Signal recovery problem arises in many regions,such as wireless communications,image processing,compressed sensing,statistical regression,and matrix completion,etc.Recovering the signal with high quality is not only of great significance to promote the application of the related technologies,but also a very important signal processing problem.Although the Bayesian framework provides a theoretical basis for the optimal mean square error solution to such problem,the increase of the user data results in the increase of the signal dimension and high-dimensional integral in the exact Bayesian estimator,whose computational complexity is exponential in the signal dimension.The message passing based Bayesian approximate algorithm is one kind of important method for solving this high-dimensional integral problem.However,due to the restriction of the system model,the existing message passing algorithms cannot be applied to some significant scenarios,such as user activity detection of the grant-free massive multiple-input multiple-output(MIMO)system,high-dimensional structured sparse signal recovery,high-dimensional multiple measurement vector(MMV),as well as multi-layer bilinear inverse problem.In view of this,we propose message passing based algorithms to solve these problems,and analyze the dynamic performance of the algorithms.The main contents and contributions of this paper include:1.To detect the user activity of the grant-free massive MIMO system,we propose a joint user activity and channel estimation method by combining loopy belief propagation(LBP)and generalized approximate message passing(GAMP),and a hybrid bilinear GAMP(HyBiGAMP)algorithm.Firstly,the channel from user to base station in massive connectivity scenario is modeled as conditioned Gaussian distribution by separating user activity factor from the channel,then a joint estimation is proposed by combining LBP and GAMP.Secondly,we propose a HyBiGAMP algorithm by combining LBP and bilinear GAMP,and then propose a joint user activity,channel,and data estimation based on the HyBiGAMP algorithm,which efficiently reduces the pilot symbols.2.We propose an expectation maximum(EM)-aided hybrid generalized expectation consistent(HyGEC)algorithm for recovering the high-dimensional structured sparse signal.Firstly,we propose a HyGEC algorithm by combining LBP and generalized expectation consistent for generalized linear inference problem with structured sparse signal.Under the ill-conditioned measurement matrix,the proposed HyGEC is more stable than the competitive hybrid GAMP.Secondly,for an unknown sparse ratio,we propose an EM-aided HyGEC algorithm,where the sparse ratio is estimated in M-step while the HyGEC is carried out to obtain the posterior distribution in E-step.3.We derive an analytical expression of the input-output mutual information of the high-dimensional MMV channel and propose a message passing solution for the signal recovery.As compared to the classical MMV problem,the high-dimensional MMV problem is extended in two aspects:row independent identically distributed inputs and row-wise outputs.Firstly,in the large-system limit,we derive an analytic expression of the input-output mutual information of the high-dimensional MMV channel by the replica method,where the fixed-point equations of the exact minimum mean square error(MMSE)estimator can be obtained.Secondly,we propose a message passing algorithm for the high-dimensional MMV problem using vector message passing rules.Finally,we obtain the state evolution equations of the proposed algorithm by performing state evolution analysis,which reveals the Bayes-optimality of the proposed algorithm.4.We propose the multi-layer BiG-AMP(ML-BiGAMP)to solve the multilayer generalized bilinear inverse problem.Firstly,we extend the existing BiG-AMP algorithm to multi-layer region and propose ML-BiGAMP algorithm.Secondly,in the large-system limit,we give the state evolution analysis,which predicts the dynamic mean square error of the proposed MLBiGAMP.Thirdly,we derive the fixed-point equations of the exact MMSE estimator by the replica method.The consistency between the state evolution equations and the fixed-point equations of the exact MMSE estimator verifies the Bayes-optimality of the ML-BiGAMP algorithm.Finally,we propose a joint user channel and data estimation method based on MLBiGAMP for an amplify-and-forward relay communications.This method can efficiently reduce the pilot symbols. |