An efficient block-based addressing scheme for the nearly optimum shaping of multidimensional signal spaces

We introduce an efficient addressing scheme for the nearly optimum shaping of a multidimensional signal constellation. The 2-D (two-dimensional) subspaces are partitioned into K energy shells of equal cardinality. The average energy of a 2-D shell can be closely approximated by a linear function of its index. In an N=2n-D space, we obtain Kⁿ shaping clusters of equal cardinality. Shaping is achieved by selecting T⩽Kⁿ of the N-D clusters with the least sum of the 2-D indices. This results in a set of T integer n-tuples with the components in the range [0, K-1] and the sum of the components being at most a given number L. The problem of addressing is to find a one-to-one mapping between the set of such n-tuples and the set of integers [0, T-1] such that the mapping and its inverse can be easily implemented. In the proposed scheme, the N-D clusters are grouped into blocks of identical binary weight vectors. This results in a simple rule for the addressing of points within the blocks. The addressing of the blocks is based on some recursive relationship which allows us to decompose the problem into simpler parts. The overall scheme requires a modest amount of memory and has a small computational complexity