Constellation design via capacity maximization

Traditional constellations are uniformally spaced. By giving up uniform spacing, constellations can be designed to have larger joint (i.e. overall) capacity or parallel decoding capacity. In this paper non-uniformally spaced (i.e. 'geometrically' shaped) constellations are designed to maximize either of these quantities. By way of numerical capacity computations we show that except in special cases, there are no universally optimal geometrically shaped constellations across all code rates, and that the optimization of a constellation has to target a specific code rate. Unlike joint capacity, optimizing for parallel decoding capacity is label dependent. For PAM and PSK constellations, we found the maximum parallel decoding capacity to be achieved using gray (not necessarfly binary reflective gray) labels. However, for PAM constellations, not all gray labels can yield the highest parallel decoding capacity. Besides the conventional use of a (log(2)(M)-1)/log(2) (M) code rate with an M-point constellation for bandwidth efficient communications, the optimized constellations could offer further non-trivial gains at lower code rates (unlike traditional constellations). An optimized constellation is used with a state-of-the-art LDPC code and simulation results are presented. This paper also draws a distinction between probabilistic shaping and geometric shaping and in fact proves under broad conditions, that any gain in capacity which can be found via probabilistic shaping can also be achieved or exceeded solely through geometric shaping.