Optimal QAM constellations with nonuniform sources on AWGN and Rayleigh fading channels

In this paper we show how to optimize QAM signal constellation which minimizes bit error probabilities with nonuniform sources on AWGN and Rayleigh fading channel. The optimal signal constellation is not equally spaced. We show that the optimal system has a lower error probability than a conventional system, i.e., a system with source coding, equiprobable symbols, and equal space constellation. In several cases presented in this paper the error probability of the optimal system is significantly lower than in the conventional system. We prove that even with equal symbol probabilities the equal space constellation is not optimal for 16QAM, but only asymptotically optimal. Finally, we evaluate the robustness of constellations to estimation error in symbol probabilities.