Analysis of the iterative decoding of LDPC and product codes using the Gaussian approximation

We propose a novel density evolution approach to analyze the iterative decoding algorithms of low-density parity-check (LDPC) codes and product codes, based on Gaussian densities. Namely, for these classes of codes we derive a one-dimensional (1D) map whose iterates directly represent the error probability both for the additive white Gaussian noise (AWGN) and the Rayleigh-fading channel. These simple models allow a qualitative analysis of the nonlinear dynamics of the decoding algorithm. As an application, we compute the decoding thresholds and show that they are consistent with the simulation results available in the literature.