Decentralized message-passing for semi-blind channel estimation in cell-free systems based on Bethe free energy optimization

In this work, we investigate uplink communication in Semi-Blind Cell-Free (CF) Massive Multiple-Input Multiple-Output (MaMIMO) systems. One of the major challenges in CF MaMIMO systems is pilot contamination, where multiple user terminals (UTs) may use the same pilot sequence due to an imbalance between the number of UTs and the length of the pilot sequence. Semi-blind approaches have been proposed to address this issue, where access points (APs) jointly estimate both the channel and user data. This joint estimation leads to a bilinear problem. Channel estimation in bilinear systems with Gaussian input has been studied in prior work, with expectation propagation (EP)- based algorithms, such as variable-level (VL)-EP and hybrid expectation maximization (EM)-EP, being proposed. However, in this paper, the user data follows a categorical distribution. To develop a tractable algorithm that leverages the finite alphabet of the user data, we investigate the Bethe free energy (BFE) of the bilinear system and propose a message-passing algorithm by minimizing the BFE. The resulting algorithm combines variational Bayes (VB), belief propagation (BP), and EP.