This is a question from Chapter 4 of Time Series: Theory and Methods by Brockwell and Davis (1991).
Question: Let $\{Z_t\}$ and $\{W_t\}$ be white noise processes with mean zero and variance $\sigma^2$. If $\{X_t\}$ and $\{Y_t\}$ are stationary processes satisfying $$ X_t - \alpha X_{t-1} = W_t; $$ and $$ Y_t - \alpha Y_{t-1} = X_t + Z_t, $$ where $|\alpha| < 1$ and $\{W_t\}$ and $\{Z_t\}$ are uncorrelated, find the spectral density of $\{Y_t\}$.
Attempt: Write $\alpha(z) = 1-\alpha z$. Then I have that $$|\alpha(e^{-i\omega})|^2 f_X(\omega) = \sigma^2/(2\pi),$$ using that the spectral density of $\{W_t\}$ is $\sigma^2/(2\pi)$ aand denoting the spectral density of $\{X_t\}$ by $f_X$.
Since $\alpha < 1 $, $\{X_t\}$ is a causal function of $\{W_t\}$ . Hence we can find $\psi_j$ such that $X_t = \sum_{j=0}^\infty \psi_j W_{t-j}$. This means that $\{X_t\}$ and $\{Z_t\}$ are uncorrelated. So the spectral density of $\{X_t + Z_t\}$ is $f_X + f_Z$. Then, I obtain that
$$|\alpha(e^{-i\omega})|^2 f_Y(\omega) = f_X(\omega) + f_Z(\omega).$$
Finally, I see that $$f_Y(\omega) = \frac{\sigma^2}{2\pi} \frac{1}{(1- 2 \alpha \cos \omega + \alpha^2)^2} + \frac{\sigma^2}{2\pi} \frac{1}{(1- 2 \alpha \cos \omega + \alpha^2)} .$$
Is this correct? I have tried verifying my answer using R code, but the estimated frequency didn't match up with what I got. Thank you!