Complexity versus performance of capacity-achieving irregular repeat-accumulate codes on the binary erasure channel

We derive upper and lower bounds on the encoding and decoding complexity of two capacity-achieving ensembles of irregular repeat-accumulate (IRA1 and IRA2) codes on the binary erasure channel (BEC). These bounds are expressed in terms of the gap between the channel capacity and the rate of a typical code from this ensemble for which reliable communications is achievable under message-passing iterative (MPI) decoding. The complexity of the ensemble of IRA1 codes grows like the negative logarithm of the gap to capacity. On the other hand, the complexity of the ensemble of IRA2 codes with any choice of the degree distribution grows at least like the inverse square root of the gap to capacity, and at most like the inverse of the gap to capacity.     

Structured IRA Codes: Performance Analysis and Construction

In this letter, we present design techniques for structured irregular repeat-accumulate (S-IRA) codes with low error-rate floors. These S-IRA codes need not be quasi-cyclic, permitting flexibility in code dimension, length, and rate. We present a simple ensemble estimate of the level of the error-rate floor of finite-length IRA codes on the additive white Gaussian noise channel. This performance estimate provides guidance on the choice of IRA code column weights which yield low floors. We also present two design algorithms for S-IRA codes accompanied by software- and hardware-based performance results which demonstrate their low floors. Lastly, we present two design algorithms for multirate S-IRA code families implementable by a single encoder/decoder     

Source-optimized irregular repeat accumulate codes with inherent unequal error protection capabilities and their application to scalable image transmission.

The common practice for achieving unequal error protection (UEP) in scalable multimedia communication systems is to design rate-compatible punctured channel codes before computing the UEP rate assignments. This paper proposes a new approach to designing powerful irregular repeat accumulate (IRA) codes that are optimized for the multimedia source and to exploiting the inherent irregularity in IRA codes for UEP. Using the end-to-end distortion due to the first error bit in channel decoding as the cost function, which is readily given by the operational distortion-rate function of embedded source codes, we incorporate this cost function into the channel code design process via density evolution and obtain IRA codes that minimize the average cost function instead of the usual probability of error. Because the resulting IRA codes have inherent UEP capabilities due to irregularity, the new IRA code design effectively integrates channel code optimization and UEP rate assignments, resulting in source-optimized channel coding or joint source-channel coding. We simulate our source-optimized IRA codes for transporting SPIHT-coded images over a binary symmetric channel with crossover probability p. When p = 0.03 and the channel code length is long (e.g., with one codeword for the whole 512 x 512 image), we are able to operate at only 9.38% away from the channel capacity with code length 132380 bits, achieving the best published results in terms of average peak signal-to-noise ratio (PSNR). Compared to conventional IRA code design (that minimizes the probability of error) with the same code rate, the performance gain in average PSNR from using our proposed source-optimized IRA code design is 0.8759 dB when p = 0.1 and the code length is 12800 bits. As predicted by Shannon's separation principle, we observe that this performance gain diminishes as the code length increases.     

Near-Capacity Dirty-Paper Code Design: A Source-Channel Coding Approach

This paper examines near-capacity dirty-paper code designs based on source-channel coding. We first point out that the performance loss in signal-to-noise ratio (SNR) in our code designs can be broken into the sum of the packing loss from channel coding and a modulo loss, which is a function of the granular loss from source coding and the target dirty-paper coding rate (or SNR). We then examine practical designs by combining trellis-coded quantization (TCQ) with both systematic and nonsystematic irregular repeat-accumulate (IRA) codes. Like previous approaches, we exploit the extrinsic information transfer (EXIT) chart technique for capacity-approaching IRA code design; but unlike previous approaches, we emphasize the role of strong source coding to achieve as much granular gain as possible using TCQ. Instead of systematic doping, we employ two relatively shifted TCQ codebooks, where the shift is optimized (via tuning the EXIT charts) to facilitate the IRA code design. Our designs synergistically combine TCQ with IRA codes so that they work together as well as they do individually. By bringing together TCQ (the best quantizer from the source coding community) and EXIT chart-based IRA code designs (the best from the channel coding community), we are able to approach the theoretical limit of dirty-paper coding. For example, at 0.25 bit per symbol (b/s), our best code design (with 2048-state TCQ) performs only 0.630 dB away from the Shannon capacity.     

基于高斯近似的IRA码构造算法研究

介绍了非规则重复累积码(IRA)的结构,对其Tanner图和校验矩阵进行了分析。IRA码的构造分为优化度分布和设计奇偶校验矩阵2部分。在AWGN信道下,对给定的噪声方差,采用高斯近似的方法优化度分布并得出优化结果。根据度分布和相应规则设计奇偶校验矩阵,给出了设计步骤。对设计出的码进行计算机仿真,结果表明这类码相对于计算机随机构造的LDPC码能带来性能上的提高,且随码长增加,码的性能有明显改善。     

Scalable image and video transmission using irregular repeat-accumulate codes with fast algorithm for optimal unequal error protection

This paper considers designing and applying punctured irregular repeat-accumulate (IRA) codes for scalable image and video transmission over binary symmetric channels. IRA codes of different rates are obtained by puncturing the parity bits of a mother IRA code, which uses a systematic encoder. One of the main ideas presented here is the design of the mother code such that the entire set of higher rate codes obtained by puncturing are good. To find a good unequal error protection for embedded bit streams, we employ the fast joint source-channel coding algorithm in Hamzaoui et al. to minimize the expected end-to-end distortion. We test with two scalable image coders (SPIHT and JPEG-2000) and two scalable video coders (3-D SPIHT and H.26L-based PFGS). Simulations show better results with IRA codes than those reported in Banister et al. with JPEG-2000 and turbo codes. The IRA codes proposed here also have lower decoding complexity than the turbo codes used by Banister et al.     

Accumulate-Repeat-Accumulate Codes

In this paper, we propose an innovative channel coding scheme called accumulate-repeat-accumulate (ARA) codes. This class of codes can be viewed as serial turbo-like codes or as a subclass of low-density parity check (LDPC) codes, and they have a projected graph or protograph representation; this allows for high-speed iterative decoding implementation using belief propagation. An ARA code can be viewed as precoded repeat accumulate (RA) code with puncturing or as precoded irregular repeat accumulate (IRA) code, where simply an accumulator is chosen as the precoder. The amount of performance improvement due to the precoder will be called precoding gain. Using density evolution on their associated protographs, we find some rate-1/2 ARA codes, with a maximum variable node degree of 5 for which a minimum bit SNR as low as 0.08 dB from channel capacity threshold is achieved as the block size goes to infinity. Such a low threshold cannot be achieved by RA, IRA, or unstructured irregular LDPC codes with the same constraint on the maximum variable node degree. Furthermore, by puncturing the inner accumulator, we can construct families of higher rate ARA codes with thresholds that stay close to their respective channel capacity thresholds uniformly. Iterative decoding simulation results are provided and compared with turbo codes. In addition to iterative decoding analysis, we analyzed the performance of ARA codes with maximum-likelihood (ML) decoding. By obtaining the weight distribution of these codes and through existing tightest bounds we have shown that the ML SNR threshold of ARA codes also approaches very closely to that of random codes. These codes have better interleaving gain than turbo codes     

IRA码密度进化方法分析与门限判决

针对IRA(非规则重复累积)码的和积译码算法,深入研究了其密度进化方法的原理,并推导了2种密度进化实现算法,即IRA码的离散密度进化和高斯近似算法。在AWGN(加性高斯白噪声)信道中利用2种算法对DVB-S2标准中的IRA码进行门限判决,并对其中的一类码的译码性能进行仿真,为研究和应用DVB-S2标准提供了参考数据。     

Design of Efficiently Encodable Rate-Compatible LDPC Codes Using Vandermonde Extension Matrices

A low-complexity algorithm for the design of efficiently-encodable rate-compatible (RC) low-density parity-check (LDPC) codes by deterministically extending an irregular repeat-accumulate (IRA) is introduced. The extending structure is based on circulants shifted according a truncated Vandermonde matrix (VM) and therefore termed as “extended VM” (eVM). The novel extending algorithm is significantly less computationally complex than other known similar methods since it does not require any optimization of the extending profile or any post-construction girth conditioning. To improve the codes’ properties and correcting capabilities in low code rate applications, the optimal proportions of degree-1 and degree-2 parity bits for the extended nodes are investigated and, in contrast to existing deterministic extending approaches for RC-IRA codes, an extending increment step equal to half the information block length is chosen. Various bit error rate (BER) and frame error rate (FER) have been obtained for different code rates, R, and information block length k ₀ = 512 and 1024 bits considering an additive white Gaussian noise (AWGN) channel. The results have demonstrated that the proposed eVM RC-LDPC codes, despite their very simple structure and very low computational complexity, exhibit excellent performance only slightly inferior to both dedicated IRA and previously known RC-IRA codes for different data block sizes.     

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     

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。     

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     

Improved low-density parity-check codes using irregular graphs

We construct new families of error-correcting codes based on Gallager's (1973) low-density parity-check codes. We improve on Gallager's results by introducing irregular parity-check matrices and a new rigorous analysis of hard-decision decoding of these codes. We also provide efficient methods for finding good irregular structures for such decoding algorithms. Our rigorous analysis based on martingales, our methodology for constructing good irregular codes, and the demonstration that irregular structure improves performance constitute key points of our contribution. We also consider irregular codes under belief propagation. We report the results of experiments testing the efficacy of irregular codes on both binary-symmetric and Gaussian channels. For example, using belief propagation, for rate 1/4 codes on 16000 bits over a binary-symmetric channel, previous low-density parity-check codes can correct up to approximately 16% errors, while our codes correct over 17%. In some cases our results come very close to reported results for turbo codes, suggesting that variations of irregular low density parity-check codes may be able to match or beat turbo code performance     

IRA码在体全息存储系统中的应用研究

在全息存储系统中,噪声分布是不均匀的,为了对存储的信息位进行不等保护,文中采用能纠不均匀错误的IRA码为系统的纠错码。通过高斯估计方法设计出适合全息存储系统的IRA码,并对其在系统中的应用进行了仿真,结果表明全息信道中IRA码的性能要远优于RS码。     

Finite-length rate-compatible LDPC codes: a novel puncturing scheme - [transactions letters]

In this paper, we study rate-compatible puncturing of finite-length low-density parity-check (LDPC) codes. We present a novel rate-compatible puncturing scheme that is easy to implement. Our scheme uses the idea that the degradation in performance is reduced by selecting a puncturing pattern wherein the punctured bits are far apart from each other in the Tanner graph of the code. Although the puncturing scheme presented is tailored to regular codes, it is also directly applicable to irregular parent ensembles. By simulations, the proposed rate-compatible puncturing scheme is shown to be superior to the existing puncturing methods for both regular and irregular LDPC codes over the binary erasure channel (BEC) and the additive white Gaussian noise (AWGN) Channel.     

Extended Hamming accumulate codes and modified irregular repeataccumulate codes

Two classes of codes, called extended Hamming accumulate codes and modified nonsystematic irregular repeat accumulate (IRA) codes, are introduced. Simulation results show that the performance of modified nonsystematic IRA codes are slightly superior to turbo codes of comparable complexity on an additive white Gaussian noise channel     

Design and analysis of simple irregular LDPC codes with finite‐lengths and their application in CDMA systems

In this paper, we design and optimize simple irregular low‐density parity‐check (LDPC) codes for finite‐length applications where the asymptotic noise threshold of the channel cannot play as a dominant optimization factor. Our design procedure is based on some observations resulted from analytical study of these codes. Although we present our design procedure for some specified rates but it can generally be used for any rate. Specifically, we design a simple irregular LDPC code for IS‐95 and compare its performance with the other reported codes 1 - 3 for this application. Our results show a 3.7‐fold increase in the capacity at bit error rate (BER) equal to 10⁻⁵ compared to the low‐rate orthogonal convolutional codes and 1.2 times increase compared to a high performance LDPC code of Reference 3 . Copyright © 2004 John Wiley & Sons, Ltd.     

基于循环中国剩余定理和改进的PEG算法的IRA码

为了避免交织器产生的时延,通过改进的渐进边增长(PEG)算法和循环中国剩余定理构造了一种不规则重复累积(IRA)码.与常规的IRA码相比,提出的码字具有半随机半结构化形式,不需要设计交织器,且码长选择更加灵活.仿真结果显示,在码率为1/2的条件下,当误码率为10-6时,构造的IRA(1 000,500)码与PEG-IRA(1 000,500)码和基于剩余类数对的IRA(1 000,500)码相比,在对应的相同条件下分别取得了0.2 dB和0.1 dB左右的净编码增益提升;且在码率为3/4时,所构造的IRA(16 200,11 880)码比相同码长和码率的DVB-S2标准LDPC码净编码增益提高了约0.1 dB左右.     

On the asymptotic input-output weight distributions of some accumulate-based codes

In this letter, we show how to compute the asymptotic growth rate of input-output weight enumerator (AGR-IOWE) for some accumulate-based codes by using the sharp tools already developed. Numerical results on the AGR-IOWE for irregular repeat-accumulate (IRA) codes, systematic regular RA (SRA) codes, and concatenated zigzag codes are reported. It is observed that the SRA code has the same AGR-IOWE as a comparable concatenated zigzag code. For both SRA and concatenated zigzag codes, if keeping the code rate fixed, the increase of the grouping factor for the component punctured accumulate code may result in better asymptotic performance under maximum-likelihood decoding, but often worse performance under iterative sum-product decoding.     

On CS Image Reconstruction Using LDPC Code Over Radio Mobile Channel

In this work, the effect of additive white Gaussian noise and fading channel on the compressed sensing or compressive sampling (CS) image reconstruction process are demonstrated. First, the work suggests encoding of the sensed samples by low density parity check code (LDPC) before transmission. It is well known that longer is the length of the LDPC codes better (lower) is the bit error rate (BER) performance. Thus to improve CS reconstruction a method to construct a larger length but 4 cycle free irregular LDPC code structure is also proposed. The code construction is based on the LDPC code in IEEE WiMAX standard. The proposed CS-LDPC structure is then extended for \(4^n\)-QAM to find an optimal set of thresholds by minimizing BER (equivalently symbol error rate) using differential evolution (DE). The algorithm works on the log likelihood ratio values obtained by LDPC decoding. Extensive simulation results show the efficacy of the use of LDPC codes and the trade-off in code rate and measurements on reconstruction quality. Improved performance with the proposed DE based demodulation is also demonstrated.