Graph-based iterative decoding algorithms for parity-concatenated trellis codes

We construct parity-concatenated trellis codes in which a trellis code is used as the inner code and a simple parity-check code is used as the outer code. From the Tanner-Wiberg-Loeliger (1981, 1996) graph representation, several iterative decoding algorithms can be derived. However, since the graph of the parity-concatenated code contains many short cycles, the conventional min-sum and sum-product algorithms cannot achieve near-optimal decoding. After some simple modifications, we obtain near-optimal iterative decoders. The modifications include either (a) introducing a normalization operation in the min-sum and sum-product algorithms or (b) cutting the short cycles which arise in the iterative Viterbi algorithm (IVA). After modification, all three algorithms can achieve near-optimal performance, but the IVA has the least average complexity. We also show that asymptotically maximum-likelihood (ML) decoding and a posteriori probability (APP) decoding can be achieved using iterative decoders with only two iterations. Unfortunately, this asymptotic behavior is only exhibited when the bit-energy-to-noise ratio is above the cutoff rate. Simulation results show that with trellis shaping, iterative decoding can perform within 1.2 dB of the Shannon limit at a bit error rate (BER) of 4×10^-5 for a block size of 20000 symbols. For a block size of 200 symbols, iterative decoding can perform within 2.1 dB of the Shannon limit