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Calculation of the error function.

  1. Apr 5, 2013 #1

    MathematicalPhysicist

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    Gold Member

    I have the next two signals:

    X(t) and G(t) and a random process Y(t)=G(t)X(t) where X(t) and G(t) are wide sense stationary with expectation values: E(X)=0, E(G)=1.

    Now, it's also given that ##G(t)=\cos(3t+\psi)## where ##\psi## is uniformly distributed on the interval ##(0,2\pi]## and is statistically independent of X(t).

    The signal X(t) is transfered through a low pass filter, given in the frequency domain as ##H(\Omega)=1## when ##\Omega \leq 4\pi## and otherwise zero.

    I am given that ##Y(\Omega)=X(\Omega)H(\Omega)##, and I want to calculate:

    ##\epsilon = E((X(t)-Y(t))^2)##

    I guess I can go to the frequency domain, but I also need to use the http://en.wikipedia.org/wiki/Law_of_total_expectation

    But I am not sure how exactly to condition this, thanks in advance.
     
    Last edited by a moderator: Apr 5, 2013
  2. jcsd
  3. Apr 5, 2013 #2

    Mark44

    Staff: Mentor

    For LaTeX, use either $$ on each end or ## (for inline LaTeX) on each end of your expressions. The $ pair is pretty much equivalent to [ tex ] and the # pair is equivalent to [ itex ] (all without extra spaces inside the brackets).
     
  4. Apr 5, 2013 #3
    Hmmmmm... I am not that expert but isn't it

    epsilon = E((X(t)-Y(t))^2) = E((X-G X)^2)= E((X(1-G))^2) = E(X^2 (1-G)^2) =
    E (X^2)*E((1-G)^2) = E^2(X)*E^2(1-G) = 0
     
  5. Apr 5, 2013 #4

    MathematicalPhysicist

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    Gold Member

    Sorry, I have abuse of notation, I should have denoted:
    $$Z(\Omega)=Y(\Omega)H(\Omega)$$

    And I am looking for $$E((X(t)-Z(t))^2)$$

    I was tired yesterday evening, a long exam that day.
     
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