Belief Propagation Decoding Research Articles

Window-constrained interconnect-efficient progressive edge growth LDPC codes

One of the attractive features of low-density parity-check (LDPC) codes is the parallel iterative nature of their iterative belief propagation decoding, making them amenable to efficient hardware implementation. However, for an arbitrary code construction, the random-like connections between the code's Tanner graph variable and check nodes makes fully-parallel implementation a difficult task as this leads to complex interconnect wiring and routing congestion. In this paper, we present a novel LDPC code design approach, based on the progressive edge growth (PEG) Tanner graph construction, to solve the problem of dense connections between processing nodes. The approach is based on controlling the maximum connection length between processing nodes in order to make fully parallel implementation feasible. The proposed algorithm offers a good compromise between error correction performance and decoder complexity. Simulation results and FPGA-based implementation comparisons are presented to demonstrate the advantages of the proposed LDPC code constructions, and it is shown that, with proper window-constrained node placement design, an improvement of up to 40% in interconnect efficiency is achievable without any significant degradation in error correction capability.

Distributed Compression for Condition Monitoring of Wind Farms

A good understanding of individual and collective wind farm operation is necessary for improving the overall performance of the wind farm “grid,” as well as estimating in real time the amount of energy that can be generated for effectively managing demand and supply over the smart grid. This paper proposes a scheme for compressing wind speed measurements exploiting both temporal and spatial correlation between the readings via distributed source coding. The proposed scheme relies on a correlation model based on true measurements. Two compression schemes are proposed, both of low encoding complexity, as well as a particle-filtering-based belief propagation decoder that adaptively estimates the nonstationary noise of the correlation model. Simulation results using realistic models show significant performance improvements compared to the scheme that does not dynamically refine correlation.

Effects of Single-Cycle Structure on Iterative Decoding of Low-Density Parity-Check Codes

We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding. For fixed numbers of BP iterations, the bit error probability approaches a limit as blocklength tends to infinity, and the limit is obtained via density evolution. On the other hand, the difference between the bit error probability of codes with blocklength $n$ and that in the large blocklength limit is asymptotically $\alpha(\epsilon,t)/n + \Theta(n^{-2})$ where $\alpha(\epsilon,t)$ denotes a specific constant determined by the code ensemble considered, the number $t$ of iterations, and the erasure probability $\epsilon$ of the BEC. In this paper, we derive a set of recursive formulas which allows evaluation of the constant $\alpha(\epsilon,t)$ for standard irregular ensembles. The dominant difference $\alpha(\epsilon,t)/n$ can be considered as effects of cycle-free and single-cycle structures of local graphs. Furthermore, it is confirmed via numerical simulations that estimation of the bit error probability using $\alpha(\epsilon,t)$ is accurate even for small blocklengths.

Iterative Linear Programming Decoding of Nonbinary LDPC Codes With Linear Complexity

The problem of low-complexity linear programming (LP) decoding of nonbinary low-density parity-check (LDPC) codes is considered, and an iterative LP decoding algorithm is presented. Results that were previously derived for binary LDPC codes are extended to the nonbinary case. Both simple and generalized nonbinary LDPC codes are considered. It is shown how the algorithm can be implemented efficiently using a finite-field fast Fourier transform. Then, the convergence rate of the algorithm is analyzed. The complexity of the algorithm scales linearly in the block length, and it can approximate, up to an arbitrarily small relative error, the objective function of the exact LP solution. When applied to a typical code from an appropriate nonbinary LDPC code ensemble, the algorithm can correct a constant fraction of errors in linear (in the block length) computational complexity. Computer experiments with the new iterative LP decoding algorithm show that, in the error floor region, it can have better performance compared to belief propagation decoding, with similar computational requirements.

A Parallel IRRWBF LDPC Decoder Based on Stream-Based Processor

Low-density parity check (LDPC) codes have gained much attention due to their use of various belief-propagation (BP) decoding algorithms to impart excellent error-correcting capability. The BP decoders are quite simple; however, their computation-intensive and repetitive process prohibits their use in energy-sensitive applications such as sensor networks. Bit flipping-based decoding algorithms, especially implementation-efficient, reliability ratio-based, weighted bit-flipping (IRRWBF) decoding; have shown an excellent tradeoff between error-correction performance and implementation cost. In this paper, we show that with IRRWBF, iterative re-computation can be replaced by iterative selective updating. When compared with the original algorithm, simulation results show that, decoding speed can be increased by 200 to 600 percent , as the number of decoding iterations is increased from 5 to 1,000. The decoding steps are broken down into various stages such that the update operations are mostly of the single-instruction, multiple-data (SIMD) type. In this paper, we show that by using Intel Wireless MMX 2 accelerating technology in the proposed algorithm, the speed increased by 500 to 1,500 percent. The results of implementing the proposed scheme using an Intel/Marvell PXA320 (806 MHz) CPU are presented. The proposed scheme can be used effectively in real-time LDPC codes for energy-sensitive mobile devices.

A Novel Informed Dynamic Scheduling Algorithm for Belief Propagation Decoding of LDPC Codes

A Novel Informed Dynamic Scheduling Algorithm for Belief Propagation Decoding of LDPC Codes

Performance improvement of short-length regular low-density parity-check codes with low-complexity post-processing

It is well known that extremely long low-density parity-check (LDPC) codes perform exceptionally well for error correction applications, short-length codes are preferable in practical applications. However, short-length LDPC codes suffer from performance degradation owing to graph-based impairments such as short cycles, trapping sets and stopping sets and so on in the bipartite graph of the LDPC matrix. In particular, performance degradation at moderate to high Eb/N0 is caused by the oscillations in bit node a posteriori probabilities induced by short cycles and trapping sets in bipartite graphs. In this study, a computationally efficient algorithm is proposed to improve the performance of short-length LDPC codes at moderate to high Eb/N0. This algorithm makes use of the information generated by the belief propagation (BP) algorithm in previous iterations before a decoding failure occurs. Using this information, a reliability-based estimation is performed on each bit node to supplement the BP algorithm. The proposed algorithm gives an appreciable coding gain as compared with BP decoding for LDPC codes of a code rate equal to or less than 1/2 rate coding. The coding gains are modest to significant in the case of optimised (for bipartite graph conditioning) regular LDPC codes, whereas the coding gains are huge in the case of unoptimised codes. Hence, this algorithm is useful for relaxing some stringent constraints on the graphical structure of the LDPC code and for developing hardware-friendly designs.

Doubly multistage decoding of low-density parity-check codes over ℤ2m

A doubly multistage decoder (DMD) for low-density parity-check codes over ℤ2m, m>1, which fuses the multistage decoding approaches of Armand et al. and Varnica et al., is proposed. Two variants of the DMD are considered. The first (respectively, second) performs belief-propagation (BP) [respectively, offset min-sum (OMS)] decoding in each decoding stage and is referred to as DMD - BP (respectively, DMD - OMS). For the moderate-length codes considered in this study, the computer simulations show the DMD - BP (respectively, DMD - OMS) achieving coding gains of up to 0.43 dB (respectively, 0.67 dB) over standard BP decoding at a bit error rate of 10−6 on an additive-white-Gaussian-noise channel, while requiring significantly less computational power. Remarkably, DMD - OMS outperforms DMD - BP, yet has lower computational complexity than DMD - BP. Snapshots of the log-likelihood ratio densities of the decoded bits midway through the decoding process explain the superiority of the DMD over standard BP decoding.

Nonuniform Encoding and Hybrid Decoding Schemes for Equal Error Protection of Rateless Codes

Messages are generally selected with the same probability in the encoding scheme of rateless codes for equal error protection. In addition, a belief propagation (BP) decoding scheme is generally used because of the low computational complexity. However, the probability of recovering a new message by BP decoding is reduced if both the recovered and unrecovered messages are selected uniformly. Thus, more codeword symbols than expected are required for the perfect recovery of message symbols. Therefore, a new encoding scheme with a nonuniform selection of messages is proposed in this paper. In addition, a BP-Gaussian elimination hybrid decoding scheme that complements the drawback of the BP decoding scheme is proposed. The performances of the proposed schemes are analyzed and compared with those of the conventional schemes.

Tree-Structure Expectation Propagation for LDPC Decoding Over the BEC

We present the tree-structure expectation propagation (Tree-EP) algorithm to decode low-density parity-check (LDPC) codes over discrete memoryless channels (DMCs). EP generalizes belief propagation (BP) in two ways. First, it can be used with any exponential family distribution over the cliques in the graph. Second, it can impose additional constraints on the marginal distributions. We use this second property to impose pair-wise marginal constraints over pairs of variables connected to a check node of the LDPC code's Tanner graph. Thanks to these additional constraints, the Tree-EP marginal estimates for each variable in the graph are more accurate than those provided by BP. We also reformulate the Tree-EP algorithm for the binary erasure channel (BEC) as a peeling-type algorithm (TEP) and we show that the algorithm has the same computational complexity as BP and it decodes a higher fraction of errors. We describe the TEP decoding process by a set of differential equations that represents the expected residual graph evolution as a function of the code parameters. The solution of these equations is used to predict the TEP decoder performance in both the asymptotic regime and the finite-length regime over the BEC. While the asymptotic threshold of the TEP decoder is the same as the BP decoder for regular and optimized codes, we propose a scaling law (SL) for finite-length LDPC codes, which accurately approximates the TEP improved performance and facilitates its optimization.

Dithered Belief Propagation Decoding

We introduce two dithered belief propagation decoding algorithms to lower the error floor with a minimal hardware overhead. One of the algorithms can additionally improve the decoding performance in the waterfall region using a large iteration limit but with a negligible increase in the average time complexity.

Design and Analysis of Non-Binary LDPC and IRA Modulation Codes Using Average Zero-Word Probability

It has been shown that, under belief-propagation (BP) decoding, random-coset GF(q) low-density parity-check (LDPC) codes and irregular repeat-accumulate (IRA) codes with q-ary nonuniform signal constellations approach the unrestricted Shannon limit. In previous works, extrinsic information transfer (EXIT) charts were employed in the design of random-coset GF(q) LDPC and IRA modulation codes. However, in the EXIT charts, there is no closed-form expression for check node decoder (CND) curves. The CND curves for random-coset GF(q) LDPC and IRA modulation codes rely on Monte Carlo simulations, resulting in a high design complexity. This study presents new design methods of random-coset GF(q) LDPC and IRA modulation codes based on the average zero-word probability. The average zero-word probability serves as a surrogate for LLR messages, just as the mutual information acts as a surrogate for LLR messages in EXIT charts. Based on the average zero-word probability, closed-form expressions of CND input-output relations are derived for random-coset GF(q) LDPC and IRA modulation codes. Simple convergence criteria for random-coset LDPC and IRA modulation codes are proposed. Based on the proposed convergence criteria, six codes are designed with nonuniform signal constellations. Simulation results show that the proposed codes have near-capacity performances and are comparable with those designed based on EXIT charts.

Finding the Exhaustive List of Small Fully Absorbing Sets and Designing the Corresponding Low Error-Floor Decoder

This work provides an efficient exhaustive search algorithm for finding all small fully absorbing sets (FASs) of any arbitrary low-density parity-check (LDPC) code. The proposed algorithm is based on the branch-&-bound principle for solving NP-complete problems. In particular, given any LDPC code, the problem of finding all FASs of size less than t is formulated as an integer programming problem, for which a new branch-&-bound algorithm is devised with new node selection and tree-trimming mechanisms. The resulting algorithm is capable of finding all FASs of size <; 7 for LDPC codes of length <; 1000. When limiting the FASs of interest to those with the number of violated parity-check nodes <; 3, the proposed algorithm is capable of finding all such FASs of size <; 14 for LDPC codes of lengths <; 1000. The resulting exhaustive list of small FASs is then used to devise a new efficient post-processing low-error floor LDPC decoder. The numerical results show that by exploiting the exhaustive list of small FASs, the proposed post-processing decoder can significantly lower the error-floor performance of a given LDPC code. For various example codes of length <; 3000, the proposed post-processing decoder lowers the error floor by a couple of orders of magnitude when compared to the standard belief propagation decoder and by an order of magnitude when compared to other existing low error-floor decoders.

On the Design of LDPC-Convolutional Ensembles Using the TEP Decoder

Low-density parity-check convolutional (LDPCC) codes asymptotically achieve channel capacity under belief propagation (BP) decoding. In this paper, we decode LDPCC codes using the Tree-Expectation Propagation (TEP) decoder, recently proposed as an alternative decoding method to the BP algorithm for the binary erasure channel (BEC). We show that, for LDPCC codes, the TEP decoder improves the BP solution with a comparable complexity or, alternatively, it allows using shorter codes to achieve similar error rates. We also propose a window-sliding scheme for the TEP decoder to reduce the decoding latency.

On Pseudocodewords and Decision Regions of Linear Programming Decoding of HDPC Codes

In this paper we explore the decision regions of Linear Programming (LP) decoding. We compare the decision regions of an LP decoder, a Belief Propagation (BP) decoder and the optimal Maximum Likelihood (ML) decoder. We study the effect of minimal-weight pseudocodewords on LP decoding. We present global optimization as a method for finding the minimal pseudoweight of a given code as well as the number of minimal-weight generators. We present a complete pseudoweight distribution for the [24, 12, 8] extended Golay code, and provide justifications of why the pseudoweight distribution alone cannot be used for obtaining a tight upper bound on the error probability.

New stopping criteria for iterative decoding of LDPC codes in H‐ARQ systems

SUMMARYUnder severely unreliable channel, decoding of error‐correcting codes frequently fails, which requires a lot of computational complexity, especially, in the iterative decoding algorithm. In hybrid automatic repeat request systems, most of computation power is wasted on failed decoding if a codeword is retransmitted many times. Therefore, early stopping of iterative decoding needs to be adopted. In this paper, we propose a new stopping algorithm of iterative belief propagation decoding for low‐density parity‐check codes, which is effective on both high and low signal‐to‐noise ratio ranges and scalable to variable code rate and length. The proposed stopping algorithm combines several good stopping criteria. Each criterion is extremely simple and will not be a burden to the overall system. With the proposed stopping algorithm, it is shown via numerical analysis that the decoding complexity of hybrid automatic repeat request system with adaptive modulation and coding scheme can be fairly reduced. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Analysis of Error Floors of Non-binary LDPC Codes over BEC

In this paper, we investigate the error floors of the non-binary low-density parity-check codes transmitted over the binary erasure channels under belief propagation decoding. We propose a method to improve the decoding erasure rates in the error floors by optimizing labels in zigzag cycles in the Tanner graphs of codes. Furthermore, we give lower bounds on the bit and the symbol erasure rates in the error floors. The simulation results show that the presented lower bounds are tight for the codes designed by the proposed method.

Binary Soliton-Like Rateless Coding for the Y-Network

For the binary erasure channel, Luby Transform (LT) and Raptors codes have been shown to achieve capacity by carefully designed degree distributions for multicasting scenarios. Generalizing fountain codes to multihop networks requires transport nodes to perform network coding (NC). However, if intermediate nodes perform decentralized NC blindly, the statistical properties imposed by the fountain code are lost, and thus, a Gaussian elimination decoder must be used at the sink at the cost of significant increase in complexity compared to a belief propagation (BP) decoder. Addressing this problem, in this paper, we propose a new protocol, namely Soliton-like rateless coding (SLRC), by exploiting the benefits of fountain coding and NC coding over a Y-network. Ensuring key properties of the fountain code are preserved; BP can be effectively applied when transport nodes perform NC. Additionally, the proposed coding protocol is resilient to nodes churn rates. The SLRC scheme is evaluated against buffer-and-forwarding, and the distributed LT (DLT) codes; SLRC exhibits a 5% reduction in overhead compared to the state of the art DLT code at high decoding success rates. Simulations show that the proposed scheme preserves the benefits of NC and fountain coding.

Windowed Decoding of Protograph-Based LDPC Convolutional Codes Over Erasure Channels

We consider a windowed decoding scheme for LDPC convolutional codes that is based on the belief-propagation (BP) algorithm. We discuss the advantages of this decoding scheme and identify certain characteristics of LDPC convolutional code ensembles that exhibit good performance with the windowed decoder. We will consider the performance of these ensembles and codes over erasure channels with and without memory. We show that the structure of LDPC convolutional code ensembles is suitable to obtain performance close to the theoretical limits over the memoryless erasure channel, both for the BP decoder and windowed decoding. However, the same structure imposes limitations on the performance over erasure channels with memory.

Effective Informed Dynamic Scheduling for Belief Propagation Decoding of LDPC Codes

The simultaneous flooding scheduling is popular for Low-Density Parity-Check (LDPC) Belief Propagation (BP) decoding. Non-simultaneous sequential scheduling is superior to the flooding scheduling, and asynchronous dynamic scheduling has better FER performance than the sequential scheduling. However, all strategies encounter the trouble of locating the error variable node. This paper proposes an informed dynamic scheduling strategy, which utilizes the instability of the variable node and the residual of the variable-to-check message to locate the message to be updated first. The informed dynamic scheduling overcomes the trapping sets effectively. This paper also designs an informed dynamic scheduling strategy with adaptivity to pass more messages in parallel, which effectively postpones the influence of cycles in the Tanner graph. In some sense, the strategy lengthens cycles. Simulation results show that the two informed dynamic scheduling strategies outperform other algorithms.