Estimating the PDF of the SIC-MMSE equalizer output and its applications in designing LDPC codes with turbo equalization

We consider the analysis and design of low-density parity check (LDPC) codes for intersymbol interference (ISI) channels when used with soft interference cancellation plus linear minimum mean-square error filtering (SIC-MMSE) turbo equalization. We discuss techniques to compute the probability density function (pdf) of the extrinsic information at the output of the SIC-MMSE equalizer as a function of pdf of the input extrinsic information, channel impulse response, and the signal-to-noise ratio. For static ISI channels, we show that the output pdf can be modeled as symmetric Gaussian, and show that the mean can be evaluated without simulating the equalizer. For channels with long memory, we propose to use the unscented transform technique to compute the mean, which significantly reduces the computation required. Finally, for fading channels, we model the pdf by a mixture of symmetric Gaussian densities. Using these techniques, we are able to fairly accurately compute the thresholds for LDPC codes and design good irregular LDPC codes. Simulation results are in good agreement with the computed thresholds and the designed irregular LDPC codes outperform regular ones significantly.