If I have $\mathbf x_n=[x_0, x_1,... ,x_K]^T$ and $\mathbf y_n=[y_0, y_2, ..., y_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$.

What is the distribution of the following inner product: $$\sum_{n=0}^N \mathbf x_n^H\mathbf x_n$$ Secondly, what is the distribution of $$\sum_{n=0}^N \mathbf x_n^H\mathbf y_n$$ If the answer is Gamma distribution, what are the parameters of this Gamma distribution? Note that each element in both vectors is complex, random, and independent of the other elements.

Thank you in advance.