Distribution of L-values in gray-mapped M2-QAM: closed-form approximations and applications

In this paper we develop closed form approximations for the probability density function (PDF) of the reliability metrics (L-values) in bit-interleaved coded modulation (BICM). The expressions are valid for M²-ary quadrature amplitude modulations (M²-QAM) with binary reflected Gray mapping when the metrics are calculated using the so-called max-log approximation. Based on the developed expressions, we also propose two simple Gaussian mixture approximations that are analytically tractable. We apply our developments to efficiently calculate the BICM capacity, and to develop bounds on the coded bit-error rate when a convolutional code is used. The coded performance of a hybrid automatic repeat request (HARQ) based on constellation rearrangement is also evaluated.