If $\mathbf x_n=[x_{n,1}, ... ,x_{n,K}]^T$ and $\mathbf y_n=[y_{n,1}, ..., y_{n,K}]^T$, where $x\sim\mathcal C\mathcal N(\mathbf 0,\sigma_x^2\mathbf I)$, and $y\sim\mathcal C\mathcal N(\mathbf 0,\sigma_y^2\mathbf I)$.

And consider: $$I=\left|\sqrt{(1-\tau^2)}\sum_{n=1}^N \mathbf x_n^H\mathbf x_n + \tau\sum_{n=1}^N \mathbf x_n^H\mathbf y_n\right|^2$$ What is $E[I]$? and what is $E[I^2]$. Note that $\tau$ is a constant such that $0\lt\tau\lt1$.


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