 Solution

For the causal case $|\varphi |<1$, expandHence the white noise coefficients are ${\psi }_0=1$ and ${\psi }_j=(\varphi +\theta ){\varphi }^{j-1}$ for $j\ge 1$. Orthogonality of white noise gives ${\gamma }_k={\sigma }^2{\sum }_{j\ge 0}{\psi }_j{\psi }_{j+k}$ for $k\ge 0$. Summing the geometric series yields the autocovariance of a causal ARMA(1,1) process:For $\varphi =0$, this means ${\gamma }_1={\sigma }^2\theta$ and ${\gamma }_k=0$ for $k\ge 2$, together with ${\gamma }_0={\sigma }^2(1+{\theta }^2)$. For $\theta =-\varphi$, all positive-lag covariances vanish and ${\gamma }_0={\sigma }^2$, as expected from cancellation of the two filter polynomials.