# Homework Help: Determining the autocorrelation function

1. Apr 8, 2012

### L.Richter

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. Apr 8, 2012

### RoshanBBQ

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

3. Apr 8, 2012

### 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?

4. Apr 8, 2012

### RoshanBBQ

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.

5. Apr 8, 2012

### 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.