The Three/Two Gaussian Parametric LDLC Lattice Decoding Algorithm and Its Analysis

Abstract

Low density lattice codes (LDLCs) are high-dimensional lattices with a sparse inverse generator matrix that can be decoded efficiently using iterative decoding. In the iterative LDLC decoder, the messages are Gaussian mixtures, and for any implementation, the Gaussian mixtures must be approximated. This paper describes a parametric LDLC decoding algorithm, where internally at the variable node, infinite Gaussian mixtures are approximated with three or two Gaussians, while the messages between nodes are single Gaussians. Strengths of the algorithm include its simplicity and suitability for analysis. Analysis is performed by evaluating the Kullback–Leibler divergence between the true messages and the three/two Gaussian approximation. The approximation using three or two Gaussians is more accurate than previously proposed approximations. Also, noise thresholds for the proposed LDLC decoder are presented, and the proposed decoder reduces the noise thresholds 0.05 dB compared with previous parametric decoders. The numerical results show that for $n=100$ and $n=1000$ , the two-Gaussian approximation is the same as the full-complexity decoder. But when the dimension is $n=10\,000$ , a three-Gaussian approximation is needed.