Pairwise optimization of modulation constellations for non-uniform sources

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 at a time, with all 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 (SER) performance that is substantially better than rectangular quadrature amplitude modulation signal sets used with either Gray mapping or more recently developed maps. SER gains as high as 5 dB in E(b)/N(o) SNR are obtained for highly non-uniform sources. Symbol mappings are also developed for the new constellations using a similar pairwise optimization method whereby we assign and compare a weighted score for each pair. These maps, when compared to the mappings used in conjunction with the standard rectangular QAM constellation, again achieve considerable performance gains in terms of bit error rate (BER). Gains as high as 4 dB were achieved over rectangular QAM with Gray mapping, or more than I dB better than previously improved mappings. Finally, the uncoded pairwise optimized system is compared to a standard tandem (separate) source and channel coding system. Although neither system is universally better, the uncoded system with optimized constellations outperforms the tandem coding system for low-to-mid SNRs. Performance/complexity trade-offs between the two systems are also discussed.