitude category and is useful in modeling propagation phenomena such as multi-path 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 second-order 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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- Winter '09