Phase distribution of a non-zero mean complex Gaussian random variable

Question

We know that if $X\sim CN(0,\sigma^2\mathbf{I}_{2\times 2})$, then its phase has a uniform distribution over $[0,2\pi]$. Now what is the distribution of its phase if $X\sim CN(\mu,\sigma^2\mathbf{I}_{2\times 2})$? Obviously it is not uniform.

Answer 1

Such distribution is covered in "Analysis on Functions and Characteristics of the Rician Phase Distribution" by Zhongtao Luo, Yanmei Zhan and Edmond Jonckheere (see here: https://ieeexplore.ieee.org/document/9238805)

For small variance, the distribution can be approximated by Gaussian. For higher variance, the distribution becomes more like a von-mises or wrapped-gaussian distribution. An exact expression for the distribution is given in above referenced paper.