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 quantizer for memoryless sources, and results in N-sphere shaping of N-dimensional cubic lattice-based constellations. Because V-sphere shaping is optimal in Ndimensions, shaping gains higher than those of N-dimensional Voronoi constellations can be realized. While optimal shaping for a large N can realize most of the 1.53 dB total shaping gain, it has the undesirable effect of increasing the size and the peak-to-average power ratio of the constituent 2D constellation. This limits its usefulness for many real world channels which have nonlinearities. The proposed scheme alleviates this problem by achieving optimal constellation shapes for a given limit on the constellation expansion ratio or the peak-to-average power ratio of the constituent 2D constellation. Results of Calderbank and Ozarow on nonequiprobable signaling are used to 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.