Spatially coupled turbo-coded continuous phase modulation: asymptotic analysis and optimization

For serially or parallel concatenated communication systems, spatial coupling techniques enable to improve the threshold of these systems under iterative decoding using belief propagation (BP). For the case of low-density parity-check (LDPC) codes, it has been shown that, under some asymptotic assumptions, spatially coupled ensembles have BP thresholds that approach the bitwise maximum a posteriori (MAP) threshold of the related uncoupled ensemble. This phenomenon is often referred to as threshold saturation , and it has sometimes very important consequences. For example, in the case of regular LDPC code ensembles, spatial coupling enables to achieve asymptotically the capacity for any class of binary memoryless symmetric channels. Since then, this threshold saturation has been conjectured or proved for several other types of concatenations. In this work, we consider a serially concatenated scheme which is the serial concatenation of a simple outer convolutional code and a continuous phase modulator (CPM) separated by an interleaver. Then, we propose a method to do the spatial coupling of several replicas of this serially concatenated scheme, aiming to improve the asymptotic convergence threshold. First, exploiting the specific structure of the proposed system, an original procedure is proposed in order to terminate the spatially coupled turbo-coded CPM scheme. In particular, the proposed procedure aims to ensure the continuity of the transmitted signal among spatially coupled replicas, enabling to keep one of the core characteristics and advantages of coded CPM schemes. Then, based on an asymptotic analysis, we show that the proposed scheme has very competitive thresholds when compared to carefully designed spatially coupled LDPC codes. Furthermore, it is shown how we can accelerate the convergence rate of the designed systems by optimizing the connection distributions in the coupling matrices. Finally, by investigating on different continuous phase modulation schemes, we corroborate the conjecture stating that spatially coupled turbo-coded CPM schemes saturate to a lower bound very close to the threshold given by the extrinsic information transfer (EXIT) area theorem.