Shaping multidimensional signal spaces. II. Shell-addressed constellations

For Pt. I, see ibid., pp. 1799-1808, Nov. 1993. By appropriately selecting the boundary of a multidimensional signal constellation used for data transmission, the average energy of the constellation can be reduced. Reduction in the average energy (shaping gain) is obtained at the price of increasing the constellation-expansion ratio (CER_s ) and the peak-to-average-power ratio (PAR). The authors describe some practical means for selecting the boundary so as to achieve a point with low addressing complexity near the knee of the corresponding tradeoff curves (shaping gain versus CER_s or PAR). One class of addressing schemes is based on using a lookup table. A method to facilitate the realization of the addressing lookup table is introduced. This method is based on the decomposition of addressing into a hierarchy of addressing steps, each of a low complexity. This avoids exponential growth of the complexity. Using this addressing decomposition and a memory of a practical size, one can move along a tradeoff curve which has negligible suboptimality. Another class of addressing schemes is based on using a Voronoi constellation in a space of half the original dimensionality