A survey of FPGA-based LDPC decoders

Low-Density Parity Check (LDPC) error correction decoders have become popular in communications systems, as a benefit of their strong error correction performance and their suitability to parallel hardware implementation. A great deal of research effort has been invested into LDPC decoder designs that exploit the flexibility, the high processing speed and the parallelism of Field-Programmable Gate Array (FPGA) devices. FPGAs are ideal for design prototyping and for the manufacturing of small-production-run devices, where their in-system programmability makes them far more cost-effective than Application-Specific Integrated Circuits (ASICs). However, the FPGA-based LDPC decoder designs published in the open literature vary greatly in terms of design choices and performance criteria, making them a challenge to compare. This paper explores the key factors involved in FPGA-based LDPC decoder design and presents an extensive review of the current literature. In-depth comparisons are drawn amongst 140 published designs (both academic and industrial) and the associated performance trade-offs are characterised, discussed and illustrated. Seven key performance characteristics are described, namely their processing throughput, latency, hardware resource requirements, error correction capability, processing energy efficiency, bandwidth efficiency and flexibility. We offer recommendations that will facilitate fairer comparisons of future designs, as well as opportunities for improving the design of FPGA-based LDPC decoder

Non-Uniform Window Decoding Schedules for Spatially Coupled LDPC Codes

Spatially coupled low-density parity-check codes can be decoded using a graph-based message passing algorithm applied across the total length of the coupled graph. However, considering practical constraints on decoding latency and complexity, a sliding window decoding approach is normally preferred. In order to reduce decoding complexity compared with standard parallel decoding schedules, serial schedules can be applied within a decoding window. However, uniform serial schedules within a window do not provide the expected reduction in complexity. Hence, we propose non-uniform schedules (parallel and serial) based on measured improvements in the estimated bit error rate (BER). We show that these non-uniform schedules result in a significant reduction in complexity without any loss in performance. Furthermore, based on observations made using density evolution, we propose a non-uniform pragmatic decoding schedule (parallel and serial) that does not require any additional calculations (e.g., BER estimates) within the decoding process

낸드플래시 메모리 오류정정을 위한 고성능 LDPC 복호방법 연구

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2013. 8. 성원용.반도체 공정의 미세화에 따라 비트 에러율이 증가하는 낸드 플래시 메모리에서 고성능 에러 정정 방법은 필수적이다. Low-density parity-check (LDPC) 부호와 같은 연판정 에러 정정 부호는 뛰어난 에러 정정 성능을 보이지만, 높은 구현 복잡도로 인해 플래시 메모리 시스템에 적용되기 힘든 단점이 있다. 본 논문에서는 LDPC 부호의 효율적인 복호를 위해 고성능 메시지 전파 스케줄링 방법과 저 복잡도 복호 알고리즘을 제안한다. 특히 finite geometry (FG) LDPC 부호에 대한 효율적인 디코더 아키텍쳐를 제안하며, 구현된 디코더를 이용하여 낸드 플래시 메모리에 대해 연판정 복호시의 에너지 소모량에 대해 연구한다. 본 논문의 첫 번째 부분에서는 동적 스케줄링 (informed dynamic scheduling, IDS) 알고리즘의 성능향상 방법에 대해 연구한다. 이를 위해 우선 기존의 가장 빠른 수렴 속도를 보이는 IDS 알고리즘인 레지듀얼 신뢰 전파 (residual belief propagation, RBP) 알고리즘의 동작 특성을 분석하고, 이를 바탕으로 특정 노드에 메시지 갱신이 집중되는 것을 방지하여 RBP 알고리즘의 수렴속도를 증가시킨 improved RBP (iRBP) 알고리즘을 제안한다. 또한 iRBP의 뛰어난 수렴속도와 기존의 NS 알고리즘의 우수한 에러 정정 능력을 모두 갖춘 신드롬 기반의 혼합 스케줄링 (mixed scheduling) 방법을 제안한다. 끝으로 다양한 부호율의 LDPC 부호에 대한 모의실험을 통해 제안된 신드롬 기반의 혼합 스케줄링 방법이 본 논문에서 시험된 다른 모든 스케줄링 알고리즘의 성능을 능가함을 확인하였다. 논문의 두 번째 부분에서는 복호 실패시 많은 비트 에러를 발생시키는 a posteriori probability (APP) 알고리즘의 개선 방안에 방안을 제안한다. 또한 빠른 수렴속도와 우수한 에러 마루 (error-floor) 성능으로 데이터 저장장치에 적합한 FG-LDPC 부호에 대해 제안된 알고리즘이 적용된 하드웨어 아키텍처를 제안하였다. 제안된 아키텍처는 높은 노드 가중치를 가지는 FG-LDPC 부호에 적합하도록 쉬프트 레지스터 (shift registers)와 SRAM 기반의 혼합 구조를 채용하며, 높은 처리량을 얻기 위해 파이프라인된 병렬 아키텍처를 사용한다. 또한 메모리 사용량을 줄이기 위해 세 가지의 메모리 용량 감소 기법을 적용하며, 전력 소비를 줄이기 위해 두 가지의 저전력 기법을 제안한다. 본 제안된 아키텍처는 부호율 0.96의 (68254, 65536) Euclidean geometry LDPC 부호에 대해 0.13-um CMOS 공정에서 구현하였다. 마지막으로 본 논문에서는 연판정 복호가 적용된 낸드 플래시 메모리 시스템의 에너지 소모를 낮추는 방법에 대해 제안한다. 연판정 기반의 에러 정정 알고리즘은 높은 성능을 보이지만, 이는 플래시 메모리의 센싱 수와 에너지 소모를 증가 시키는 단점이 있다. 본 연구에서는 앞서 구현된 LDPC 디코더가 채용된 낸드 플래시 메모리 시스템의 에너지 소모를 분석하고, LDPC 디코더와 BCH 디코더 간의 칩 사이즈와 에너지 소모량을 비교하였다. 이와 더불어 본 논문에서는 LDPC 디코더를 이용한 센싱 정밀도 결정 방법을 제안한다. 본 연구를 통해 제안된 복호 및 스케줄링 알고리즘, VLSI 아키텍쳐, 그리고 읽기 정밀도 결정 방법을 통해 낸드 플래시 메모리 시스템의 에러 정정 성능을 극대화 하고 에너지 소모를 최소화 할 수 있다.High-performance error correction for NAND flash memory is greatly needed because the raw bit error rate increases as the semiconductor geometry shrinks for high density. Soft-decision error correction, such as low-density parity-check (LDPC) codes, offers high performance but their implementation complexity hinders wide adoption to consumer products. This dissertation proposes two high-performance message-passing schedules and a low-complexity decoding algorithm for LDPC codes. In particular, an efficient decoder architecture for finite geometry (FG) LDPC codes is proposed, and the energy consumption of soft-decision decoding for NAND flash memory is analyzed. The first part of this dissertation is devoted to improving the informed dynamic scheduling (IDS) algorithms. We analyze the behavior of the residual belief propagation (RBP), which is the fastest IDS algorithm, and develop an improved RBP (iRBP) by avoiding the concentration of message updates at a particular node. We also study the syndrome-based mixed scheduling of the iRBP and the node-wise scheduling (NS). The proposed mixed scheduling outperforms all other scheduling methods tested in this work. The next part of this dissertation is to develop a conditional variable node update scheme for the a posteriori probability (APP) algorithm. The developed algorithm is robust to decoding failures and can reduce the dynamic power consumption by lowering switching activities in the LDPC decoder. To implement the developed algorithm, we propose a memory-efficient pipelined parallel architecture for LDPC decoding. The architecture employs FG-LDPC codes that not only show fast convergence speed and good error-floor performance but also perform well with iterative decoding algorithms, which is especially suitable for data storage devices. We also developed a rate-0.96 (68254, 65536) Euclidean geometry LDPC code and implemented the proposed architecture in 0.13-um CMOS technology. This dissertation also covers low-energy error correction of NAND flash memory through soft-decision decoding. The soft-decision-based error correction algorithms show high performance, but they demand an increased number of flash memory sensing operations and consume more energy for memory access. We examine the energy consumption of a NAND flash memory system equipping an LDPC code-based soft-decision error correction circuit. The sum of energy consumed at NAND flash memory and the LDPC decoder is minimized. In addition, the chip size and energy consumption of the decoder were compared with those of two Bose-Chaudhuri-Hocquenghem (BCH) decoding circuits showing the comparable error performance and the throughput. We also propose an LDPC decoder-assisted precision selection method that needs virtually no overhead. This dissertation is intended to develop high-performance and low-power error correction circuits for NAND flash memory by studying improved decoding and scheduling algorithms, VLSI architecture, and a read precision selection method.1 Introduction 1 1.1 NAND Flash Memory 1 1.2 LDPC Codes 4 1.3 Outline of the Dissertation 6 2 LDPC Decoding and Scheduling Algorithms 8 2.1 Introduction 8 2.2 Decoding Algorithms for LDPC Codes 10 2.2.1 Belief Propagation Algorithm 10 2.2.2 Simplified Belief Propagation Algorithms 12 2.3 Message-Passing Schedules for Decoding of LDPC Codes 15 2.3.1 Static Schedules 15 2.3.2 Dynamic Schedules 17 3 Improved Dynamic Scheduling Algorithms for Decoding of LDPC Codes 22 3.1 Introduction 22 3.2 Improved Residual Belief Propagation Algorithm 23 3.3 Syndrome-Based Mixed Scheduling of iRBP and NS 26 3.4 Complexity Analysis and Simulation Results 28 3.4.1 Complexity Analysis 28 3.4.2 Simulation Results 29 3.5 Concluding Remarks 33 4 A Pipelined Parallel Architecture for Decoding of Finite-Geometry LDPC Codes 36 4.1 Introduction 36 4.2 Finite-Geometry LDPC Codes and Conditional Variable Node Update Algorithm 38 4.2.1 Finite-Geometry LDPC codes 38 4.2.2 Conditional Variable Node Update Algorithm for Fixed-Point Normalized APP-Based Algorithm 40 4.3 Decoder Architecture 46 4.3.1 Baseline Sequential Architecture 46 4.3.2 Pipelined-Parallel Architecture 54 4.3.3 Memory Capacity Reduction 57 4.4 Implementation Results 60 4.5 Concluding Remarks 64 5 Low-Energy Error Correction of NAND Flash Memory through Soft-Decision Decoding 66 5.1 Introduction 66 5.2 Energy Consumption of Read Operations in NAND Flash Memory 67 5.2.1 Voltage Sensing Scheme for Soft-Decision Data Output 67 5.2.2 LSB and MSB Concurrent Access Scheme for Low-Energy Soft-Decision Data Output 72 5.2.3 Energy Consumption of Read Operations in NAND Flash Memory 73 5.3 The Performance of Soft-Decision Error Correction over a NAND Flash Memory Channel 76 5.4 Hardware Performance of the (68254, 65536) LDPC Decoder 81 5.4.1 Energy Consumption of the LDPC Decoder 81 5.4.2 Performance Comparison of the LDPC Decoder and Two BCH Decoders 83 5.5 Low-Energy Error Correction Scheme for NAND Flash Memory 87 5.5.1 Optimum Precision for Low-Energy Decoding 87 5.5.2 Iteration Count-Based Precision Selection 90 5.6 Concluding Remarks 91 6 Conclusion 94 Bibliography 96 Abstract in Korean 110 감사의 글 112Docto

Short-length Low-density Parity-check Codes: Construction and Decoding Algorithms

Error control coding is an essential part of modern communications systems. LDPC codes have been demonstrated to offer performance near the fundamental limits of channels corrupted by random noise. Optimal maximum likelihood decoding of LDPC codes is too complex to be practically useful even at short block lengths and so a graph-based message passing decoder known as the belief propagation algorithm is used instead. In fact, on graphs without closed paths known as cycles the iterative message passing decoding is known to be optimal and may converge in a single iteration, although identifying the message update schedule which allows single-iteration convergence is not trivial. At finite block lengths graphs without cycles have poor minimum distance properties and perform poorly even under optimal decoding. LDPC codes with large block length have been demonstrated to offer performance close to that predicted for codes of infinite length, as the cycles present in the graph are quite long. In this thesis, LDPC codes of shorter length are considered as they offer advantages in terms of latency and complexity, at the cost of performance degradation from the increased number of short cycles in the graph. For these shorter LDPC codes, the problems considered are: First, improved construction of structured and unstructured LDPC code graphs of short length with a view to reducing the harmful effects of the cycles on error rate performance, based on knowledge of the decoding process. Structured code graphs are particularly interesting as they allow benefits in encoding and decoding complexity and speed. Secondly, the design and construction of LDPC codes for the block fading channel, a particularly challenging scenario from the point of view of error control code design. Both established and novel classes of codes for the channel are considered. Finally the decoding of LDPC codes by the belief propagation algorithm is considered, in particular the scheduling of messages passed in the iterative decoder. A knowledge-aided approach is developed based on message reliabilities and residuals to allow fast convergence and significant improvements in error rate performance

New Identification and Decoding Techniques for Low-Density Parity-Check Codes

Error-correction coding schemes are indispensable for high-capacity high data-rate communication systems nowadays. Among various channel coding schemes, low-density parity-check (LDPC) codes introduced by pioneer Robert G. Gallager are prominent due to the capacity-approaching and superior error-correcting properties. There is no hard constraint on the code rate of LDPC codes. Consequently, it is ideal to incorporate LDPC codes with various code rate and codeword length in the adaptive modulation and coding (AMC) systems which change the encoder and the modulator adaptively to improve the system throughput. In conventional AMC systems, a dedicated control channel is assigned to coordinate the encoder/decoder changes. A questions then rises: if the AMC system still works when such a control channel is absent. This work gives positive answer to this question by investigating various scenarios consisting of different modulation schemes, such as quadrature-amplitude modulation (QAM), frequency-shift keying (FSK), and different channels, such as additive white Gaussian noise (AWGN) channels and fading channels. On the other hand, LDPC decoding is usually carried out by iterative belief-propagation (BP) algorithms. As LDPC codes become prevalent in advanced communication and storage systems, low-complexity LDPC decoding algorithms are favored in practical applications. In the conventional BP decoding algorithm, the stopping criterion is to check if all the parities are satisfied. This single rule may not be able to identify the undecodable blocks, as a result, the decoding time and power consumption are wasted for executing unnecessary iterations. In this work, we propose a new stopping criterion to identify the undecodable blocks in the early stage of the iterative decoding process. Furthermore, in the conventional BP decoding algorithm, the variable (check) nodes are updated in parallel. It is known that the number of iterations can be reduced by the serial scheduling algorithm. The informed dynamic scheduling (IDS) algorithms were proposed in the existing literatures to further reduce the number of iterations. However, the computational complexity involved in finding the update node in the existing IDS algorithms would not be neglected. In this work, we propose a new efficient IDS scheme which can provide better performance-complexity trade-off compared to the existing IDS ones. In addition, the iterative decoding threshold, which is used for differentiating which LDPC code is better, is investigated in this work. A family of LDPC codes, called LDPC convolutional codes, has drawn a lot of attentions from researchers in recent years due to the threshold saturation phenomenon. The IDT for an LDPC convolutional code may be computationally demanding when the termination length goes to thousand or even approaches infinity, especially for AWGN channels. In this work, we propose a fast IDT estimation algorithm which can greatly reduce the complexity of the IDT calculation for LDPC convolutional codes with arbitrary large termination length (including infinity). By utilizing our new IDT estimation algorithm, the IDTs for LDPC convolutional codes with arbitrary large termination length (including infinity) can be quickly obtained

LDPC Codes with Local and Global Decoding

This paper presents a theoretical study of a new type of LDPC codes motivated by practical storage applications. LDPCL codes (suffix L represents locality) are LDPC codes that can be decoded either as usual over the full code block, or locally when a smaller sub-block is accessed (to reduce latency). LDPCL codes are designed to maximize the error-correction performance vs. rate in the usual (global) mode, while at the same time providing a certain performance in the local mode. We develop a theoretical framework for the design of LDPCL codes. Our results include a design tool to construct an LDPC code with two data-protection levels: local and global. We derive theoretical results supporting this tool and we show how to achieve capacity with it. A trade-off between the gap to capacity and the number of full-block accesses is studied, and a finite-length analysis of ML decoding is performed to exemplify a trade-off between the locality capability and the full-block error-correcting capability.Comment: 41 page

Iterative graphical algorithms for phase noise channels.

Doctoral Degree. University of KwaZulu-Natal, Durban.This thesis proposes algorithms based on graphical models to detect signals and charac- terise the performance of communication systems in the presence of Wiener phase noise. The algorithms exploit properties of phase noise and consequently use graphical models to develop low complexity approaches of signal detection. The contributions are presented in the form of papers. The first paper investigates the effect of message scheduling on the performance of graphical algorithms. A serial message scheduling is proposed for Orthogonal Frequency Division Multiplexing (OFDM) systems in the presence of carrier frequency offset and phase noise. The algorithm is shown to have better convergence compared to non-serial scheduling algorithms. The second paper introduces a concept referred to as circular random variables which is based on exploiting the properties of phase noise. An iterative algorithm is proposed to detect Low Density Parity Check (LDPC) codes in the presence of Wiener phase noise. The proposed algorithm is shown to have similar performance as existing algorithms with very low complexity. The third paper extends the concept of circular variables to detect coherent optical OFDM signals in the presence of residual carrier frequency offset and Wiener phase noise. The proposed iterative algorithm shows a significant improvement in complexity compared to existing algorithms. The fourth paper proposes two methods based on minimising the free energy function of graphical models. The first method combines the Belief Propagation (BP) and the Uniformly Re-weighted BP (URWBP) algorithms. The second method combines the Mean Field (MF) and the URWBP algorithms. The proposed methods are used to detect LDPC codes in Wiener phase noise channels. The proposed methods show good balance between complexity and performance compared to existing methods. The last paper proposes parameter based computation of the information bounds of the Wiener phase noise channel. The proposed methods compute the information lower and upper bounds using parameters of the Gaussian probability density function. The results show that these methods achieve similar performance as existing methods with low complexity