The constellation figure of merit (CFM)

in terms of lattice theory

can be separated into the coding gain of a lattice and the shaping gain of the boundary of a constellation.In this paper

maximizing the CFM of multi-dimensional signal constellations is formulated as a series of optimization problems.The geometric characteristics of signal constellations are taken as the constraints of such problems.Since the desirable signal constellations can be achieved by solving the optimization problems

our approach can serve as a general method of the construction of multi-dimensional signal constellations.In comparison with the fact that the existing methods can only be applied when the number of signal points is small

the proposed approach can construct large size constellations with ease.The simulation results show that the symbol error probabilities (SEP) of small size constellations built by our approach are very close to the optimum and that the proposed large size constellations have better SEPs than the traditional constellations generated by integer lattices.