Iterative Decoding Thresholds Research Articles

Construction of protograph LDPC codes based on the convolution neural network

This paper presents an intelligent protograph construction algorithm. Protograph LDPC codes have shown excellent error correction performance and play an important role in wireless communications. Random search or manual construction are often used to obtain a good protograph, but the efficiency is not high enough and many experience and skills are needed. In this paper, a fast searching algorithm is proposed using the convolution neural network to predict the iterative decoding thresholds of protograph LDPC codes effectively. A special input data transformation rule is applied to provide stronger generalization ability. The proposed algorithm converges faster than other algorithms. The iterative decoding threshold of the constructed proto-graph surpasses greedy algorithm and random search by about 0.53 dB and 0.93 dB respectively under 100 times of density evolution. Simulation results show that quasi-cyclic LDPC (QC-LDPC) codes constructed from the proposed algorithm have competitive performance compared to other papers.

Hybrid Coupled Serially Concatenated Codes

In this paper, we present a class of hybrid coupled serially concatenated codes (HC-SCC), in which coupling is achieved by re-encoding partial outputs from both the outer and the inner encoders. Firstly, we derive the exact density evolution (DE) equations over the binary erasure channels (BECs) for the HC-SCCs, which can be used to analyze the impact of the coupling ratios and the coupling memories on the performances of the HC-SCCs. Secondly, we present a genie-aided lower bound for the proposed HC-SCCs to estimate their error-floors. Thirdly, we use the DE equations to design randomly punctured HC-SCCs, including rate-compatible HC-SCCs. The analytical and the numerical results show that: 1) for comparable coupling memories, the iterative decoding thresholds of the proposed HC-SCC ensembles are better than those of the recently introduced spatially coupled serially concatenated code (SC-SCC) ensembles and the partially parity-coupled turbo code (PPC-TC) ensembles; 2) with outer coupling memory five and inner coupling memory five, the iterative decoding thresholds of the rate-compatible HC-SCCs over the BECs are 0.0003 away from the corresponding channel capacities for a wide range of coding rates; 3) simulation results over the additive white Gaussian noise (AWGN) channels and the BECs are provided to confirm the performance advantages of the HC-SCCs. Particularly, the simulation results show that the HC-SCCs admit lower error-floor than the SC-SCCs and the rate-compatible HC-SCCs perform better than the PPC-TCs, which are consistent with the results of DE analysis.

Spatially Coupled Generalized LDPC Codes: Asymptotic Analysis and Finite Length Scaling

Generalized low-density parity-check (GLDPC) codes are a class of LDPC codes in which the standard single parity check (SPC) constraints are replaced by constraints defined by a linear block code. These stronger constraints typically result in improved error floor performance, due to better minimum distance and trapping set properties, at a cost of some increased decoding complexity. In this paper, we study spatially coupled generalized low-density parity-check (SC-GLDPC) codes and present a comprehensive analysis of these codes, including: (1) an iterative decoding threshold analysis of SC-GLDPC code ensembles demonstrating capacity approaching thresholds via the threshold saturation effect; (2) an asymptotic analysis of the minimum distance and free distance properties of SC-GLDPC code ensembles, demonstrating that the ensembles are asymptotically good; and (3) an analysis of the finite-length scaling behavior of both GLDPC block codes and SC-GLDPC codes based on a peeling decoder (PD) operating on a binary erasure channel (BEC). Results are compared to GLDPC block codes, and the advantages and disadvantages of SC-GLDPC codes are discussed.

Analysis and Design of Partially Information- and Partially Parity-Coupled Turbo Codes

In this paper, we study a class of spatially coupled turbo codes, namely partially information- and partially parity-coupled turbo codes. This class of codes enjoy several advantages such as flexible code rate adjustment by varying the coupling ratio and the encoding and decoding architectures of the underlying component codes can remain unchanged. For this work, we first provide the construction methods for partially coupled turbo codes with coupling memory m and study the corresponding graph models. We then derive the density evolution equations for the corresponding ensembles on the binary erasure channel to precisely compute their iterative decoding thresholds. Rate-compatible designs and their decoding thresholds are also provided, where the coupling and puncturing ratios are jointly optimized to achieve the largest decoding threshold for a given target code rate. Our results show that for a wide range of code rates, the proposed codes attain close-to-capacity performance and the decoding performance improves with increasing the coupling memory. In particular, the proposed partially parity-coupled turbo codes have thresholds within 0.0002 of the BEC capacity for rates ranging from 1/3 to 9/10, yielding an attractive way for constructing rate-compatible capacity-approaching channel codes.

Optimized Uniform Scalar Quantizers of Ternary ADCs for Large-Scale MIMO Communications

Large-scale multiple-input multiple-output (LS-MIMO), a promising technology for the future wireless networks to boost the network capacity, unavoidably faces hardware cost and power consumption issues due to a large number of high-resolution power-hungry analog-to-digital converters (ADCs) at the receiver side. The low-resolution ADCs have recently been proven to be an effective solution to cut down the system cost. This paper studies the LS-MIMO communication systems where three-level ADCs, so-called Ternary ADCs (T-ADCs), are employed to achieve better energy efficiency in some LS-MIMO configurations. The truncation limit of the uniform scalar quantizer of the T-ADCs is optimized to minimize the quantization distortion on the LS-MIMO received signal, and thus its performance approaches that of the well-known Lloyd-Max algorithm while offering lower complexity and better robustness. Furthermore, our proposed solution is independent of the number of antennas. The LS-MIMO systems' performance with the proposed T-ADCs is substantially improved over the conventional 3- σ uniform scalar quantizer in three different performance metrics - achievable rates, iterative decoding thresholds of pragmatic protograph low-density parity-check codes (LDPC), and the frame error rate (FER)/bit error rate (BER). Our energy efficiency investigation reveals that proposed T-ADCs attain better energy efficiency than that of the 1-bit ADCs and 2-bit ADCs when the ratio between the receiving antennas and the transmitting antennas is low and medium as the power consumption of ADCs and the transmission power are better balanced in comparison with one-bit ADCs and two-bit ADCs.

Error Propagation Mitigation in Sliding Window Decoding of Braided Convolutional Codes

We investigate error propagation in sliding window decoding of braided convolutional codes (BCCs). Previous studies of BCCs have focused on iterative decoding thresholds, minimum distance properties, and their bit error rate (BER) performance at small to moderate frame length. Here, we consider a sliding window decoder in the context of large frame length or one that continuously outputs blocks in a streaming fashion. In this case, decoder error propagation, due to the feedback inherent in BCCs, can be a serious problem. To mitigate the effects of error propagation, we propose several schemes: a window extension algorithm where the decoder window size can be extended adaptively, a resynchronization mechanism where we reset the encoder to the initial state, and a retransmission strategy where erroneously decoded blocks are retransmitted. In addition, we introduce a soft BER stopping rule to reduce computational complexity, and the tradeoff between performance and complexity is examined. Simulation results show that, using the proposed window extension algorithm, resynchronization mechanism, and retransmission strategy, the BER performance of BCCs can be improved by up to four orders of magnitude in the signal-to-noise ratio operating range of interest, and the soft BER stopping rule can be employed to reduce computational complexity.

Doubly-Recursive Block Markov Superposition Transmission: A Low-Complexity and Flexible Coding Scheme

In this paper, we introduce a novel method, called doubly-recursive block Markov superposition transmission (DrBMST), to construct high-performance spatially coupled codes. An important characteristic of DrBMST codes is that the degrees of the constraint nodes in their normal graphs are at most three. As a result, DrBMST codes can be decoded with low complexity. We first prove that the error probability of an enlarged DrBMST code ensemble can be made arbitrarily small under windowed maximum-likelihood decoding (MLD) by increasing the decoding window size. This result partially explains the superior performances of DrBMST codes. Then we propose to use the extrinsic information transfer (EXIT) chart analysis to estimate the iterative windowed decoding thresholds of DrBMST codes. The EXIT chart analyses show that, with such a simple structure, DrBMST codes are comparable to BMST codes with large encoding memories in terms of decoding thresholds. Finally, we carry out comparisons to validate the advantages of DrBMST codes in terms of error performances and decoding complexities. In particular, for a decoding latency of 20,000 bits, the DrBMST code performs better than the (4, 8)-regular spatially coupled low-density parity-check (SC-LDPC) code, but with lower computational complexity. Hence, DrBMST codes can be used in communication systems with limited computational resources. In addition, we show that DrBMST can be used to construct multiple-rate codes.

Design of protograph codes for additive white symmetric alpha‐stable noise channels

The protograph low-density parity-check (LDPC) codes possess many attractive properties, such as the low encoding/decoding complexity and better error floor performance, and hence have been successfully applied to different types of communication and data storage channels. In this study, the authors design protograph LDPC codes for communication systems corrupted by the impulsive noise, which are modelled as additive white symmetric alpha-stable noise (AWS α SN) channels. They start by presenting a novel simulation-based protograph extrinsic information transfer analysis to derive the iterative decoding threshold of the protograph codes. By further applying the asymptotic weight distribution analysis, the authors design new protograph codes for the AWS α SN channel. Both theoretical analysis and simulation results demonstrate that the proposed protograph codes can provide better error rate performance than the prior art AR4JA code, the irregular codes optimised for the AWGN channel, as well as the irregular codes optimised for the AWS α SN channel.

Tail-Biting Globally-Coupled LDPC Codes

This paper presents a new type of globally-coupled low-density parity-check (GC-LDPC) codes whose base matrix has a cyclic structure in the global part. Therefore, the resulting codes are referred to as tail-biting GC-LDPC (TB-GC-LDPC) codes. We propose two methods to construct TB GC quasi-cyclic LDPC (TB-GC-QC-LDPC) codes. For the first method, we extract a replicated version of a constructed base matrix and mask it with a designed masking matrix. Compared to the conventional construction methods, this method provides more flexibility in code length for TB-GC-QC-LDPC codes. The second method is based on designing the incidence matrix of a special type of packings. Examples show that the constructed TB-GC-QC-LDPC codes perform well over the additive white Gaussian noise channel (AWGNC) and the binary erasure channel (BEC). The asymptotic performance of TB-GC-LDPC ensembles over BECs are also analyzed by resorting to density evolution. Moreover, numerical results show that TB-GC-LDPC ensembles can achieve better flooding-schedule decoding (FSD) thresholds than the corresponding GC-LDPC ensembles with a similar structure. With sufficient decoding iterations in the global phase, the local/global two-phase iterative decoding (TPD) thresholds of the TB-GC-LDPC ensembles significantly outperform those of the corresponding GC-LDPC ensembles as well.

Code Design Based on Connecting Spatially Coupled Graph Chains

A novel code construction based on spatially coupled low-density parity-check (SC-LDPC) codes is presented. The proposed code ensembles are comprised of several protograph-based chains characterizing individual SC-LDPC codes. We demonstrate that the code ensembles obtained by connecting appropriately chosen individual SC-LDPC code chains at specific points have improved iterative decoding thresholds. In addition, the connected chain ensembles have a smaller decoding complexity required to achieve a specific bit error probability compared to the individual code chains. Moreover, we demonstrate that, like the individual component chains, the proposed constructions have a typical minimum distance that grows linearly with block length. Finally, we show that the improved asymptotic properties of the connected chain ensembles also translate into improved finite length performance.

Quasi-Cyclic Protograph-Based Raptor-Like LDPC Codes for Short Block-Lengths

Protograph-based Raptor-like low-density parity-check codes (PBRL codes) are a family of easily encodable rate-compatible low-density parity-check (LDPC) codes. PBRL codes have an excellent performance across all design rates. Quasi-cyclic (QC) PBRL code families permit high-speed decoder implementations. PBRL codes designed thus far, for both long and short block-lengths, have been based on optimizing the iterative decoding threshold of the protograph of the PBRL family at various design rates. This paper introduces a design method to obtain better QC PBRL code families at short block-lengths (of a few hundred bits) for low frame error rate (FER) requirements. We first select a protomatrix for the highest design rate. To add a new row to lower the rate, we keep all the previously obtained rows of the PBRL protomatrix fixed and select the new row that maximizes an upper bound on the minimum distance of any QC-LDPC code that can be obtained from the protomatrix. The new QC PBRL code families outperform the original PBRL codes at short block-lengths by providing a significantly better low-FER performance. The standard approach to computing the aforementioned upper bounds requires complexity that grows exponentially with the size of the protomatrix. However, we show that the structure of the PBRL protomatrix lets us obtain the upper bounds with complexity that grows only linearly with the size of the PBRL protomatrix. Using the complexity reduction results, we also establish an equivalence between the exhaustive search to design a new row for the PBRL protomatrix according to the new design method and an integer linear program.

Protograph-Based Globally-Coupled LDPC Codes Over the Gaussian Channel With Burst Erasures

This paper presents two types of protograph-based globally-coupled low-density parity-check (GC-LDPC) codes formed by a new edge spreading operation. This operation is called the global edge spreading. The Gaussian approximation (GA) and the protograph-based extrinsic information transfer (P-EXIT) analysis are then generalized over a special type of burst-erasure channels (BuECs). Such channel incorporates both Gaussian noise and burst erasures, and is denoted by the Gaussian channel with burst erasures (BuEC-G). Furthermore, the stability condition for BuECs-G is proved and an edge spreading optimization method is proposed to design the structured GC-LDPC codes by predicting the iterative decoding thresholds of corresponding protographs. Simulation results show that the optimized GC-LDPC codes can achieve better thresholds and error performances than existing well-designed GC-LDPC codes, and provide near-capacity performances over BuECs-G.

Rate‐compatible ARACA codes

A design method for rate-compatible (RC) accumulate repeat accumulate check accumulate (ARACA) codes as protograph-based RC low-density parity-check codes is presented. The proposed RC-ARACA codes provide not only good iterative decoding thresholds with the linear minimum distance growth property but also efficient encoder/decoder structures with low complexities, so that they are appropriate to ultra-reliable and low latency communication services for fifth generation.

Improving the Decoding Threshold of Tailbiting Spatially Coupled LDPC Codes by Energy Shaping

We show how the iterative decoding threshold of tailbiting spatially coupled (SC) low-density parity-check (LDPC) code ensembles can be improved over the binary input additive white Gaussian noise channel by allowing the use of different transmission energies for the codeword bits. We refer to the proposed approach as energy shaping. We focus on the special case where the transmission energy of a bit is selected among two values, and where a contiguous portion of the codeword is transmitted with the largest one. Given these constraints, an optimal energy boosting policy is derived by means of a protograph extrinsic information transfer analysis. We show that the threshold of tailbiting SC-LDPC code ensembles can be made close to that of terminated code ensembles while avoiding the rate loss (due to termination). The analysis is complemented by Monte Carlo simulations, which confirm the viability of the approach.

Accumulate Repeat Accumulate Check Accumulate Codes

In this paper, a novel accumulate-repeat-accumulate-check-accumulate (ARACA) code is proposed as a subclass of protograph-based low-density parity-check (LDPC) codes. The key feature of the proposed ARACA code is represented by the outer connection doping in the protograph. This feature can provide the linear minimum distance growth (LMDG) property at a good iterative decoding threshold while maintaining an efficient encoder structure. The effect of the outer connection doping on the typical minimum distance, the iterative decoding threshold, and the LMDG property is discussed and analyzed by comparing case examples and using the asymptotic protograph ensemble weight enumerator. Some good ARACA code protographs are provided for a wide range of code rates. In addition, an efficient and universal encoding procedure and the corresponding encoder structure are provided for them. The performance of the proposed ARACA code is evaluated and compared with well-known good LDPC codes. The simulation results confirm the superiority of the proposed ARACA codes in terms of encoding complexity and frame error rate performance, especially at low-rates in an ultra-reliable regime.

On Frame Asynchronous Coded Slotted ALOHA: Asymptotic, Finite Length, and Delay Analysis

We consider a frame asynchronous coded slotted ALOHA (FA-CSA) system for uncoordinated multiple access, where users join the system on a slot-by-slot basis according to a Poisson random process, and in contrast to standard frame synchronous CSA (FS-CSA), users are not frame-synchronized. We analyze the performance of FA-CSA in terms of packet loss rate and delay. In particular, we derive the (approximate) density evolution that characterizes the asymptotic performance of FA-CSA when the frame length goes to infinity. We show that, if the receiver can monitor the system before anyone starts transmitting, a boundary effect similar to that of spatially coupled codes occurs, which greatly improves the iterative decoding threshold. Furthermore, we derive tight approximations of the error floor (EF) for the finite frame length regime, based on the probability of occurrence of the most frequent stopping sets. We show that, in general, FA-CSA provides better performance in both the EF and waterfall regions as compared to FS-CSA. Moreover, FA-CSA exhibits better delay properties than FS-CSA.

Convergence Analysis of Iterative Threshold Decoding Process

Today the error correcting codes are present in all the telecom standards, in particular the low density parity check (LDPC) codes. The choice of a good code for a given network is essentially linked to the decoding performance obtained by the bit error rate (BER) curves. This approach requires a significant simulation time proportional to the length of the code, to overcome this problem Exit chart was introduced, as a fast technique to predict the performance of a particular class of codes called Turbo codes. In this paper, we success to apply Exit chart to analyze convergence behavior of iterative threshold decoding of one step majority logic decodable (OSMLD) codes. The iterative decoding process uses a soft-input soft-output threshold decoding algorithm as component decoder. Simulation results for iterative decoding of simple and concatenated codes transmitted over a Gaussian channel have shown that the thresholds obtained are a good indicator of the Bit Error Rate (BER) curves.

Spatially Coupled LDPC Codes Constructed by Parallelly Connecting Multiple Chains

A particular class of spatially coupled low-density parity-check (SC-LDPC) codes are constructed by parallelly connecting multiple different coupled chains. The connection structure is realized by edge exchanges between adjacent chains, which can help these chains support each other and improve the iterative decoding thresholds. By varying the number of the connected chains and the degree of each chain, a family of SC-LDPC codes with wide rate range can be obtained. Density evolution analysis shows that the decoding thresholds of the proposed codes are very close to Shannon limits and are slightly better than that constructed by a single chain over the binary erasure channel.

Non-Binary Protograph-Based LDPC Codes: Enumerators, Analysis, and Designs

This paper provides a comprehensive analysis of nonbinary low-density parity check (LDPC) codes built out of protographs. We consider both random and constrained edge-weight labeling, and refer to the former as the unconstrained nonbinary protograph-based LDPC codes (U-NBPB codes) and to the latter as the constrained nonbinary protograph-based LDPC codes (C-NBPB codes). Equipped with combinatorial definitions extended to the nonbinary domain, ensemble enumerators of codewords, trapping sets, stopping sets, and pseudocodewords are calculated. The exact enumerators are presented in the finite-length regime, and the corresponding growth rates are calculated in the asymptotic regime. An EXIT chart tool for computing the iterative decoding thresholds of protograph-based LDPC codes is presented, followed by several examples of finite-length U-NBPB and C-NBPB codes with high performance. Throughout this paper, we provide accompanying examples, which demonstrate the advantage of nonbinary protograph-based LDPC codes over their binary counterparts and over random constructions. The results presented in this paper advance the analytical toolbox of nonbinary graph-based codes.

The Design of Rate-Compatible Protograph LDPC Codes

This paper presents a simple yet effective method for designing nested families of LDPC codes. Rate compatible codes are essential for many communication applications, e.g. hybrid automatic repeat request (HARQ) systems, and their design is nontrivial due to the difficulty of simultaneously guaranteeing the quality of several related codes. Puncturing can be used to generate rate-compatible LDPC codes, but it produces a gap to capacity that, in practice, often significantly exceeds the gap of the mother code. We propose an alternative method based on successively extending a high-rate protograph. The resulting codes not only inherit the advantages of protograph codes, namely low encoding complexity and efficient decoding algorithms, but also cover a wide range of rates and have very good performance with iterative decoding thresholds that are within 0.2 dB of their capacity limits.