Decentralized joint channel estimation and hierarchical data detection in cell-free massive MIMO under Bethe free energy optimization framework

Cell-Free (CF) Massive Multiple-Input Multiple-Output (MaMIMO) system is a promising architecture for nextgeneration networks. Without the cellular boundary, every User Equipment (UE) in a region is jointly served by all the Access Points (APs) scattered in the same region. In those CF scenarios, the number of users may exceed the length of pilot sequence. As a result, the information obtained from pilot is insufficient

to treat the channel as deterministic. In Bayesian estimation, constrained Bethe Free Energy (BEF) optimization leads to a system of equations describing the approximate marginal posteriors. Although the system of equations cannot be solved analytically, a message-passing–like method can be used to solve it iteratively. If the constraints of the BFE optimization are strict consistency constraints, solving the optimization problem

iteratively leads to Belief Propagation (BP), which is intractable due to the product of Gaussian Mixtures. On the other hand, with relaxed moments constraints, the BFE optimization leads to Expectation Propagation (EP). Although EP avoid the evaluation

of product of Gaussian Mixtures, the iterative method will sometimes contain invalid operations (computing the moments of a function that has infinite integral value). In this paper, besides the strict consistency constraints, we also propose an additional constraint that leads to a Variational-Bayes-(VB)-like operation to avoid the intractable operations. Furthermore, we also exploit the hierarchical UE symbol structure to prevent the invalid operations.