Graph-Based Decoding in the Presence of ISI

We propose a new graph representation for ISI channels that can be used for combined equalization and decoding by linear programming (LP) or iterative message-passing (IMP) decoding algorithms. We derive this graph representation by linearizing the ML detection metric, which transforms the equalization problem into a classical decoding problem. We observe that the performance of LP and IMP decoding on this model are very similar in the uncoded case, while IMP decoding significantly outperforms LP decoding when low-density parity-check (LDPC) codes are used. In particular, in the absence of coding, for certain classes of channels, both LP and IMP algorithms always find the exact ML solution using the proposed graph representation, without complexity that is exponential in the size of the channel memory. This applies even to certain two-dimensional ISI channels. However, for some other channel impulse responses, both decoders have nondiminishing probability of failure as SNR increases. We provide analytical explanations for many of these observations. In addition, we study the error events of LP decoding in the uncoded case, and derive a measure that can be used to classify ISI channels in terms of the performance of the proposed detection scheme.