Let $\mathbf{u}=[u_1,u_2,\dots,u_N]^T |\in \mathbb{C}^{N\times 1}$ and $\mathbf{v}=[v_1,v_2,\dots,v_N]^T \in \mathbb{C}^{N\times 1} $ are complex random vectors whose entries $u_i,v_i \sim \mathcal{CN}(0,1)$ .
I am trying to find the MGF of the following expression: $$X = \frac{||\mathbf{u}-\mathbf{v}||^2}{||\mathbf{u}||^2}$$ where MGF is defined as $$\mathcal{M}_X(s) = \int_{0}^\infty e^{sx}f_X(x)\,dx.$$
where $f_X(x)$ is the pdf of $X$. I am looking for the MGF of $X$.
Thank you :)