Moments of Gaussian Random variable with zero mean and σ^2 variance, is given by is given by
Let X be zero mean Gaussian with variance $ σ^2 $
Then the moments $ E[H^k] $ are as follows:
as X is complex so $ H = X + jY $
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$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E {\{H^k}\} = \ 1,3 …\ (k-1) \sigma^k \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k \ \ \ \ even$
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k\ \ \ \ \ \ \ odd$
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$E[H^4]=E[X^4-6X^2Y^2+Y^4]=0 $
$E[|H|^4]=E[X^4+2X^2Y^2+Y^4]=8\sigma^4. $
How is it equal to $ 8\sigma^4 \ \ ? $ This is magnitude of $ \ |E| $
Best Answer
$EX^{4}=(1)(2)(3)\sigma ^{4}=6 \sigma ^{4}$. Also $EY^{4}=6 \sigma ^{4}$ and $2EX^{2}Y^{2}=EX^{2}EY^{2}=2\sigma^{4}$. Adding these we get $14 \sigma^{4}$.