Signaling in Multi-dimensional Signal spacest

In selecting the boundary of a signal constellation used for data transmission, the objective is to minimize the average energy of the set for a given number of points from a given packing. Reduction in the average energy because of using the region C as the boundary instead of a hypercube is called the shape gain of C. The price to be paid for shaping is: (i) an increase in the factor CER, (Constellation-Expansion-Ratio), (ii) an increase in the factor PAR (Peak-bAverage-power-Ratio), and (iii) an increasein the addressing complexity. The structure of the region which optimizes the tradeoff between the shape gain and the CER, and also between the shape gain and the PAR in a finite dimensional space is discussed. Examples of the optimum tradeoff curves are given. The optimum shaping region is mapped to a hypercube truncated within a simplex. This mapping has properties which facilitate the addressing of the signal points. We discuss two addressing schemes with low complexity and good performance. In spectral shaping, the rate of the constellation is muimized subject to some constraints on its power spectrum. This results in a shaping region which has different values of power along different dimensions (unsymmetrical shaping). This spectral shaping also involves the selection of an appropriate basis (modulating waveform) for the space. Finally, we discuss the selection of a signal cons~ellation for signaling over a partial-response channel using both continuous approximation and discrete analysis. We also present a closed form formula for the weight distribution of the scaled, D4 and E8 lattices.