Design of constellation labelling maps for iteratively demapped modulation schemes based on the assumption of hard-decision virtual channels

A bit-interleaved coded modulation system can be regarded as a serially concatenated (SC) system with a modulation scheme as a full rate inner code. The labelling map of the modulation scheme is very important for the performance of iterative demapping and decoding, which is usually designed with the help of the extrinsic information transfer (EXIT) chart. In the paper an analytical approximate expression for the EXIT function (EF) of a maximum a posterior probability demapper over memoryless channels is derived, where the real channel is replaced with a hard-decision virtual channel to make the EF easy to be calculated. The hard-decision virtual channel is selected to have the same capacity as the real channel. Then a closed-form EXIT function is defined for each signal point, i.e. the single signal EXIT function (SSEF). A useful property is proved that the area under the SSEF for each signal is determined by the constellation itself and has nothing to do with the labelling map. Based on the property and the binary switch algorithm, a general method is proposed to design the labelling maps of modulation schemes with desirable EFs matching channel codes. SC systems can then be built with the desired convergence and asymptotic behaviour.