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$\boldsymbol{h_1}$ and $\boldsymbol{h_2}$ are i.i.d. circularly symmmetric complex Gaussian random vectors with zero mean and covariance matrix $\boldsymbol{K}$.

$ \boldsymbol{h_1} = \left [h_1(0),\cdots,h_1(N-1) \right ]^T$ and $ \boldsymbol{h_2} = \left [h_2(0),\cdots,h_2(N-1) \right ]^T$

Need help in computing the expectation given below

$\mathbb{E} \left[ \frac{\left| \sum_{i=0}^{i=N-1}h_1(i)\right|^2 h_1(m)h_1^*(n)}{\left| \sum_{i=0}^{i=N-1}h_1(i)\right|^2 + \left| \sum_{i=0}^{i=N-1}h_2(i)\right|^2}\right ]$

Since a CSCGRV contains independent real and imaginary parts, $\left| \sum_{i=0}^{i=N-1}h_1(i)\right|^2 $ is the sum of two independent random variables, where each one is the square of the sum of real/imaginary parts and it has a distribution of the square of a Gaussian random variable.

How do I proceed further?

kjetil b halvorsen
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user36
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  • Unless $K$ itself is spherical--that is, it is a multiple of the identity matrix--it looks like you would have to resort to numerical integration. Do you need to consider the most general $K,$ then? – whuber Dec 23 '21 at 15:58
  • $K$ cannot be an identity matrix. Using conditional expectation sequentially, can we get a closed form solution? – user36 Dec 24 '21 at 16:39
  • I doubt it. It doesn't even seem to exist in the real case with $N=2.$ – whuber Dec 24 '21 at 16:46

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