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]$X[t]$ that is:
$$ Y[t] = \frac 12 (X[t] + X[t-1]) $$$$ Y[t] = \frac 12 \left(X[t] + X[t-1]\right) $$
I have to find the autocorrelation $r_Y [t+\tau,t]$.
Here's what I got so far:
$$ r_Y [t+\tau,t] = E(Y[t+\tau]Y[t]) $$ $$ = \frac 14 E((X[t+\tau] + X[t+\tau-1])(X[t] + X[t-1])) $$\begin{align} r_Y [t+\tau,t]& = E\left(Y[t+\tau]Y[t]\right)\\ &= \frac 14 E\left\{\left(X[t+\tau] + X[t+\tau-1]\right)\left(X[t] + X[t-1]\right)\right\}\\ &= \frac 14 \left(r_X[t+\tau,t]+r_X[t+\tau,t-1]+r_X[t+\tau-1,t]+r_X[t+\tau-1,t-1]\right) \end{align} $$ = \frac 14 (r_X[t+\tau,t]+r_X[t+\tau,t-1]+r_X[t+\tau-1,t]+r_X[t+\tau-1,t-1]) $$
FromFrom 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.
