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Let x and y be complex Gaussian random vectors with length L, and $\mathbb{E}[x]=a$ and $\mathbb{E}[y]=b$, and they are correlated such that $c = \mathbb{E}\left[\left|\mathbf{\mathit{\mathbf{x}}}^{T}\mathbf{y}\right|\right]$

Can we write $\mathbb{E}[|\mathbf{x}^T\mathbf{y}|^2]=Var(\mathbf{x}^T\mathbf{y})+\mathbb{E}[|\mathbf{x}^T\mathbf{y}|]^2$ $?$ Hence,

$$\mathbb{E}|\mathbf{x}^T\mathbf{y}|^2]=Var(\mathbf{x}^T\mathbf{y})+c^2$$

Then, can we derive an analytical expression for $Var(\mathbf{x}^T\mathbf{y})$ or $\mathbb{E}|\mathbf{x}^T\mathbf{y}|^2]$) ?

Shumpei
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    Have you first attempted the simpler problem of the expected value of the product of two univariate random variables? If not, I believe you will find that helpful. – Glen_b Jan 07 '20 at 02:02
  • Thank you Glen, I have tried. Nevertheless, I derived an approximation based on this answer: https://stats.stackexchange.com/questions/76961/variance-of-product-of-2-independent-random-vector and it seems working according to simulation results. – Shumpei Jan 08 '20 at 23:16

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