SHAPING MULTIDIMENSIONAL SIGNAL SPACES .2. SHELL-ADDRESSED CONSTELLATIONS

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). In this paper, we describe some practical means to select 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. We introduce a method to facilitate the realization of the addressing lookup table. This method is based on the decomposition of the addressing into a hierarchy of addressing steps, each of a low complexity. This avoids the exponential growth of the complexity. Based on this addressing decomposition, by using a memory of a practical size, we 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. We also introduce hybrid multilevel addressing schemes which combine the two classes. These schemes provide single points with moderate addressing complexity near the knee of the optimum tradeoff curves.