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Multi-Gb/s LDPC Code Design and Implementation

IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS 2009年第2期17卷 262-268页

作者： Sha, Jin Wang, Zhongfeng Gao, minglun Li, Li Nanjing Univ KLAPEM Inst VLSI Design Nanjing 210093 Peoples R China Oregon State Univ Sch Elect Engn & Comp Sci Corvallis OR 97331 USA

Low-density parity-check (LDPC) code, a very promising near-optimal error correction code (ECC), is being widely considered in next generation industry standards. The VLSI implementation of high-speed LDPC decoder remains a big challenge. This paper presents the construction of a new class of implementation-oriented LDPC codes, namely shift-LDPC codes. With girth optimization, this kind of codes can perform as well as computer generated random codes. More importantly, the decoder can be efficiently implemented to obtain very high decoding speeds. In addition, more than 50% of message memory can be generally saved over conventional partially parallel decoder architectures. We demonstrate the benefits of the proposed techniques with an application-specific integrated circuit (ASIC design (in 0.18-mu m CMOS) for a 8192-bit regular LDPC code, which can achieve 5 Gb/s throughput at 15 iterations.

关键词： Error correction codes (ECC) low-density parity-check (LDPC) codes min-sum algorithm parallel processing VLSI

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High-Throughput Layered Decoder Implementation for Quasi-Cyclic LDPC Codes

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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 2009年第6期27卷 985-994页

作者： Zhang, Kai Huang, Xinming Wang, Zhongfeng Worcester Polytech Inst Dept Elect & Comp Engn Worcester MA 01609 USA Broadcom Corp Irvine CA 92617 USA

This paper presents a high-throughput decoder design for the Quasi-Cyclic (QC) Low-Density Parity-Check (LDPC) codes. Two new techniques are proposed, including parallel layered decoding architecture (PLDA) and critical path splitting. PLDA enables parallel processing for all layers by establishing dedicated message passing paths among them. The decoder avoids crossbar-based large interconnect network. Critical path splitting technique is based on articulate adjustment of the starting point of each layer to maximize the time intervals between adjacent layers, such that the critical path delay can be split into pipeline stages. Furthermore, min-sum and loosely coupled algorithms are employed for area efficiency. As a case study, a rate-1/2 2304-bit irregular LDPC decoder is implemented using ASIC design in 90nm CMOS process. The decoder can achieve the maximum decoding throughput of 2.2Gbps at 10 iterations. The operating frequency is 950MHz after synthesis and the chip area is 2.9mm(2).

关键词： Low-density parity-check codes quasi-cyclic codes parallel architecture layered decoding critical path splitting min-sum algorithm loosely coupled algorithm VLSI

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MAP estimation via agreement on trees: Message-passing and linear programming

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IEEE TRANSACTIONS ON INFORMATION THEORY 2005年第11期51卷 3697-3717页

作者： Wainwright, MJ Jaakkola, TS Willsky, AS Univ Calif Berkeley Dept Elect Engn & Comp Sci Berkeley CA 94720 USA Univ Calif Berkeley Dept Stat Berkeley CA 94720 USA MIT Dept Elect Engn & Comp Sci Cambridge MA 02139 USA

We develop and analyze methods for computing provably optimal maximum a posteriori probability (MAP) configurations for a subclass of Markov random fields defined-on graphs with cycles. By decomposing the original distribution into a convex combination of tree-structured distributions, we obtain an upper bound on the optimal value of the original problem (i.e., the log probability of the MAP assignment) in terms of the combined optimal values of the tree problems. We prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common. An important implication is that any such shared configuration must also be a MAP configuration for the original distribution. Next we develop two approaches to attempting to obtain tight upper bounds: a) a tree-related linear program (LP), which is derived from the Lagrangian dual of the upper bounds;and b) a tree-reweighted max-product message-passing algorithm that is related to but distinct from the max-product algorithm. In this way, we establish a connection between a certain LP relaxation of the mode-finding problem and a reweighted form of the max-product (min-sum) message-passing algorithm.

关键词： approximate inference integer programming iterative decoding linear programming (LP) relaxation Markov random fields marginal polytope maximum a posteriori probability (MAP) estimation max-product algorithm message-passing algorithms min-sum algorithm

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Max-product for maximum weight matching: Convergence, correctness, and LP duality

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IEEE TRANSACTIONS ON INFORMATION THEORY 2008年第3期54卷 1241-1251页

作者： Bayati, Mobsen Shah, Devavrat Sharma, Mayank Microsoft Res Redmond WA 98052 USA MIT Dept Elect Engn & Comp Sci Cambridge MA 02139 USA IBM Corp Thomas J Watson Res Ctr Yorktown Hts NY 10598 USA

Max-product "belief propagation" (BP) is an iterative, message-passing algorithm for finding the maximum a posteriori (MAP) assignment of a discrete probability distribution specified by a graphical model. Despite the spectacular success of the algorithm in many application areas such as iterative decoding and combinatorial optimization, which involve graphs with many cycles, theoretical results about both the correctness and convergence of the algorithm are known in only a few cases (see Section I for references). In this paper, we will prove the correctness and convergence of max-product for finding the maximum weight matching (MWM) in bipartite graphs. Even though the underlying graph of the MWM problem has many cycles, somewhat surprisingly we show that the max-product algorithm converges to the correct MWM as long as the MWM is unique. We provide a bound on the number of iterations required and show that for a graph of size n, the computational cost of the algorithm scales as O(n(3)), which is the same as the computational cost of the best known algorithms for finding the MWM. We also provide an interesting relation between the dynamics of the max-product algorithm and the auction algorithm, which is a well-known distributed algorithm for solving the MWM problem.

关键词： auction algorithm belief propagation (BP) distributed optimization linear programming Markov random fields maximum weight matching (MWM) max-product algorithm message-passing algorithms min-sum algorithm

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High Throughput and Hardware Efficient Hybrid LDPC Decoder Using Bit-Serial Stochastic Updating

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS 2023年第9期70卷 3653-3664页

作者： Hu, Shuai Han, Kaining Zhu, Yubin Shen, Guodong Wang, Fujie Hu, Jianhao Univ Elect Sci & Technol China Natl Key Lab Sci & Technol Commun Chengdu 610054 Peoples R China

Hybrid low-density parity-check (LDPC) decoding combines conventional Belief-Propagation (BP) algorithm with stochastic decoding to achieve high performance and low complexity simultaneously. However, lossy and inefficient stochastic to-binary (S2B) conversion brings extra performance degradation and decoding latency. In this paper, a bit-serial stochastic updating based hybrid decoding (BSSU-HD) is proposed, which employs fully correlated stochastic (FCS) check nodes (CNs) and probability tracers assisted variable nodes (VNs) to accomplish accurate and efficient S2B conversion. Two strategies, including random source selection and tracing speed switching, are proposed to further improve performance and convergence. A BSSU LDPC decoder for IEEE 802.3an is designed in a 65-nm CMOS process, which occupies 4.6 mm(2) silicon area and achieves a throughput of 200.8 Gb/s at E-b/N-0 = 4.4 dB with 500 MHz clock frequency from a 1.2 V supply voltage. The power and energy efficiency are 2.933 W and 14.61 pJ/bit, respectively. To the best of our known, it achieves the best decoding performance, the highest throughput and hardware efficiency among state-of-the-art IEEE 802.3an LDPC decoders. We also verify that the BSSU-HD can achieve better performance for multi-rate 5th generation (5G) New Ratio (NR) LDPC codes than conventional algorithm, which greatly extends the application of the stochastic decoding.

关键词： Decoding Iterative decoding Hardware Throughput IEEE 802 3 Standard EPON Stochastic processes LDPC code probability tracer fully correlated stochastic min-sum algorithm

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Low-Complexity First-Two-minimum-Values Generator for Bit-Serial LDPC Decoding

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II-EXPRESS BRIEFS 2016年第5期63卷 483-487页

作者： Lee, Jea Hack Sunwoo, Myung Hoon Ajou Univ Dept Elect & Comp Engn Suwon 443749 South Korea

This brief proposes a low-complexity first-two-minimum-values generator for a bit-serial scheme. Since the hardware complexity of generators utilizes a significant portion of the min-sum low-density parity-check decoder, a low-complexity generator is crucially important. To reduce hardware complexity, an existing bit-serial generator that finds only one minimum value instead of two has been proposed;however, it can cause bit error rate (BER) degradation. By contrast, the proposed low-complexity bit-serial generator can find the exact first two minimum values and thus can improve the BER performance. Moreover, the proposed generator does not suffer from any throughput loss since its latency is almost the same as that of the existing generator.

关键词： Area-time complexity bit-serial scheme low-complexity design low-density parity-check (LDPC) decoder minimum-value generator min-sum algorithm

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A Two-Bit Precision Decoding algorithm for LDPC Codes Over the Binary Symmetric Channel

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IEEE WIRELESS COMMUNICATIONS LETTERS 2019年第1期8卷 109-112页

作者： Cho, Sunghye Yang, Kyeongcheol Pohang Univ Sci & Technol POSTECH Dept Elect Engn Pohang 37673 South Korea Samsung Elect Co Ltd DRAM Design Team Memory Business Suwon 16677 South Korea

The min-sum algorithm as an approximation of the sum-product algorithm (SPA) for decoding of LDPC codes has much lower computational complexity, but slightly worse performance than the SPA. However, severe performance degradation might occur when it is realized in hardware due to the finite precision effect, especially in two-bit precision. In order to overcome this problem, we propose a two-hit precision decoding algorithm. The proposed algorithm employs one of several decoders adaptively, based on the syndrome weight of the estimated decoder output every iteration. Numerical results show that it has similar performance to the SPA over the binary symmetric channel.

关键词： Binary symmetric channel low-density parity-check code min-sum algorithm error floor syndrome weight

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Convergence Properties of Message-Passing algorithm for Distributed Convex Optimisation With Scaled Diagonal Dominance

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2021年 69卷 3868-3877页

作者： Zhang, Zhaorong Fu, minyue Univ Newcastle Sch Elect Engn & Comp Univ Dr Callaghan NSW 2308 Australia

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.

关键词： Convergence Optimization Signal processing algorithms Linear programming Couplings Belief propagation Task analysis Distributed optimisation message passing belief propagation min-sum algorithm convex optimisation

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Concatenated codes:: Serial and parallel

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IEEE TRANSACTIONS ON INFORMATION THEORY 2005年第5期51卷 1625-1634页

作者： Barg, A Zémor, G Ecole Natl Super Telecommun Bretagne F-75634 Paris France

An analogy is examined between serially concatenated codes and parallel concatenations whose interleavers;are described by bipartite graphs with good expanding properties. In particular, a modified expander code construction is shown to behave very much like Forney's classical concatenated codes, though with improved decoding complexity. It is proved that these new codes achieve the Zyablov bound delta(Z) on the minimum distance. For these codes, a soft-decision, reliability-based, linear-time decoding algorithm is introduced, that corrects any fraction of errors up to almost delta(Z)/2. For the binary-symmetric channel, this algorithm's error exponent attains the Forney bound previously known only for classical (serial) concatenations.

关键词： bipartite-graph codes concatenated codes expander codes Forney bound min-sum algorithm soft-decision decoding Zyablov bound

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Message-Passing algorithms and Improved LP Decoding

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IEEE TRANSACTIONS ON INFORMATION THEORY 2012年第12期58卷 7260-7271页

作者： Arora, Sanjeev Daskalakis, Constantinos Steurer, David Princeton Univ Dept Comp Sci Princeton NJ 08540 USA MIT Dept Elect Engn & Comp Sci Cambridge MA 02139 USA Cornell Univ Dept Comp Sci Ithaca NY 14850 USA

Linear programming (LP) decoding for low-density parity-check codes (and related domains such as compressed sensing) has received increased attention over recent years because of its practical performance-coming close to that of iterative decoding algorithms-and its amenability to finite-blocklength analysis. Several works starting with the work of Feldman et al. showed how to analyze LP decoding using properties of expander graphs. This line of analysis works for only low error rates, about a couple of orders of magnitude lower than the empirically observed performance. It is possible to do better for the case of random noise, as shown by Daskalakis et al. and Koetter and Vontobel. Building on work of Koetter and Vontobel, we obtain a novel understanding of LP decoding, which allows us to establish a 0.05 fraction of correctable errors for rate-1/2 codes;this comes very close to the performance of iterative decoders and is significantly higher than the best previously noted correctable bit error rate for LP decoding. Our analysis exploits an explicit connection between LP decoding and message-passing algorithms and, unlike other techniques, directly works with the primal linear program. An interesting byproduct of our method is a notion of a "locally optimal" solution that we show to always be globally optimal (i.e., it is the nearest codeword). Such a solution can in fact be found in near-linear time by a "reweighted" version of the min-sum algorithm, obviating the need for LP. Our analysis implies, in particular, that this reweighted version of the min-sum decoder corrects up to a 0.05 fraction of errors.

关键词： Linear programming (LP) decoding low-density parity-check (LDPC) codes message-passing algorithms min-sum algorithm

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