非规则LDPC码的不等错误保护性能研究

提出了一种具有不等错误保护性能的非规则低密度校验(LDPC,low-density parity-check)码信道编码方案, 构造了重量递增校验(weight-increasing parity-check)矩阵,系统编码时,重要信息比特映射到LDPC码的“精华”比特上。AWGN和Rayliegh衰落信道的仿真结果表明,与随机构造的非规则LDPC码相比,WICP-LDPC码具有好的UEP性能。     

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     

The design of efficiently-encodable rate-compatible LDPC codes - [transactions papers]

We present a new class of irregular low-density parity-check (LDPC) codes for moderate block lengths (up to a few thousand bits) that are well-suited for rate-compatible puncturing. The proposed codes show good performance under puncturing over a wide range of rates and are suitable for usage in incremental redundancy hybrid-automatic repeat request (ARQ) systems. In addition, these codes are linear-time encodable with simple shift-register circuits. For a block length of 1200 bits the codes outperform optimized irregular LDPC codes and extended irregular repeat-accumulate (eIRA) codes for all puncturing rates 0.6~0.9 (base code performance is almost the same) and are particularly good at high puncturing rates where good puncturing performance has been previously difficult to achieve.     

Correcting capabilities of binary irregular LDPC code under low-complexity iterative decoding algorithm

This paper deals with the irregular binary low-density parity-check (LDPC) codes and two iterative low-complexity decoding algorithms. The first one is the majority error-correcting decoding algorithm, and the second one is iterative erasure-correcting decoding algorithm. The lower bounds on correcting capabilities (the guaranteed corrected error and erasure fraction respectively) of irregular LDPC code under decoding (error and erasure correcting respectively) algorithms with low-complexity were represented. These lower bounds were obtained as a result of analysis of Tanner graph representation of irregular LDPC code. The numerical results, obtained at the end of the paper for proposed lower-bounds achieved similar results for the previously known best lower-bounds for regular LDPC codes and were represented for the first time for the irregular LDPC codes.     

Quasi-cyclic LDPC codes using overlapping matrices and their layered decoders

Quasi-cyclic (QC) low-density parity-check (LDPC) codes have the parity-check matrices consisting of circulant matrices. Since QC LDPC codes whose parity-check matrices consist of only circulant permutation matrices are difficult to support layered decoding and, at the same time, have a good degree distribution with respect to error correcting performance, adopting multi-weight circulant matrices to parity-check matrices is useful but it has not been much researched. In this paper, we propose a new code structure for QC LDPC codes with multi-weight circulant matrices by introducing overlapping matrices. This structure enables a system to operate on dual mode in an efficient manner, that is, a standard QC LDPC code is used when the channel is relatively good and an enhanced QC LDPC code adopting an overlapping matrix is used otherwise. We also propose a new dual mode parallel decoder which supports the layered decoding both for the standard QC LDPC codes and the enhanced QC LDPC codes. Simulation results show that QC LDPC codes with the proposed structure have considerably improved error correcting performance and decoding throughput.     

改进型多元QC-LDPC码的构造及其在PDM-CO-OFDM系统中的应用

位长度相同的多元LDPC(NB-LDPC)码优于相应的二 元LDPC(B-LDPC)码,但是它的实现复杂度相对较高。为了降低NB- LDPC码的实现复杂度,提高系统的编码增益,利用置换多项式的方法对一般多元准循 环LDPC(NB-QC-LDPC)码进行改进,并将改进后的NB-QC-LDPC码应用于基于偏振复用的 相干光正交频分复用(PDM-CO-OFDM)系统中,详细研究了其传输性能。仿真结果表明:基于GF(4) QC-LDPC 编码的系统性能 明显优于相应的B-QC-LDPC编码的系统性能,而且基于改进型GF(4) QC-LDPC编码的 系统与 一般GF(4) QC-LDPC编码的系统相比,其误码性能可改善0.65dB, 频谱效率提高了2.16bit/s/Hz,抑制信道色散能力和运转复杂度也 均得到了改善。     

HIGH-PERFORMANCE SIMPLE-ENCODING GENERATOR-BASED SYSTEMATIC IRREGULAR LDPC CODES AND RESULTED PRODUCT CODES

Low-Density Parity-Check (LDPC) code is one of the most exciting topics among the coding theory community.It is of great importance in both theory and practical communications over noisy channels.The most advantage of LDPC codes is their relatively lower decoding complexity compared with turbo codes,while the disadvantage is its higher encoding complexity.In this paper,a new ap- proach is first proposed to construct high performance irregular systematic LDPC codes based on sparse generator matrix,which can significantly reduce the encoding complexity under the same de- coding complexity as that of regular or irregular LDPC codes defined by traditional sparse parity-check matrix.Then,the proposed generator-based systematic irregular LDPC codes are adopted as con- stituent block codes in rows and columns to design a new kind of product codes family,which also can be interpreted as irregular LDPC codes characterized by graph and thus decoded iteratively.Finally, the performance of the generator-based LDPC codes and the resultant product codes is investigated over an Additive White Gaussian Noise (AWGN) and also compared with the conventional LDPC codes under the same conditions of decoding complexity and channel noise.     

误码条件下LDPC码参数的盲估计

针对非合作信号处理中LDPC码（Low-Density Parity-Check）的盲识别问题,提出了一种容错能力较强的开集识别算法.该算法通过对码字矩阵进行高斯约旦消元找到汉明重量较小的"相关列",并根据"相关列"中所包含的约束关系求得LDPC码的校验向量,然后剔除"相关列"中为"1"位置对应的错误码字.若根据高斯约旦消元求校验向量和剔除错误码字进行迭代无法得到更多校验向量,则对得到的这些校验向量进行稀疏化,再进行译码纠错.最后,综合利用校验向量的求解,错误码字的剔除,校验向量稀疏化,LDPC码译码进行迭代,实现LDPC码校验矩阵的有效重建.仿真结果表明,对于IEEE 802.16e标准中的（576,288）LDPC码,在误比特率为0.0022时,本文算法仍可以达到较好的识别效果.     

Quasi-cyclic LDPC codes for fast encoding

In this correspondence we present a special class of quasi-cyclic low-density parity-check (QC-LDPC) codes, called block-type LDPC (B-LDPC) codes, which have an efficient encoding algorithm due to the simple structure of their parity-check matrices. Since the parity-check matrix of a QC-LDPC code consists of circulant permutation matrices or the zero matrix, the required memory for storing it can be significantly reduced, as compared with randomly constructed LDPC codes. We show that the girth of a QC-LDPC code is upper-bounded by a certain number which is determined by the positions of circulant permutation matrices. The B-LDPC codes are constructed as irregular QC-LDPC codes with parity-check matrices of an almost lower triangular form so that they have an efficient encoding algorithm, good noise threshold, and low error floor. Their encoding complexity is linearly scaled regardless of the size of circulant permutation matrices.     

Joint iterative decoding for pragmatic irregular LDPC-coded multi-relay cooperations

In this article, a new kind of pragmatic simple-encoding irregular systematic low-density parity-check (LDPC) code for multi-relay coded cooperation is designed, where the introduced joint iterative decoding is performed in the destination based on a proposed joint Tanner graph for all the constituent LDPC codes used by the source and relays in multi-relay cooperation. The theoretical analysis and numerical results show that the coded cooperations outperform the coded non-cooperation under the same code rate, and also achieve a good trade-off between the performance and the decoding complexity associated with the number of relays. This performance gain can be credited to the additional exchange of extrinsic information from the LDPC codes used by the source and the relays in both ideal and non-ideal cooperations.     

LDPC block and convolutional codes based on circulant matrices

A class of algebraically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes and their convolutional counterparts is presented. The QC codes are described by sparse parity-check matrices comprised of blocks of circulant matrices. The sparse parity-check representation allows for practical graph-based iterative message-passing decoding. Based on the algebraic structure, bounds on the girth and minimum distance of the codes are found, and several possible encoding techniques are described. The performance of the QC LDPC block codes compares favorably with that of randomly constructed LDPC codes for short to moderate block lengths. The performance of the LDPC convolutional codes is superior to that of the QC codes on which they are based; this performance is the limiting performance obtained by increasing the circulant size of the base QC code. Finally, a continuous decoding procedure for the LDPC convolutional codes is described.     

Construction of Near-Optimum Burst Erasure Correcting Low-Density Parity-Check Codes

In this paper, a simple and effective tool for the design of low-density parity-check (LDPC) codes for iterative correction of bursts of erasures is presented. The design method consists of starting from the parity-check matrix of an LDPC code and developing an optimized parity-check matrix, with the same performance over the memoryless erasure channel, and suitable also for the iterative correction of single erasure bursts. The parity-check matrix optimization is performed by an algorithm called pivot searching and swapping (PSS) algorithm. It executes permutations of carefully chosen columns of the parity-check matrix, after a local analysis of particular variable nodes called stopping set pivots. This algorithm can be in principle applied to any LDPC code. If the input parity-check matrix is designed to achieve a good performance over the memoryless erasure channel, then the code obtained after the application of the algorithm provides a good joint correction of independent erasures and single erasure bursts. Numerical results are provided in order to show the algorithm effectiveness when applied to different categories of LDPC codes.     

LDPC码在802.16a协议中的应用

基于LDPC码的优越性能，找出一组优秀的非正则LDPC码应用于IEEE802．16a OFDM环境中，并对其性能进行仿真。仿真结果表明，LDPC码在衰落信道下具有良好的纠错能力，适用于WMAN等采用OFDM的无线通信系统。     

Design of Irregular LDPC Codes for BIAWGN Channels with SNR Mismatch

Belief propagation (BP) algorithm for decoding lowdensity parity-check (LDPC) codes over a binary input additive white Gaussian noise (BIAWGN) channel requires the knowledge of the signal-to-noise ratio (SNR) at the receiver to achieve its ultimate performance. An erroneous estimation or the absence of a perfect knowledge of the SNR at the decoder is referred to as ?SNR mismatch?. SNR mismatch can significantly degrade the performance of LDPC codes decoded by the BP algorithm. In this paper, using extrinsic information transfer (EXIT) charts, we design irregular LDPC codes that perform better (have a lower SNR threshold) in the presence of mismatch compared to the conventionally designed irregular LDPC codes that are optimized for zero mismatch. Considering that min-sum (MS) algorithm is the limit of BP with infinite SNR over-estimation, the EXIT functions generated in this work can also be used for the efficient analysis and design of LDPC codes under the MS algorithm.     

Early detection of successful decoding for dual-diagonal block-based LDPC codes

Low-density parity-check (LDPC) codes [1] have attracted much attention in the last decade owing to their capacityapproaching performance. LDPC codes with a dual-diagonal blockbased structure can be encoded in linear time with lower encoder hardware complexity [2]. This class of LDPC codes is adopted by a number of standards such as wireless LAN (IEEE 802.11n) [3], wireless MAN (IEEE 802.16e, WiMAX) [4] and satellite TV (DVB-S2) [5]. LDPC codes are commonly decoded by the iterative belief-propagation (BP) algorithm. The decoder checks the parity-check equations to detect successful decoding at the end of the iteration. The Tanner graph of an irregular LDPC code consists of nodes with different degrees such that coded bits have unequal error protection [6]. Coded bits associated with higher degree nodes tend to converge to the correct answer more quickly. Hence, in order to give better protection to the transmitted data, data bits are always mapped to higher degree nodes whereas parity bits are mapped to lower degree nodes in the encoding process. The commonly used parity-check equations Hc t ? 0t will be satisfied after all the coded bits are correctly decoded. However, as discussed above, data bits converge to the correct answer much more quickly than parity bits, so some unnecessary iterations are wasted waiting for the parity bits to be decoded. In this Letter, a new set of low-complexity check equations are derived for dual-diagonal block-based LDPC codes. Early detection of successfully decoded data can be achieved by exploiting the structure and degree of distribution of the dual-diagonal parity check matrix. The decoder power, speed and complexity can be improved by adopting these equations. Simulation shows that the coding gain performance is little changed.     

Construction of Regular and Irregular LDPC Codes: Geometry Decomposition and Masking

Two algebraic methods for systematic construction of structured regular and irregular low-density parity-check (LDPC) codes with girth of at least six and good minimum distances are presented. These two methods are based on geometry decomposition and a masking technique. Numerical results show that the codes constructed by these methods perform close to the Shannon limit and as well as random-like LDPC codes. Furthermore, they have low error floors and their iterative decoding converges very fast. The masking technique greatly simplifies the random-like construction of irregular LDPC codes designed on the basis of the degree distributions of their code graphs     

Estimation of Bit and Frame Error Rates of Finite-Length Low-Density Parity-Check Codes on Binary Symmetric Channels

A method for estimating the performance of low-density parity-check (LDPC) codes decoded by hard-decision iterative decoding algorithms on binary symmetric channels (BSCs) is proposed. Based on the enumeration of the smallest weight error patterns that cannot be all corrected by the decoder, this method estimates both the frame error rate (FER) and the bit error rate (BER) of a given LDPC code with very good precision for all crossover probabilities of practical interest. Through a number of examples, we show that the proposed method can be effectively applied to both regular and irregular LDPC codes and to a variety of hard-decision iterative decoding algorithms. Compared with the conventional Monte Carlo simulation, the proposed method has a much smaller computational complexity, particularly for lower error rates.     

Hybrid Hard-Decision Iterative Decoding of Irregular Low-Density Parity-Check Codes

Time-invariant hybrid (Hscr_TI) decoding of irregular low-density parity-check (LDPC) codes is studied. Focusing on Hscr_TI algorithms with majority-based (MB) binary message-passing constituents, we use density evolution (DE) and finite-length simulation to analyze the performance and the convergence properties of these algorithms over (memoryless) binary symmetric channels. To apply DE, we generalize degree distributions to have the irregularity of both the code and the decoding algorithm embedded in them. A tight upper bound on the threshold of MB Hscr_TI algorithms is derived, and it is proven that the asymptotic error probability for these algorithms tends to zero, at least exponentially, with the number of iterations. We devise optimal MB Hscr_TI algorithms for irregular LDPC codes, and show that these algorithms outperform Gallager's algorithm A applied to optimized irregular LDPC codes. We also show that compared to switch-type algorithms, such as Gallager's algorithm B, where a comparable improvement is obtained by switching between different MB algorithms, MB Hscr_TI algorithms are more robust and can better cope with unknown channel conditions, and thus can be practically more attractive     

High-Rate Nonbinary Regular Quasi-Cyclic LDPC Codes for Optical Communications

The parity-check matrix of a nonbinary (NB) low-density parity-check (LDPC) code over Galois field GF(q) is constructed by assigning nonzero elements from GF(q) to the 1s in corresponding binary LDPC code. In this paper, we state and prove a theorem that establishes a necessary and sufficient condition that an NB matrix over GF(q), constructed by assigning nonzero elements from GF(q) to the 1s in the parity-check matrix of a binary quasi-cyclic (QC) LDPC code, must satisfy in order for its null-space to define a nonbinary QC-LDPC (NB-QC-LDPC) code. We also provide a general scheme for constructing NB-QC-LDPC codes along with some other code construction schemes targeting different goals, e.g., a scheme that can be used to construct codes for which the fast-Fourier-transform-based decoding algorithm does not contain any intermediary permutation blocks between bit node processing and check node processing steps. Via Monte Carlo simulations, we demonstrate that NB-QC-LDPC codes can achieve a net effective coding gain of 10.8 dB at an output bit error rate of 10^-12. Due to their structural properties that can be exploited during encoding/decoding and impressive error rate performance, NB-QC-LDPC codes are strong candidates for application in optical communications.     

On construction of rate-compatible low-density Parity-check codes

In this letter, we present a framework for constructing rate-compatible low-density parity-check (LDPC) codes. The codes are linear-time encodable and are constructed from a mother code using puncturing and extending. Application of the proposed construction to a type-II hybrid automatic repeat request (ARQ) scheme with information block length k=1024 and code rates 8/19 to 8/10, using an optimized irregular mother code of rate 8/13, results in a throughput which is only about 0.7 dB away from Shannon limit. This outperforms existing similar schemes based on turbo codes and LDPC codes by up to 0.5 dB.