Gaussian Message Passing Research Articles

Decentralized Baseband Processing With Gaussian Message Passing Detection for Uplink Massive MU-MIMO Systems

Decentralized baseband processing (DBP) architecture, which partitions the base station antennas into multiple antenna clusters, has been recently proposed to alleviate the excessively high interconnect bandwidth, chip input/output data rates, and detection complexity for massive multi-user multiple-input multiple-output (MU-MIMO) systems. In this paper, we propose a novel decentralized Gaussian message passing (GMP) detection for the DBP architecture. Based on the message passing rule, each antenna cluster iteratively calculated the local means and variances which are fused to generate the global symbol beliefs for demodulation. The state evolution framework of the decentralized GMP algorithm is presented under the assumptions of large-system limit and Gaussian sources. Analytical results corroborated by simulations demonstrate that the nonuniform antenna cluster partition scheme exhibits a higher convergence rate than the uniform counterpart. Simulation results illustrate that the proposed decentralized GMP detection outperforms the recently proposed decentralized algorithms.

Joint Channel Estimation and Equalization for Index-Modulated Spectrally Efficient Frequency Division Multiplexing Systems

Spectrally efficient frequency division multiplexing (SEFDM) relying on index modulation (IM) has emerged as a promising multicarrier technique. In this paper, we develop a joint channel estimation and equalization method based on factor graphs for SEFDM-IM signaling over frequency-selective fading channels. By approximating the interference in the frequency domain, we reformulate the problem to obey a linear state-space model and construct a multi-layer factor graph. To support a reconfigurable architecture, non-orthogonal demodulation is adopted and the colored noise encountered is approximated by a complex auto-regressive (CAR) model. For deriving a low-complexity parametric Gaussian message passing (GMP)-based method, we exploit an expectation propagation (EP)-based technique for approximating the discrete a posteriori distributions of the transmitted symbols in a Gaussian form. To further simplify the result, variational message passing (VMP) is applied to an equivalent soft node to obtain a Gaussian form. Moreover, we also derive the Cramér-Rao lower bound (CRLB) in closed-form. The overall complexity only grows linearly with the number of subcarriers and logarithmically with the length of the channel's memory. Compared to its Nyquist signaling based counterpart, SEFDM-IM signaling relying on the proposed algorithm exhibits up to 25% higher bandwidth efficiency without any bit error rate (BER) performance degradation.

Turbo Message Passing-Based Receiver Design for Time-Varying OFDM Systems

In this paper, we study time-varying orthogonal frequency division multiplexing (OFDM) systems, and propose a joint channel-and-signal estimation receiver based on turbo message passing (TMP) to efficiently suppress inter-carrier interference (ICI). We establish a factor graph representation of the problem and divide the whole factor graph into two parts, one for channel estimation and the other for signal detection. For the first part, we use Gaussian message passing (GMP) for channel estimation; for the second part, a discrete state space (DSS) model is employed to describe the transition of signal states, and a forward-backward algorithm is adopted for message passing over the transition trellis in signal detection. The resulting algorithm is referred to as DSS-GMP. The complexity of DSS-GMP quickly becomes the bottleneck as the increase of the signal constellation size and the ICI width. To address this issue, we further develop a continuous-state-space (CSS) model based turbo message passing algorithm, where the messages of modulated signals are approximated as continuous Gaussian messages. Numerical results demonstrate that the TMP based scheme significantly outperforms the state-of-the-art schemes.

Deep neural network-aided Gaussian message passing detection for ultra-reliable low-latency communications

Ultra-reliable low-latency communications (URLLC) is a key technology in 5G supporting real-time multimedia services, which requires a low-cost signal recovery technology in the physical layer. A kind of well-known low-complexity signal detection is message passing algorithm (MPA) based on factor graph. However, reliability and robustness of MPA are deteriorated when there are cycles in factor graph. To address this issue, we propose two novel Gaussian message passing (GMP) algorithms with the aid of deep neural network (DNN), in which the network architectures consist of two DNNs associated with detections for mean and variance of the signal. Particularly, the network architecture is constructed by transforming the factor graph and message update functions of the original GMP algorithm from node-type into edge-type. Then, weights and bias parameters are assigned in the network architecture. With the aid of deep learning methods, the optimal weights and bias parameters are obtained. Numerical results demonstrate that two proposed DNN-aided GMP algorithms can significantly improve the convergence of original GMP algorithm and also achieve robust performances in the cases without prior information.

Gaussian Message Passing for Overloaded Massive MIMO-NOMA

This paper considers a low-complexity Gaussian message passing (GMP) Multi-User Detection (MUD) scheme for a coded massive multiple-input multiple-output (MIMO) system with non-orthogonal multiple access (massive MIMO-NOMA), in which a base station with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{s}$ </tex-math></inline-formula> antennas serves <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}$ </tex-math></inline-formula> sources simultaneously in the same frequency. Both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{s}$ </tex-math></inline-formula> are large numbers, and we consider the overloaded cases with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}&gt;N_{s}$ </tex-math></inline-formula> . The GMP for MIMO-NOMA is a message passing algorithm operating on a fully-connected loopy factor graph, which is well understood to fail to converge due to the correlation problem. The GMP is attractive as its complexity order is only linearly dependent on the number of users, compared to the cubic complexity order of linear minimum mean square error (LMMSE) MUD. In this paper, we utilize the large-scale property of the system to simplify the convergence analysis of the GMP under the overloaded condition. We prove that the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">variances</i> of the GMP definitely converge to the mean square error (MSE) of the LMMSE multi-user detection. Second, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">means</i> of the traditional GMP will fail to converge when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}/N_{s}&lt; (\sqrt {2}-1)^{-2}\approx 5.83$ </tex-math></inline-formula> . Therefore, we propose and derive a new convergent GMP called scale-and-add GMP (SA-GMP), which always converges to the LMMSE multi-user detection performance for any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}/N_{s}&gt;1$ </tex-math></inline-formula> , and show that it has a faster convergence speed than the traditional GMP with the same complexity. Finally, the numerical results are provided to verify the validity and accuracy of the theoretical results presented.

FTN Signaling-Aided Space-Time Multi-Mode Index Modulation Systems With a GMP-Based Receiver

In this paper, we propose a faster-than-Nyquist (FTN) signaling-aided space-time multi-mode index modulation (IM) scheme with a low-complexity Gaussian message passing (GMP) based receiver. The advantages of spatial domain IM, time-domain IM relying on multiple modes and FTN signaling are explored. Specifically, multi-mode IM is employed in time domain by activating all time slots, hence the spectral efficiency is improved substantially compared to its single-mode counterpart. Meanwhile, the transmission rate is further increased by utilizing FTN signaling. At the receiver, based on the state-space model of IM symbols, a two-layer Forney style factor graph (FG) representation for the system is constructed to exploit the truncated inter-symbol interference structure imposed by FTN. Belief propagation is invoked to update the messages over the FG. The discrete messages from the channel decoder are properly approximated to be Gaussian for reducing the complexity of joint detection of multidimensional index modulated symbols. Moreover, the extrinsic information from the equalizer is derived with a concise representation. A low-complexity log-likelihood ratio calculation method is further proposed, where the search space is significantly reduced using a Gaussian mixture model. Simulation results demonstrate that the proposed scheme outperforms the existing schemes in terms of bit error rate given the same spectral efficiency. The outstanding performance and low complexity of the proposed GMP-based receiver are also verified by comparisons with existing methods.

Notice of Violation of IEEE Publication Principles: A Robust Signal Detection Scheme for 5G Massive Multiuser MIMO Systems

Notice of Violation of IEEE Publication Principles A Robust Signal Detection Scheme for 5G Massive Multiuser MIMO Systems by Imran Khan in IEEE Transactions on Vehicular Technology, July 2018, 9597-9604 After careful and considered review of the content and authorship of this paper by a duly constituted expert committee, this paper has been found to be in violation of IEEE's Publication Principles. This paper copied portions of text from the following paper with no credit to the original work (including appropriate references to the original author(s) and/or paper title) and without permission. Convergence Analysis and Assurance for Gaussian Message Passing Iterative Detector in Massive MU-MIMO Systems by Lei Liu, Chau Yuen, Yong Liang Guan, Ying Li, and Yuping Su in IEEE Transactions on Wireless Communications, June 2016, pp.6487-6501 Aiming at the problem of high detection complexity at the base station in massive multiuser multiple-input multiple-output system, this paper proposes a low complexity variable node full information Gaussian message passing iterative detection based on forced convergence (VFI-GMPID-FC) algorithm. First, the traditional GMPID algorithm is improved to get the VFI-GMPID algorithm. The detection performance of the VFI-GMPID algorithm approaches the minimum mean square error (MMSE) detection algorithm, but the complexity is much lower than that of the MMSE algorithm. Combined with the idea of forced convergence and VFI-GMPID, the VFI-GMPID-FC algorithm is proposed to further reduce the complexity of the algorithm and improve the detection efficiency. Finally, the simulation results show that the proposed algorithm can effectively reduce the complexity of the algorithm while ensuring improved detection performance.

Scalable Uplink Signal Detection in C-RANs via Randomized Gaussian Message Passing

Cloud Radio Access Network (C-RAN) is a promising architecture for unprecedented capacity enhancement in next-generation wireless networks thanks to the centralization and virtualization of base station processing. However, centralized signal processing in C-RANs involves high computational complexity that quickly becomes unaffordable when the network grows to a huge size. Among the first, this paper endeavours to design a scalable uplink signal detection algorithm, in the sense that both the complexity per unit network area and the total computation time remain constant when the network size grows. To this end, we formulate the signal detection in C-RAN as an inference problem over a bipartite random geometric graph. By passing messages among neighboring nodes, message passing (a.k.a. belief propagation) provides an efficient way to solve the inference problem over a sparse graph. However, the traditional message-passing algorithm is not guaranteed to converge, because the corresponding bipartite random geometric graph is locally dense and contains many short loops. As a major contribution of this paper, we propose a randomized Gaussian message passing (RGMP) algorithm to improve the convergence. Instead of exchanging messages simultaneously or in a fixed order, we propose to exchange messages asynchronously in a random order. The proposed RGMP algorithm demonstrates significantly better convergence performance than conventional message passing. The randomness of the message update schedule also simplifies the analysis, and allows the derivation of the convergence conditions for the RGMP algorithm. In addition, we propose a blockwise RGMP (B-RGMP) algorithm for practical implementation. The average computation time of B-RGMP remains constant when the network size increases.

Cooperative Joint Localization and Clock Synchronization Based on Gaussian Message Passing in Asynchronous Wireless Networks

Localization and synchronization are very important in many wireless applications such as monitoring and vehicle tracking. Utilizing the same time of arrival (TOA) measurements for simultaneous localization and synchronization is challenging. In this paper, we present a factor graph (FG) representation of the joint localization and time synchronization problem based on TOA measurements, in which the non-line-of-sight measurements are also taken into consideration. On this FG, belief propagation (BP) message passing and variational message passing (VMP) are applied to derive two fully distributed cooperative algorithms with low computational requirements. Due to the nonlinearity in the observation function, it is intractable to compute the messages in closed form and most existing solutions rely on Monte Carlo methods, e.g., particle filtering. We linearize a specific nonlinear term in the expressions of messages, which enables us to use a Gaussian representation for all messages. Accordingly, only the mean and variance have to be updated and transmitted between neighboring nodes, which significantly reduces the communication overhead and computational complexity. A message passing schedule scheme is proposed to trade off between estimation performance and communication overhead. Simulation results show that the proposed algorithms perform very close to particle-based methods with much lower complexity especially in densely connected networks.

Convergence Analysis and Assurance for Gaussian Message Passing Iterative Detector in Massive MU-MIMO Systems

This paper considers a low-complexity Gaussian Message Passing Iterative Detection (GMPID) algorithm for massive Multiuser Multiple-Input Multiple-Output (MU-MIMO) system, in which a base station with $M$ antennas serves $K$ Gaussian sources simultaneously. Both $K$ and $M$ are very large numbers, and we consider the cases that $K<M$. The GMPID is a low-complexity message passing algorithm based on a fully connected loopy graph, which is well understood to be not convergent in some cases. As it is hard to analyse the GMPID directly, the large-scale property of the massive MU-MIMO is used to simplify the analysis. Firstly, we prove that the variances of the GMPID definitely converge to the mean square error of Minimum Mean Square Error (MMSE) detection. Secondly, we propose two sufficient conditions that the means of the GMPID converge to those of the MMSE detection. However, the means of GMPID may not converge when $ K/M\geq (\sqrt{2}-1)^2$. Therefore, a new convergent GMPID called SA-GMPID (scale-and-add GMPID) , which converges to the MMSE detection in mean and variance for any $K<M$ and has a faster convergence speed than the GMPID, but has no higher complexity than the GMPID, is proposed. Finally, numerical results are provided to verify the validity and accuracy of the theoretical results.

A Factor Graph Approach to Exploiting Cyclic Prefix for Equalization in OFDM Systems

In OFDM systems, cyclic prefix (CP) insertion and removal enables the use of a set of computationally efficient single-tap equalizers at the receiver. Due to the extra transmission time and energy, the CP causes a loss in both spectrum efficiency and power efficiency. On the other hand, as a repetition of part of the data, the CP brings extra information and can be exploited for detection. Therefore, instead of discarding the CP observation as in the conventional OFDM system, we utilize all the received signals in a soft-input soft-output equalizer of a turbo equalization OFDM system. First, the models for both the CP part and the non-CP part of observation are presented in a Forney-style factor graph (FFG). Then based on the computation rules of the FFG and the Gaussian message passing (GMP) technique, we develop an equalization algorithm. With proper approximation, the complexity of the proposed algorithm is reduced to \changedmathcal O(2RNlog2N+4RGlog2G+2RG) per data block for R iterations, where N is the length of the data block and G is equal to P+L-1 with P the length of the CP and L the maximum delay spread of the channel. To justify the performance improvement, SNR analysis is provided. Simulation results show that the proposed approach achieves a significant gain over the conventional approach and the turbo equalization system converges within two iterations.

EM-Based Joint Channel Estimation and Detection for Frequency Selective Channels Using Gaussian Message Passing

It has been recently shown that the expectation-maximization (EM) algorithm may be viewed as message passing in factor graphs, and in particular, for a linear Gaussian system (with unknown coefficients), the EM algorithm may be purely implemented with Gaussian message passing. In this work, with a Gaussian assumption of the data symbols and a Forney-style factor graph representation for single-carrier transmission over frequency selective channels, a Gaussian message passing EM approach for joint channel estimation and detection is developed. The complexity of the proposed approach grows logarithmically with the length of the observation vector, enabling an efficient handling of (quasi-static and time-varying) frequency selective channels with a large number of channel taps.

GMP-Based Channel Estimation for Single-Carrier Transmissions over Doubly Selective Channels

We present a graph-based channel estimation approach for SC-IFDE (single-carrier transmissions with iterative frequency domain equalization) without CP (cyclic prefix) over doubly selective channels using the recently developed Gaussian message passing (GMP) technique. A direct application of the GMP updating rules in the FFG (Forney-style factor graph) of the SC-IFDE system model incurs high complexity. Approximate updating rules are therefore developed to overcome this problem. The proposed GMP-based channel estimation approach has similar complexity as the low-complexity Kalman-filtering based frequency domain channel estimation approach in the literature, but significantly outperforms the latter due to its enhanced capability in capturing the time correlation information of doubly selective channels through bidirectional processing.

A low-complexity iterative channel estimation and detection technique for doubly selective channels

In this paper, we propose a low-complexity iterative joint channel estimation, detection and decoding technique for doubly selective channels. The key to the proposed technique is a segment-by-segment processing strategy under the assumption that the channel is approximately static within a short segment of a data block. Through a virtual zero-padding technique, the proposed segment-by-segment equalization approach inherits the low-complexity advantage of the conventional frequency domain equalization (FDE), but does not need the assistance of guard interval (for cyclic-prefixing or zero-padding), thereby avoiding the spectral and power overheads. Furthermore, we develop a low-complexity bidirectional channel estimator, where the Gaussian message passing (GMP) technique is used to exploit the channel correlation information, and the intermediate channel estimation results in the iterative process are employed to perform inter-tap interference cancellation. Simulation results demonstrate the effectiveness of the proposed detection and channel estimation algorithms.