Irregular LDPC Research Articles

Performance and Complexity Trade-Off between Short-Length Regular and Irregular LDPC

Performance and Complexity Trade-Off between Short-Length Regular and Irregular LDPC

Nonbinary LDPC Coded QAM Signals With Optimized Mapping: Bounds and Simulation Results

This paper studies specific properties of nonbinary low-density parity-check (NB LDPC) codes when used in coded modulation systems. The paper is focused on the practically important NB LDPC codes over the Galois extension fields GF(2m) with m ≤ 6 used with QAM signaling. Performance of NB LDPC coded transmission strongly depends on the mapping of nonbinary symbols to signal constellation points. We obtain a random coding bound on the maximum-likelihood decoding error probability for an ensemble of random irregular NB LDPC codes used with QAM signaling for specific symbol-to-signal point mappings. This bound is based on the ensemble average squared Euclidean distance spectra derived for these mappings. The simulation results for the belief-propagation decoding in the coded modulation schemes with the NB quasi-cyclic (QC)-LDPC codes under different mappings are given. Comparisons with the optimized binary QC-LDPC codes in the WiFi and 5G standards, as well as with the new bound, are performed.

Trapping and Absorbing Set Enumerators for Irregular Generalized Low-Density Parity-Check Code Ensembles

The definitions of (elementary) trapping and (fully) absorbing sets are extended to generalized low-density parity-check (GLDPC) codes. In particular, the fully absorbing sets are stable under the bit flipping algorithm. The finite-length (elementary) trapping and (fully) absorbing set enumerators for unstructured irregular GLDPC code ensembles are derived using generating functions. An efficient method to evaluate the asymptotic trapping and (fully) absorbing set distributions is presented, which requires solving a system of equations. Using this method, the asymptotic distribution of trapping and (fully) absorbing sets are evaluated for some example GLDPC code ensembles. Experimental results show that the proposed definitions yield graph structures that are harmful for bit flipping decoders. The generating function approach is also used to derive the finite-length and asymptotic trapping and (elementary) absorbing set enumerators for non-binary irregular LDPC code ensembles.

Energy consumption modeling and analysis for short-reach optical transmissions using irregular LDPC codes.

As the transmission rate of data center (DC) related short-reach optical transmissions quickly increases to keep up with internet services' data demand pace, advanced FEC technology starts to establish its key role in short reach optical transmission systems. Coding strategy using Quasi-Cyclic LDPC (QC-LDPC) codes with hard-decision (HD) decoders stands out due to its excellent error correction ability and simple hardware implementation. Meanwhile, with the introduction of green communication concept and increasingly extensive applications of DC, energy consumption of short-reach links has become an urgent problem to be studied. Based on tanner-graph, energy consumption models of regular LDPC codes' HD decoding circuits over AWGN channel have been proposed. In this work, for the first time, we introduce energy consumption of optical transmitters and characteristics of short reach optical channels into energy consumption modeling. Besides, probability distributions of flipping operation as well as bit errors during irregular QC-LDPC codes' HD decoding iterations are analytically derived using density evolution algorithm. Based on our proposed model and post-layout circuit simulation, systems' total power when using four QC-LDPC codes in IEEE and 3GPP with Gallager A/B decoders together with the case of using no FEC are analyzed in two typical scenarios. Numerical results show that, in short range scenario with slight bandwidth limit, uncoded strategy is much more energy-saving than all adopted LDPC-coded strategies in 50Gbaud OOK and 25Gbaud PAM4 transmissions. While in severely band-limited case when FEC is necessary, a tradeoff between energy consumption of transmitters and decoders occurs in 70Gbaud OOK and 35Gbaud PAM4 transmissions. Besides, LDPC codes with longer length at similar rate are proved to consume less power for their better error correction ability. And in long range scenario, power differences among coding strategies grow significantly when pre-bit error rate (BER) approaches their error correction threshold, reflecting the great influence of error correction threshold on the energy consumption of LDPC decoders. Among all adopted coding strategies, a (19200, 15872) LDPC code with Gallager B decoder consumes minimum power in both two scenarios. However, it is also found that decoding algorithm minimizing the total power differs at different pre-BER if we concentrate on decoding algorithm, justifying the significance of our modeling in minimizing system energy consumption.

Design and Analysis of NB QC-LDPC Codes Over Small Alphabets

We propose a novel approach to optimization of irregular nonbinary (NB) quasi-cyclic (QC)-LDPC codes over small alphabets. In this approach, first, the base parity-check matrices are constructed by a simulated annealing method, and then these matrices are labeled by the field elements while maximizing the so- called generalized girth of the Tanner graph. In order to analyze the performance of the constructed irregular NB LDPC codes, a new ensemble of irregular NB LDPC codes over the extensions of the binary Galois field is introduced. A finite-length random coding bound on the error probability of the maximum-likelihood (ML) decoding over the binary phase shift keying (BPSK) input AWGN channel for the new code ensemble is derived. The frame error rate (FER) performance of the sum-product belief-propagation (BP) decoding of the constructed NB QC-LDPC block codes is compared to that of both the optimized binary QC-LDPC block codes in the 5G standard and the best known NB QC-LDPC codes as well as to the derived random coding bound on the ML decoding error probability. It is shown that the obtained bound predicts the behavior of BP decoding performance of practical NB QC-LDPC codes more accurately than the BP decoding thresholds do.

A Number Theoretic Approach to Cycles in LDPC Codes

LDPC codes constructed from permutation matrices have recently attracted the interest of many researchers. A crucial point when dealing with such codes is trying to avoid cycles of short length in the associated Tanner graph, i.e. obtaining a possibly large girth. In this paper, we provide a framework to obtain constructions of such codes. We relate criteria for the existence of cycles of a certain length with some number-theoretic concepts, in particular with the so-called Sidon sets. In this way we obtain examples of LDPC codes with a certain girth. Finally, we extend our constructions to also obtain irregular LDPC codes.

Improved Performance of M-Ary PSK Modulation over AWGN Channel Using Channel Coding Technique

Spectral efficiency improvement with guaranteed communication consistency using channel coding techniques is the direction in which communication systems evolves. This paper presents M-ary PSK modulation performance over AWGN channel for higher M values. In this paper we simulate the performance of M-ary PSK modulation for higher orders of modulation applying irregular LDPC channel coding technique and obtain improved SNR as compared to uncoded M-ary Phase Shift Keying for M values varying from 4 to 4096.

Finite-Length Performance Analysis of LDPC Coded Continuous Phase Modulation

Serial concatenation of LDPC codes and continuous phase modulation (CPM) has recently gained significant attention due to its capacity-approaching performance, efficient detection as well as owing to its constant-envelope nature. Most of the previous contributions on LDPC coded CPM were devoted to the design of LDPC codes and their asymptotic performance analysis. However, there is a paucity of work on the finite-length performance estimation of LDPC coded CPM, primarily because existing performance estimation techniques cannot be readily applied to the LDPC coded CPM. To fill this gap, we conceive an analytical bit error probability estimation technique for finite-length LDPC coded CPM in the waterfall region. Numerical results are provided both for regular and irregular LDPC codes having different codeword lengths, demonstrating that the estimated performances are closely matched by the simulated ones.

Design and Construction of Irregular LDPC Codes for Channels with Synchronization Errors: New Aspect of Degree Profiles

Design and Construction of Irregular LDPC Codes for Channels with Synchronization Errors: New Aspect of Degree Profiles

A Novel Algorithm to Design Rate-Adaptive Irregular LDPC Codes

This paper proposes an algorithm to design rate-adaptive irregular LDPC codes with improved Bit error rate (BER) performance. It focuses on achieving better BER results for higher order coding rates which are in high demand for 5G mobile communications. The algorithm generates a parity check matrix based on the message block size. The encoding complexity of the irregular LDPC code is comparatively reduced due to the proposed parity check matrix. Simulation results compare the BER performances of irregular LDPC codes for various code rates and modulation schemes with the Parity check matrix (PCM) generated by the proposed algorithm. The BER performance of improved irregular LDPC code is analyzed in a real-time scenario using Software Defined Radio platform as well.

Improved EXIT Algorithm Based on Gaussian Mixture Model and its Application to LDPC Construction in Coding Cooperative Systems With Hybrid Fading

In this study, an EXIT algorithm based on Gaussian mixture coding and multi-dimensional node information calculation is proposed. The objective is to solve the problem of performance analysis of cooperative communication and LDPC code construction under hybrid fading and impulse noise. First, Gaussian mixture model estimation of the decoding initialization information is performed by the expectation maximized algorithm. In addition, key parameters, such as weight, mean, and variance of the Gaussian mixture model are obtained. Then, the multi-dimensional information processing is performed on the variable node and the check node information in the traditional EXIT algorithm based on the multi-dimensional Gaussian mixture model and system symmetry analysis. Finally, the objective function of LDPC code optimization is constructed, the constraints are given, and the optimal LDPC code degree distribution is obtained by the differential evolution algorithm. The simulation results show that compared with the traditional regular and other irregular LDPC codes, the newly constructed degree distribution has better performance in waterfall characteristics and error floor characteristics. Such distribution provides a necessary method to analyze and optimize the performance of cooperative coding system under hybrid media or hybrid fading.

Modeling and Analysis of Chaos-based Spread Spectrum Scheme using Irregular LDPC Code and Non-Coherent 16-DCSK under Fading and Jamming

Modeling and Analysis of Chaos-based Spread Spectrum Scheme using Irregular LDPC Code and Non-Coherent 16-DCSK under Fading and Jamming

Design of Irregular SC-LDPC Codes With Non-Uniform Degree Distributions by Linear Programming

In this paper, we propose new design algorithms of irregular spatially-coupled low-density parity-check (SC-LDPC) codes with non-uniform degree distributions using linear programming (LP). In general, irregular SC-LDPC codes with non-uniform degree distributions are difficult to design with low complexity because their density evolution equations are multi-dimensional. To overcome this problem, proposed design algorithms are based on three main ideas: a local design of degree distributions, pre-computation of the input/output message relationship, and selection of a proper objective function. These ideas make it possible to design degree distributions of irregular SC-LDPC codes by solving low-complexity LP problems over the binary erasure channel (BEC). It is shown that the proposed irregular SC-LDPC codes designed by the proposed algorithms are superior to regular SC-LDPC codes in terms of both asymptotic and finite-length performances over the BEC. We also confirm that the proposed irregular SC-LDPC code achieves better performance compared with an optimized irregular block LDPC code in the same blocklength, which implies that the proposed design algorithms also provide a new way to construct capacity-approaching block LDPC codes.

High performance short-block binary regular LDPC codes

LDPC code shows a good performance with long-block codes. However, certain channels are constrained to use short-block codes due to latency. Therefore, concatenated LDPC codes with iterative decoding is a good choice to get a good performance. Concatenated binary LDPC codes were introduced as a class of concatenated codes in which the LDPC codes are irregular codes having different parameters. Although irregular LDPC codes are more efficient than regular codes, irregular LDPC codes have an error floor and a higher encoding complexity than regular code. In this paper, in order to get a good performance/complexity trade-off with a short-block code, we investigate a parallel concatenation of two identical regular binary LDPC codes, using an interleaver. Simulation results show that the proposed code outperforms a single LDPC code. The proposed code needs less time decoding delay than a single LDPC code with the same iteration number. Also, the proposed code needs a less number of iterations to achieve the same performance of a single LDPC code that leads to less decoding complexity and decoding delay. A simplified algorithm, the min-sum algorithm, that is used to decode the component codes shows a small performance loss with respect to the sum-product algorithm.

ILDPC coded RCM scheme with optimised interleaver

In this study, an irregular low-density parity-check (iLDPC) coded rate compatible modulation (iLDPC-RCM) is proposed. A particularly designed interleaver is inserted between LDPC and RCM such that a fast convergence and a low bit-error rate (BER) can be achieved. The combination of strong error correction capability of irregular LDPC code and the seamless adaptation of RCM enables the proposed scheme to achieve a robust and spectrally efficient transmission over time-varying channels. In addition, a joint belief propagation algorithm is proposed to lower the BER of iLDPC-RCM and speed up the convergence of iterative decoding. Simulation results show that the proposed iLDPC-RCM improves the BER and throughput performance while maintaining an acceptable computational complexity at the same time, validating its advantages over the regular LDPC coded RCM with the same coding rate.

Rate-Adaptive Protograph LDPC Codes for Multi-Level-Cell NAND Flash Memory

The multi-level-cell (MLC) NAND flash memory exhibits a diversity of the raw bit error rate (BER) over different program/erase (P/E) cycles and different types of bits within a memory cell. In this letter, we first apply a protograph-based extrinsic information transfer chart analysis to the MLC flash channel and design novel rate-adaptive protograph low-density parity-check (RAP-LDPC) codes by using a code extension approach. The proposed RAP-LDPC code has multiple code rates, which can be adapted dynamically to different P/E cycles. To mitigate the unbalanced raw BERs between different types of bits within a memory cell, we further propose an optimum mapping between the variable nodes of the protograph and different types of bits of the memory cell. Since the proposed RAP-LDPC codes are based on the same parity-check matrix with specific structure, a single protograph encoder/decoder is sufficient to handle all the code rates. Simulation results demonstrate that the proposed RAP-LDPC codes with optimum mapping outperform the irregular LDPC codes for all the code rates with a faster decoding convergence speed for the MLC flash channel.

Analytical Lower Bounds on the Size of Elementary Trapping Sets of Variable-Regular LDPC Codes With Any Girth and Irregular Ones With Girth 8

In this paper, we give lower bounds on the size of $(a,b)$ elementary trapping sets (ETSs) of variable-regular LDPC codes with any girth, $g$ , and irregular ones with girth 8, where $a$ is the size and $b$ is the number of degree-one check nodes. Our proposed lower bounds are analytical, applicable to all values of $g$ and $b$ , based on graph theories and tighter than the existing ones in the literature. Our results mostly depend on the girth. We also propose results, which are independent of the girth and rely on the variables $a$ , $b$ , $\gamma $ , and the column weight value. We obtain the tightest lower bounds on the size of ETSs of variable-regular LDPC codes with girth eight. These results provide us with a chance to present a method to achieve the minimum size of ETSs of irregular LDPC codes with girth, especially those whose column weight values are a subset of {2, 3, 4, 5, 6} and fulfill the inequality $(b/a) . Moreover, we present a range of numerical results about $(a,b)$ ETSs with girths 8 and 10 to compare the tightness between our proposed lower bounds and the existing bounds in the literature.

UEP Constructions of Quasi-Cyclic Low-Density Parity-Check Codes via Masking

In this paper, the algebraic constructions of quasi-cyclic low-density parity-check (QC-LDPC) codes with the unequal error protection (UEP) property are considered. A criterion for constructing such codes via the masking technique is proposed, based on which explicit conditions on the base parity-check matrices and masking matrices to achieve UEP are provided. We also give three specific constructions of UEP QC-LDPC codes. Furthermore, a sufficient condition to ensure strict UEP is presented. Simulation results demonstrate the superiority of our constructed codes over time-sharing schemes. The constructed codes also have competitive error performance against randomly constructed equal-error-protection LDPC codes and irregular UEP LDPC codes designed based on the degree distribution.

Spatially coupled low-density parity-check error correction for holographic data storage

The spatially coupled low-density parity-check (SC-LDPC) was considered for holographic data storage. The superiority of SC-LDPC was studied by simulation. The simulations show that the performance of SC-LDPC depends on the lifting number, and when the lifting number is over 100, SC-LDPC shows better error correctability compared with irregular LDPC. SC-LDPC is applied to the 5:9 modulation code, which is one of the differential codes. The error-free point is near 2.8 dB and over 10−1 can be corrected in simulation. From these simulation results, this error correction code can be applied to actual holographic data storage test equipment. Results showed that 8 × 10−2 can be corrected, furthermore it works effectively and shows good error correctability.

A Hybrid Statistical and Prioritised Unequal Error Protection Scheme for IEEE 802.11n LDPC Codes

The combination of powerful error correcting codes such as (LDPC) codes and Quadrature Amplitude Modulation (QAM) has been widely deployed in wireless communication standards such as the IEEE 802.11n and DVB-T2. Recently, several Unequal Error Protection schemes which exploit non-uniform degree distribution of bit nodes in irregular LDPC codes have been proposed. In parallel, schemes that exploit the inherent UEP characteristics of the QAM constellation have also been developed. In this paper, a hybrid UEP scheme is proposed for LDPC codes with QAM. The scheme uses statistical distribution of source symbols to map the systematic bits of the LDPC encoded symbols to the QAM constellation. Essentially, systematic symbols having highest probabilities of occurrence are mapped onto the low power region of the QAM constellation and those with a low probability of occurrence are mapped onto the high power region. The decrease in overall transmission power allows for an increased spacing between the QAM constellation points. Additionally, the scheme uses the distribution of the bit node degree of the LDPC code-word to map the parity bits having the highest degree onto prioritised QAM constellation points. Simulations with the IEEE 802.11n LDPC codes revealed that the proposed scheme can provide gains of up to 0.91 dB in Eb/No compared with other UEP schemes for a range of Bit Error Rate (BER) values