How to derive the characteristic function of a polar coordinates representation of a bivariate normal

Suppose to have a bivariate normal variable $\mathbf{x}=(x_1,x_2)$ with mean $\mu$ and covariance matrix $\Sigma$. I move from $\mathbf{x}$ to $(\theta,r)$ where $x_1 = r \cos \theta$ and $x_2 = r \sin \theta$: the representation in polar coordinates of $\mathbf{x}$. My question is: is it possible to derive the characteristic function of $(\theta,r)$? and the characteristic function of $\theta$?