White-noise addition to an ARMA(1,1) process ($A=v(1+{\theta }^2)+w(1+{\varphi }^2),C=v\theta -w\varphi$)

Adding independent white noise of variance $w$ to a process with transfer function $(1+\theta z)/(1-\varphi z)$ and driving variance $v$ gives numerator $A+2C\cos \omega$ in its rational time-series spectral density. Put $P=A+2C$ and $Q=A-2C$. If both are positive, the invertible moving-average factor has coefficient $\alpha =(\sqrt{P}-\sqrt{Q})/(\sqrt{P}+\sqrt{Q})$ and driving variance $\lambda =(\sqrt{P}+\sqrt{Q}{)}^2/4$. Filtering the actual sum by $(1+\alpha B{)}^{-1}(1-\varphi B)$ produces its weak white noise driver.