*Bounty: 150*

*Bounty: 150*

Considering the following random vectors:

begin{align}

textbf{h} &= [h_{1}, h_{2}, ldots, h_{M}]^{T} sim mathcal{CN}left(textbf{0}*{M},dtextbf{I}*{M times M}right), [8pt]

textbf{w} &= [w_{1}, w_{2}, ldots, w_{M}]^{T} sim mathcal{CN}left(textbf{0}*{M},frac{1}{p}textbf{I}*{M times M}right), [8pt]

textbf{y} &= [y_{1}, y_{2}, ldots, y_{M}]^{T} sim mathcal{CN}left(textbf{0}*{M},left(d + frac{1}{p}right)textbf{I}*{M times M}right),

end{align}

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.