Find the autocorrelation function of $v_t=\frac{1}{9}(\omega_{t-1}+\omega_{t}+\omega_{t+1})$
What I have tried:
$$\rho(h) = \frac{\gamma(t+h, t)}{\sqrt{\gamma(t+h, t+h)\gamma(t,t)}}=\frac{\gamma(h)}{\gamma(0)}$$
I have actually got this for $h=0$, however, I am struggling with $h=1$, for example:
$$\rho(1) = \frac{\gamma(t+1,t)}{\sqrt{\gamma(t+1, t+1)\gamma(t,t)}}$$
Therefore,
$$\gamma(t+1,t) = \frac{1}{9}cov([\omega_t+\omega_{t+1}+\omega_{t+2}],[\omega_{t-1}+\omega_{t}+\omega_{t+1}])$$
I am unsure with simplifying this.