Blind Channel Estimation and Joint Symbol Detection with Data-Driven Factor Graphs

TL;DR

This work tackles blind joint channel estimation and symbol detection over time-variant ISI channels by integrating belief propagation on a factor graph with the EM algorithm (EMBP). It introduces a data-driven variant (EMBP$^\star$) that uses momentum in BP updates and learned EM update scheduling to boost performance while reducing complexity, aided by a VAELE-based initialization. The proposed approach achieves competitive, and often superior, performance to coherent BP and pilot-based detectors in high-$\mathsf{snr}$ regimes, while remaining practical for short-block transmissions. The combination of model-based inference with offline learning yields a robust, low-overhead detector suitable for rapidly varying channels and future communication systems.

Abstract

We investigate the application of the factor graph framework for blind joint channel estimation and symbol detection on time-variant linear inter-symbol interference channels. In particular, we consider the expectation maximization (EM) algorithm for maximum likelihood estimation, which typically suffers from high complexity as it requires the computation of the symbol-wise posterior distributions in every iteration. We address this issue by efficiently approximating the posteriors using the belief propagation (BP) algorithm on a suitable factor graph. By interweaving the iterations of BP and EM, the detection complexity can be further reduced to a single BP iteration per EM step. In addition, we propose a data-driven version of our algorithm that introduces momentum in the BP updates and learns a suitable EM parameter update schedule, thereby significantly improving the performance-complexity tradeoff with a few offline training samples. Our numerical experiments demonstrate the excellent performance of the proposed blind detector and show that it even outperforms coherent BP detection in high signal-to-noise scenarios.

Figures (13)

• Figure 1: Factor graph representation of \ref{['eq:Ungerboeck_factorization']} for a channel with memory $L=2$.
• Figure 2: Squared error of the channel estimation $\hat{\bm{h}}$ for the EMBP and VAE-LE algorithm at ${\mathsf{snr}=10}\,\text{dB}$ for $10^5$ random channels with ${L=5}$, based on different initializations $\hat{\bm{h}}_\text{init}$: (a) ${\hat{\bm{h}}_\text{init} = \bm{h} + \sqrt{\gamma} \bm{h}_w}$ with ${\bm{h}_w \sim \mathcal{CN}(0,\bm{I}_{L+1}), \gamma \in \mathbb{R}^+}$ (solid line: mean, dashed line: median, dotted lines: 25/75 percentiles), (b) ${\hat{\bm{h}}_\text{init} = \bm{h}_{\bm{\delta}}}$ (circles: mean), (c) initialization of the EMBP algorithm using the result $\hat{\bm{h}}_\text{VAE}$ of the VAELE with ${\hat{\bm{h}}_\text{init} = \bm{h}_{\bm{\delta}}}$ (star: mean). The horizontal axis indicates the MSE of the initialization ${\hat{\bm{h}}_\text{init}}$.
• Figure 3: MSE versus $\mathsf{snr}$ for $10^7$ random channels with ${L=5}$ for various algorithms and initialization methods (solid line: mean, dashed line: median).
• Figure 4: BER over $\mathsf{snr}$ for various detection schemes, averaged over $10^7$ random channels with ${L=5}$.
• Figure 5: BER and ELBO ${\mathcal{L}(Q,\bm{\theta})}$ of the exact APP distribution $Q_\text{APP}$ and the approximate distribution $Q_\text{BP}$ for different channel assumptions ${\bm{h}=\alpha \bm{h}_0}$.

Theorems & Definitions (1)

• Theorem 1