On the ratio of independent complex Gaussian random variables

In this paper, we derive a closed form equation for the joint probability distribution f R z , Θ z ( r z , θ z ) of the amplitude R z and phase Θ z of the ratio Z = X Y of two independent non-zero mean Complex Gaussian random variables X ∼ C N ( ν x e j ϕ x , σ x 2 ) and Y ∼ C N ( ν y e j ϕ y , σ y 2 ) . The derived joint probability distribution only contains a confluent hypergeometric function of the first kind 1 F 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.