Here is what is given: We have $X[t] = W[t] + 0.5*W[t-1]$ and we know that $W[k]$ is a stationary white gaussian noise process with mean equal to 0 and variance equal to 1.
We are looking for the autocorrelation function of $X$ which we name $Rxx$. Of course, I know how to numerically calculate the autocorrelation function. But we are trying to do it without actual values of the process $W$.
My approach: $Rxx[k] = E{X[k]*X[0]'}$ with $E$ being the expected-value-operator and $X'$ meaning complex-conjugated (but let's assume that $W$ is a real-valued process, so $X' == X$). Since $W$ is stationary and therefore $X$ is stationary too, we can write down the autocorrelation only depending on k, which is the time difference t1 - t2. By setting $X[0]$ in the second argument of the expected-value operator we automatically have the time difference k - 0 = k on which $Rxx$ depends. No we write: $Rxx[k] = E{(W[k] + 0.5W[k-1])(W[0] + 0.5W[1])}$ which yields: $Rxx[k] = E{W[0]W[0]} + 0.5*E{W[k]W[-1]} + 0.5*E{W[k-1]W[0]} + 0.25*E{W[k-1]W[-1]}$
Furthermore, I know that there can only be values for the $Rxx$ other than 0 for k = -1, 0, 1. The value for -1 and 1 is the same of course since $Rxx$ is symmetric to 0.
I am stuck now in how I calculate $Rxx[0]$ and $Rxx[1], Rxx[-1]$ because I do not know how to evaluate e.g. this: $E{W[0]W[0]}$ (<- should be 0, shouldn't it?), or any other of the $E{}$ - formulas.
Please tell me how I can bring my approach to an end or how to do it differently from the start!