Multidimensional constellations. I. Introduction, figures of merit, and generalized cross constellations

The authors discuss the major attributes desired in signal constellations, such as signal-to-noise ratio (SNR) efficiency, simplicity of mapping bits to points and vice versa, compatibility with coded modulation schemes, and compatibility with quadrature amplitude modulation (QAM). The capability of supporting a so-called opportunistic secondary channel, often used for internal control signaling, is considered. The gain in SNR efficiency of a multidimensional constellation (lattice code) consisting of the points from a lattice Λ within a region R compared to a cubic constellation is shown to be approximately separable into the coding gain of Λ and the shape gain of R, for large constellations. Similarly, the expansion of the associated constituent 2-D constellation is shown to be approximately separable into a constellation expansion ratio (CER) coding component CER_c(Λ) and a shaping component CER _s(R). The N sphere is the region R with the best shape gain, but N also has large constellation expansion. Bounds for the best possible shape gain versus CER_s(R) or peak-to-average-power ratio (PAR) are given. Generalized cross constellations are discussed. These constellations yield a modest shape gain with very low CER_s(R) or PAR, are easily implemented, are well suited for use with coded QAM modems, and can be readily adapted to support an opportunistic secondary channel