Graph Neural Network-based Joint Equalization and Decoding

TL;DR

The paper addresses equalization and decoding over ISI channels by learning message-passing on graph-structured representations. It builds GNNs on Forney factor graphs (FFG) for equalization and connects them to Tanner-graph–based decoders to realize joint equalization and decoding (JED) in an end-to-end framework, with a flooding schedule that reduces latency. Empirical results show that FFG-based GNN equalizers approach MAP performance and that flooding JED achieves a BER gain of $2.25$ dB at a fixed low latency compared to a BCJR-BP baseline, while offering significant latency reductions relative to sequential or traditional iterative schemes. Overall, the work demonstrates the practicality and scalability of end-to-end GNN receivers for communications, with promising directions for generalization and larger, higher-order systems.

Abstract

This paper proposes to use graph neural networks (GNNs) for equalization, that can also be used to perform joint equalization and decoding (JED). For equalization, the GNN is build upon the factor graph representations of the channel, while for JED, the factor graph is expanded by the Tanner graph of the parity-check matrix (PCM) of the channel code, sharing the variable nodes (VNs). A particularly advantageous property of the GNN is the robustness against cycles in the factor graphs which is the main problem for belief propagation (BP)-based equalization. As a result of having a fully deep learning-based receiver, joint optimization instead of individual optimization of the components is enabled, so-called end-to-end learning. Furthermore, we propose a parallel flooding schedule that further reduces the latency, which turns out to improve also the error correcting performance. The proposed approach is analyzed and compared to state-of-the-art baselines in terms of error correcting capability and latency. At a fixed low latency, the flooding GNN for JED demonstrates a gain of 2.25 dB in bit error rate (BER) compared to an iterative Bahl--Cock--Jelinek--Raviv (BCJR)-BP baseline.

Figures (7)

• Figure 1: Block diagram of the system model with an ISI channel and GNN for iterative/joint equalization and decoding.
• Figure 2: Forney factor graph with GNN elements are presented for an exemplary pair of VN/FN for a channel with memory $L=2$. Blue nodes represent MLP.
• Figure 3: Uncoded BER, BMI and latency of different equalizers for the Proakis-C channel and block length $N=132$. For the BER and BMI plot, $N_\mathrm{It}$ is selected such that increasing it does not improve the performance. Latency is given at $E_\mathrm{b}/N_0 = \qty{14}{dB}$, and graphed over different values of $N_\mathrm{It}$. Note that the latency of the BCJR is the only value depending on the sequence length due to its sequential nature.
• Figure 4: JED system with a decoder GNN based on the Tanner graph of the channel code and an equalizer GNN based on the FFG of the channel. Note that the Tanner graph of the channel code code has been rearranged according to the interleaver to directly match the equalizer VN.
• Figure 5: Schedules of the GNN-based JED