The complex Double Gaussian distribution

We present the complex Double Gaussian distribution that describes the product of two independent, non-zero mean, complex Gaussian random variables, a doubly-infinite summation of terms. This distribution is useful in a wide array of problems. We discuss its application to blind TR detection systems by deriving the Neyman-Pearson optimal detector when the channel is modeled as the product of two independent complex Gaussian random variables, such as in a Time Reversal scenario. We show that near-optimal detection performance can be achieved with as few as 25 summation terms. Theoretical analysis and Monte Carlo simulations illustrate our results.