Distribution of L-values in gray-mapped M²-QAM signals : exact expressions and simple approximations

In this paper we develop formulas for the probability density function (PDF) of the reliability metrics in bit- interleaved coded modulation (BICM) for arbitrary M2-ary quadrature amplitude modulation (M2-QAM) with Gray mapping. The advantage of this approach over an entirely general one proposed previously, is that the resulting formulas are obtained without resorting to any algorithmic steps. We also propose Gaussian mixture approximations that are analytically tractable, and we analyze their accuracy when evaluating uncoded bit error rate (BER) and BICM capacity