Non-circular combined expectation propagation and variational Bayes for semi-blind channel estimation in cell-free systems

In this work, we investigate uplink communication in Semi-Blind Cell-Free (CF) Massive Multiple-Input Multiple-Output (MaMIMO) systems. A key challenge in CF MaMIMO systems is pilot contamination, which arises when multiple user terminals (UTs) share 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 by enabling access points (APs) to jointly estimate both the channel and user data. This joint estimation, however, leads to a bilinear problem. To develop a tractable algorithm that exploits the finite alphabet of the user data, we analyze the constrained Bethe Free Energy (BFE) of the bilinear system and propose a message-passing algorithm based on minimizing the constrained BFE. We decompose complex quantities into in-phase and quadrature components, removing circular Gaussian constraints. Additionally, we find that replacing normal covariance constraints with average covariance constraints significantly reduces computational complexity. Simulation results indicate that this simplification incurs negligible performance loss.