Probability density function of reliability metrics in BICM with arbitrary modulation: Closed-form through algorithmic approach

In the popular bit-interleaved coded modulation (BICM) the output of the channel encoder and the input of the modulator are separated by a bit-level interleaver. From the decoder's point of view, the modulator, the transmission channel, and the demodulator (calculating bits' reliability metrics) become a memoryless BICM channel with binary inputs and real outputs. In unfaded channels, the BICM channel's outputs (reliability metrics) are known to be Gaussian for binary- or quaternary phase shift keying but their probability density function (PDF) is not known for higher-order modulation. We fill this gap by presenting an algorithmic method to calculate closed-form expressions for the PDF of reliability metrics in BICM with arbitrary modulation and bits-to-symbol mapping when the so-called max-log approximation is applied. Such probabilistic description of BICM channel is useful to analyze, from an information- theoretic point of view, any BICM constellation/mapping design.