I am trying to get an understanding of autocorrelation and I am having some issues with trying to understand the process.

I have a Bernoulli process called $X[t]$. In this process, $P(X[t] = 1) = p$ and $P(X[t] = 0) = 1-p.$

We have a new process formed from X[t] that is:

$$ Y[t] = \frac 12 (X[t] + X[t-1]) $$

I have to find the autocorrelation $r_Y [t+\tau,t]$.

Here's what I got so far:

$$ r_Y [t+\tau,t] = E(Y[t+\tau]Y[t]) $$ $$ = \frac 14 E((X[t+\tau] + X[t+\tau-1])(X[t] + X[t-1])) $$ $$ = \frac 14 (r_X[t+\tau,t]+r_X[t+\tau,t-1]+r_X[t+\tau-1,t]+r_X[t+\tau-1,t-1]) $$

From there, I can get $r_x[t+\tau,t]$:

$$ r_X[t+\tau,t] = \begin{cases} p^2 & \quad \text{if } \tau \neq 0\\ p & \quad \text{if } \tau = 0\\ \end{cases} $$

But the wall that I am running into (and maybe it's because it's not making sense to me because of lack of sleep) is how you substitute that expression back into $r_Y[t+\tau,t]$? I know you end up with three different answers, but trying to substitute it back in seems to cause a mess for me. I am hoping someone will help me with that part.