Pairwise optimization of modulation constellations

The design of two-dimensional signal constellations for the transmission of binary non-uniform memoryless sources over additive white Gaussian noise channels is investigated. The main application of this problem is the implementation of improved constellations where transmitted data is highly non-uniform. A simple algorithm, which optimizes a constellation by re-arranging its points in a pairwise fashion (i.e., two points are modified a time with the other points remaining fixed), is presented. In general, the optimized constellations depend on both the source statistics and the signal-to-noise ratio (SNR) in the channel. We show that constellations designed with source statistics considered can yield symbol error rate performance that is substantially better than rectangular quadrature amplitude modulation signal sets used with either gray mapping or more recently developed maps. Gains as high as 5 dB in E_b/N₀ SNR are obtained for highly non-uniform sources.