= Solution
For a real <weakly stationary process>, extend its <autocovariance> by $\gamma_{-k}=\gamma_k$. The exact <existence of a time-series spectral density> condition is that its <spectral measure of a stationary time series> be <absolutely continuous with respect to> <Lebesgue measure>. Absolute summability $\gamma_0+2\sum_{k\geq1}|\gamma_k|<\infty$ is a useful sufficient condition, not a necessary one.
We use the conventional angular-frequency density on $[-\pi,\pi]$, restricted to $[0,\pi]$ by symmetry. Under the absolute-summability condition, the <Fourier series> and its inverse relation are
$$
\boxed{f(\omega)=\frac1{2\pi}\left(\gamma_0+2\sum_{k\geq1}\gamma_k\cos(k\omega)\right),\qquad
\gamma_k=2\int_0^\pi f(\omega)\cos(k\omega)\,d\omega.}
$$
Thus $2\int_0^\pi f=\gamma_0$. If a one-sided density is instead normalized to integrate to the full <variance>, use $g=2f$ and omit the factor two in the inverse formula. This is the <positive-frequency spectral normalization> convention difference. With merely an integrable spectral density, the inverse relation remains valid; one must not assume pointwise convergence of the unweighted Fourier series. Its <Fejér sums> recover the density in $L^1$:
$$
f_N(\omega)=\frac1{2\pi}\left[\gamma_0+2\sum_{k=1}^N\left(1-\frac{k}{N+1}\right)\gamma_k\cos(k\omega)\right]\longrightarrow f.
$$
If $X$ and $Y$ are <independent> stationary processes, the cross <covariances> vanish, so
$$
\gamma_Z(k)=\operatorname{Cov}(X_{t+k}+Y_{t+k},X_t+Y_t)=\gamma_X(k)+\gamma_Y(k).
$$
Linearity of the inverse <Fourier series> relation, or addition of the <spectral measures of a stationary time series>, gives \b[$f_Z=f_X+f_Y$]. Independence can in fact be weakened to zero cross-covariances at every lag.
For the <ARMA>$(1,1)$ representation, away from an uncancelled unit root the transfer function gives
$$
\boxed{f_X(\omega)=\frac v{2\pi}\frac{1+\theta^2+2\theta\cos\omega}{1+\phi^2-2\phi\cos\omega}.}
$$
The usual causal case has $|\phi|<1$; the same expression holds for the two-sided stationary solution when $|\phi|>1$. We work in the nondegenerate case $v,w>0$ and $\phi\ne\pm1$; cancellations and zero-noise cases are obtained by the appropriate reduced representation or limits. Since <independent> <white noise> of <variance> $w$ contributes $w/(2\pi)$, put
$$
A=v(1+\theta^2)+w(1+\phi^2),\qquad C=v\theta-w\phi.
$$
Then
$$
\boxed{f_Z(\omega)=\frac1{2\pi}\frac{A+2C\cos\omega}{1+\phi^2-2\phi\cos\omega}.}
$$
The <white-noise addition to an ARMA(1,1) process> problem is therefore the factorization
$$
A+2C\cos\omega=\lambda(1+\alpha^2+2\alpha\cos\omega).
$$
Define
$$
P=A+2C=v(1+\theta)^2+w(1-\phi)^2,\qquad
Q=A-2C=v(1-\theta)^2+w(1+\phi)^2.
$$
Both are positive under the stated nondegeneracy conditions. An invertible choice is
$$
\boxed{\alpha=\frac{\sqrt P-\sqrt Q}{\sqrt P+\sqrt Q},\qquad
\lambda=\frac{(\sqrt P+\sqrt Q)^2}{4}.}
$$
These satisfy $|\alpha|<1$, $\lambda(1+\alpha)^2=P$, and $\lambda(1-\alpha)^2=Q$. Hence
$$
\boxed{\left(\frac{1+\alpha}{1-\alpha}\right)^2=
\frac{v(1+\theta)^2+w(1-\phi)^2}{v(1-\theta)^2+w(1+\phi)^2}.}
$$
In particular, it is the autoregressive coefficient $\phi$ that enters the terms from the added observation noise.
To obtain a representation of the actual $Z$, define
$$
\xi_t=(1+\alpha B)^{-1}(1-\phi B)Z_t.
$$
The stable inverse exists because $|\alpha|<1$, and spectral filtering gives $f_\xi=\lambda/(2\pi)$. Thus $\xi$ is <weak white noise>, and
$$
\boxed{Z_t=\phi Z_{t-1}+\alpha\xi_{t-1}+\xi_t.}
$$
Its <variance> can equivalently be written
$$
\boxed{\lambda=\frac{v(1+\theta^2)+w(1+\phi^2)}{1+\alpha^2}
=\frac{A+\sqrt{A^2-4C^2}}2.}
$$
When $\alpha\ne0$, also $\lambda=(v\theta-w\phi)/\alpha$; when $\alpha=0$, necessarily $C=0$ and $\lambda=A$, so the latter quotient should not be used. The model may reduce in order through cancellation. Boundary limits with $P=0$ or $Q=0$ give $|\alpha|=1$ and need not be stably invertible; the displayed stable reconstruction applies to the nondegenerate case above.
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