Shaping of multi-dimensional signal constellations using a lookup table

Shaping relates to the selection of the boundary of a signal constellation to reduce its average energy. Addressing is the assignment of the data bits to the constellation points. A major concern about the shaping regions is their addressing complexity. The authors use a lookup table for addressing. The method is based on partitioning the two-dimensional subconstellations into shaping shells of equal size and increasing average energy. A lookup table is used to select a subset of the Cartesian product of the partitions. This partitioning is compatible with a multidimensional trellis coded modulation scheme. As part of the calculations, a closed-form formula is presented for the weight distribution of the half integer lattice Z^N+(1/2)^N, for dimensionally N=4,8