Shaping multidimensional signal spaces. I. Optimum shaping, shell mapping

The structure of the regions which provide the optimum tradeoff between γ_s (shaping gain) and CER (constellation-expansion ratio) and between γ_s and PAR (peak to average power ratio) in a finite dimensional space is introduced. Analytical expressions are derived for the corresponding tradeoff curves. In general, the initial parts of the curves have a steep slope. This means that an appreciable portion of the maximum shaping gain, corresponding to a spherical region, can be achieved with a small value of CER_s, PAR. The technique of shell mapping is introduced. This is a change of variable which maps the optimum shaping region to a hypercube truncated within a simplex. This mapping is a useful tool in computing the performance, and also in facilitating the addressing of the optimum shaping region. Using the shell mapping, a practical addressing scheme is presented that achieves a point on the optimum tradeoff curves. For dimensionalities around 12, the point achieved is located near the knee of the corresponding tradeoff curve. For larger dimensionalities, a general shaping region with two degrees of freedom is used. This region provides more flexibility in selecting the tradeoff point