Abstruct-Variable-rate data transmission schemes in which constellation points are selected according to a nonuniform probability distribution are studied. When the criterion is one of minimizing the average transmitted energy for a given average bit rate, the best possible distribution with which to select constellation points is a Maxwell-Bdtzmann distribution. In principle, when constellation points are selected according to a Maxwell-Boltzmann distribution, the ultimate shaping gain (7re/6 or 1.53 dB) can be achieved in any dimension. Nonuniform signaling schemes can be designed by mapping simple variable-length prefix codes onto the constellation. Using the Huffman procedure, prefix codes can be designed that approach the optimal performance. These schemes provide a fixed-rate primary channel and a variable-rate secondary channel, and are easily incorporated into standard lattice-type coded modulation schemes.
Indeed, by selecting constellation points properly based on the MB distribution at any dimension, the ultimate shaping gain (1.53 dB) can be achieved REF .
Optimal nonuniform signaling for Gaussian channels
IEEE Trans. Inf. Theory
Computer Science