SHAPING MULTIDIMENSIONAL SIGNAL SPACES .1. OPTIMUM SHAPING, SHELL MAPPING

In selecting the boundary of a signal constellation used for data transmission, one tries to minimize the average energy of the constellation for a given number of points from a given packing. The reduction in the average energy per two dimensions due to the use of a region C as the boundary instead of a hypercube is called the shaping gain gamma(s) of C. The price to be paid for shaping involves: 1) an increase in the factor constellation-expansion ratio (CER(s)), 2) an increase in the factor peak-to-average-power ratio (PAR), and 3) an increase in the addressing complexity. In general, there exists a tradeoff between gamma(s) and CER(s), PAR. In this work, the structure of the region which simultaneously optimizes both of these tradeoffs is introduced. In an N-D (N-dimensional) optimum shaping region (N even), a 2-D sphere is the boundary of the 2-D subspaces and an N-D sphere is the boundary of the whole space. Analytical expressions are derived for the optimum tradeoff curves. By applying a change of variable denoted as shell mapping, the optimum shaping region is mapped to a hypercube truncated within a simplex. This mapping facilitates the performance computation, and also the addressing of the optimum shaping region. Using shell mapping, we introduce an addressing scheme with low complexity to achieve a point on the optimum tradeoff curves. To obtain more flexibility in selecting the tradeoff point, a second shaping method with more degrees of freedom is used. In this method, a 2-D sphere is the boundary of the 2-D subspaces, an N'-D sphere, N' greater-than-or-equal-to 2, is the boundary of the N'-D subspaces, and an N-D sphere is the boundary of the whole space.