Mapping Optimization for a MAP Turbo Detector Over a Frequency-Selective Channel

In this paper, we consider a bit interleaved coded modulation (BICM) scheme over a frequency-selective channel. At the receiver, a turbo detector composed of a maximum a posteriori (MAP) detector and a MAP decoder is used. We propose to optimize the mapping choice in this context since it is well known that it has a significant impact on the performance of iterative receivers. We consider fixed and Rayleigh fading channels. Based on the Gaussian approximation, we give an analytical upper bound of the bit error probability (BEP) at the output of the detector for fixed channels. Then, we derive an approximate analytical expression of the BEP for both fixed and Rayleigh fading channels when the a priori information provided by the decoder is reliable, and then, most errors at the output of the detector are isolated. We find that the mapping minimizing this expression corresponds to a move away from the symbols differing by only one bit, unlike the Gray mapping. Since an exhaustive search to find this mapping is complicated for high-order modulations, we propose to use the binary switching algorithm (BSA) to approach the solution. Then, based on these results, we propose to adapt the mapping according to the signal-to-noise ratio (SNR) of the transmission.