Does the Performance of LDPC Codes Depend on the Channel?

In this letter, we discuss the performance of low-density parity-check (LDPC) codes on memoryless channels. Using a recently proposed analysis technique based on extrinsic information transfer (EXIT) charts, we present an interpretation of the known fact that the bit-error rate (BER) performance of an ensemble of LDPC codes shows little dependence on the specific memoryless channel. This result has been partially observed in the literature for symmetric channels and is here extended to asymmetric channels. We conjecture and demonstrate that the performance of an ensemble of LDPC codes depends primarily and solely on the mutual information (MI) between the input and the output of the channel. As a validation of this conjecture, we compare the performance of a few LDPC codes with various rates for five representative memoryless (both symmetric and asymmetric) channels, obtaining results in excellent agreement with the EXIT chart-based prediction     

Design of multi-input multi-output systems based on low-density Parity-check codes

We design serial concatenated multi-input multi-output systems based on low-density parity-check (LDPC) codes. We employ a receiver structure combining the demapper/detector and the decoder in an iterative fashion. We consider the a posteriori probability (APP) demapper, as well as a suboptimal demapper incorporating interference cancellation with linear filtering. Extrinsic information transfer (EXIT) chart analysis is applied to study the convergence behavior of the proposed schemes. We show that EXIT charts match very well with the simulated decoding trajectories, and they help explain the impact of different mappings and different demappers. It is observed that if the APP demapper transfer characteristics are almost flat, the LDPC codes optimized for binary-input channels are good enough to achieve performance close to the channel capacity. We also present a simple code-optimization method based on EXIT chart analysis, and we design a rate-1/2 LDPC code that achieves very low bit-error rates within 0.15 dB of the capacity of a two-input two-output Rayleigh fading channel with 4-pulse amplitude modulation. We next propose to use a space-time block code as an inner code of our serial concatenated coding scheme. By means of a simple example scheme, using an Alamouti inner code, we demonstrate that the design/optimization of the outer code (e.g., LDPC code) is greatly simplified.     

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.     

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.     

Extrinsic information transfer functions: model and erasure channel properties

Extrinsic information transfer (EXIT) charts are a tool for predicting the convergence behavior of iterative processors for a variety of communication problems. A model is introduced that applies to decoding problems, including the iterative decoding of parallel concatenated (turbo) codes, serially concatenated codes, low-density parity-check (LDPC) codes, and repeat-accumulate (RA) codes. EXIT functions are defined using the model, and several properties of such functions are proved for erasure channels. One property expresses the area under an EXIT function in terms of a conditional entropy. A useful consequence of this result is that the design of capacity-approaching codes reduces to a curve-fitting problem for all the aforementioned codes. A second property relates the EXIT function of a code to its Helleseth-Klove-Levenshtein information functions, and thereby to the support weights of its subcodes. The relation is via a refinement of information functions called split information functions, and via a refinement of support weights called split support weights. Split information functions are used to prove a third property that relates the EXIT function of a linear code to the EXIT function of its dual.     

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。     

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.     

纠删信道中LDPC码EXIT曲线匹配设计

基于单栓码和重复码EXIT图面积特性的分析,介绍了在二元纠删信道中,采用EXIT曲线匹配设计LDPC码度分布序列的最佳性:当内外码的EXIT曲线完全匹配时,所设计的度分布序列对应的速率能够达到信道容量;而当内外码EXIT曲线之间有间隔时,所设计的度分布序列对应的速率严格小于信道容量,并且间隔面积越大,所损失的速率越大.     

Simple and Accurate Design of Low-Density Parity-Check Codes for Multi-Input Multi-Output Systems

In this paper we design an irregular low-density parity-check (LDPC) code for multiple-input multiple-output (MIMO) systems, using a simple extrinsic information transfer (EXIT) chart method. The MIMO systems considered are the optimal maximum a posteriori probability (MAP) detector and the suboptimal minimum mean square error soft-interference cancellation (MMSE-SIC) detector. The MIMO detector and the LDPC decoder exchange soft information and form a turbo iterative receiver. The EXIT charts are used to obtain the edge degree distribution of the irregular LDPC code which is optimized for the MIMO detector. It is shown that the performance of the designed LDPC code is better than that of conventional LDPC code which was optimized for either the Additive White Gaussian Noise (AWGN) channel or the MIMO channel without an explicit consideration of the given detector structure.     

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.     

LDPC-coded cooperative relay systems: performance analysis and code design

We treat the problem of designing low-density parity-check (LDPC) codes to approach the capacity of relay channels. We consider an efficient analysis framework that decouples the factor graph (FG) of a B-block transmission into successive partial FGs, each of which denotes a two-block transmission. We develop design methods to find the optimum code ensemble for the partial FG. In particular, we formulate the relay operations and the destination operations as equivalent virtual MISO and MIMO systems, and employ a binary symmetric channel (BSC) model for the relay node output. For AWGN channels, we further develop a Gaussian approximation for the detector output at the destination node. Jointly treating the relay and the destination, we analyze the performance of the LDPC-coded relay system using the extrinsic mutual information transfer(EXIT) chart technique. Furthermore, differential evolution is employed to search for the optimum code ensemble. Our results show that the optimized codes always outperform the regular LDPC codes with a significant gain; in the AWGN case, when Protocol-II is employed and the relay is close to the source, the optimized code performs within 0.1dB to the capacity bound.     

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.     

EXIT-chart properties of the highest-rate LDPC code with desired convergence behavior

We consider uniparametric LDPC decoding schemes, i.e., the class of decoding algorithms for which an extrinsic information transfer (EXIT) chart analysis of the decoder is exact. We treat the general case of code design for a desired convergence behavior and provide necessary conditions and sufficient conditions that the EXIT chart of the maximum rate low-density parity-check code must satisfy. Our results generalize some of the existing results for the binary erasure channel: our results apply to all uniparametric decoding schemes and they apply to any desired convergence behavior.     

Design and analysis of nonbinary LDPC codes for arbitrary discrete-memoryless channels

We present an analysis under the iterative decoding of coset low-density parity-check (LDPC) codes over GF(q), designed for use over arbitrary discrete-memoryless channels (particularly nonbinary and asymmetric channels). We use a random- coset analysis to produce an effect that is similar to output symmetry with binary channels. We show that the random selection of the nonzero elements of the GF(q) parity-check matrix induces a permutation-invariance property on the densities of the decoder messages, which simplifies their analysis and approximation. We generalize several properties, including symmetry and stability from the analysis of binary LDPC codes. We show that under a Gaussian approximation, the entire q-1-dimensional distribution of the vector messages is described by a single scalar parameter (like the distributions of binary LDPC messages). We apply this property to develop extrinsic information transfer (EXIT) charts for our codes. We use appropriately designed signal constellations to obtain substantial shaping gains. Simulation results indicate that our codes outperform multilevel codes at short block lengths. We also present simulation results for the additive white Gaussian noise (AWGN) channel, including results within 0.56 dB of the unrestricted Shannon limit (i.e., not restricted to any signal constellation) at a spectral efficiency of 6 bits/s/Hz.     

Design and Analysis of Successive Decoding With Finite Levels for the Markov Channel

This paper proposes a practical successive decoding scheme with finite levels for the finite-state Markov channels (FSMCs) where there is no a priori state information at the transmitter or the receiver. The design employs either a random interleaver or a deterministic interleaver with an irregular pattern and an optional iterative estimation and decoding procedure within each level. The interleaver design criteria may be the achievable rate or the extrinsic information transfer (EXIT) chart, depending on the receiver type. For random interleavers, the optimization problem is solved efficiently using a pilot-utility function, while for deterministic interleavers, a good construction is given using empirical rules. Simulation results demonstrate that the new successive decoding scheme combined with irregular low-density parity-check (LDPC) codes can approach the identically and uniformly distributed (i.u.d.) input capacity on the Markov-fading channel using only a few levels.     

DVB-S2中LDPC码的阈值分析

DVB-S2的核心编码LDPC码采用不同的码率来适应不同的信道条件,阈值分析是信道条件和码率选择的关键,介绍了用EXIT曲线图法来进行阈值分析,该方法比密度进化法计算量大大减少,使得阈值分析变得更加简单,还介绍了可变码率信道编码在DVB中的应用.     

Surrogate-channel design of universal LDPC codes

A universal code is a code that may be used across a number of different channel types or conditions with little degradation relative to a good single-channel code. The explicit design of universal codes, which simultaneously seeks to solve a multitude of optimization problems, is a daunting task. This letter shows that a single channel may be used as a surrogate for an entire set of channels to produce good universal LDPC codes. This result suggests that sometimes a channel for which LDPC code design is simple may be used as a surrogate for a channel for which LDPC code design is complex. We explore here the universality of LDPC codes over the BEC, AWGN, and flat Rayleigh fading channels in terms of decoding threshold performance. Using excess mutual information as a performance metric, we present design results which support the contention that an LDPC code designed for a single channel can be universally good across the three channels.     

一种用于BICM-ID系统的新颖8APSK映射方案

为了能够通过高阶调制信号增加信道容量,提高编码增益和频谱效率,对8阶振幅移相键控(APSK)星座映射方案进行优化.基于欧氏距离设计准则提出一种新颖(2,6)-scheme 8APSK映射方案,并应用于联合准循环构造法构造的低密度奇偶校验(LDPC)(4599,4307)码的比特交织编码调制迭代译码(BICM-ID)系统中.信道容量仿真表明,所提方案在高、低信噪比区域都具有非常优越的互信息性能.误码率(BER)性能仿真表明,在BER为10-7时,联合LDPC(4599,4307)码的(2,6)-scheme 8APSK映射方案较(4,4)-scheme 8APSK映射方案、8PSK调制的格雷(Gray)映射、集分割(SP)映射、半集分割(SSP)映射分别提高了约0.45 dB、1.10 dB、1.62 dB、2.13 dB的编码增益.外附信息转移(EXIT)图仿真说明,所提方案能够更早地打开译码通道,从而更早地通过迭代来实现无错译码.     

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.     

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.