Let $\mathbf{x}, \mathbf{y} \in \mathbb{C}^N$ be two random vectors, where $\mathbf{x}$ has elements distributed as $\mathcal{CN}(0,\sigma_{\mathbf{x}}^{2})$ and $\mathbf{y}$ has elements distributed as $\mathcal{CN}(0,\sigma_{\mathbf{y}}^{2})$.
Let $\mathbf{z} = \frac{\mathbf{x} + \mathbf{y}}{\|\mathbf{x} + \mathbf{y}\|}$. How can I compute $\mathbb{E} \big[ |\mathbf{z}^{\mathrm{H}} \mathbf{x}|^{2} \big]$?
Tentative solution:
- $|\mathbf{z}^{\mathrm{H}} \mathbf{x}|^{2} = \| \mathbf{z} \|^2 \| \mathbf{x} \|^2 \cos^{2} \theta_{\mathbf{z} \mathbf{x}} = \| \mathbf{x} \|^2 \cos^{2} \theta_{\mathbf{z} \mathbf{x}}$, where $\theta_{\mathbf{z} \mathbf{x}}$ is the angle between the two vectors. Hence, $\mathbb{E} \big[ |\mathbf{z}^{\mathrm{H}} \mathbf{x}|^{2} \big] = \mathbb{E} \big[ \|\mathbf{x}\|^{2} \big] \mathbb{E} \big[ \cos^{2} \theta_{\mathbf{z} \mathbf{x}} \big] = N \ \mathbb{E} \big[ \cos^{2} \theta_{\mathbf{z} \mathbf{x}} \big]$. Still, I don't know how to compute the expectation of $\cos^{2} \theta_{\mathbf{z} \mathbf{x}}$.