Deriving performance bounds for ISI channels using Gallager codes

We develop density evolution methods for evaluating the performance of Gallager's low-density parity-check (LDPC) codes over binary inter-symbol interference (ISI) channels. In contrast to previous work on memoryless channels, the noise tolerance thresholds for ISI channels depend on the transmitted sequences. The concentration statements are appropriately adjusted to reflect this difference. We compare the thresholds of regular Gallager codes over the 1 - D partial response channel to the i.i.d. information rate, showing that at high code rates, regular Gallager codes are asymptotically optimal.