Bethe Free Energy Minimization Based Message Passing Detection Algorithms For Massive MIMO Systems

With the rapid growth of various types of intelligent mobile terminals and the rapid development of the Internet of Things,the number of users and data traffic in mobile communication systems are experiencing exponential growth.In order to support various application scenarios of future mobile communications,a-cademia and industry have conducted long-term research and exploration on the key technologies of wireless communications such as massive multiple input multiple output(MIMO)technology.Traditional estimation and detection algorithms have the problems of complexity and applicability when used in new scenarios of future mobile communication.We focus on the large-dimensional signal estimation and detection problems in various scenarios of future mobile communication systems.On the basis of exploring the unified Bayesian inference framework based on Bethe free energy minimization,we consider the method of Bayesian inference and carry out research on novel and efficient message passing algorithms for massive MIMO with large-scale antennas at the base station,grant-free transmission and non-orthogonal multiple access(NOMA)based on perfect energy spread transform(PEST).Firstly,the variational method is introduced for the general detection and estimation problem,that is,belief function is introduced into the Bayesian inference to approximate the posterior probability so that de-tection and estimation problem can be transformed into the constrained variational free energy minimization.On the basis of variational Bayesian inference,we derive a unified Bayesian inference framework under the constrained Bethe free energy minimization.With the design of constraints including factorization,region-based approximation and exponential family distribution,we derive different message passing algorithms using the method of Lagrangian multipliers.Specifically,mean field(MF)algorithm,variational message passing(VMP)and expectation maximization(EM)are obtained from the use of factorization,belief prop-agation(BP)is obtained from the use of region-based approximation,and expectation propagation(EP)is obtained from the use of exponential family distribution.Further,for the generalized linear model(GLM),with the constraint of Gaussian distribution which is a special case of exponential family distribution,we derive generalized linear expectation propagation(GLEP)and generalized linear approximate expectation propagation(GLAEP).The algorithm of traditional generalized approximate message passing(GAMP)is the special case of GLAEP in the large-scale system.Secondly,for massive MIMO detection with large-scale antennas at the base station,with the reason that traditional MIMO detection algorithms have high computational complexity and conventional GAMP is not suitable for specially correlated channels,we propose GAMP detection with linear preprocessing,which is also called row-orthogonal GAMP(RO-GAMP)detection.The received signals in massive MIMO are performed with linear preprocessing at the base station and the output of preprocessing is denoted as obser-vation signals of detection.Under the framework of Bethe free energy minimization,we put forward four design principles for linear preprocessing and the optimal conditions on these principles are derived.Then,we investigate two kinds of preprocessing matrices.When these two matrices are used for massive MIMO transmission employing orthogonal frequency division multiplexing(OFDM)with slow-varying channels,a low-complexity preprocessing method is presented finally by sharing preprocessing matrix over several time and frequency resource elements.Numerical results demonstrate the advantage of RO-GAMP and low com-plexity RO-GAMP over GAMP and GAMP based on beam domain channel,in terms of symbol error rate and convergence rate.Then,we propose hybrid message passing algorithms for joint user detection and channel estimation in the grant-free uplink transmission owing to high computational complexity and poor performance of active user detection in the signal reconstruction algorithms of compressed sensing(CS).We consider two kinds of typical massive MIMO channels:independent identically distribution(i.i.d)channel and spatially sparse channel,and for each channel we consider two scenarios where prior probability of user activity is known and unknown at the BS respectively.Under the framework of Bethe free energy minimization,the constraints of marginalization consistency are reasonably relaxed as the hybrid constraints including marginalization consis-tency,mean and variance matching and factorization,such that the joint user detection and channel estimation is modeled as Bethe free energy minimization with hybrid constraints.Then,by computing the stationary points of Bethe free energy minimization,hybrid message passing algorithms are derived and damping is introduced to improve the convergence of hybrid message passing algorithms.Simulation results show that the performance of hybrid message passing algorithms are much better that traditional signal reconstruction algorithms in the CS,in terms of activity detection and channel estimation.Finally,for uplink multiuser detection in NOMA based on PEST,with the reason that traditional detec-tion algorithms have the problem of high computational complexity,we propose message passing algorithm with lower complexity.We regard multiuser detection of NOMA based on PEST as multi-level GLM and under the framework of Bethe free energy minimization,we select the mean and variance matching as con-straints and investigate the algorithm of approximate message passing(AMP)based PEST.What's more,linear preprocessing at the receiver is proposed to solve the poor convergence of PEST-AMP in correlated channels.Specially,for slowly-varying channels,preprocessing matrices are shared over multiple time and frequency resource elements to reduce the preprocessing complexity.Then,we propose soft input soft output(SISO)detection for the iterative structure of detection and decoding.Simulation results show that compared to minimum mean square error(MMSE)detection with successive interference cancellation(SIC),AMP with preprocessing and it's low-complexity form have better performance in terms of bit error rate performance,where performance of AMP with preprocessing shared is very close to AMP with preprocessing.