Shaping the boundary of a multi-dimensional signal constellation with a non-flat power spectrum

The author considers the selection of the boundary (shaping region) of a multi-dimensional signal constellation with the objective of minimising the constellation energy for a given rate and subject to some constraints on the resulting power spectrum. This shaping region is selected such that the efficient addressing technique of `shell mapping´ is applicable. Numerical results are presented for the asymptotic performance in an infinite dimensional space, and also for the finite dimensional case. These results show that, by using the proposed method in a space of moderate dimensionality (say 16), one can obtain: a substantial saving in energy (about 1 dB), and a decrease in the PAR (peak-to-average-power-ratio) by a factor of about 2-3