Bilinear hybrid expectation maximization and expectation propagation for semi-blind channel estimation

This paper discusses channel estimation during uplink transmission in Cell-Free (CF) Massive MIMO (MaMIMO) systems. We model the problem as a semi-blind estimation problem with independent and identically distributed (i.i.d.) Gaussian input. Two hybrid Expectation Maximization (EM) and Expectation Propagation (EP) algorithms are proposed to improve convergence behavior. The first algorithm, EM-EP, adopts a vector-level EP approach by treating the per-user channel coefficients and data sequence as EP variables. To make the algorithm tractable, we use the central limit theorem (CLT) to approximate the interference terms and employ EM to construct a majorizer function for the likelihood of the received data, leading to majorization minimization. To further enhance convergence behavior, we propose a matrixlevel loop-free EM-EP algorithm. In this algorithm, we treat the channel coefficients and data sequences corresponding to users using the same pilot as EP variables. This method is an alternating minimization algorithm, ensuring convergence. Our simulations verify the effectiveness of the two proposed algorithms.