Threshold Saturation for Nonbinary SC-LDPC Ensembles on Discrete-Input Memoryless Channels

Abstract

In this letter, we investigate the threshold saturation effect of nonbinary spatially coupled low-density parity-check (SC-LDPC) ensembles defined over GF( $ {q}$ ) when the transmission takes place over discrete-input memoryless channels. To this end, we extend the duality rule for entropy established in the analysis of binary LDPC ensembles to the nonbinary case. This enables us to derive potential functions associated with log-likelihood ratio vector densities. Finally, we prove the existence of the threshold saturation effect following the approach introduced by Yedla et al. The analysis in this letter generalizes the existing work on nonbinary SC-LDPC ensembles and implies that the proof technique based on potential functions is also applicable to nonbinary sparse graphical codes over nonbinary discrete-input channels under belief propagation decoding.