Analysis of spatially coupled LDPC codes on the binary erasure channel

The finite-length performance of spatially coupled low-density parity-check (SC-LDPC) codes constructed from protographs is analyzed. The equivalence of peeling decoding (PD) and belief propagation (BP) for LDPC codes over the binary erasure channel (BEC) is examined. Modifying the scheduling for PD, it is shown that exactly the same variable nodes (VNs) are resolved in every iteration as with BP. The decoding process is then modeled as a random stochastic process to obtain estimates for the error probability in the waterfall region. Given the protograph used to generate a SC-LDPC code ensemble, a set of scaling parameters is computed to characterize the average finite-length performance in the waterfall region. The error performance of structured SC-LDPC code ensembles is shown to follow a scaling law similar to that of unstructured randomly-constructed SC-LDPC codes. Several SC-LDPC protograph structures are compared. The analysis reveals significant differences in their finite-length scaling behavior, which is corroborated by simulation. In the waterfall region, spatially-coupled repeat-accumulate codes outperform other SC-LDPC codes that have the same rate but better asymptotic thresholds. The decrease in the number of erased VNs during the decoding process is analyzed instead of the number of resolvable equations. The former quantity can also be tracked with much less complexity using density evolution. Finally, a scaling law using this quantity is established for SC-LDPC codes.