Some new results on the design of codes for inter-symbol interference channels based on convergence of turbo equalization

It is well-known that iterative equalization/decoding offers good coding gains for intersymbol interference (ISI) channels. Recently, precoded ISI channels have been considered with turbo equalization which results in significant interleaving gains due to the recursive nature of precoded ISI channels. This paper presents new results pertaining to the design of codes for precoded ISI channels. An analytical proof is presented to show that precoding results in a loss in reliability during the first iteration for all 2-tap ISI channels. We compute the ratio of probability of error between nonprecoded and precoded ISI channels, which is closely related to the reliability, via a recursion. Based on this and the convergence properties of precoded and non-precoded channels, it is shown that by using a mixture of precoded and non-precoded parts, better performance can be achieved. Due to the recursiveness introduced from precoding, the codes required to achieve good performance are very simple codes and, hence, the resulting schemes provide good bit error rate performance at low receiver complexity.