Distribution of the sum of a complex Gaussian and the product of two complex Gaussians

The probability density function of sum of a complex Gaussian and the product of two complex Gaussians is derived. This distribution occurs in wireless communications where Gaussian signals are transmitted over Rayleigh channels. The result is validated using the Kolmogorov-Smirnov distance and the accuracy of several approximations are compared, including fitted gamma and Nakagami distributions.