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Low-density parity-check (LDPC) codes are widely used as error correction codes in new generation digital TV standards, such as the second generation of terrestrial digital video broadcasting standard (DVB-T2), Advanced Television Systems Committee (ATSC) 3.0, etc. The nonlinear belief propagation (BP) algorithm has excellent decoding performance for LDPC codes, but is often simplified in hardware implementations by linear min-sum (MS) algorithm due to its high complexity. This simplification also leads to over-estimation problems, which can be corrected by adding factors in conventional algorithms (e.g., normalized min-sum (NMS), offset min-sum (OMS), and variable scaling normalized min-sum (VMS) algorithms). However, the correction factors of these modified MS algorithms cannot adapt to different channels and modulations, and the performance needs further improvement. In this paper, the concepts of over-estimation value (OEV) and over-estimation rate (OER) are introduced to describe the over-estimation problem of the MS algorithm. Then, under the guidance of OEV and OER, a polygonal line min-sum (PMS) algorithm with correction factors adapted to different channels and modulations is proposed according to LLR distribution. Following the properties of OEV and OER, PMS algorithm is further simplified into Simplified PMS (SPMS) algorithm. LDPC codes from ATSC 3.0 are adopted in this paper to evaluate SPMS algorithm in comparison with the conventional algorithms. Extensive simulation results show that the SPMS algorithm for ATSC 3.0 LDPC decoder has at most 1.61dB, 0.24dB and 0.36dB gain over NMS, OMS and VMS algorithms respectively when frame error rate (FER) is at 10⁻⁴ level over additive white Gaussian noise (AWGN) channel with QPSK modulation. More importantly, the simulation results show that the SPMS algorithm can achieve much better performance than these modified MS algorithms over AWGN and Rayleigh channel with higher-order modulations or under limited maximum iteration number.

The integration of terrestrial and satellite communications (Satcom) is advocated for satisfying the challenging requirements of seamless, high-performance services. However, both the bandwidth and the power available are limited over satellite channels. In this paper, we propose index modulation (IM) and code-aided Satcom by conveying information by a pair of distinguishable constellation modes and their permutations. In order to combat both the linear and nonlinear distortionimposed by satellite channels, we conceive a factor graph (FG)-based iterative detection algorithm for Satcom relying on dualmode (DM) IM (Sat-DMIM). The correlation amongst Sat-DMIM symbols imposed by both the channel-induced dispersion and the mode-selection mapping is explicitly represented by the FGconstructed. Then the amalgamated belief propagation (BP) and mean field (MF) message passing algorithm is derived over this FG for detecting both the IM bits and the classic constellation mapping bits, while eliminating both the linear and nonlineardistortions. The complexity of the iterative detection algorithm is reduced by linearizing some high-order terms appearing in nonlinear distortion components using the a posteriori estimates of the Sat-DMIM symbols obtained from the previous iteration. Our simulation results demonstrate the power of the proposed amalgamated BP-MF-based and partial linearization approximation-based iterative detection algorithms.

In this paper, we design a kind of robust rateless multi-user code, which is called spatially coupled repeat-accumulate-repeat (SC-RA-R) code, for the interleave-division multiple-access (IDMA) system. The code is constructed by serially concatenating a fixed outer spatially coupled repeat-accumulate (SC-RA) code with an adjustable inner irregular repetition code. It achieves rateless property by simply adjusting the parameters of the inner repetition code. The IDMA system with the proposed SC-RA-R multi-user codes can adapt to varying channel qualities and varying number of users via rateless property while keeping encoder and decoder implementation unchanged. Analytical extrinsic information transfer (EXIT) functions are derived to determine belief propagation (BP) decoding thresholds of SC-RA-R codes on the IDMA system. The optimization expression is also given under the constraint that repeat parameters are monotonically non-decreasing with the sum rate decreasing. The numerical results show that the maximum achievable sum rates of the SC-RA-R codes can be close to the Shannon bound for arbitrary channel qualities. Besides, our proposed SC-RA-R codes have better BP thresholds and bit-error-rate (BER) performances than the conventional optimal parallel-concatenate codes (PCC) and the optimal partially repeated spatially coupled low-density parity-check (PR-SC-LDPC) codes on IDMA systems.

A novel rapid code acquisition scheme is proposed for low spreading factor satellite communication systems relying on message passing algorithms (MPA), where the acquisition of pseudonoise (PN) codes is converted to a sequence estimation problem. Due to the presence of data transitions in low spreading factor systems, conventional acquisition methods based on correlation would show poor capture performance. To overcome the impact of data transitions, this letter proposes a joint data chip and PN sequence estimation technique with the aid of factor graph and belief propagation (BP) algorithm. Meanwhile, in view of the short-term invariance of data chips, this letter models data chips as Markov chains to improve prior knowledge. Simulation results demonstrate that the proposed algorithm can save the mean acquisition time (MAT) about three orders of magnitude and improve the acquisition performance by 2 dB at 99% detection probability compared with serial search.

This correspondence discusses the estimation of direction-of-arrival (DOA) in a magnitude-aided antenna array (MA-AA), where magnitude-only radio frequency (RF) chains are introduced into the classical AA to acquire magnitude measurements. DOAs are initially estimated by the multiple signal classification (MUSIC) algorithm based on the complex-valued measurements from the conventional antennas. After griding the neighborhoods of these initial DOAs, the DOA estimation problem with the hybrid observations in MA-AA is converted as the recovery problem of sparse signals, which can be resolved by generalized approximate message passing (GAMP). Due to the “spatial leakage” effect, non-zero clusters appear around true DOAs. Their positions (i.e., non-zero supports) provide DOAs estimations. Moreover, DOAs and supports remain unchanged in several snapshots, then common supports are shared. Therefore, the cluster-sparse property of sparse signals is exploited by modeling a hidden Markov-tree (HMT) in the shared supports, on which belief propagation (BP) is executed to recover the binary probabilities of supports. Some unknown hyper-parameters in GAMP and BP are learned by expectation-maximization (EM). In comparison to existing estimators, EM-BP-GAMP shows advantages on DOA estimation, computational complexity, and DOA resolution. With the EM-BP-GAMP estimator, MA-AA is more energy-efficient than the classical AA. These advantages are successfully validated by experimental results.

Belief propagation (BP) has attracted increasing attention due to its high parallelism. In this letter, a correction method is proposed to improve the block error rate (BLER) performance of the BP decoder for polar codes so that it is comparable to that of the successive cancellation list (SCL) decoder. Compared with BP flipping (BPF) decoding, the proposed decoder focuses on prior knowledge of the code bits in the factor graph. Two strategies have been proposed to construct the correction set. The simulation results show that the BPC decoder exhibits a BLER similar to that of BPF decoder while reducing the average latency and complexity by half. In addition, the BLER of the BPC decoder can approach that of the cyclic redundancy check (CRC)-aided SCL (CA-SCL) decoder in regions with low and medium signal-to-noise ratios (SNRs).

Layered decoding (LD) facilitates a partially parallel architecture for performing belief propagation (BP) algorithm for decoding low-density parity-check (LDPC) codes. Such a schedule for LDPC codes has, in general, reduced implementation complexity compared to a fully parallel architecture and higher convergence rate compared to both serial and parallel architectures, regardless of the codeword length or code-rate. In this paper, we introduce a modified shuffling method which shuffles the rows of the parity-check matrix (PCM) of a quasi-cyclic LDPC (QC-LDPC) code, yielding a PCM in which each layer can be produced by the circulation of its above layer one symbol to the right. The proposed shuffling scheme additionally guarantees the columns of a layer of the shuffled PCM to be either zero weight or single weight. This condition has a key role in further decreasing LD complexity. We show that due to these two properties, the number of occupied look-up tables (LUTs) on a field programmable gate array (FPGA) reduces by about 93% and consumed on-chip power by nearly 80%, while the bit error rate (BER) performance is maintained. The only drawback of the shuffling is the degradation of decoding throughput, which is negligible for low values of$$E_b/N_0$$Eb/N0until the BER of 1e−6.

This paper focuses on the distributed static estimation problem and a Belief Propagation (BP) based estimation algorithm is proposed. We provide a complete analysis for convergence and accuracy of it. More precisely, we offer conditions under which the proposed distributed estimator is guaranteed to converge and we give concrete characterizations of its accuracy. Our results not only give a new algorithm with good performance but also provide a useful analysis framework to learn the properties of a distributed algorithm. It yields better theoretical understanding of the static distributed state estimator and may generate more applications in the future.

In this paper, we propose a channel reliability compensation factor to enhance the performance of belief propagation polar decoders on flat Rayleigh fading channels with noisy channel estimates. By including the error variance of the channel estimate in the derivation of the channel intrinsic information, the formula for calculating the value of channel reliability compensation factor is provided. Simulation results show that a BP polar decoder with the proposed compensation factor achieves a gain of 1.5 dB at a BER of $$10^{-3}$$ compared to the one without using the compensation factor. This gain is obtained with no additional complexity.

The Low-Density Parity Check (LDPC) codes of Euclidean Geometry (EG) are encrypted and decrypted in numerous ways, namely Soft Bit Flipping (SBF), Sequential Peeling Decoder (SPD), Belief Propagation Decoder (BPD), Majority Logic Decoder/Detector (MLDD), and Parallel Peeling Decoder (PPD) decoding algorithms. These algorithms provide aextensive range of trade-offs between latency decoding, power consumption, hardware complexity-required resources, and error rate performance. Therefore, the problem is to communicate a sophisticated technique specifying the both soft and burst errors for effective information transmission. In this research, projected a technique named as Hybrid SBF (HSBF) decoder for EG-LDPC codes, which reduces the decoding complexity and maximizes the signal transmission and reception. In this paper, HSBF is also known as Self Reliability based Weighted Soft Bit Flipping (SRWSBF) Decoder. It is obvious from the outcomes that the proposed technique is better than the decoding algorithms SBF, MLDD, BPD, SPD and PPD. Using Xilinx synthesis and SPARTAN 3e, a simulation model is designed to investigate latency, hardware utilization and power consumption. Average latency of 16.65 percent is found to be reduced. It is observed that in considered synthesis parameters such as number of 4-input LUTs, number of slices, and number of bonded IOBs, excluding number of slice Flip-Flops, hardware utilization is minimized to an average of 4.25 percent. The number of slices Flip-Flops resource use in the proposed HSBF decoding algorithm is slightly higher than other decoding algorithms, i.e. 1.85%. It is noted that, over the decoding algorithms considered in this study, the proposed research study minimizes power consumption by an average of 41.68%. These algorithms are used in multimedia applications, processing systems for security and information.

The success of Multi-hypothesis tracking(MHT) lies in the use of multiple scans, which can often yield improved tracking performance over single scan based data association methods in radar-centric multi-target tracking(MTT) systems. However, with the increase of the number of targets associated with measurements, there is an exponentially increasing need for formulating potential hypotheses, of which the computational cost may be prohibitively expensive. In this paper, a multi-hypothesis fractional belief propagation (MHFBP) based data association algorithm is proposed by the use of a probabilistic graph model for both the previous trajectories and the current measurements. To achieve this idea, there are two main steps. First, the trajectory-related indicators associated state variables are created by making use of a convex fractional free energy (FFE)function. Second, the convex optimization algorithm is used for the objective function and the fractional belief propagation (FBP) is exploited to obtain the best marginal belief of measurement for target association. In this way, incorrect hypotheses with extremely low probability can be eliminated. Finally, the proposed method is applied to multiple scenarios for MTT by indoor radar system. From the results, we can observe that it provides higher tracking performance compared with the classical MHT, feature-aided MHT(FA-MHT) and MHT-belief propagation (MHT-BP). In addition, we see that the computational burden of the proposed method is reduced significantly to avoid the phenomenon of exponential growth with increasing the number of targets.

The state estimation algorithm estimates the values of the state variables based on the measurement model described as the system of equations. Prior to applying the state estimation algorithm, the existence and uniqueness of the solution of the underlying system of equations is determined through the observability analysis. If a unique solution does not exist, the observability analysis defines observable islands and further defines an additional set of equations (measurements) needed to determine a unique solution. For the first time, we utilise factor graphs and Gaussian belief propagation algorithm to define a novel observability analysis approach. The observable islands and placement of measurements to restore observability are identified by following the evolution of variances across the iterations of the Gaussian belief propagation algorithm over the factor graph. Due to sparsity of the underlying power network, the resulting method has the linear computational complexity (assuming a constant number of iterations) making it particularly suitable for solving large-scale systems. The method can be flexibly matched to distributed computational resources, allowing for determination of observable islands and observability restoration in a distributed fashion. Finally, we discuss performances of the proposed observability analysis using power systems whose size ranges between 1354 and 70000 buses., Comment: 9 pages, 9 figure, version of the journal paper submitted for publication

In this letter, a double polar code (D-Polar) for joint source and channel coding (JSCC) is proposed, in which the source compress is implemented by a polar code, the channel error correction is performed by a systematic polar code. Furthermore, a turbo-like belief propagation (TL-BP) decoder consisted of a channel BP decoder and a source BP decoder is proposed for joint source and channel decoding in the receiver. In this TL-BP decoder, the soft information is exchanged between the channel BP decoder and the source BP decoder so as to improve the efficiency of channel decoding in utilizing the source information residual in the compressed bits. Simulation results show that the performance of the proposed D-Polar JSCC scheme with TL-BP decoder is significantly improved compared with the existing source-channel joint polarization scheme.

How to observe the state of a network from a limited number of measurements has become an important issue in complex networks, engineering, communication, epidemiology, etc. Under some scenarios, it is neither unfeasible nor unnecessary to observe the entire network. Therefore, we investigate the target observation of a network in this paper. We propose a target minimal dominating set problem corresponding to target observation, which is a natural generalization of classical minimal dominating set problem. Three algorithms are proposed to approximate the minimum set of occupied nodes sufficient for target observation. Extensive numerical results on computer-generated random networks and real-world networks demonstrate that the proposed algorithms offer superior performance in identification of a target minimal dominating set., 20 pages, 7 figures

In previous papers, decoding schemes which did not use machine learning considered additive white Gaussian noise or memoryless impulse noise. The decoding methods applying deep learning to reduce computational complexity and decoding latency didn’t consider the impulse noise. Here, we apply the Long Short-Term Memory (LSTM) neural network (NN) decoder for Polar codes under the Markov Gaussian memory impulse noise channel, and compare its bit error rate with the existing Polar code decoders like Successive Cancellation (SC), Belief Propagation (BP) and Successive Cancellation List (SCL). In the simulation results, we first find the optimal training SNR value 4.5 dB in the Markov Gaussian memory impulse noise channel for training the proposed LSTM based Polar code decoder. The optimal training SNR value is different from that 1.5 dB in the AWGN channel. The bit error rate of the propose LSTM based Polar code decoder is one third that of the previous non-deep-learning-based decoder SC/BP/SCL in Markov Gaussian memory impulse noise channels. The execution time of the proposed LSTM-based method is 5 ~ 12 times less and thus has much less decoding latency than that of SC/BP/SCL methods because the proposed LSTM-based method has inherent parallel structure and has one shot operation.

In the context of the non-Gaussian delay model, the fully distributed time synchronization approach based on Gaussian belief propagation will lead to the decline of synchronization accuracy. This paper proposes a distributed time synchronization algorithm based on sequential belief propagation (SBP-DTS), which assumes that the network delay is unknown. SBP-DTS first establishes a factor graph (FG) model for the time synchronization problem of wireless sensor networks (WSNs), and then uses sequential belief propagation (BP) algorithms to estimate node clock parameters under an unknown random delay model. At the same time, in order to reduce the amount of data exchanged between nodes during the execution of sequential belief propagation algorithm, the weighted expectation–maximization (EM) algorithm is used to reduce the number of Gaussian mixture components in the message. At last, the performance of SBP-DTS is evaluated under asymmetric Gaussian and exponential delay models.

For polar codes, the statistical breakdown of belief propagation (BP) decoding errors is firstly proposed by Sun et al. The errors are classified into three categories: unconverged errors, false converged errors and oscillation errors, which are corrected by three different post-processing methods. As for the BP List (BPL) decoding, we discover that one type of error can be transformed to another type by different factor graphs in the failed BP decoding. Besides, false converged error is usually easier to be detected and modified. Therefore, we only target false converged error to detect and correct. In this work, we propose a two-level detection rule for false converged errors and its corresponding post-processing algorithm to modify these errors in order to improve the performance of the BPL decoding. Numerical results show that the error-correction performance of the proposed decoder is more than 0.85dB better than that of the permuted BPL (PBPL) decoder with slight extra computation complexity at the frame error rate (FER) of $10^{-5}$ for 5G (1024,512) polar code.

For the scenarios with high throughput requirements, the belief propagation (BP) decoding is one of the most promising decoding strategies for polar codes. By pruning the redundant variable nodes (VNs) and check nodes (CNs) in the original factor graph, the graph is condensed to a sparse bipartite graph that is similar to the graph for low-density parity-check (LDPC) codes. In this paper, we introduce the bit flipping scheme into the LDPC-like BP (L-BP) decoding and propose two methods to identify the error-prone VNs. By additional decoding attempts, the L-BP flip (L-BPF) decoding improves the error-correction performance with a similar average complexity for high Eb/N0 values. The simulation results show that the L-BPF decoding achieves 0.25 dB gain compared with the L-BP decoding. Finally, a parallel decoder with the proposed L-BPF algorithm for an (256,128) polar code is implemented using 65nm CMOS technology, and it delivers a throughput of 1877.3 Mbps.

The recent development of deep learning methods provides a new approach to optimize the belief propagation (BP) decoding of linear codes. However, the limitation of existing works is that the scale of neural networks increases rapidly with the codelength, thus they can only support short to moderate codelengths. From the point view of practicality, we propose a high-performance neural min-sum (MS) decoding method that makes full use of the lifting structure of protograph low-density parity-check (LDPC) codes. By this means, the size of the parameter array of each layer in the neural decoder only equals the number of edge-types for arbitrary codelengths. In particular, for protograph LDPC codes, the proposed neural MS decoder is constructed in a special way such that identical parameters are shared by a bundle of edges derived from the same edge-type. To reduce the complexity and overcome the vanishing gradient problem in training the proposed neural MS decoder, an iteration-by-iteration (i.e., layer-by-layer in neural networks) greedy training method is proposed. With this, the proposed neural MS decoder tends to be optimized with faster convergence, which is aligned with the early termination mechanism widely used in practice. To further enhance the generalization ability of the proposed neural MS decoder, a codelength/rate compatible training method is proposed, which randomly selects samples from a set of codes lifted from the same base code. As a theoretical performance evaluation tool, a trajectory-based extrinsic information transfer (T-EXIT) chart is developed for various decoders. Both T-EXIT and simulation results show that the optimized MS decoding can provide faster convergence and up to 1dB gain compared with the plain MS decoding and its variants with only slightly increased complexity. In addition, it can even outperform the sum-product algorithm for some short codes., To appear in the IEEE JSAC Series on Machine Learning in Communications and Networks

International audience; This paper describes a practical Slepian-Wolf source coding scheme based on Low Density Parity Check (LDPC) codes. It considers the realistic setup where the parameters of the statistical model between the source and the side information are unknown. A novel Self-Corrected Belief-Propagation (SC-BP) algorithm is proposed in order to make the coding scheme robust to incorrect model parameters by introducing some memory inside the LDPC decoder. A Two Dimensional Density Evolution (2D-DE) analysis is then developed to predict the theoretical performance of the SC-BP decoder. Both the 2D-DE analysis and Monte-Carlo simulations confirm the robustness of the SC-BP decoder. The proposed solution allows for an important complexity reduction and shows a performance very close to existing methods which jointly estimate the model parameters and the source sequence.

In this paper, we propose a design technique for the construction of variable-regular time-invariant spatially-coupled low-density parity-check (SC-LDPC) codes with small constraint length and low error floor. The proposed technique reduces the error floor by imposing simple constraints on the short cycles in the code’s Tanner graph, which in turn, result in the elimination of the most dominant trapping sets of the code. In some cases, we also derive lower bounds on the syndrome former memory for satisfying such constraints. The designed codes are superior to the state-of-the-art in terms of error floor performance and/or decoding complexity and latency.

In this work, we consider the decoding of short sparse graph-based channel codes via reinforcement learning (RL). Specifically, we focus on low-density parity-check (LDPC) codes, which for example have been standardized in the context of 5G cellular communication systems due to their excellent error correcting performance. LDPC codes are typically decoded via belief propagation on the corresponding bipartite (Tanner) graph of the code via flooding, i.e., all check and variable nodes in the Tanner graph are updated at once. We model the node-wise sequential LDPC scheduling scheme as a Markov decision process (MDP), and obtain optimized check node (CN) scheduling policies via RL to improve sequential decoding performance as compared to flooding. In each RL step, an agent decides which CN to schedule next by observing a reward associated with each choice. Repeated scheduling enables the agent to discover the optimized CN scheduling policy which is later incorporated in our RL-based sequential LDPC decoder. In order to reduce RL complexity, we propose a novel graph-induced CN clustering approach to partition the state space of the MDP in such a way that dependencies between clusters are minimized. Compared to standard decoding approaches from the literature, some of our proposed RL schemes not only improve the decoding performance, but also reduce the decoding complexity dramatically once the scheduling policy is learned. By concatenating an outer Hamming code with an inner LDPC code which is decoded based on our learned policy, we demonstrate significant improvements in the decoding performance compared to other LDPC decoding policies.

Linear nested codes, where two or more sub-codes are nested in a global code, have been proposed as candidates for reliable multi-terminal communication. In this paper, we consider nested array-based spatially coupled low-density parity-check (SC-LDPC) codes and propose a line-counting based optimization scheme for minimizing the number of dominant absorbing sets in order to improve its performance in the high signal-to-noise ratio regime. Since the parity-check matrices of different nested sub-codes partially overlap, the optimization of one nested sub-code imposes constraints on the optimization of the other sub-codes. To tackle these constraints, a multi-step optimization process is applied first to one of the nested codes, then sequential optimization of the remaining nested codes is carried out based on the constraints imposed by the previously optimized sub-codes. Results show that the order of optimization has a significant impact on the number of dominant absorbing sets in the Tanner graph of the code, resulting in a tradeoff between the performance of a nested code structure and its optimization sequence: the code which is optimized without constraints has fewer harmful structures than the code which is optimized with constraints. We also show that for certain code parameters, dominant absorbing sets in the Tanner graphs of all nested codes are completely removed using our proposed optimization strategy., Accepted for publication in IEEE Transactions on Communications

Tracking an unknown number of targets based on multipath measurements provided by an over-the-horizon radar (OTHR) network with a statistical ionospheric model is complicated, which requires solving four subproblems: Target detection, target tracking, multipath data association, and ionospheric heights identification. A joint solution is desired since the four subproblems are highly correlated, but suffering from the intractable inference problem of high-dimensional latent variables. In this article, a unified message passing (MP) approach, combining belief propagation (BP) and mean-field (MF) approximation, is developed for simplifying the intractable inference. Based upon the factor graph corresponding to a factorization of the joint probability density function (PDF) of the latent variables and a choice for a separation of this factorization into BP region and MF region, the posterior PDFs of target kinematic state and ionospheric heights, are approximated by MF due to its simple MP update rules for conjugate-exponential models. The posterior PDFs of target visibility state and multipath data association which contains one-to-one frame (hard) constraints (i.e., at each time, a measurement is either originated from one target via a particular path or it is clutter, and each target generates at most one measurement under a particular path) are approximated by BP. Finally, the approximated posterior PDFs are updated iteratively in a closed-loop manner, which is effective for dealing with the coupling issue among latent variables. Meanwhile, the proposed approach has the measurement-level fusion architecture due to the direct processing of the raw multipath measurements from an OTHR network, which is beneficial to improving tracking fusion performance. Its performance is demonstrated on a simulated OTHR network multitarget tracking scenario.

We consider the problem of joint channel estimation and data decoding in uplink massive multiple input multiple output systems with low resolution analog-to-digital converters (ADCs) at the base station. The nonlinearities introduced by the ADCs make the problem challenging: in particular, the existing linear detectors perform poorly. Also, the channel coding used in commercial wireless systems necessitates soft symbol detection to obtain satisfactory performance. In this paper, we present a low-complexity variational Bayesian (VB) inference procedure to jointly solve the (possibly correlated) channel estimation and soft symbol decoding problem. We present the approach in progressively more complex scenarios, including the case where even the channel statistics are not available at the receiver. Finally, we combine our proposed VB procedure with a belief propagation (BP) based channel decoder, which further enhances the performance without any additional complexity. We numerically evaluate the bit error rate (BER) and the normalized mean squared error (NMSE) in the channel estimates obtained by our algorithm as a function of various system parameters, and benchmark the performance against genie-aided and state-of-the-art receivers. The results show that VB procedure is a promising technique for the design of low-complexity advanced receivers in low resolution ADC based systems.

Thanks to its superior features of fast read/write speed and low power consumption, spin-torque transfer magnetic random access memory (STT-MRAM) has become a promising non-volatile memory (NVM) technology that is suitable for many applications. However, the reliability of STT-MRAM is seriously affected by the variation of the memory fabrication process and the working temperature, and the later will lead to an unknown offset of the channel. Hence, there is a pressing need to develop more effective error correction coding techniques to tackle these imperfections and improve the reliability of STT-MRAM. In this work, we propose, for the first time, the application of deep-learning (DL)-based algorithms and techniques to improve the decoding performance of linear block codes with short codeword lengths for STT-MRAM. We formulate the belief propagation (BP) decoding of linear block code as a neural network (NN) and propose a novel neural normalized-offset reliability-based min-sum (RB-MS) (NNORB-MS) decoding algorithm. We successfully apply our proposed decoding algorithm to the STT-MRAM channel through channel symmetrization to overcome the channel asymmetry. We also propose an NN-based soft information generation method (SIGM) to take into account the unknown offset of the channel. Simulation results demonstrate that our proposed NNORB-MS decoding algorithm can achieve significant performance gain over both the hard-decision decoding (HDD) and the regular RB-MS decoding algorithm, for cases without and with the unknown channel offset. Moreover, the decoder structure and time complexity of the NNORB-MS algorithm remain similar to those of the regular RB-MS algorithm.

Polar code is one of the channel codes and is used in the 5th generation of mobile communication system (5G). This encoding scheme is based on the operation of channel polarization, and it is possible to achieve the capacity of arbitrary binary-input symmetric discrete memoryless channels. Compared with Turbo codes and LDPC (Low-Density Parity-Check) codes, the implementation of the encoder and decoder is easier. BP (Belief Propagation) decoding, one of the decoding methods of the polar codes, can be performed at high speed because it can decode in parallel. However, the disadvantages of the BP decoding are the hardware and computational complexities. The technique of quantization can be used to reduce complexities of hardware and calculation. One of the quantization methods is the information bottleneck method, which allows an observation variable to compressed one while trying to preserve the mutual information shared with a relevant variable. As a novel approach, the information bottleneck method is used in the design of quantizers for the BP decoding of LDPC codes. In this paper, we propose a discrete BP polar decoder that can use only unsigned integers in the decoding process by using the information bottleneck method. Thus, we can replace complex calculations of BP decoding with simple lookup tables. We also investigate the minimum bit width for quantization with negligible degradation and the suboptimal Eb/N0 for designing lookup tables, where Eb and N0 denote energy per bit and noise power density, respectively. The simulation results show that the proposed method can achieve almost the same error correcting capability compared with the BP decoding without compression in the range of low Eb/N0. Besides, we show that the proposed decoder can compress both channel outputs and BP messages with small loss compared with the uniform quantization decoder.

Communication overhead is a key task for the large-scale wireless sensor networks (WSNs) in the cooperative localization problems. This paper is aimed to propose a low-overhead algorithm without sacrificing the performance of localization, which is more practical for energy-saving WSNs. In this paper, the cooperative localization problem is formulated as a variational reference problem on a factor graph (FG). Combining the high performance of particle-based algorithms and the overhead advantage of parametric ones, a parametric variational sum-product algorithm (PVSPA) is proposed for the cooperative localization of WSNs. In prediction operation, to ensure high accuracy, messages referenced inside sensor nodes are particle-based. In correction operation, Gaussian parametric representation is adopted to the observation message and the communication overhead is obviously decreased. Simulation results show that the localization accuracy of PVSPA equally matches the classic particle-based algorithms, while the communication overhead of it is far lower.

Known for their capacity-achieving abilities and low complexity for both encoding and decoding, polar codes have been selected as the control channel coding scheme for 5G communications. To satisfy the needs of high throughput and low latency, belief propagation (BP) is chosen as the decoding algorithm. However, it suffers from worse error performance than that of cyclic redundancy check (CRC)-aided successive cancellation list (CA-SCL). Recently, convolutional neural network-aided bit-flipping (CNN-BF) is applied to BP decoding, which can accurately identify the erroneous bits to achieve a better error rate and lower decoding latency than prior critical-set bit-flipping (CS-BF) mechanism. However, successive BF, having better error correction capability, has not been explored in CNN-BF since the more complicated flipping strategy is out of the scope of supervised learning. In this work, by using imitation learning, a convolutional neural network-aided tree-based multiple-bits BF (CNN-Tree-MBF) mechanism is proposed to explore the benefits of multiple-bits BF. With the CRC information as additional input data, the proposed CNN-BF model can further reduce 5 flipping attempts. Besides, a tree-based flipping strategy is proposed to avoid useless flipping attempts caused by wrongly flipped bits. From the simulation results, our approach can outperform CS-BF and reduce flipping attempts by 89% when code length is 64, code rate is 0.5 and SNR is 1 dB. It also achieves a comparable block error rate (BLER) as CA-SCL.

In this article we propose a technique for polar codes (PC) construction for any code length. By default, PC construction is limited to code length proportional to the power of two. To construction the code length arbitrary, puncturing, shortening and extension techniques must be applied. However, performance is degraded with the use of these techniques. Other ways to design polar codes with arbitrary code length but which have encoding and decoding with higher complexity such as multi-kernel, concatenated codes and specific constructions for belief propagation (BP) or successive cancellation list (SCL) decoding. The polarization theory is generalized for non-uniform channels (NUC) and with this approach we can construction rate-compatible PC and variable code length. We developed an implementation algorithm based on the of PC construction by Gaussian approximation (NUPGA). In a scenario where the transmission is over an additive white Gaussian noise (AWGN) channel and under successive cancellation (SC) decoding, the PC construction of arbitrary code length can be implemented with NUPGA. With NUPGA we re-polarize the projected synthetic channels by choosing more efficiently the positions of the information bits. In addition, we present a generalization of the Gaussian approximation (GA) for the polarization and re-polarization processes and an extension technique for PC. The PC construction based on NUPGA present better performance than the existing techniques as shown in the simulations of this work.

In this paper, we extend the bilinear generalized approximate message passing (BiG-AMP) approach, originally proposed for high-dimensional generalized bilinear regression, to the multi-layer case for the handling of cascaded problem such as matrix-factorization problem arising in relay communication among others. Assuming statistically independent matrix entries with known priors, the new algorithm called ML-BiGAMP could approximate the general sum-product loopy belief propagation (LBP) in the high-dimensional limit enjoying a substantial reduction in computational complexity. We demonstrate that, in large system limit, the asymptotic MSE performance of ML-BiGAMP could be fully characterized via a set of simple one-dimensional equations termed state evolution (SE). We establish that the asymptotic MSE predicted by ML-BiGAMP' SE matches perfectly the exact MMSE predicted by the replica method, which is well known to be Bayes-optimal but infeasible in practice. This consistency indicates that the ML-BiGAMP may still retain the same Bayes-optimal performance as the MMSE estimator in high-dimensional applications, although ML-BiGAMP's computational burden is far lower. As an illustrative example of the general ML-BiGAMP, we provide a detector design that could estimate the channel fading and the data symbols jointly with high precision for the two-hop amplify-and-forward relay communication systems., Comment: 61 pages, 16 figures. This paper has been accepted by IEEE Transaction on Signal Processing

Carrier frequency offset (CFO) and channel estimation is a classic topic with a large body of prior work using the maximum likelihood (ML) approach together with the Cramer-Rao lower bound (CRLB) analysis. We give the maximum a posteriori probability (MAP) estimation solution, which is particularly useful for tracking. Unlike the ML cases, the corresponding Bayesian CRLB (BCRLB) shows a clear relation with parameters and low complexity algorithms are provided to achieve the BCRLB in almost all SNR range. Among them, the universal algorithm takes a new approach and avoids error floors of the traditional approach. We assume that the time invariant MIMO channel within a packet can have spatial correlation and nonzero mean. The estimation is based on pilot signals. An unexpected result is that the joint MAP estimation is equivalent to an individual MAP estimation of the frequency offset first, again different from the ML results. We provide insight on the pilot/training signal design based on the BCRLB. Unlike past algorithms that trade performance and/or complexity for the accommodation of time varying channels, the MAP solution provides a different route for dealing with time variation. Within a short enough (segment of) packet where the channel and CFO are approximately time invariant, the low complexity algorithm can be employed. Similar to the belief propagation, the estimation of the previous (segment of) packet can serve as the prior knowledge for the next (segment of) packet.

This paper applies the variational methods to learn the parameters and the probability of evidence of directed graphic models (also known as Bayesian networks (BNs)) when data contains missing values. One class of variational methods, the Bethe/Kikuchi approximate algorithm, is combined with Expectation-Maximization (EM) to learn BN model parameters from data using an upper limit on cluster size. Our proposed method is novel since the Bethe/Kikuchi algorithms are typically only used to approximate marginal distributions. We review popular Bethe/Kikuchi approximate algorithms, including Belief Propagation (BP), Iterative Join Graph Propagation (IJGP), and Fundamental Cycle Base (FCB) algorithms for discrete BNs, and conduct experiments to test their learning performance using a wide range of discrete BNs in practice. Experiments show that by using IJGP or FCB algorithms, with EM, we can obtain more accurate results than the conventional approximate algorithms, such as Evidence Pre-propagation Importance Sampling and Variational Inference. When exact methods are intractable the IJGP/FCB with EM can be an effective alternative solution.

This paper studies the convergence properties of the well-known message-passing algorithm for convex optimisation. Under the assumption of pairwise separability and scaled diagonal dominance, asymptotic convergence is established and a simple bound for the convergence rate is provided for message-passing. In comparison with previous results, our results do not require the given convex program to have known convex pairwise components and that our bound for the convergence rate is tighter and simpler. When specialised to quadratic optimisation, we generalise known results by providing a very simple bound for the convergence rate.

In this paper, we apply polar codes to a modified version of OFDM systems with index modulation which is called OFDM with in-phase/quadrature index modulation (OFDM-I/Q-IM). We provide general design guidelines for the proposed polar-coded OFDM-I/Q-IM systems. In the proposed system, at the transmitter, we employ a random frozen bits appending scheme which not only makes the polar code compatible with OFDM-I/Q-IM but also improves the bit error rate (BER) performance of the system. Furthermore, at the receiver, it is shown that the a posteriori information for each index provided by the index detector is essential for the iterative decoding of polar codes by the belief propagation (BP) algorithm. Simulation results show that the proposed polar-coded OFDM-I/Q-IM system outperforms its OFDM counterpart in terms of BER performance.

Since the inherent serial nature of successive cancellation list (SCL) decoding results in a long latency, belief propagation (BP) decoding for polar codes has drawn attention for high-throughput applications. However, its error correction performance is inferior to that of SCL decoding. Therefore, the bit-flipping strategy has been recently applied to BP decoding, which can approach the SCL decoding performance through multiple additional decoding attempts. The original BP flip (BPF) decoding suffers from an inaccurate identification of erroneous bits by a fixed flip set (FS), which has been improved by the generalized BPF (GBPF) decoding. In this article, the GBPF decoding is extended to support multiple bits being flipped in one decoding attempt. In addition, for two types of decoding errors: detected errors and undetected errors, we propose two novel methods to more effectively identify erroneous bits. For detected errors, the concept of loop sets is defined and a loop-based identification method is introduced based on the study of error patterns of BP decoding. On the other hand, a method to generate a more accurate fixed FS is proposed for undetected errors, which considers the bit error distribution under BP decoding. Combining the two methods, the GBPF with merged sets (GBPF-MS) decoding can achieve the SCL-8 performance and outperforms the state-of-the-art BPF, BP list, and SC flip (SCF) decoding, for polar codes with length 1024 and information rate 1/2. Implemented by 40nm CMOS technology, the proposed GBPF-MS decoder with ten flips exhibits an average throughput of 4.19 Gbps at 2.5 dB, which is $1.6\times $ and $1.72\times $ faster than the state-of-the-art SCL-4 and SCF decoders, respectively.

In this paper, we propose a new construction of generalized root protograph (GRP) low-density parity-check (LDPC) codes which has a guaranteed diversity order. New hybrid-automatic request with incremental redundancy (H-ARQ-IR) is also proposed to ensure high diversity and rate-compatibility in wireless communication systems, where the GRP LDPC codes are employed along with a method called feedback encoding. The proposed H-ARQ-IR has advantages of low-complexity and near-optimal performance. In order to show its performance behavior, theoretical analysis for the proposed scheme is conducted for the block fading channel. Furthermore, numerical analysis shows that it achieves near maximum diversity order under the belief propagation (BP) decoding with binary phase shift keying (BPSK) modulation and high-order quadrature amplitude modulations (QAMs).

We analyze the finite-length performance of spatially coupled low-density parity-check (SC-LDPC) codes under window decoding over the binary erasure channel. In particular, we propose a refinement of the scaling law by Olmos and Urbanke for the frame error rate (FER) of terminated SC-LDPC ensembles under full belief propagation (BP) decoding. The refined scaling law models the decoding process as two independent Ornstein-Uhlenbeck processes, in correspondence to the two decoding waves that propagate toward the center of the coupled chain for terminated SC-LDPC codes. We then extend the proposed scaling law to predict the performance of (terminated) SC-LDPC code ensembles under the more practical sliding window decoding. Finally, we extend this framework to predict the bit error rate (BER) and block error rate (BLER) of SC-LDPC code ensembles. The proposed scaling law yields very accurate predictions of the FER, BLER, and BER for both full BP and window decoding., Comment: Published in IEEE Transactions on Communications (Early Access). This paper was presented in part at the IEEE Information Theory Workshop (ITW), Visby, Sweden, August 2019 (arXiv:1904.10410)

A novel concatenated coding scheme is introduced, where the Reed-Solomon (RS) code is concatenated with the spatially coupled low-density parity-check (SC-LDPC) code, namely the RS-SC-LDPC code. We first propose the locally systematic encoding of the inner code and its termination when considering the concatenated code has a finite length. A decoding scheme is then proposed, in which the belief propagation (BP) based sliding window decoding (SWD) decodes the inner code with the assistance of the outer Berlekamp-Massey (BM) decoding feedback. This research shows the RS-SC-LDPC codes can achieve a high decoding performance without yielding an error floor until the bit error rate (BER) of 10−8. They also significantly outperform the incumbent RS-convolutional concatenated (RS-CC) codes, as well as the BCH-SC-LDPC codes.

We present Tangent Space Belief Propagation (TSBP) as a method for graph denoising to improve the robustness of manifold learning algorithms. Dimension reduction by manifold learning relies heavily on the accurate selection of nearest neighbors, which has proven an open problem for sparse and noisy datasets. TSBP uses global nonparametric belief propagation to accurately estimate the tangent spaces of the underlying manifold at each data point. Edges of the neighborhood graph that deviate from the tangent spaces are then removed. The resulting denoised graph can then be embedded into a lower-dimensional space using methods from existing manifold learning algorithms. Artificially generated manifold data, simulated sensor data from a mobile robot, and high dimensional tactile sensory data are used to demonstrate the efficacy of our TSBP method.

This paper introduces a network linear block coding framework for multi-way relaying with differential MPSK modulation. We consider a system with $K$ user terminals and $L$ relays employing a selective detect-and-forward protocol. We show that such a system can be represented as a systematic $(K+L,K)$ linear block code. This framework allows the use of linear systematic block codes as network codes yielding power saving gains at user terminals. We analyse the theoretical performance of our scheme with optimal decoding, showing that our system can achieve a diversity order that equals to the minimum Hamming distance of the associated code. Then we consider a sub-optimal decoder based on log-domain belief propagation, and present numerical results for three 4-ary linear block codes and two binary LDPC codes as examples. Theoretical and simulation results demonstrate a significant performance gain of our scheme over uncoded transmission, even though some relays may be silent occasionally. We present a technique based on hard thresholding the received samples at the terminals, that makes the performance closer to the case when no relay is silent, without any significant increase in complexity. Furthermore, we also present guidelines for choosing suitable codes, and show that systematic block codes with sparse generator matrices are in particular desirable for our system.

The belief propagation (BP) decoding of short low-density parity check codes suffers from short cycles and trapping sets. A number of perturbation schemes have been proposed to overcome the limitation. This letter presents a generalized framework for multi-round BP decoding with input perturbation and proposes an improved perturbation scheme. In particular, the impulsive perturbation which is applied to only a few symbols is proposed, along with a new space selection criterion. The proposed scheme performs better than existing schemes for short codes at lower complexity and supports a parallel implementation.

In this study, a serially concatenated scheme of polar codes with convolutional codes is proposed to improve the error correction performance. The novel belief propagation (BP) decoding algorithm addresses two issues that are present in the currently available BP decoding algorithms. The first issue is the poor performance of the BP decoding algorithms, in particular the introduction of an error floor. The second is the component codes can only use systematic codes in the traditional concatenated scheme of polar codes with convolutional codes, which inhibits the effective update of the prior information of the redundant check bits. The proposed BP decoding algorithm is based on right-directed message processing, which effectively improves the decoding performance. In addition, the proposed concatenated scheme extends the selection of component codes from the systematic polar codes to the non-systematic polar codes. Hence, the areas of applications and the prior information of information bits for polar codes are expanded and more effectively updated, respectively. The simulation results show that the proposed scheme is much better than the traditional concatenated scheme, and the error floor is no longer introduced in terms of the block error rate, while the storage and computational complexities have not increased obviously.

Quantum stabilizer codes constructed from sparse matrices have good performance and can be efficiently decoded by belief propagation (BP). A conventional BP decoding algorithm treats binary stabilizer codes as additive codes over GF(4). This algorithm has a relatively complex process of handling check-node messages, which incurs higher decoding complexity. Moreover, BP decoding of a stabilizer code usually suffers a performance loss due to the many short cycles in the underlying Tanner graph. In this paper, we propose a refined BP decoding algorithm for quantum codes with complexity roughly the same as binary BP. For a given error syndrome, this algorithm decodes to the same output as the conventional quaternary BP but the passed node-to-node messages are single-valued, unlike the quaternary BP, where multivalued node-to-node messages are required. Furthermore, the techniques of message strength normalization can naturally be applied to these single-valued messages to improve the performance. Another observation is that the message-update schedule affects the performance of BP decoding against short cycles. We show that running BP with message strength normalization according to a serial schedule (or other schedules) may significantly improve the decoding performance and error floor in computer simulation.

Low-density parity-check (LDPC) codes have been chosen for the fifth-generation (5G) new radio (NR) enhanced Mobile Broad Band (eMBB) data channel. Since NR LDPC codes puncture the first two block columns of the parity-check matrix, the decoding complexity is reduced, but it slows down the convergence rate. In this letter, we propose a fixed schedule that decodes the layers with the least-punctured edges and those with the highest-degree (LPHD), which has much faster convergence speed than conventional schemes. The fixed scheduling schemes cannot take full advantage of the dynamic changes of decoding messages. As a result, the residual-based layered belief propagation (RB-LBP) is proposed to rearrange the layers dynamically among different iterations. Compared with some informed dynamic scheduling (IDS) strategies, RB-LBP reduces the decoding complexity with slight performance loss.

Belief propagation (BP) is a soft-decision (SD) decoder that is commonly used to obtain near-optimal decoding performance for linear codes defined over sparse parity-check matrix. Nevertheless, the BP yields poor performance when used in the decoding of algebraic block codes, usually described by a dense parity-check matrix. Hence, enhanced BP decoders transform the parity check matrix of such codes at each iteration for efficient decoding. In this article, the performance of the transformed parity-check matrix of the iterative SD decoder is analysed for the class of binary cyclic codes using a perfect knowledge model (PKM). The PKM computes a list of candidate matrices and selects a baseline parity-check matrix according to a distance metric. The selected matrix is optimal since it minimizes the probability of error over various choices in the list. Results show that, for a given channel condition, the conventional transformed matrix obtained by Gaussian elimination is sub-optimal and does not necessarily contain unitary weighted columns at corresponding columns of the unreliable bits. Moreover, PKM can be used to verify the performances of newly developed iterative SD decoders for binary cyclic codes based on parity-check equations instead of maximum-likelihood decoding.

In this paper, we propose a new approach to design finite alphabet iterative decoders (FAIDs) for Low-Density Parity Check (LDPC) codes over binary symmetric channel (BSC) via recurrent quantized neural networks (RQNN). We focus on the linear FAID class and use RQNNs to optimize the message update look-up tables by jointly training their message levels and RQNN parameters. Existing neural networks for channel coding work well over Additive White Gaussian Noise Channel (AWGNC) but are inefficient over BSC due to the finite channel values of BSC fed into neural networks. We propose the bit error rate (BER) as the loss function to train the RQNNs over BSC. The low precision activations in the RQNN and quantization in the BER cause a critical issue that their gradients vanish almost everywhere, making it difficult to use classical backward propagation. We leverage straight-through estimators as surrogate gradients to tackle this issue and provide a joint training scheme. We show that the framework is flexible for various code lengths and column weights. Specifically, in high column weight case, it automatically designs low precision linear FAIDs with superior performance, lower complexity, and faster convergence than the floating-point belief propagation algorithms in waterfall region.