Analysis of Coded Shaped QAM Signaling at Short and Moderate Lengths

This paper studies distance properties of coded modulation systems with shaping. The relation between Hamming spectra of a code and its squared Euclidean distance (SED) spectra is analyzed. The main focus of the paper is the so-called shaping after coding. Three shaping schemes based on different mappings of the code symbols onto the shaped QAM signals are proposed and analyzed. Examples of NB LDPC codes used in communication standards are considered. An approach to optimization of shaping books, which combines a genetic algorithm with optimization of the SED for the NB LDPC coded shaped signal sets, is studied. An approximation of the two-dimensional moment generating function for pairs of Hamming distances and SEDs for coded shaped signal sets is derived. A comparison of the estimated SED spectra for coded shaped signal sets with different mappings are presented.