How to write this state space model as an ARMA(1,1) Problem: we have two processes $x_{t}$ and $y_{t}$ as follows:
$x_{t} = y_{t} + w_{t}$
$y_{t} = \phi_{0} + \phi_{1}y_{t-1} + u_{t}$
$w_{t}, u_{t}$ iid with var: $\sigma_{w}^{2}, \sigma_{u}^{2}$, mean of $w_{t}$ is 0 (mean of $u_{t}$ not mentioned)
$\phi_{1} \neq 0$
then: $x_{t}$ is ARMA(1,1) with:
(i) AR part = $\phi_{1}$
(ii) MA part:
$\theta_{1} = \frac{-\sigma_{u}^{2}-(1+\phi_{1}^2)\sigma_{w}^{2}+\sqrt{\sigma_{u}^{4}+2(1+\phi_{1}^{2})\sigma_{w}^{2} \sigma_{u}^{2}+(1-\phi_{1}^{2})^{2} \sigma_{w}^{4}}}{2\phi_{1}\sigma_{w}^{2}}$
My approach: inserting the definition of $y_{t}$ on the RHS of $x_{t}$ and then replace $y_{t-1}$ gives:
$x_{t} = \phi_{0} + \phi_{1} x_{t-1} + u_{t} + w_{t} - \phi_{1} w_{t-1}$
which proves (i) and suggests $\theta_{1} = -\phi_{1}$. How can I derive result (ii) ?
 A: The key idea is to rewrite $u_t + w_t - \phi_1 w_{t-1}$ as an MA(1) process, that is, find an $\epsilon_t$ and $\theta_1$ such that:
$$u_t + w_t - \phi_1 w_{t-1} = \epsilon_t + \theta_1 \epsilon_{t-1}$$
More details
You're correct in obtaining:
$$ x_t = \phi_0 + \phi_1 x_{t-1} + u_t + w_t - \phi_1 w_{t-1}$$
Let $\zeta_t = u_t + w_t - \phi_1 w_{t-1}$. Based on the Wold representation, my answer here shows, you can write $\zeta_t$ as following an MA(1) structure $\zeta_t = \epsilon_t + \theta_1 \epsilon_{t-1}$ with:
$$ \frac{1}{\theta_1} + \theta_1 = -\frac{1}{\phi_1}\left(\frac{\sigma^2_u}{\sigma^2_w} + 1 \right)   - \phi_1$$
Now it's just algebra. Rewrite as a quadratic equation in $\theta_1$:
$$ \phi_1 \sigma^2_w\theta^2_1  + \left[\sigma^2_u+( 1 + \phi_1^2) \sigma^2_w\right] \theta_1  + \phi_1\sigma^2_w = 0 $$
Applying solution to the quadratic equation:
$$ \theta_1 = \frac{ - \left[\sigma^2_u+( 1 + \phi_1^2) \sigma^2_w \right] \pm \sqrt{\sigma^4_u+(1 + \phi^2_1)^2\sigma^4_w+ 2\sigma^2_u(1 + \phi_1^2)\sigma^2_w - 4\phi^2_1\sigma^4_w }}{2\phi_1\sigma^2_w}$$
A little more algebra that $(1 + \phi^2_1)^2\sigma^4_w -4\phi^2_1\sigma^4_w = (1 - 2 \phi_1^2 + \phi_1^4)\sigma^4_w = (1 - \phi^2_1)^2\sigma^4_w$
$$ \theta_1 = \frac{ - \left[\sigma^2_u+( 1 + \phi_1^2) \sigma^2_w \right] \pm \sqrt{\sigma^4_u+ 2\sigma^2_u(1 + \phi_1^2)\sigma^2_w + (1 - \phi^2_1)^2\sigma^4_w }}{2\phi_1\sigma^2_w}$$
And there you are (the argument also needs $|\theta_1| < 1$ which I believe will rule out one of the solutions). There may be a simpler, slicker way to do this, but this gets there.
