High SNR Bounds for the BICM Capacity

In this paper, different aspects of the bit-interleaved coded modulation (BICM) capacity for the Gaussian channel are analyzed. Analytical bounds for the BICM capacity are developed. These bounds suggest that the BICM capacity at high signal-to-noise ration (SNR) is determined by the multiplicity of the minimum Euclidean distance over all the subconstellations generated by the mapper. Based on this observation, we conjecture that for any constellation, the highest BICM capacity at high SNR is always obtained by a Gray code, if one exists. Ready-to-use expressions based on Gauss--Hermite quadratures to compute the coded modulation and BICM capacities for any SNR are also presented. Using these expressions, it is shown that the BICM capacity is in general a nonconvex, nonconcave function of the input bit distribution. For 8PAM and 8PSK, there exist 12 and 7 classes of mappings, respectively, with equivalent high-SNR behavior, of which the best class comprises all Gray codes