Autocovariance of a causal ARMA(1,1) process (${\gamma }_k=\frac{{\sigma }^2(\varphi +\theta )(1+\varphi \theta )}{1-{\varphi }^2}{\varphi }^{k-1}(k\ge 1)$)

With positive moving-average sign and $|\varphi |<1$, the noise coefficients are ${\psi }_0=1$ and ${\psi }_j=(\varphi +\theta ){\varphi }^{j-1}$ for $j\ge 1$. Summing their overlap gives ${\gamma }_0={\sigma }^2(1+{\theta }^2+2\varphi \theta )/(1-{\varphi }^2)$ and ${\gamma }_k={\sigma }^2(\varphi +\theta )(1+\varphi \theta ){\varphi }^{k-1}/(1-{\varphi }^2)$ for $k\ge 1$. Negative lags follow by symmetry.