Performance of Belief Propagation for Decoding LDPC Codes in the Presence of Channel Estimation Error

In this paper, we investigate the performance of the belief propagation (BP) algorithm for decoding low-density parity-check codes over the additive white Gaussian noise channel when there is an incorrect estimate of the channel signal-to-noise ratio (SNR) (referred to as "SNR mismatch") at the decoder. At the extremes for over- and underestimation of SNR, the performance of BP tends to that of min-sum algorithm and the channel bit-error rate, respectively. Our results for regular codes indicate that the sensitivity to mismatch increases by increasing the variable-node degree and by decreasing the check-node degree. The effect of variable-node degree, however, appears to be more profound, such that at a given rate, the codes with the smallest variable and check degrees are more robust against SNR mismatch. For irregular codes, by comparing the thresholds of a few ensembles, we demonstrate that the ensemble which performs better in the absence of mismatch can perform worse in the presence of it. To obtain our asymptotic results, we propose a computationally efficient method based on the Gaussian approximation of density evolution in the presence of SNR mismatch. We also show that the asymptotic results are consistent with simulation results for codes with finite block lengths     

基于EXIT图分析的BICM-ID系统SNR估计失配研究

研究了基于LDPC码的BICM-ID系统中,信道的SNR估计失配对于接收机性能的影响.提出了采用EX-IT图的分析方法,比较SNR过估计和欠估计对于解调器以及LDPC码译码器性能的影响,该方法不需要BER性能的仿真,简单直观.仿真结果表明不管是对于解调器还是LDPC码译码器,SNR过估计的影响相对较大,而欠估计则对性能的影响较小.     

LDPC Codes for Flat Rayleigh Fading Channels with Channel Side Information

In this paper, we design capacity approaching lowdensity parity-check (LDPC) codes in the low signal-to-noise ratio (SNR) regime for flat Rayleigh fading channels with channel side information at transmitter and receiver. We use the structure advocated by Caire et al, which uses a single codebook with dynamic power allocation. The extrinsic information transfer (EXIT) function method is used to design the LDPC codes which approach the channel capacities.We also study the EXIT function properties of various demappers.     

Bandwidth-Efficient Modulation Codes Based on Nonbinary Irregular Repeat–Accumulate Codes

Using nonbinary low-density parity-check (LDPC) codes with random-coset mapping, Bennatan and Burshtein constructed bandwidth-efficient modulation codes with remarkable performance under belief propagation (BP) decoding. However, due to the random nature of LDPC codes, most of the good LDPC codes found in the literature do not have a simple encoding structure. Thus, the encoding complexity of those LDPC codes can be as high as O(N ²), where N is the codeword length. To reduce the encoding complexity, in this paper, nonbinary irregular repeat-accumulate (IRA) codes with time-varying characteristic and random-coset mapping are proposed for bandwidth-efficient modulation schemes. The time-varying characteristic and random-coset mapping result in both permutation-invariance and symmetry properties, respectively, in the densities of decoder messages. The permutation-invariance and symmetry properties of the proposed codes enable the approximations of densities of decoder messages using Gaussian distributions. Under the Gaussian approximation, extrinsic information transfer (EXIT) charts for nonbinary IRA codes are developed and several codes of different spectral efficiencies are designed based on EXIT charts. In addition, by proper selection of nonuniform signal constellations, the constructed codes are inherently capable of obtaining shaping gains, even without separate shaping codes. Simulation results indicate that the proposed codes not only have simple encoding schemes, but also have remarkable performance that is even better than that constructed using nonbinary LDPC codes.     

Decoding and Design of LDPC Codes for High-Order Modulations

A concatenated code model is proposed for high-order low-density parity-check (LDPC) coded modulations. A corresponding concatenated-code belief propagation (CCBP) decoding algorithm is derived for our proposed concatenated code. Moreover, the design of LDPC codes under the CCBP decoding is developed using extrinsic information transfer (EXIT) charts. Compared with other algorithms, the CCBP method provides an excellent parallel decoding process, and the EXIT-based design method offers highly accurate LDPC code ensembles. Simulation results show that the performance of the proposed CCBP algorithm is superior to that of the conventional belief propagation decoding within a wide range of modulation orders, and the EXIT-based method can design capacity-approaching LDPC codes for high-order modulations.     

非规则LDPC码在RICE信道中的性能分析

本文对非规则LDPC码在RICE信道的性能进行了分析和仿真,修正了BP译码算法,证明了RICE信道满足对称性,给出了RICE信道译码稳定性条件,推导出了RICE信道的Shannon容量限,采用VC编程对码长N=49512和3072进行了仿真,同时与同码长的Turbo码进行了比较;仿真结果表明LDPC码在码长N＝49512、码率R＝1/3时,与Shannon限相差1dB以内、在低信噪比时其性能优于Turbo码,以及LDPC码本身有很好的交织特性和抗衰落的能力;这进一步表明了LDPC码在包括RICE信道在内的各种信道中的性能都是非常优良的.     

A Measure of the Girth Histogram for Improved Girth-Conditioning Construction of LDPC Codes

In this letter, we present a general quantitative measure of the girth histogram of low-density parity-check (LDPC) codes with joint consideration of the degrees and local girths of symbol nodes. We then propose a further girth-conditioning algorithm to optimize such a measure during the construction of LDPC codes. We show by simulation results that the algorithm improves not only the performance of irregular codes at high signal-to-noise ratios (SNR) considerably, but also the rates of regular codes     

浅海水声信道中原模图LDPC码的设计及性能分析

原模图低密度奇偶校验码相较于传统LDPC码,具有结构简单、迭代译码门限低、易于扩展及线性编译码复杂度等优点。针对强多途、长时延、窄带宽的浅海水声信道,该文研究了PG-LDPC码的设计及性能特征,提出一种码型设计方案,并采用基于原模图度分布的外部信息转移图算法,对所设计PG-LDPC码的纠错性能进行分析及预测。仿真与实验结果表明,与(3,6)随机规则LDPC码相比,所提的PG-LDPC码在低、高信噪比区域均有良好的纠错性能。     

Rice信道下LDPC码密度进化的研究

应用低密度奇偶校验(LDPC)码译码消息的密度进化可以得到码集的噪声门限,依此评价不同译码算法的性能,并可以用来优化非正则LDPC码的次数分布对。该文首先以Rice信道下正则LDPC码为例,讨论了不同量化阶数及步长时BP,BP-based 和offset BP-based 3种译码算法的DDE(Discrete Density Evolution)分析,接着在offset BP-based译码算法的DDE分析基础上,采用差分进化方法对Rice信道下非正则LDPC码的次数分布对进行了优化,得出了相应的噪声门限。最后,给出了Rice信道下码率为1/2的优化非正则LDPC码的概率聚集函数(PMF)进化曲线。     

基于FFT-BP算法的多元域LDPC码译码器设计与FPGA实现

GF(q)域上的LDPC码是二进制LDPC码的扩展,它具有比二进制LDPC码更好的纠错性能。FFT-BP算法是高效的LDPC码译码算法,本文在GF(4)域上探讨该算法的设计与实现。本文的创新之处在于,根据FFT-BP算法的特点设计了一种利用Tanner图进行信息索引的方式,简化了地址查询模块的设计。实验表明,在归一化信噪比为2.6dB时,译码器的误码率可达到10-6。     

具有不等错误保护特性的LDPC 编码调制方案

针对LDPC（low-density parity-check）编码调制系统，提出了一种新的具有不等错误保护特性的调制方案，在一个码字内，利用不同的调制方式对于重要的比特给予较强的保护，对于次要的比特给予较弱的保护，该方案既适用于非规则LDPC码，也适用于规则LDPC码。计算机仿真结果表明，新方案的性能是传统16QAM及4QAM的折衷，当采用1／2码率时，其频带利用率与8PSK相同，但是误码率性能优于8PSK。新方案的性能优于现有文献中基于比特可靠性的调制映射方案，并采用EXIT（extrinsic information transfer）图对新方案的优异性能给出了解释。     

基于EXIT图和自适应微粒群算法的度分布对优化方法

综合EXIT图法和自适应微粒群优化(APSO)算法的优点,该文提出了一种基于EXIT图和APSO算法的非正则LDPC码度分布对优化方法。该方法设计了衡量EXIT曲线匹配程度的全局代价函数,并运用APSO算法对度分布对进行快速迭代优化,迭代过程中不需要固定CND曲线,可以获得EXIT曲线更加匹配的优化度分布对,以及更高的噪声门限。仿真结果表明,该方法在码结构优化方面有着很好的性能,且优化速度较高斯逼近法有了较大提高。     

Optimization of scaling soft information in iterative decoding via density evolution methods

Density evolution has recently been used to analyze iterative decoding and explain many characteristics of iterative decoding including convergence of performance and preferred structures for the constituent codes. The scaling of extrinsic information (messages) has been heuristically used to enhance the performance in the iterative decoding literature, particularly based on the min-sum message passing algorithm. In this paper, it is demonstrated that density evolution can be used to obtain the optimal scaling factor and also estimate the maximum achievable scaling gain. For low density parity check (LDPC (codes and serially) concatenated convolutional codes (SCCC) with two-state constituent codes, the analytic density evolution technique is used, while the signal-to-noise ratio (SNR) evolution technique and the EXIT chart technique is used for SCCC with more than 2 state constituent codes. Simulation results show that the scaling gain predicted by density evolution or SNR evolution matches well with the scaling gain observed by simulation.     

Performance analysis and code optimization of low densityparity-check codes on Rayleigh fading channels

A numerical method has been presented to determine the noise thresholds of low density parity-check (LDPC) codes that employ the message passing decoding algorithm on the additive white Gaussian noise (AWGN) channel. In this paper, we apply the technique to the uncorrelated flat Rayleigh fading channel. Using a nonlinear code optimization technique, we optimize irregular LDPC codes for such a channel. The thresholds of the optimized irregular LDPC codes are very close to the Shannon limit for this channel. For example, at rate one-half, the optimized irregular LDPC code has a threshold only 0.07 dB away from the capacity of the channel. Furthermore, we compare simulated performance of the optimized irregular LDPC codes and turbo codes on a land mobile channel, and the results indicate that at a block size of 3072, irregular LDPC codes can outperform turbo codes over a wide range of mobile speeds     

Design and analysis of E2RC codes

We consider the design and analysis of the efficiently-encodable rate-compatible (E²RC) irregular LDPC codes proposed in previous work. In this work we introduce semi-structured E²RC-like codes and protograph E²RC codes. EXIT chart based methods are developed for the design of semi-structured E²RC-like codes that allow us to determine near-optimal degree distributions for the systematic part of the code while taking into account the structure of the deterministic parity part, thus resolving one of the open issues in the original construction. We develop a fast EXIT function computation method that does not rely on Monte-Carlo simulations and can be used in other scenarios as well. Our approach allows us to jointly optimize code performance across the range of rates under puncturing.We then consider protograph E²RC codes (that have a protograph representation) and propose rules for designing a family of rate-compatible punctured protographs with low thresholds. For both the semi-structured and protograph E²RC families we obtain codes whose gap to capacity is at most 0.3 dB across the range of rates when the maximum variable node degree is twenty.     

Doubly Generalized LDPC Codes over the AWGN Channel

In this paper, the design of doubly generalized low-density parity-check (DGLDPC) codes is proposed. This approach generalizes the structure of LDPC codes at both check and variable nodes. The performance of DGLDPC codes over the AWGN channel is analyzed using EXIT charts. Combined with differential evolution optimization, this analysis provides thresholds for DGLDPC codes that are better than that of LDPC and GLDPC codes with the same maximum variable degree. These theoretical thresholds are verified via simulations. Furthermore DGLDPC codes exhibit a lower error floor compared with their LDPC and GLDPC counterparts.     

Serial concatenation of LDPC codes and differential modulations

In this paper, we consider serially concatenated schemes with outer novel and efficient low-density parity-check (LDPC) codes and inner modulations effective against channel impairments. With a pragmatic approach, we show how to design LDPC codes tailored for simple and robust modulation formats, such as differentially encoded (DE) modulations. The LDPC codes are optimized through the use of a recently proposed analysis technique based on extrinsic information transfer (EXIT) charts. In particular, we optimize, through a "clever" random walk in the parametric space, the degree distributions of the outer LDPC codes, obtaining significant insights on the impact of such distributions on the performance of the proposed concatenated schemes. The optimization is carried out for transmission over both the additive white Gaussian noise channel and a noncoherent channel. The performance predicted by the EXIT chart-based optimization is confirmed by results obtained via computer simulations, considering phase-shift keying and quadrature amplitude modulation at the transmitter side, and iterative demodulation/decoding at the receiver side. The significance of the proposed optimized design of LDPC-coded schemes with DE modulations is validated by the fact that standard nonoptimized LDPC codes perform poorly when used together with inner DE modulations.     

Iterative decoding of one-step majority logic deductible codesbased on belief propagation

Previously, the belief propagation (BP) algorithm has received a lot of attention in the coding community, mostly due to its near-optimum decoding for low-density parity check (LDPC) codes and its connection to turbo decoding. In this paper, we investigate the performance achieved by the BP algorithm for decoding one-step majority logic decodable (OSMLD) codes. The BP algorithm is expressed in terms of likelihood ratios rather than probabilities, as conventionally presented. The proposed algorithm fits better the decoding of OSMLD codes with respect to its numerical stability due to the fact that the weights of their check sums are often much higher than that of the corresponding LDPC codes. Although it has been believed that OSMLD codes are far inferior to LDPC codes, we show that for medium code lengths (say between 200-1000 bits), the BP decoding of OSMLD codes can significantly outperform BP decoding of their equivalent LDPC codes. The reasons for this behavior are elaborated     

一种低码率的LDPC码在LTE-A中的性能研究

本文通过分析LTE-Advanced系统中准循环LDPC码校验矩阵的构造方法,在不改变母码矩阵的基础上,采用一种灵活的扩展方法,构造了一种低码率的LDPC码。采用一种很实用的编码算法和差分译码算法,在MATLAB仿真平台下,比较了这种LDPC码和Turbo码的性能。结果表明：在短码情况下,这种LDPC码在低信噪比下性能略低于Turbo码,但随着信噪比的增加,LDPC码性能优于Turbo码;在长码情况下,LDPC码的性能明显优于Turbo码。为LTE-Advanced系统的信道编解码器的硬件设计提供了一套有效的编译码算法方案,具有较好的实用价值。     

基于EXIT图的Turbo均衡LDPC码性能分析及优化研究

EXIT图是用来分析级联系统中外信息迭代交换的重要工具,本文将Brink的基于EXIT图的优化MIMO信道LDPC码的思想推广到用于Turbo均衡的LDPC码性能分析及码结构设计优化中。文章首先给出了基于互信息的接收机具体分析模型,接着分别讨论了接收机分析模型中两个单元的EXIT曲线:线性MMSE均衡器和VND(variable node decoder)的联合EXIT曲线,CND(check node variable)的EXIT曲线的详细计算步骤。进一步以获得的EXIT曲线为基础,提出了用于Tur- bo均衡的LDPC码的码结构优化算法,EQVAC-EXIT(EQVAC:equalizer VND and CND)算法,算法能够自动地进行码集噪声门限值计算及优化次数分布对的搜索。最后的数值仿真结果表明,(3,6)正则码及优化得到的非正则码的噪声门限值距离Narayanan的结果仅相差0．03dB左右,优化得到的边的次数分布及节点的次数分布与Narayanan的结果相比也很接近。