Constellation shaping for the Gaussian channel

Constellation shaping is method of improving the effectiveness of digital communications over noisy, bandlimited channels. We present a new view to the problem of constellation shaping. Both a new procedure and information theoretic analysis are discussed. The article presents an approach to understanding constellation shaping that avoids the “continuous approximation” analysis of performance. A unique “type-mapping” approach to shaping is derived and related to monomial orderings on a ring of polynomials. The basic methods of constellation shaping can be roughly characterized as forms of coset coding (i.e., codes for which messages are associated with cosets of a subgroup such as a linear subspace) and enumerative coding (i.e., codes for which messages are enumerations of vectors)