From BP to MAP via Spatially Coupling
It is well-known that belief-propagation (BP) decoding of low-density parity-check (LDPC) codes is suboptimal and that the noise threshold of maximum-a-posteriori (MAP) decoding can be larger than the BP threshold. Recently, Kudekar et al. proved that regular LDPC ensembles can be spatially coupled (SC) so that the BP noise threshold saturates to the MAP noise threshold of the original ensemble. These SC ensembles are instances of LDPC convolutional (LDPCC) codes and the new proof explains an earlier observation by Lentmaier et al. that terminated LDPCC codes allow reliable communication at rates very close to capacity.The main benefit of SC codes is that code optimization is not required for near-optimal performance. For a variety of problems, this allows SC codes to approach capacity universally (e.g., like random codes with MAP decoding). This talk will give an overview of these recent results and describe a simple proof technique that allows one prove threshold saturation for a broad class of spatially-coupled systems.