ON OPTIMAL SHAPING OF MULTIDIMENSIONAL CONSTELLATIONS

A scheme for the optimal shaping of multidimensional constellations is proposed. This scheme is motivated by a type of structured vector qua ntizer for memoryless sources, and results in N-sphere shaping of N-di mensional cubic lattice-based constellations. Because N-sphere shaping is optimal in N dimensions, shaping gains higher than those of N-dime nsional Voronoi constellations can be realized. While optimal shaping for a large N can realize most of the 1.53 dB total shaping gain, it h as the undesirable effect of increasing the size and the peak-to-avera ge power ratio of the constituent 2D constellation. This limits its us efulness for many real world channels which have nonlinearities. The p roposed scheme alleviates this problem by achieving optimal constellat ion shapes for a given limit on the constellation expansion ratio or t he peak-to-average power ratio of the constituent 2D constellation. Re sults of Calderbank and Ozarow on nonequiprobable signaling are used t o reduce the complexity of this scheme and make it independent of the data rate with essentially no effect on the shaping gain. Comparisons with Forney's trellis shaping scheme are also provided.