# The expected value of the power of the sum of inner products of two independent complex normal vectors

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 $$0\lt\tau\lt1$$.