A Novel Variable-Order Threshold Serial Belief Propagation-Based Decoding Algorithm for LT Codes

In this letter, a new early termination method is proposed for Luby transform (LT) belief propagation (BP) decoder. The proposed method is decided whether BP decoder output converges to original data bits by observing only sign alterations of log-likelihood ratio (LLR) messages in BP decoder structure. Simulation results and complexity analyzes show that proposed method has low computational complexity and small average iteration amounts compared to conventional early termination methods in literature. In addition to this, the proposed method doesn't cause any performance degradation in BP decoder. The method can be easily applied to code families which can be decoded by BP algorithm such as low density parity check (LDPC) codes, polar codes and Raptor codes.

This paper presents the performance of a hard decision belief propagation (HDBP) decoder used for Luby transform (LT) codes over additive white Gaussian noise channels; subsequently, three improved HDBP decoders are proposed. We first analyze the performance improvement of the sorted ripple and delayed decoding process in a HDBP decoder; subsequently, we propose ripple-sorted belief propagation (RSBP) as well as ripple-sorted and delayed belief propagation (RSDBP) decoders to improve the bit error rate (BER). Based on the analysis of the distribution of error encoded symbols, we propose a ripple-sorted and threshold-based belief propagation (RSTBP) decoder, which deletes low-reliability encoded symbols, to further improve the BER. Degree distribution significantly affects the performance of LT codes. Therefore, we propose a method for designing optimal degree distributions for the proposed decoders. Through simulation results, we demonstrate that the proposed RSBP and RSDBP decoders provide significantly better BER performances than the HDBP decoder. RSDBP and RSTDP combined with the proposed degree distributions outperformed state-of-the-art degree distributions in terms of the number of encoded symbols required to recover an input symbol correctly (NERRIC) and the frame error rate (FER). For a hybrid decoder formulated by combining RSDBP with a soft decision belief propagation decoder, the proposed degree distribution outperforms the other degree distributions in terms of decoding complexity.

Since their invention Luby Transform (LT) codes have been regarded as an efficient capacity achieving channel code over binary erasure channels. Majority of the previous research is focused on designing the degree distribution of LT codes which are asymptotically optimal. Nevertheless, deigning the optimal degree distribution of LT codes for finite and small message lengths remains as an open problem. This work addresses the degree distribution optimization of LT codes over binary erasure channel. Based on AND-OR tree analysis of Belief Propagation (BP) algorithm, a new formulation of the problem in the form of standard semidefinite programming is presented. The new formulation is free of any approximation. The obtained semidefinite program is reduced to a linear program where its numerical solution is feasible for reasonable message lengths. Simulations confirm that the optimized degree distributions outperform Robust Soliton distribution, both in terms of overhead and Encoding/Decoding complexity.

In this paper, we discuss the belief propagation (BP) decoding of polar codes. The performance of polar codes for short lengths is not satisfactory. Therefore, motivated by this we propose a novel technique to improve the performance of polar codes under BP decoding. In order to enhance the reliability of the variable nodes’ propagated messages in BP decoding, the messages are multiplied by previous messages of variable nodes. It is also shown that the BP decoding can be performed on polar codes if the factor graph is constructed according to the parity check matrix of polar codes. Simulation results in binary-input additive white Gaussian noise channel show that using the proposed method a gain of 1–2 dB can be achieved. We also show that the complexity of proposed decoder is same as the complexity of standard BP decoder. Furthermore, we show that the proposed decoder requires about 10–25 iterations less than the standard BP decoder.

The cache-enabled D2D network supports direct interconnection among users and can help to offload the data traffic through user caching. In this paper, we consider a reasonable caching strategy design in the cache-enabled D2D network to improve its performance. First we formulate an optimization problem aiming to minimize the average file download time of the users. Next we transform the original problem to a max-product form that can be distributed solved. Then we introduce the factor graph and the conventional belief propagation (BP) algorithm. Finally, we show how we design a BP algorithm with Randomized Gaussian Message Passing scheme and Weighting mechanism (BP-RW) to improve its convergence and performance. The distributed BP-RW algorithm can assign almost all computations to each user in the network, which will help to reduce the heavy computational overhead of the central controller. Meanwhile, simulation results show that the BP-RW algorithm performs better than the conventional BP algorithm, and performs very closely to the centralized greedy algorithm with much lower computational complexity.

Distributed Luby transform (DLT) codes achieve a significant performance improvement in multi-source relay networks compared with the individual Luby transform (LT) code designed for each source. Since the DLT codes preserve the properties of the LT codes, the error floor is dominated by the low variable-node degrees in the bipartite graph, developed for iterative message passing decoding. Therefore, we analyze the error floor of the DLT codes for a multi-source, single relay, and a single destination network over additive white Gaussian noise (AWGN) channels. We modify the encoding process at the sources and propose a new relay combining scheme at the relay. The encoding process at the sources and the combining scheme at the relay are coordinated with the aim of improving the error floor performance of the proposed DLT codes. We derive a lower bound of the bit error probability of the DLT codes over AWGN channels. Furthermore, we optimize the relay degree distribution in terms of the overhead by using the framework of extrinsic information transfer chart analysis. The numerical results confirm that the proposed DLT coding scheme significantly outperforms the conventional scheme, especially in the error floor region.

Rateless coding such as Luby transform (LT) codes is representative of rate adaptive solutions to improve the frequency efficiency of transmitting data over a time-varying wireless channel. Also, these type of codes use belief propagation (BP) decoding which searches for encoded packets with degree 1 among the large numbers of encoded packets. This method fails if no encoded packets with degree 1 exist. However, in reality, small numbers of input packets are often encountered. In this paper, we have proposed a decoding algorithm that increases the performance of these codes for the small number of source packets. After the failure of BP decoding, to recover the remaining source packets, we use a method which keeps on decoding using the encoded packets whose degrees are not one. This algorithm continues until a degree 1 packet emerges in one of the iterations of this decoding process. Then, decoding algorithm switches back to the traditional BP decoding.

Luby Transform (LT) codes, the first realization of rateless codes are widely used in wireless communication mainly for its adaptability to varying channel conditions and its capacity approaching performance. In spite of the above advantages and its simplicity in implementation, LT codes suffer from a bottle neck of premature termination due to the poor design of degree distribution. In this study, a novel degree distribution called Joint Degree Distribution (JDD) is proposed for the successful completion of LT encoder/decoder processes. The efficient utilization of the bandwidth is tried to be achieved by using only 'k' encoded symbols for 'k' source symbols, unlike in traditional systems. JDD is carefully designed to ensure that the encoding/decoding delay does not exceed that which is existent in the traditional systems. The performance of JDD for throughput and bit error rate, experimented over Additive White Gaussian Noise (AWGN) channel compared to conventional degree distribution was observed to be much better.

Low density parity-check (LDPC) codes are a class of linear block codes that are decoded by running belief propagation (BP) algorithm or log-likelihood ratio belief propagation (LLR-BP) over the factor graph of the code. One of the disadvantages of LDPC codes is the onset of an error floor at high values of signal to noise ratio caused by trapping sets. In this paper, we propose a two stage decoder to deal with different types of trapping sets. Oscillating trapping sets are taken care by the first stage of the decoder and the elementary trapping sets are handled by the second stage of the decoder. Simulation results on the regular PEG (504,252,3,6) code and the irregular PEG (1024,518,15,8) code shows that the proposed two stage decoder performs significantly better than the standard decoder.

In this paper, we compare detection algorithms for additive white Gaussian noise (AWGN) channels and correlated/signal-dependent noise channels for a serially concatenated recording system from the viewpoint of probability propagation. It has been proven that the classical turbo decoding algorithm is an instance of Pearl's belief propagation algorithm, and the decoding algorithm for serially concatenated convolutional codes also can be derived from a belief propagation viewpoint. This result is also valid for correlated/signal-dependent noise channels, such as magnetic recording channels. Using the relationship of probability dependency, Bayesian networks and their corresponding probability propagation schemes for a rate-16/17 serially concatenated recording system are obtained for both AWGN and correlated/signal-dependent noise channels. Noise predictive turbo systems (NPTS) are also discussed from a graphical perspective.

It has been shown that the stable fixed points of belief propagation (BP) algorithms correspond to extrema of the Bethe free energy. In this paper, we describe the dual problem for the minimization of the Bethe free energy and solve it using simple nonlinear block Gauss-Seidel and Jacobi algorithms. The use of the nonlinear block Gauss-Seidel algorithm corresponds to serial scheduling for the BP algorithm. In addition, it is shown that applying the nonlinear block Jacobi algorithm on the dual of the Bethe free energy corresponds to the parallel BP algorithm.

In this paper, we analyze the performance of neural belief propagation (BP) decoding on the additive white Gaussian noise (AWGN) channel, compared to the traditional BP algorithm. Previous investigations have shown that assigning pre-trained weights to BP messages can significantly improve the decoding performance in case of high-density parity-check (HDPC) codes, by reducing the negative impact of short cycles. These weights are trained by a neural network whose structure matches the trellis of the decoder. Specifically, we show that medium-density paritycheck (MDPC) codes decoded with neural BP algorithm can achieve lower bit error rate versus HDPC codes with the same codeword length and the same code rate.

In this paper, we propose two novel generalized belief propagation (BP) algorithms to improve the convergence behavior of the conventional BP algorithm. By incorporating a dynamic temperature into the free energy formulation, message passing is performed on a dynamic surface of energy cost. The proposed cooling process helps BP converge to a stable fixed point with a lower energy value that leads to better estimations. For decoding turbo-like error correcting codes, we adopt a parametric Gaussian approximation to relax the binary parity check constraints and generalize the conventional binary networks as well. Both the computational complexity and the convergence rate of the proposed annealed BP algorithms are almost the same as those of the conventional BP algorithm. Simulated performance of the proposed algorithms when they are used to decode a low density parity check (LDPC) code and the (23,12) Golay code is presented to validate our proposals.

Severe intersymbol interference (ISI) is one of the main obstacles for reliable signal reception in ATSC DTV systems. Decision feedback equalizers (DFEs) are commonly used to suppress the ISI. However, DFEs may suffer from error propagation due to incorrect symbol decisions from the symbol slicer. This phenomenon deteriorates the performance even more when the post-cursor ISI is strong. In order to reduce error propagation, we present a novel hybrid equalization scheme for ATSC channels. The proposed scheme consists of an adaptive decision feedback sparsening filter (DFSF), and an iterative maximum a posteriori (MAP) equalizer based on the belief propagation (BP) algorithm. In the first stage, instead of removing all the ISI from post cursors, the DFSF employs a modified feedback filter which leaves the strongest post-cursor ISI taps uncorrected. As a result, a long ISI channel is equalized to a sparse channel having only a small number of nonzero taps. In the second stage, a belief propagation algorithm is applied to mitigate the residual ISI. Since the channel is typically time-varying and suffers from Doppler fading, the DFSF is adapted using the least mean square (LMS) algorithm, such that the amplitude and the locations of the nonzero taps of the equalized sparse channel appear to be fixed. As such, the channel appears to be static during the second stage of equalization which consists of the BP detector. Simulation results demonstrate that the proposed scheme outperforms the traditional DFE in symbol error rate, under both static channels and dynamic ATSC channels.

In this paper, we develop a Belief Propagation (BP) algorithm for similarity computation to improve the recommendation accuracy of the neighborhood method, which is one of the most popular Collaborative Filtering (CF) recommendation algorithms. We formulate a probabilistic inference problem as to compute the marginal posterior distributions of similarity variables from their joint posterior distribution given the observed ratings. However, direct computation is prohibitive in large-scale recommender systems. Therefore, we introduce an appropriate chosen factor graph to express the factorization of the joint distribution function, and utilize the BP algorithm that operates in the factor graph to exploit the factorization for efficient inference. In addition, since the high degree at the factor node incurs an exponential increase in computational complexity, we also propose a complexity-reduction technique. The overall complexity of the proposed BP algorithm on a factor graph is linear in the number of variables, which ensures scalability. Finally, through experiments on the MovieLens dataset, we show the superior prediction accuracy of the proposed BP-based similarity computation algorithm for recommendation.