Mixed Gamma Approximation for Check Node Updates in Density Evolution of LDPC Codes

To assist the design and optimization of low-density parity-check (LDPC) codes via density evolution (DE) on binary input additive white Gaussian noise (BIAWGN) channels, we propose a novel mixed Gamma approximation (MGA) scheme to obtain more accurate distribution of messages updated and output by the check nodes during DE iterations. Firstly, we highlight the inaccuracy of existing Gaussian approximation (GA) methods in approximating the distribution of check node output messages, especially when the messages from variable nodes are small with high probability (i.e. low signal-to-noise ratio), and the check nodes have a large degree, which leads to inexact results in GA methods. Then, we establish the MGA scheme by utilizing the statistical properties of Gamma distribution and combine it with GA, which outperforms the existing GA methods in the metrics of error of output mean and Kullback-Leibler (KL) divergence of output distribution for a wide range of parameters. Simulation and analysis validate that our MGA scheme has the potential for the design and optimization of LDPC codes, which can provide adequately accurate estimation of check node outputs with moderate complexity for a variety of approximation methods, such as Gaussian capacity approximation, and significantly reduce the computational complexity by sacrificing minor accuracy.