Many fading channels follow a Rayleigh distribution. FIGS. 22a-24b are locus plots of PAM-4, 8, and 16 constellations that have been optimized for PD capacity on a Rayleigh fading channel. Locus plots versus user bit rate per dimension and versus SNR are provided. Similar processes can be used to obtain capacity optimized constellations that are optimized using other capacity measures, such as joint capacity, and/or using different modulation schemes.
Geometric PAM-8, PAM-16, and PAM-32 Constellations
As described above, geometric constellations can be obtained that are optimized for joint or PD capacity at specific SNRs. In addition, ranges can be specified for the constellation points of a geometric constellation that are probabilistically likely to result in geometric constellations that provide at least a predetermined performance improvement relative to a constellation that maximizes dmin. Turning now to FIGS. 25-95, geometric PAM-8, PAM-16, and PAM-32 constellations optimized for joint and PD capacity over the Rayleigh fading channel at specific SNRs are listed. The performances of the optimal constellations are compared to the performances of traditional constellations that maximize dmin. Ranges for the constellation points are also defined at specific SNRs, where constellations having constellation points selected from within the ranges are probabilistically likely (with probability close to one) to result in at least a predetermined performance improvement at the specified SNR relative to a traditional constellation that maximizes dmin.
