Expectation of $\left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2$, with complex Gaussians?

Question

Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ are independent, what would be the expectation

$$\mathbb{E} \left[ \left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2 \right], $$ where $\mathcal{CN}(.,.)$ is the complex normal random variable.

Answer 1

Using the same trick from another answer, as well as the trace trick and $E[1/\Vert z\Vert^2]$ from yet another answer, we find

$$ \frac{\sigma_x^2 \, \sigma_y^2}{(\sigma_x^2+\sigma_y^2)^2} \, \frac{1}{M-1} + \frac{(\sigma_x^2)^2}{(\sigma_x^2+\sigma_y^2)^2} \;. $$

Answer 2

By using the first term of Taylor's expansion of the expectation, it can be approximated as

$$\mathbb{E} \left[ \left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2 \right] \approx \frac{\mathbb{E} \left[ \|(\textbf{x}+\textbf{y})^{H} \textbf{x} \|^2 \right]}{ \mathbb{E} \left[ \| \textbf{x} + \textbf{y} \|^4 \right]} = \frac{(M+1)(\sigma_{x}^2)^2 +\sigma_{x}^2 \sigma_{y}^2}{(M+1)(\sigma_{x}^2 + \sigma_{y}^2)^2}, $$ which produces results close to the simulated expectation.