On Optimal Shaping of Multidimensional Constellations - An Alternative Approach to Lattice-Bounded (Voronoi) Constellations

Abstract

A scheme for the optimal shaping of multidimensional constellations is proposed. This scheme uses some of the ideas from a type of structured vector quantizer originally proposed for the quantization of memoryless sources, and results in N- sphere shaping of N-dimensional cubic lattice based constellations. Its implementation complexity is very reasonable. Because N - sphere shaping is optimal in N dimensions, shaping gains higher than those of N - dimensional Voronoi constellations can be realized. Optimal shaping for a large N however has the undesirable effect of increasing the size and the peak-to-average power ratio of the constituent 2D constellation, thus limiting its usefulness in practical implementation over QAM modems. It is shown that 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. Finally, compatibility with trellis-coded modulation is demonstrated for the realization of both shaping and coding gain, giving this scheme a distinct edge over lattice- bounded constellations.