Let $\mathbf{x},\mathbf{y} \in \mathbb{C}^{N}$ be two independent vectors, with elements distributed independently as $\mathcal{C}\mathcal{N}(0,\sigma_{\mathbf{x}}^{2})$ and $\mathcal{C}\mathcal{N}(0,\sigma_{\mathbf{y}}^{2})$, respectively.
I need to derive the PDF of $Z = |\mathbf{x}^{\mathrm{H}} \mathbf{y}|^{2}$, denoted $f_{Z}(z)$.
Tentative solution:
- I know that when $\sigma_{\mathbf{x}}^{2} = \sigma_{\mathbf{x}}^{2} = 1$ and one of the two vectors is normalized (e.g., $\|\mathbf{x}\|_{2}=1$), we have $f_{Z}(z) = \exp(-z)$, but I can't derive the distribution for the general case.
- I can write $|\mathbf{x}^{\mathrm{H}} \mathbf{y}|^{2} = \mathbf{x}^{\mathrm{H}} \mathbf{y} \mathbf{y}^{\mathrm{H}} \mathbf{x} = \big| \sum_{n=1}^{N} x_{n}^{*} y_{n} \big|^{2}$, where $x_{n}$ and $y_{n}$ are the elements of $\mathbf{x}$ and $\mathbf{y}$, respectively, but here I get stuck.