Graph-based message-passing schedules for decoding LDPC codes

We study a wide range of graph-based message-passing schedules for iterative decoding of low-density parity-check (LDPC) codes. Using the Tanner graph (TG) of the code and for different nodes and edges of the graph, we relate the first iteration in which the corresponding messages deviate from their optimal value (corresponding to a cycle-free graph) to the girths and the lengths of the shortest closed walks in the graph. Using this result, we propose schedules, which are designed based on the distribution of girths and closed walks in the TG of the code, and categorize them as node based versus edge based, unidirectional versus bidirectional, and deterministic versus probabilistic. These schedules, in some cases, outperform the previously known schedules, and in other cases, provide less complex alternatives with more or less the same performance. The performance/complexity tradeoff and the best choice of schedule appear to depend not only on the girth and closed-walk distributions of the TG, but also on the iterative decoding algorithm and channel characteristics. We examine the application of schedules to belief propagation (sum-product) over additive white Gaussian noise (AWGN) and Rayleigh fading channels, min-sum (max-sum) over an AWGN channel, and Gallager's algorithm A over a binary symmetric channel.     

Joint message-passing decoding of LDPC codes and partial-responsechannels

Ideas of message passing are applied to the problem of removing the effects of intersymbol interference (ISI) from partial-response channels. Both bit-based and state-based parallel message-passing algorithms are proposed. For a fixed number of iterations less than the block length, the bit-error rate of the state-based algorithm approaches a nonzero constant as the signal-to-noise ratio (SNR) approaches infinity. This limitation can be removed by using a precoder. It is well known that low-density parity-check (LDPC) codes can be decoded using a message-passing algorithm. Here, a single message-passing detector/decoder matched to the combination of a partial-response channel and an LDPC code is investigated     

Low-complexity decoding of LDPC codes

The offset BP-based algorithm is a reduced-complexity derivative of the belief-propagation algorithm for decoding of low-density parity- check codes. It uses a constant offset term to simplify decoding, but at the cost of performance. A method to obtain a 'variable' offset parameter is presented here, to improve the performance of the algorithm.     

Memory-efficient sum-product decoding of LDPC codes

Low-density parity-check (LDPC) codes perform very close to capacity for long lengths on several channels. However, the amount of memory (fixed-point numbers that need to be stored) required for implementing the message-passing algorithm increases linearly as the number of edges in the graph increases. In this letter, we propose a decoding algorithm for decoding LDPC codes that reduces the memory requirement at the decoder. The proposed decoding algorithm can be analyzed using density evolution; further, we show how to design good LDPC codes using this. Results show that this algorithm provides almost the same performance as the conventional sum-product decoding of LDPC codes.     

一种改进的LDPC码硬判决译码算法

LDPC码的硬判决译码通常是利用比特翻转算法(BF)以及在其基础上改进的加权比特翻转算法(WBF)来实现的,但是前者算法性能较差,而后者的复杂度较高,为了让译码算法能够兼顾其性能和复杂度,针对之前的BF以及WBF算法,提出了一种改进的LDPC码硬判决译码算法,该算法能够在前两次迭代中完成多个比特位的翻转.仿真结果表明,这种改进的算法可以在性能损失较小的条件下,大大降低算法的复杂度,从而提高译码的效率,减轻硬件的负担.     

Computationally efficient decoding of LDPC codes

A computationally efficient algorithm for the decoding of low-density parity check codes is introduced. Instead of updating all bit and check nodes at each decoding iteration, the developed algorithm only updates unreliable check and bit nodes. A simple reliability criteria is developed to determine the active bit and check nodes per decoding iteration. Based on the developed technique, significant computation reductions are achieved with very little or no loss in the BER performance of the LDPC codes. The proposed method can be implemented with a slight modification to the sum-product algorithm with negligible additional hardware complexity.     

Fast decoding of LDPC codes using quantisation

A very fast quantised belief propagation algorithm is proposed for decoding low-density parity-check codes. This algorithm enables flexible quantisation schemes with low complexity and good performance near to or even better than that of continuous decoding. It can lower the error floor at high signal-to-noise ratios     

A decoding algorithm for finite-geometry LDPC codes

In this paper, we develop a new low-complexity algorithm to decode low-density parity-check (LDPC) codes. The developments are oriented specifically toward low-cost, yet effective, decoding of (high-rate) finite-geometry (FG) LDPC codes. The decoding procedure updates iteratively the hard-decision received vector in search of a valid codeword in the vector space. Only one bit is changed in each iteration, and the bit-selection criterion combines the number of failed checks and the reliability of the received bits. Prior knowledge of the signal amplitude and noise power is not required. An optional mechanism to avoid infinite loops in the search is also proposed. Our studies show that the algorithm achieves an appealing tradeoff between performance and complexity for FG-LDPC codes.     

窗口大小自适应的空间耦合LDPC码译码算法

空间耦合低密度奇偶校验（Spatially-Coupled Low-Density Parity-Check,SC-LDPC）码具有接近香农限性能,基于置信传播译码算法,窗口译码（Windowed Decoding,WD）能够获得较小延时的同时也存在一定的局限性。为了进一步提高WD的译码性能,对SC-LDPC码的窗口译码算法提出了提前终止译码和动态调整窗口大小相结合的改进方法。该方法监测窗口大小的动态变化及相应窗口的平均迭代次数,通过加性高斯白噪声信道下的仿真分析,与传统窗口译码相比,其误码率降低,且计算复杂度更低。     

LDPC码的不可检测译码错误分析

纠错性能良好的LDPC码同时具有良好的检错性能。本文介绍了LDPC码的不可检测译码错误概率上限，讨论了不可检测译码错误与码参数的关系，提出对于特定构造的LDPC码，在基于AWGN信道传输和置信传播迭代译码时，可以应用迭代过程中连续消息的密度进化来分析不可检测译码错误概率，并给出了一些短码长高码率情况下的仿真结果。     

LDPC码BP译码算法研究

本文研究了LDPC码的译码。深入研究了概率域BP算法和LLRBP算法。通过计算机仿真,比较了不同码长和不同迭代次数对译码性能的影响。通过计算机仿真得到在LDPC译码过程中,LDPC码的码字越长,构造的校验矩阵中短周期就越短,性能提高就越为明显。同时得到在LDPC译码过程中,迭代次数越多,各节点获得的信息的准确性就越高,性能提高就越为明显。     

An improved decoding algorithm for finite-geometry LDPC codes

In this letter, an improved bit-flipping decoding algorithm for high-rate finite-geometry low-density parity-check (FG-LDPC) codes is proposed. Both improvement in performance and reduction in decoding delay are observed by flipping multiple bits in each iteration. Our studies show that the proposed algorithm achieves an appealing tradeoff between performance and complexity for FG-LDPC codes.     

一种动态加权的LDPC译码方法

为了加快低密度奇偶校验（LDPC）码的译码速度,有效改善LDPC码的译码性能,针对校验节点更新过程中的对数似然比（LLR）值的大小,设计了一种LDPC码的动态加权译码方法。以IEEE 802.16e标准的奇偶校验矩阵为例,根据LLR值的变化规律,利用增长因子和抑制因子对和积译码算法和最小和译码算法进行动态加权。仿真结果显示,基于动态加权的译码方法相对于传统译码方法误码率都有明显改进,译码复杂度也有所降低。     

Two-dimensional correction for min-sum decoding of irregular LDPC codes

Two-dimensional (2-D) correction schemes are proposed to improve the performance of conventional min-sum (MS) decoding of irregular low density parity check codes. An iterative procedure based on parallel differential optimization is presented to obtain the optimal 2-D factors. Both density evolution analysis and simulation show that the proposed method provides a comparable performance as belief propagation (BP) decoding while requiring less complexity. Interestingly, the new method exhibits a lower error floor than that of BP decoding. With respect to conventional MS and 1-D normalized MS decodings, the 2-D normalized MS offers a better performance. The 2-D offset MS decoding exhibits a similar behavior.     

Comments on successive relaxation for decoding of LDPC codes

The application of successive relaxation (SR) to the fixed-point problem associated with the iterative decoding of low-density parity-check (LDPC) codes was proposed by Hemati et al.. The simulation results presented by Hemati et al. for the SR version of belief propagation (BP) in the likelihood ratio (LR) domain and that of min-sum (MS) in the log-likelihood ratio (LLR) domain are based on the assumption of all-zero codeword transmission. This assumption however results in erroneous error rates when SR is applied in the LR domain. Here, we correct the simulation results reported by Hemati et al. for SR-BP in the LR domain. Furthermore, we investigate the performance of SR-BP and SR-MS in the LLR and LR domains, respectively. The results for a binary input additive white Gaussian noise (BIAWGN) channel show that for both BP and MS, the application of SR in the two domains of LR and LLR results in different error correcting performance. In particular, for the tested codes, it is shown that among the four algorithms, SR-MS-LLR has the best performance. It outperforms standard MS and BP by up to about 0.6 dB and 0.3 dB, respectively, offering an attractive solution in terms of performance/complexity tradeoff.     

LDPC码的自适应补偿最小和译码算法

LDPC码置信传播算法由于复杂度过高而无法实际应用,最小和算法虽然能降低复杂度但却带来了较大的性能损失。补偿最小和算法通过在最小和算法中引入固定修正因子,在几乎不增加算法复杂度的条件下获得接近置信传播算法的性能。为了进一步提升补偿最小和算法的性能,给出了补偿最小和算法的自适应修正因子的计算方法并结合层译码调度策略,提出层自适应补偿最小和算法。仿真表明,所提算法具有更优的性能和更快的收敛速度。     

一种适用于大数逻辑可译LDPC码的自适应译码算法

大数逻辑可译低密度奇偶校验(LDPC)码是一类具有较大列重的码,针对此类特殊的LDPC码,提出了一种基于整数可靠度的低复杂度自适应译码算法。在译码的过程中,算法对每个校验节点分别引入不同的自适应修正因子对外信息进行修正。仿真表明提出的自适应译码算法的性能与和积译码算法的性能相当,在误码率(BER)约为10-5时两种算法性能之间仅有0.1 d B的差异。所提算法具有复杂度低、可并行操作、全整数的信息传递等优点,十分有利于工程实现。     

LDPC码的信道自适应迭代译码算法

中短长度LDPC码采用BP迭代译码时,在低信噪比区,大量的经过多次迭代仍不能纠正的错误帧造成了平均迭代次数的大大增加;在中高信噪比区,比特对数似然比值的振荡造成了译码性能的降低。为了减少低信噪比区的平均迭代次数,提高中高信噪比区的译码性能,本文提出了一种LDPC码的信道自适应迭代译码算法。该算法采用基于校验和错误模式的预判决机制和消息加权平均算法,通过低信噪比区预判决机制的主导作用来减少平均迭代次数,通过中高信噪比区消息加权平均算法的主导作用来抑制因比特LLR值振荡而形成的错误帧,从而实现译码算法与信道变化特征的自适应,提高了译码效率。仿真结果显示,相对于BP译码算法,该译码算法在低信噪比区减少了平均迭代次数而译码性能没有显著变化,在中高信噪比区提高了译码性能而平均迭代次数没有明显增加。