Assume that $z$ is a random vector. Also, assume that $$z = A(\|y\|)$$ where A is whitening function and $\|\cdot\|$ is complex modulus (norm) such that if
$$y_j = a_j + ib_j,$$ then $$\|y_j\|=\sqrt{a_j^2+b_j^2}$$ and $y$ is a complex vector which is derived from a real vector $x$ using some invertible projection matrix $W$ $$x = Wy$$ Considering $$z\sim\mathcal{N}(0, I),$$ what can we say about distribution of $x|z$?
My problem is with inverting the modulus operator. If that was not present, I could have said
$$x|y\sim\mathcal{N}(Wy, \sigma^2I)$$ Any suggestions would be appreciated.