Considering the following random vectors:
$\textbf{h} = [h_{1}, h_{2}, \ldots, h_{M}]^{T} \sim \mathcal{CN}(\textbf{0}_{M},d\textbf{I}_{M \times M})$,
$\textbf{w} = [w_{1}, w_{2}, \ldots, w_{M}]^{T} \sim \mathcal{CN}(\textbf{0}_{M},(\frac{1}{p})\textbf{I}_{M \times M})$,
$\textbf{y} = [y_{1}, y_{2}, \ldots, y_{M}]^{T} \sim \mathcal{CN}(\textbf{0}_{M},(d + \frac{1}{p})\textbf{I}_{M \times M})$,
where $\textbf{y} = \textbf{h} + \textbf{w}$ and therefore, $\textbf{y}$ and $\textbf{h}$ are not independent.
I'm trying to find the following expectation:
$\mathbb{E} \left[ \frac{\textbf{h}^{H} \textbf{y}\textbf{y}^{H} \textbf{h}}{ \| \textbf{y} \|^{4} } \right]$,
where $\| \textbf{y} \|^{4} = (\textbf{y}^{H} \textbf{y}) (\textbf{y}^{H} \textbf{y}$).
In order to find the desired expectation, I'm applying the following approximation:
$\mathbb{E} \left[ \frac{\textbf{x}}{\textbf{z}} \right] \approx \frac{\mathbb{E}[\textbf{x}]}{\mathbb{E}[\textbf{z}]} - \frac{\text{cov}(\textbf{x},\textbf{z})}{\mathbb{E}[\textbf{z}]^{2}} + \frac{\mathbb{E}[\textbf{x}]}{\mathbb{E}[\textbf{z}]^{3}}\text{var}(\mathbb{E}[\textbf{z}])$.
However, applying this approximation to the desired expectation is time consuming and prone to errors as it involves expansions with lots of terms .
I have been wondering if there is a more direct/smarter way of finding the desired expectation.