Conditionally Gaussian Distributions and their Application in the Performance of Maximum Likelihood Decode and Forward Cooperative Systems

The statistics of gamma conditionally Gaussian (CG) random variables are derived in closed form. These variables can be loosely defined to be normally distributed with mean and variance proportional to a gamma random variable. In this paper, we provide exact expressions for the bit error rate (BER) for single relay maximum likelihood (ML) decode and forward (DF) cooperative systems in Nakagami-m fading for binary phase shift keying (BPSK). This is done by expressing the ML decision variable in terms of functions of gamma CG random variables. For the piecewise linear (PL) approximation to the ML detector, a closed form expression for the BER is obtained. Simulation results are provided to verify the validity of the derived analytical expressions.