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Determining the autocorrelation function

  1. Apr 8, 2012 #1
    1. The problem statement, all variables and given/known data
    Stress described as:

    S(t) = a0 + a1X(t) + a2X2(t)

    where X(t) is a the random displacement, a Gaussian random process and is stationary.

    Determine the autocorrelation function of S(t) (hint: remember a nice formula for the evaluation of high order moments of Gaussian random variables).

    2. Relevant equations

    Rxx(T) = E[X(t)X(t+T)]


    3. The attempt at a solution

    I am unsure of how to approach the problem using the above equation. Please advise. I can do the math I just need to see the setup with the proper functions plugged in.
     
    Last edited: Apr 8, 2012
  2. jcsd
  3. Apr 8, 2012 #2
    Maybe you are confused by having the equation be in terms of x when in this case it is in terms of s?
    [tex]R_{SS}(\tau)= E\{S(t)S(t+\tau)\}[/tex]
     
  4. Apr 8, 2012 #3
    Am I correct in thinking that I have to calculate the product of the mean of S(t) and S(t+T)?

    Also since the X(t) is stationary, can I assume that S(t) is also stationary and that the expectation would be S(t)2?
     
  5. Apr 8, 2012 #4
    You are correct in that you must calculate the mean of S(t) times S(t + T). You are also correct that S(t) is stationary. You are incorrect, however, in thinking you would compute the expectation of S(t)2. A WSS process will have an autocorrelation that is a function of a time difference. Computing the expectation of S(t)2 would give you the autocorrelation of S evaluated at a time difference of zero. You want a function for all time differences. The time difference is T the way you write it.

    Insert the definition of S(t) into that expectation. Distribute the terms on each other. Use the linearity of the expectation operator.
     
  6. Apr 8, 2012 #5
    Thank you so much for your help!

    This is part 4 of one problem. I still have 2 more complete problems to do! I will keep you in mind for any further help, if that's ok.
     
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