EBP-GEXIT Charts for M-ary AWGN Channel for Generalized LDPC and Turbo Codes

TL;DR

This paper introduces a method to estimate the MAP threshold of various sparse-graph codes over M-ary AWGN channels using an approximation of the EBP-GEXIT function, extending previous results to non-binary channels.

Contribution

It extends the EBP-GEXIT analysis to non-binary channels and provides a practical way to estimate MAP thresholds for multiple code families, including LDPC and turbo codes.

Findings

BP threshold of spatially-coupled SC-TC saturates to the MAP threshold

Extended GEXIT charts effectively estimate MAP thresholds in non-binary channels

The method applies to LDPC, generalized LDPC, doubly generalized LDPC, and turbo codes.

Abstract

This work proposes a tractable estimation of the maximum a posteriori (MAP) threshold of various families of sparse-graph code ensembles, by using an approximation for the extended belief propagation generalized extrinsic information transfer (EBP-GEXIT) function, first proposed by Measson et al. We consider the transmission over non-binary complex-input additive white Gaussian noise channel and extend the existing results to obtain an expression for the GEXIT function. We estimate the MAP threshold by applying the Maxwell construction to the obtained approximate EBP-GEXIT charts for various families of low-density parity-check (LDPC), generalized LDPC, doubly generalized LDPC, and serially concatenated turbo codes (SC-TC). When codewords of SC-TC are modulated using Gray mapping, we also explore where the spatially-coupled belief propagation (BP) threshold is located with respect to the previously computed MAP threshold. Numerical results indicate that the BP threshold of the spatially-coupled SC-TC does saturate to the MAP threshold obtained via EBP-GEXIT chart.