Optimal Constellation for General Rectangular PAM/QAM with Arbitrary Code Mapping

Recently the exact expression for the BER of rectangular QAM with arbitrary code mapping was obtained. It is then natural to ask for the optimal constellation design which minimizes the BER. We develop an iterative algorithm to find such optimal solution for any one or higher dimensional rectangular PAM/QAM with arbitrary code mapping. As it turns out, the uniform constellation is not optimal even for Gray codes, especially in the low SNR region. And the performance improvement for some code mapping schemes after the optimization can be as much as over 20%. Furthermore, as bit rate gets lower, an interesting clustering phenomenon results in the optimal solution. Simulation results back up the advantages of our optimization method. The result obtained in this paper can be applied to the design of practical systems.