Robust Approximate Message Passing Detection Based on Minimizing Bethe Free Energy for Massive MIMO Systems

Owing to the advantage of low-complexity, message passing algorithms have been extensively studied for massive multiple-input multiple-output (MIMO) detection. Most of message passing algorithms assume that the channel state information (CSI) is completely known by receiver, which is unrealistic in wireless communication. In this paper, we investigate a robust approximate message passing (RAMP) detection algorithm with imperfect CSI based on minimizing Bethe free energy. For given pilot structure and channel estimation methods, the results of channel estimation should be denoted by a probability density function of CSI rather than its estimation values. Based on such observations, the MIMO detection issue in the presence of channel estimation error is formulated as a Bethe free energy minimization subject to appropriately imposed constraints and the given statistical model of CSI. The Lagrange multiplier theory is employed to identify the stationary points of the constrained Bethe free energy, which give us back the fixed-point equations. This results in an iterative algorithm for detection in massive MIMO systems. Numerical experiments corroborate its superiority in terms of SER performance when the CSI is imperfect.