p.d.f. of $\left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2$, where $\textbf{x}$ and $\textbf{y}$ are complex Gaussians?

Question

Given that the 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 p.d.f. of

$$Z = \left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2, $$ where $\mathcal{CN}(.,.)$ is the complex normal random variable.

Which kind of variable change could be applied here?

Answer 1

You can use the rotational invariance of the Gaussian distribution to take $\mathbf{y}=2^{-1/2}\sigma_y(\sqrt \xi_{2M},0,0,\ldots,0)$, with $\xi_{2M}$ distributed independently of $\mathbf{x}$ according to a chi-squared distribution with $2M$ degrees of freedom. Then $$Z=\tfrac{1}{2}\sigma_y^2 \xi_{2M}|z|^2\;\;\text{with}\;\; z=\frac{x_1}{\sum_{n=1}^M |x_n|^2}.$$ This can be further reduced to $$Z=\frac{\sigma_y^2}{\sigma_x^2} \frac{\xi_{2M}\xi_{2}}{(\xi_{2}+\xi_{2M-2})^2},$$ with $\xi_2$ and $\xi_{2M-2}$ independently chi-squared distributed with $2$, respectively, $2M-2$ degrees of freedom.

I don't think the distribution of this rational function of three independent chi-squared distributions has a closed form expression. For $M\gg 1$ one has simply $$Z\rightarrow \frac{\sigma_y^2}{2M\sigma_x^2}\xi_2,$$ so $Z$ has for large $M$ a chi-squared distribution with 2 degrees of freedom.