On the ratio of independent complex Gaussian random variables

Abstract

In this paper, we derive a closed form equation for the joint probability distribution $${{f_{{R}_{z}}},{\varTheta _{z}}}({r_{z}},{\theta _{z}})$$ of the amplitude $${R_{z}}$$ and phase $${\varTheta _{z}}$$ of the ratio $${Z=\frac{X}{Y}}$$ of two independent non-zero mean Complex Gaussian random variables $$X\sim CN(\nu _{x} \mathrm {e}^{j\phi _{x}},{\sigma ^{2}_{x}})$$ and $$Y\sim CN(\nu _{y} \mathrm {e}^{j\phi _{y}},{\sigma ^{2}_{y}})$$ . The derived joint probability distribution only contains a confluent hypergeometric function of the first kind $${_1F_{1}}$$ without infinite summations resulting in computational efficiency. We further derive the probability distribution for the ratio of two non-zero mean independent real Rician random variables containing an infinite summation generated by the estimation of the Cauchy product of equivalent series of two modified Bessel functions.