Determining the autocorrelation function

[L.Richter]

Homework Statement

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).

Homework Equations

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

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.

 

[RoshanBBQ]

Maybe you are confused by having the equation be in terms of x when in this case it is in terms of s?
$$R_{SS}(\tau)= E\{S(t)S(t+\tau)\}$$

 

[L.Richter]

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?

 

[RoshanBBQ]

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?

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.

 

[L.Richter]

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.