Irregular Low-density Parity-check Research Articles

Enhancing Design Space Exploration by Extending CPU/GPU Specifications onto FPGAs

The design cycle for complex special-purpose computing systems is extremely costly and time-consuming. It involves a multiparametric design space exploration for optimization, followed by design verification. Designers of special purpose VLSI implementations often need to explore parameters, such as optimal bitwidth and data representation, through time-consuming Monte Carlo simulations. A prominent example of this simulation-based exploration process is the design of decoders for error correcting systems, such as the Low-Density Parity-Check (LDPC) codes adopted by modern communication standards, which involves thousands of Monte Carlo runs for each design point. Currently, high-performance computing offers a wide set of acceleration options that range from multicore CPUs to Graphics Processing Units (GPUs) and Field Programmable Gate Arrays (FPGAs). The exploitation of diverse target architectures is typically associated with developing multiple code versions, often using distinct programming paradigms. In this context, we evaluate the concept of retargeting a single OpenCL program to multiple platforms, thereby significantly reducing design time. A single OpenCL-based parallel kernel is used without modifications or code tuning on multicore CPUs, GPUs, and FPGAs. We use SOpenCL (Silicon to OpenCL), a tool that automatically converts OpenCL kernels to RTL in order to introduce FPGAs as a potential platform to efficiently execute simulations coded in OpenCL. We use LDPC decoding simulations as a case study. Experimental results were obtained by testing a variety of regular and irregular LDPC codes that range from short/medium (e.g., 8,000 bit) to long length (e.g., 64,800 bit) DVB-S2 codes. We observe that, depending on the design parameters to be simulated, on the dimension and phase of the design, the GPU or FPGA may suit different purposes more conveniently, thus providing different acceleration factors over conventional multicore CPUs.

Threshold Saturation for Spatially Coupled LDPC and LDGM Codes on BMS Channels

Spatially-coupled low-density parity-check (LDPC) codes, which were first introduced as LDPC convolutional codes, have been shown to exhibit excellent performance under low-complexity belief-propagation decoding. This phenomenon is now termed threshold saturation via spatial coupling. Spatially-coupled codes have been successfully applied in numerous areas. In particular, it was proven that spatially-coupled regular LDPC codes universally achieve capacity over the class of binary memoryless symmetric (BMS) channels under belief-propagation decoding. Recently, potential functions have been used to simplify threshold saturation proofs for scalar and vector recursions. In this paper, potential functions are used to prove threshold saturation for irregular LDPC and low-density generator-matrix (LDGM) codes on BMS channels, extending the simplified proof technique to BMS channels. The corresponding potential functions are closely related to the average Bethe free entropy of the ensembles in the large-system limit. These functions also appear in statistical physics when the replica method is used to analyze optimal decoding.

A Simple Proof of Maxwell Saturation for Coupled Scalar Recursions

Low-density parity-check (LDPC) convolutional codes (or spatially coupled codes) were recently shown to approach capacity on the binary erasure channel (BEC) and binary-input memoryless symmetric channels. The mechanism behind this spectacular performance is now called threshold saturation via spatial coupling. This new phenomenon is characterized by the belief-propagation threshold of the spatially coupled ensemble increasing to an intrinsic noise threshold defined by the uncoupled system. In this paper, we present a simple proof of threshold saturation that applies to a wide class of coupled scalar recursions. Our approach is based on constructing potential functions for both the coupled and uncoupled recursions. Our results actually show that the fixed point of the coupled recursion is essentially determined by the minimum of the uncoupled potential function and we refer to this phenomenon as Maxwell saturation. A variety of examples are considered including the density-evolution equations for: irregular LDPC codes on the BEC, irregular low-density generator-matrix codes on the BEC, a class of generalized LDPC codes with BCH component codes, the joint iterative decoding of LDPC codes on intersymbol-interference channels with erasure noise, and the compressed sensing of random vectors with independent identically distributed components.

Channel Hardening-Exploiting Message Passing (CHEMP) Receiver in Large-Scale MIMO Systems

In this paper, we propose a multiple-input multiple-output (MIMO) receiver algorithm that exploits channel hardening that occurs in large MIMO channels. Channel hardening refers to the phenomenon where the off-diagonal terms of the H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sup> H matrix become increasingly weaker compared to the diagonal terms as the size of the channel gain matrix H increases. Specifically, we propose a message passing detection (MPD) algorithm which works with the real-valued matched filtered received vector (whose signal term becomes H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> Hx, where x is the transmitted vector), and uses a Gaussian approximation on the off-diagonal terms of the H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> H matrix. We also propose a simple estimation scheme which directly obtains an estimate of H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> H (instead of an estimate of H), which is used as an effective channel estimate in the MPD algorithm. We refer to this receiver as the channel hardening-exploiting message passing (CHEMP) receiver. The proposed CHEMP receiver achieves very good performance in large-scale MIMO systems (e.g., in systems with 16 to 128 uplink users and 128 base station antennas). For the considered large MIMO settings, the complexity of the proposed MPD algorithm is almost the same as or less than that of the minimum mean square error (MMSE) detection. This is because the MPD algorithm does not need a matrix inversion. It also achieves a significantly better performance compared to MMSE and other message passing detection algorithms using MMSE estimate of H. Further, we design optimized irregular low density parity check (LDPC) codes specific to the considered large MIMO channel and the CHEMP receiver through EXIT chart matching. The LDPC codes thus obtained achieve improved coded bit error rate performance compared to off-the-shelf irregular LDPC codes.

GPU Accelerated Belief Propagation Decoding of Non-Binary LDPC Codes with Parallel and Sequential Scheduling

Low-Density Parity-Check (LDPC) codes are very powerful channel coding schemes with a broad range of applications. The existence of low complexity (i.e., linear time) iterative message passing decoders with close to optimum error correction performance is one of the main strengths of LDPC codes. It has been shown that the performance of these decoders can be further enhanced if the LDPC codes are extended to higher order Galois fields, yielding so called non-binary LDPC codes. However, this performance gain comes at the cost of rapidly increasing decoding complexity. To deal with this increased complexity, we present an efficient implementation of a signed-log domain FFT decoder for non-binary irregular LDPC codes which exploits the inherent massive parallelization capabilities of message passing decoders. We employ Nvidia's Compute Unified Device Architecture (CUDA) to incorporate the available processing power of state-of-the-art Graphics Processing Units (GPUs). Furthermore, we present a CUDA implementation of the signed-log domain FFT decoder using the so-called layered update rule, in which check nodes are updated one after another. This sequential updating of nodes has been shown to converge about twice as fast as the traditional flooding scheme. To achieve a high speedup of the layered CUDA implementation, we employ quasi-cyclic non-binary LDPC codes since they allow to update multiple neighboring check nodes in parallel without any performance loss.

Combined Coding Algorithm of LDPC Codes in Bit-Patterned Media Recording Channels

This paper presents a new coding algorithm of irregular low-density parity-check (LDPC) codes. This algorithm consists of two parts of block codes; parity bits was encoded using the magic square based algorithm (MSBA) which suitable for the short block (less than 500 bits) and the data bits, the rest part of all block codes, was encoded using the random number table. This proposed coding algorithm is called ‘Combined Coding Algorithm LDPC codes (CCA-LDPC codes)’ which can be used in bit-patterned media recording (BPMR) channels. The performance of the constructed codes was numerically simulated in BPMR channels model. The obtained performance of the proposed codes was compared with the comparable existing codes. The simulation results show that, at a block length of 4110 bit and code rate of 0.9, the proposed CCA-LDPC codes yields better performance than the existing ones, especially at high signal-to-noise ratio scenario.

Optimized geometric LDPC codes with quasi-cyclic structure

This paper presents methods to the construction of regular and irregular low-density parity-check (LDPC) codes based on Euclidean geometries over the Galois field. Codes constructed by these methods have quasi-cyclic (QC) structure and large girth. By decomposing hyperplanes in Euclidean geometry, the proposed irregular LDPC codes have flexible column/row weights. Therefore, the degree distributions of proposed irregular LDPC codes can be optimized by technologies like the curve fitting in the extrinsic information transfer (EXIT) charts. Simulation results show that the proposed codes perform very well with an iterative decoding over the AWGN channel.

New constructions of LDPC codes based on circulant permutation matrices for AWGN channel

Two new design approaches for constructing Low-Density Parity-Check (LDPC) codes are proposed. One is used to design regular Quasi-Cyclic LDPC (QC-LDPC) codes with girth at least 8. The other is used to design irregular LDPC codes. Both of their parity-check matrices are composed of Circulant Permutation Matrices (CPMs). When iteratively decoded with the Sum-Product Algorithm (SPA), these proposed codes exhibit good performances over the AWGN channel.

High performance error correction for quantum key distribution using polar codes

We study the use of polar codes for both discrete and continuous variables Quantum Key Distribution (QKD). Although very large blocks must be used to obtain the efficiency required by quantum key distribution, and especially continuous variables quantum key distribution, their implementation on generic x86 Central Processing Units (CPUs) is practical. Thanks to recursive decoding, they exhibit excellent decoding speed, much higher than large, irregular Low Density Parity Check (LDPC) codes implemented on similar hardware, and competitive with implementations of the same codes on high-end Graphic Processing Units (GPUs).

Distributed joint source-channel code design for GMAC using irregular LDPC codes

Separate source and channel coding is known to be sub-optimal for communicating correlated sources over a Gaussian multiple access channel (GMAC). This paper presents an approach to designing distributed joint source-channel (DJSC) codes for encoding correlated binary sources over a two-user GMAC, using systematic irregular low-density parity check (LDPC) codes. The degree profile defining the LDPC code is optimized for the joint source probabilities using extrinsic information transfer (EXIT) analysis and linear programming. A key issue addressed is the Gaussian modeling of log-likelihood ratios (LLRs) generated by nodes representing the joint source probabilities in the combined factor graph of the two LDPC codes, referred to as source-channel factor (SCF) nodes. It is shown that the analytical expressions based on additive combining of incoming LLRs, as done in variable nodes and parity check nodes of the graph of a single LDPC code, cannot be used with SCF nodes. To this end, we propose a numerical approach based on Monte-Carlo simulations to fit a Gaussian density to outgoing LLRs from the SCF nodes, which makes the EXIT analysis of the joint decoder tractable. Experimental results are presented which show that LDPC codes designed with the proposed approach outperforms previously reported DJSC codes for GMAC. Furthermore, they demonstrate that when the sources are strongly dependent, the proposed DJSC codes can achieve code rates higher than the theoretical upper-bound for separate source and channel coding.

EXIT-Based Side Information Refinement in Wyner–Ziv Video Coding

The accuracy of the side information (SI) is critical in the performance of distributed video coding algorithms. The SI is typically built at a decoder based on the reconstructed data and on channel coding parity bits transmitted by the encoder. The optimal encoding rate is generally difficult to compute precisely due to the dynamics of video content with varying correlation. Effective methods for the refinement of imprecise SI are therefore important for improved decoding quality. In this paper, we propose to exploit the intrinsic property of channel coding algorithms in Wyner-Ziv video coding. The SI is refined via both the information-plane and the parity-plane bits, which rapidly increases the accuracy of refined SI. We use extrinsic information transfer chart analysis in order to estimate the variations of the mutual information in the iterative decoding. In particular, we characterize mutual information variations for punctured regular and irregular rate-compatible low-density parity-check codes. Tracking the mutual information changes permits to decrease the coding rate of the information and parity bitstreams, while preserving the decoding quality. Simulation results confirm that our method improves on the decoding quality of recent distributed video coding algorithms, especially for high-motion sequences or at high-coding rate regimes.

Comparison of parallelization strategies for min-sum decoding of irregular LDPC codes

Low-Density Parity-Check (LDPC) codes are powerful error correcting codes. LDPC decoders have been implemented as efficient error correction codes on dedicated VLSI hardware architectures in recent years. This paper describes two strategies to parallelize min-sum decoding of irregular LDPC codes. The first implements min-sum LDPC decoders on multicore platforms using OpenMP, while the other uses the Compute Unified Device Architecture (CUDA) to parallelize LDPC decoding on Graphics Processing Units (GPUs). Empirical studies on data with various scales show that the performance of these decoding processes is improved by these parallel strategies and the GPUs provide more efficient, fast implementation decoder.

이중 편파 MIMO를 쓰는 DVB-T2 시스템의 비트 매퍼 성능 분석

초고해상도의 비디오와 다채널 오디오를 통한 극사실적인 방송 서비스를 지향하는 UHDTV (Ultra-High Definition TeleVision)가 차세대 방송 표준으로 논의되고 있으나 기존의 지상파 방송 시스템의 데이터 전송률이 UHDTV 방송 서비스의 요구 전송률에 크게 미치지 못하기 때문에 새로운 지상파 방송 시스템의 개발 또는 현재 시스템의 전송률 증가에 대한 연구가 필요한 상황이다. 이를 위해 기존의 DVB-T2 (Digital Video Broadcasting-2nd generation Terrestrial) 시스템에 고차 성상 변조 및 이중 편파 안테나를 이용한 다중 입출력 시스템을 적용하여 데이터 전송량을 증가시키는 연구가 진행되고 있다. 비정규 LDPC (Low-Density Parity Check) 부호를 사용하는 DVB-T2에 기반한 다중 입출력 시스템의 성능을 최적화하기 위해, 다중 입출력 시스템을 위한 최적의 비트 매퍼 설계에 대한 연구가 필요하다. 그러나 현재까지 진행되어 온 비트 매퍼 설계는 단일 입출력 시스템에 대해 국한되어 있다. 따라서 본 논문에서는 다중 입출력 시스템의 위한 최적의 비트 매퍼를 설계하기 위한 사전 연구로, 이중 편파 다중 입출력 시스템의 VND (Variable Node Degree) 분포를 나타내는 파라미터를 새롭게 정의하고 정의된 파라미터에 따른 시스템의 수신 성능을 분석하였다. The UHDTV system, which provides realistic service with ultra-high definite video and multi-channel audio, has been studied as a next generation broadcasting service. Since the conventional digital terrestrial transmission system is not capable to cover the increased transmission data rate of the UHDTV service, there are great necessity of researches about increase of data rate. Accordingly, the researches has been studied to increase the transmission data rate of the DVB-T2 system using dual-polarized MIMO technique and high order modulation. In order to optimize the MIMO DVB-T2 system where irregular LDPC codes are used, it is necessary to study the design of the bit mapper that matches the LDPC code and QAM symbols in MIMO channel. However, the research related to the design of the bit mapper has been limited to the SISO system. Therefore, this paper defines a new parameter that indicates the VND distribution of MIMO DVB-T2 system and performs the performance analysis according to the parameter which will be helpful for designing a MIMO bit mapper.

Code Design for the Noisy Slepian-Wolf Problem

We consider a noisy Slepian-Wolf problem where two correlated sources are separately encoded (using codes of fixed rate) and transmitted over two independent binary memoryless symmetric channels. The capacity of each channel is characterized by a single parameter that is not known at the transmitter. System performance is evaluated by computing the set of channel parameters for which the system can successfully decode. This set is called the achievable channel parameter region (ACPR). The goal is to design systems whose ACPRs are as large as possible. The main result is the design of irregular low-density parity-check (LDPC) ensembles whose ACPRs are significantly larger than previous designs. Some previous attempts to achieve large ACPRs with LDPC codes failed because systematic codes were used. In this work, we start with systematic encoders but puncture all the systematic bits before transmission. We also show that additional gains are possible using a staggered structure which enables codes optimized for single-user channels to perform well under symmetric channel conditions. The main analysis tool is a generic density-evolution framework for the analysis of joint iterative decoding for this problem.

LDPC-Coded Cooperation with Receive Multi-Antenna and Unknown CSI in the Destination

This paper studies a coded cooperation scheme with receive multi-antenna in the destination. A kind of simple-encoding irregular systematic low-density-parity-check (LDPC) codes is applied to the scheme. For Rayleigh block fading channels, in the unknown channel state information scenario, space-based adaptive least-mean square (LMS) transversal filters are adopted in the destination to realize receive diversity gain. The filtered signals from the source and relay are decoded by a joint "Min-Sum" iterative decoding algorithm. This algorithm agrees with the bilayer Tanner graph that can be used to fully characterize two distinct irregular systematic LDPC codes employed by the source and relay. Simulation results verify the effectiveness of the space-based adaptive LMS transversal filters in the coded cooperation scheme. Meanwhile, theoretical analysis and numerical simulation show that the proposed coded cooperation scheme can well combine cooperation diversity, multi-receive diversity and channel coding gains, and clearly outperforms the coded non-cooperation one under the same conditions.

Asymptotic Distribution of Absorbing Sets and Fully Absorbing Sets for Regular Sparse Code Ensembles

It is well recognized that low-density parity-check (LDPC) codes can suffer from an error floor when decoded iteratively. This performance degradation is often attributed to the class of objects known as trapping sets. Past work has focused on characterizing the distribution of trapping sets for a variety of code ensembles, including regular, irregular and structured LDPC codes. As a subset of the trapping set collection, there exists a class of graphical structures called the absorbing sets. An absorbing set is a combinatorially-defined object; in particular a fully absorbing set is stable under bit-flipping decoding. By construction, there can exist trapping sets that are not stable under such a decoder. As a result, for finite-precision, iterative decoding algorithms used over additive channels, absorbing sets can describe decoding errors more accurately than the broader class of trapping sets. In this paper, we compute the normalized logarithmic asymptotic distributions of absorbing sets and fully absorbing sets, including elementary (fully) absorbing sets. The calculations are based on the trapping set enumeration method proposed by Milenkovic, Soljanin, and Whiting in [1]. We compare distributions of absorbing and trapping sets for representative code parameters of interest, and quantify the (lack of) discrepancies between the two. Good absorbing set properties are implied for known structured LDPC codes, including repeat accumulate codes and protograph-based constructions. Establishing the distribution of fully absorbing sets (especially when the discrepancy with the trapping set distribution is significant) allows one to further refine the estimates of the error rates under bit-flipping and related decoders.

EXIT Chart Based Design of Irregular LDPC Codes for Large-MIMO Systems

In this letter, we characterize the extrinsic information transfer (EXIT) behavior of a factor graph based message passing algorithm for detection in large multiple-input multiple-output (MIMO) systems with tens to hundreds of antennas. The EXIT curves of a joint detection-decoding receiver are obtained for low density parity check (LDPC) codes of given degree distributions. From the obtained EXIT curves, an optimization of the LDPC code degree profiles is carried out to design irregular LDPC codes matched to the large-MIMO channel and joint message passing receiver. With low complexity joint detection-decoding, these codes are shown to perform better than off-the-shelf irregular codes in the literature by about 1 to 1.5 dB at a coded BER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> in 16 × 16, 64 × 64 and 256 × 256 MIMO systems.

Minimum Distance and Trapping Set Analysis of Protograph-Based LDPC Convolutional Codes

Low-density parity-check (LDPC) convolutional codes have been shown to be capable of achieving capacity-approaching performance with iterative message-passing decoding. In the first part of this paper, using asymptotic methods to obtain lower bounds on the free distance to constraint length ratio, we show that several ensembles of regular and irregular LDPC convolutional codes derived from protograph-based LDPC block codes have the property that the free distance grows linearly with respect to the constraint length, i.e., the ensembles are asymptotically good. In particular, we show that the free distance to constraint length ratio of the LDPC convolutional code ensembles exceeds the minimum distance to block length ratio of the corresponding LDPC block code ensembles. A large free distance growth rate indicates that codes drawn from the ensemble should perform well at high signal-to-noise ratios under maximum-likelihood decoding. When suboptimal decoding methods are employed, there are many factors that affect the performance of a code. Recently, it has been shown that so-called trapping sets are a significant factor affecting decoding failures of LDPC codes over the additive white Gaussian noise channel with iterative message-passing decoding. In the second part of this paper, we study the trapping sets of the asymptotically good protograph-based LDPC convolutional codes considered earlier. By extending the theory presented in part one and using similar bounding techniques, we show that the size of the smallest non-empty trapping set grows linearly with the constraint length for these ensembles.

Performance Analysis of Ensemble of Long Irregular LDPC Code over various Channels with Cut off Rate

ABSTRACT A long Irregular LDPC code that performs at rates extremely close to the Shannon capacity has been taken. The code has carefully chosen degree patterns. Simulations has been done with Hard and Soft-decision decoding to compare the performance of this code with the rates 1/2 ,1/3, 1/4 ,2/3 ,2/5, 3/4, 3/5, 4/5, 5/6, 8/9 and 9/10 over various channels like AWGN channel, Rayleigh fading channel and Rician fading channel. The little dependence of BER performance on various channels is explored here along with the conjecture of the concept of computational cutoff rate that represents an upper limit on rate of transmission for practically instrumentable reliable communications . General Terms Orthogonal frequency division multiplexing (OFDM), Quadrature Amplitude Modulation (QAM) Keywords LDPC (Low Density parity check), Upper Bound, Cut off Rate, Bhattacharyya bound, Hard-decision, Soft-decision 1. INTRODUCTION For any channel, the bandwidth, data rate, noise and error rate are related to each other. The greater the bandwidth, the greater will be the cost. All transmission channels of any practical interest are limited in bandwidth due to the constrained physical properties of the transmission medium. For digital data transmission, in order to use bandwidth efficiently it is required to get higher possible data rate at a particular limit of error rate for a given bandwidth and to which the main constraint is the noise. If binary signals are transmitted then the supported data rate will be twice the bandwidth. But, using multilevel signaling the data rate can be increased by a factor of log

Untainted Puncturing for Irregular Low-Density Parity-Check Codes

Puncturing is a well-known coding technique widely used for constructing rate-compatible codes. In this paper, we consider the problem of puncturing low-density parity-check codes and propose a new algorithm for intentional puncturing. The algorithm is based on the puncturing of untainted symbols, i.e. nodes with no punctured symbols within their neighboring set. It is shown that the algorithm proposed here performs better than previous proposals for a range of coding rates and short proportions of punctured symbols.