Irregular Repeat-accumulate Codes Research Articles

Capacity-Achieving MIMO-NOMA: Iterative LMMSE Detection

This paper considers a low-complexity iterative Linear Minimum Mean Square Error (LMMSE) multi-user detector for the Multiple-Input and Multiple-Output system with Non-Orthogonal Multiple Access (MIMO-NOMA), where multiple single-antenna users simultaneously communicate with a multiple-antenna base station (BS). While LMMSE being a linear detector has a low complexity, it has suboptimal performance in multi-user detection scenario due to the mismatch between LMMSE detection and multi-user decoding. Therefore, in this paper, we provide the matching conditions between the detector and decoders for MIMO-NOMA, which are then used to derive the achievable rate of the iterative detection. We prove that a matched iterative LMMSE detector can achieve (i) the optimal capacity of symmetric MIMO-NOMA with any number of users, (ii) the optimal sum capacity of asymmetric MIMO-NOMA with any number of users, (iii) all the maximal extreme points in the capacity region of asymmetric MIMO-NOMA with any number of users, (iv) all points in the capacity region of two-user and three-user asymmetric MIMO-NOMA systems. In addition, a kind of practical low-complexity error-correcting multiuser code, called irregular repeat-accumulate code, is designed to match the LMMSE detector. Numerical results shows that the bit error rate performance of the proposed iterative LMMSE detection outperforms the state-of-art methods and is within 0.8dB from the associated capacity limit.

On the Design of Multi-Dimensional Irregular Repeat-Accumulate Lattice Codes

Most multi-dimensional (more than two dimensions) lattice partitions only form additive quotient groups and lack multiplication operations. This prevents us from constructing lattice codes based on multi-dimensional lattice partitions directly from non-binary linear codes over finite fields. In this paper, we design lattice codes from Construction A lattices where the underlying linear codes are non-binary irregular repeat-accumulate (IRA) codes. Most importantly, our codes are based on multi-dimensional lattice partitions with finite constellations . We propose a novel encoding structure that adds randomly generated lattice sequences to the encoder’s messages, instead of multiplying lattice sequences to the encoder’s messages. We prove that our approach can ensure that the decoder’s messages exhibit permutation-invariance and symmetry properties. With these two properties, the densities of the messages in the iterative decoder can be modeled by Gaussian distributions described by a single parameter. With Gaussian approximation, extrinsic information transfer charts for our multi-dimensional IRA lattice codes are developed and used for analyzing the convergence behavior and optimizing the decoding thresholds. Simulation results show that our codes can approach the unrestricted Shannon limit within 0.46 dB and outperform the previously designed lattice codes with 2-D lattice partitions and existing lattice coding schemes for large codeword length.

A Low-Complexity Multiuser Coding Scheme With Near-Capacity Performance

Nonorthogonal multiuser coding is an essential technique for superimposed transmission and improving spectrum efficiency of future wireless communication networks. In this paper, a low-complexity nonorthogonal coding scheme, called multiuser repetition-aided irregular repeat-accumulate (IRA) code, is proposed for a multiple access channel (MAC), and the scheme can be applied to fifth-generation (5G) nonorthogonal multiple access systems. The main idea is that not only parity checks, which are generated by an IRA encoder, but repetitions as well, are used in each user's codeword to reduce the coding and decoding complexities. Repetition is a simple way to construct a low-rate code and is shown to be beneficial for multiuser decoding iteration. With a deliberate control of the fraction of repetitions in the codeword and a degree distribution optimization of the IRA encoder, near MAC capacity performance is achieved. It is shown that as the number of users increases, more repetitions can be used, and therefore, very low encoding and decoding complexities are required.

Simplified multiuser code design for MIMO–NOMA

Combination of multiple-input-multiple-output and non-orthogonal multiple access (MIMO–NOMA) is a promising multiple-access technology, which can greatly improve spectral efficiency and reduce latency. Among major topics of MIMO–NOMA, an interesting one is how to construct suitable multiuser codes for MIMO–NOMA. In previous works, multiuser codes require to be redesigned when the number of users, transmit antennas, or receive antennas change. Therefore, the previous design methods are too complicated to be applied to MIMO–NOMA. To solve this problem, in this study, the authors first propose a simple multiuser detector (MUD) that detects the signal for each user by regarding the superimposed signal from the other users as interference. Then, based on extrinsic information transfer analysis for the MUD, they propose three criteria to simplify the code design, which copes with the changes of user number and antenna configuration. Moreover, when user number is large, each user requires a low-rate code to overcome the severe multiuser interference. On the basis of the proposed criteria, they design a low-rate repetition-aided irregular repeat-accumulate (Rep-IRA) code for MIMO–NOMA with different numbers of users and antennas, which can achieve low complexity with the aid of repetition.

EXIT Chart-Based IRA Code Design for TDMR Turbo-Equalization System

We present an extrinsic information transfer (EXIT) chart-based design technique for irregular repeat-accumulate (IRA) codes used in 2-D magnetic recording (TDMR) turbo-equalization systems. The channel model includes Voronoi magnetic grains, 2-D intersymbol interference (2D-ISI) and additive white Gaussian noise (AWGN). The receiver uses a 2D-ISI BCJR equalizer and an IRA decoder. For one outer equalizer-decoder iteration, we propose theory and simulation-based methods for computing EXIT curves. The simulation method calculates experimental EXIT curves for the check node decoder (CND) and the combination of the variable node decoder (VND) and an equalizer. The theoretical approach recursively calculates CND and VND Gaussian mixture model parameters in order to calculate EXIT curves. We then fit the VND and CND EXIT curves to find optimized variable node degree distributions. Simulation results show that the TDMR-optimized IRA codes achieve up to a 6.2% density increase in user bits/grain (U/G) compared with IRA codes designed for AWGN channels. The theory-based code designs achieve the same or better U/G as the simulation-based designs, but require 98% less design computation time. We also derive optimized IRA codes for iterative turbo-equalization; these codes can achieve simultaneous U/G gains and SNR savings compared with AWGN-optimized codes.

Partial parallel encoding and algorithmic construction of non-binary structured IRA codes

The non-binary (NB) Irregular Repeat Accumulate (IRA) codes, as a subclass of NB LDPC codes, potentially have an excellent error-correcting performance. They are also known to provide linear complexity of encoding, but the basic encoding method with the serial rate-1 accumulator significantly limits the encoder throughput. Then the objective of the research presented in this paper is to develop an encoding method providing significantly increased throughput of an NB-IRA encoder altogether with a flexible code construction methods for the structured (S-NB-IRA) codes eligible for the proposed encoding method. For this purpose, we reformulate the classic encoding algorithm to fit into the partial parallel encoder architecture. We propose the S-NB-IRA encoder block diagram and show that its estimated throughput is proportional to the submatrix size of the parity check matrix, which guarantees a wide complexity-throughput tradeoff. Then, in order to facilitate the S-NB-IRA coding systems design, we present a computer search algorithm for the construction of good S-NB-IRA codes. The algorithm aims at optimizing the code graph topology along with selecting an appropriate non-binary elements in the parity check matrix. Numerical results show that the constructed S-NB-IRA codes significantly outperform the binary IRA and S-IRA codes, while their performance is similar to the best unstructured NB-LDPC codes.

Achieving the Near-Capacity of Two-Way Relay Channels With Modulation-Coded Physical-Layer Network Coding

We propose and design a practical modulation-coded (MC) physical-layer network coding (PNC) scheme to approach the capacity limits of Gaussian and fading two-way relay channels (TWRCs). In the proposed scheme, an irregular repeat–accumulate (IRA) MC over $\textrm{GF}(q)$ with the same random coset is employed at two users, which directly maps the message sequences into coded PAM or QAM symbol sequences. The relay chooses appropriate network coding coefficients and computes the associated finite-field linear combinations of the two users' message sequences using an iterative belief propagation algorithm. For a symmetric Gaussian TWRC, we show that, by introducing the same random coset vector at the two users and a time-varying accumulator in the IRA code, the MC-PNC scheme exhibits symmetry and permutation-invariant properties for the soft information distribution of the network-coded message sequence (NCMS). We explore these properties in analyzing the convergence behavior of the scheme and optimizing the MC to approach the capacity limit of a TWRC. For a block fading TWRC, we present a new MC linear PNC scheme and an algorithm used at the relay for computing the NCMS. We demonstrate that our developed schemes achieve near-capacity performance in both Gaussian and Rayleigh fading TWRCs. For example, our designed codes over GF(7) and GF(3) with a code rate of 3/4 are within 1 and 1.2 dB of the TWRC capacity, respectively. Our method can be regarded as a practical embodiment of the notion of compute-and-forward with a good nested lattice code, and it can be applied to a wide range of network configurations.

Design of Irregular Repeat Accumulate Codes for Finite Decoder Iterations

This paper deals with the design of non-systematic irregular repeat-accumulate codes that are optimised for a finite number of decoding iterations. In many cases of practical interest, the usual approach for sparse-graph codes, which aims to approach capacity (with arbitrarily many decoder iterations), may not be suitable. This is of particular importance for systems with complexity or delay constraints. In this paper, we provide a design methodology which constrains the number of decoder iterations, as well as other measures of implementation complexity. Our approach uses extrinsic information transfer analysis, and our main contribution is the formulation of code optimisation problems which directly incorporate the number of iterations into the constraints. We focus on the single user binary erasure channel and the two-user binary adder channel, where this transfer analysis is exact. Generalisation to other sparse graph codes and other channels (under usual approximations) is straightforward.

Local-Optimality Guarantees Based on Paths for Optimal Decoding

This paper presents a unified analysis framework that captures recent advances in the study of local-optimality characterizations for codes on graphs. These local-optimality characterizations are based on combinatorial structures embedded in the Tanner graph of the code. Local optimality implies both unique maximum likelihood optimality and unique linear programming (LP) decoding optimality. Also, an iterative message-passing decoding algorithm is guaranteed to find the unique locally optimal codeword if one exists. We demonstrate an instance of this proof technique by considering a definition of local optimality that is based on the simplest combinatorial structures in Tanner graphs, namely, paths of length $h$. We apply the technique of local optimality to binary Tanner codes (including any low-density parity-check code, and in particular any irregular repeat-accumulate code with both even and odd repetition factors). Inverse polynomial bounds in the code length are proved on the word error probability of LP decoding for binary Tanner codes. When the local codes are restricted to single parity-check codes, these bounds also hold for decoding by a certain iterative message-passing decoding algorithm.

QC-IRA-d Codes Based on Circulant Permutation Matrices

This paper presents a new kind of quasi-cyclic irregular repeat accumulate (IRA) codes based on circulant permutation matrices called QC-IRA-rf codes. The presented QC IRA-rf codes have lower encoding complexity than traditional IRA codes and can be decoded in a partially parallel decoder like QC-LDPC codes. Simulation results show that QC-IRA-rf codes have good error-correcting performance.

Power-efficient Irregular Repeat-Accumulate Encoded BICM-ID for 16-ary Signal Constellations

The design of power-efficient forward error correction techniques over additive white Gaussian noise (AWGN) and frequency non-selective fading channels is considered in this work. The aim of this paper is to investigate the application of BICM with iterative demodulation and decoding (BICM-ID) system with irregular repeat-accumulate (IRA) codes to minimize the bit error rate (BER) and determine the most suitable constellation and mapping pairs for designing power-efficient BICM-ID systems over AWGN and Rayleigh fading channels using the concept of channel capacity limit. Simulation results have confirmed that a remarkable BER improvement can be achieved using the proposed design criterion. Based on the IRA-encoded BICM-ID technique considered, the BER improvement gained is exhibited either as an additional coding gain, error floor elimination, or both. The code components include the IRA codes, irregular low-density parity-check (ILDPC) codes and convolutional codes (CC). The modulation technique considered is the widely-used 16-ary constellations with various bit mappings .

Performance Comparisons and Improvements of Channel Coding Techniques for Digital Satellite Broadcasting to Mobile Users

Digital satellite multimedia broadcasting services in Europe are specified in the Mobile Satellite Services (MSS) frequency band by two recently developed standards, namely the European Telecommunications Standards Institute (ETSI) Satellite Digital Radio (SDR) and the Digital Video Broadcasting-Satellite services to Handhelds (DVB-SH) standards. Commercial deployment of operational systems based on these standards is foreseen in the coming years targeting mainly six large European markets. For these standards, a state-of-the-art channel code is deployed based on the 3rd Generation Partnership Project 2 (3GPP2) specifications having a wide range of coding rates. The purpose of this paper is twofold: (i) It studies and compares, for the first time, the Bit Error Rate (BER) performance of the 3GPP2 turbo code used in the aforementioned systems, in order to serve as a benchmark for system design engineers; and (ii) It investigates novel alternative channel coding schemes, including other state-of-the-art turbo code configurations and also Low-Density Parity-Check (LDPC) codes with linear-time encoding, such as the Irregular Repeat-Accumulate (IRA) codes, in order to improve the performance and/or reduce the complexity in future mobile satellite broadcasting systems. Extensive performance evaluation results in the Additive White Gaussian Noise (AWGN) and uncorrelated Rayleigh/Rician fading channels have shown that the use of turbo codes with higher number of states improves performance against the 3GPP2 turbo codes of up to 0.25 dB but the decoding complexity is almost doubled. Furthermore, the use of duo-binary turbo codes reduces decoding latency and makes them more robust in puncturing with performance improvement against the 3GPP2 turbo codes of up to 0.3 dB but performance degrades at very low coding rates. On the other hand, the use of Rate Compatible (RC)-IRA codes results in smaller performance improvement against the 3GPP2 turbo codes, i.e. up to 0.1 dB, but rate compatibility is used to obtain different coding rates with simple puncturing/extending method.

Rate-compatible IRA codes using quadratic congruential extension sequences and puncturing

A novel algorithm for the design of rate-compatible (RC) irregular repeat-accumulate (IRA) codes through deterministic extending based on quadratic congruential extension sequences is introduced. To maintain low-complexity, algebraic operations are used without any post-construction girth conditioning. The proposed extending algorithm is general and can be applied to any IRA mother code of information block length, k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , producing different sequences of code rates, R. Furthermore, a hybrid class of Deterministically Designed RC-IRA (D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> RC-IRA) codes is obtained by combining the proposed extending algorithm with puncturing. Simulation results have shown that for k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> = 1024 and R = 8/24, 8/23, . . . , 8/9, the proposed codes outperform equivalent deterministic coding schemes in both error rate and throughput by up to 0.62 dB and 0.6 dB, respectively.

Capacity-Achieving Codes With Bounded Graphical Complexity and Maximum Likelihood Decoding

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, the existence of capacity-achieving codes for memoryless binary-input output-symmetric (MBIOS) channels under maximum-likelihood (ML) decoding with bounded graphical complexity is investigated. Graphical complexity of a code is defined as the number of edges in the graphical representation of the code per information bit and is proportional to the decoding complexity per information bit per iteration under iterative decoding. Irregular repeat-accumulate (IRA) codes are studied first. Utilizing the asymptotic average weight distribution (AAWD) of these codes and invoking Divsalar's bound on the binary-input additive white Gaussian noise (BIAWGN) channel, it is shown that simple nonsystematic IRA ensembles outperform systematic IRA and regular low-density parity-check (LDPC) ensembles with the same graphical complexity, and are at most 0.124 dB away from the Shannon limit. However, a conclusive result as to whether these nonsystematic IRA codes can really achieve capacity cannot be reached. Motivated by this inconclusive result, a new family of codes is proposed, called low-density parity-check and generator matrix (LDPC-GM) codes, which are serially concatenated codes with an outer LDPC code and an inner low-density generator matrix (LDGM) code. It is shown that these codes can achieve capacity on any MBIOS channel using ML decoding and also achieve capacity on any BEC using belief propagation (BP) decoding, both with bounded graphical complexity. Moreover, it is shown that, under certain conditions, these capacity-achieving codes have linearly increasing minimum distances and achieve the asymptotic Gilbert–Varshamov bound for all rates. </para>

Coded DS-CDMA systems with iterative channel estimation and no pilot symbols

In this paper, we describe direct-sequence code-division multiple-access (DS-CDMA) systems with quadriphase-shift keying in which channel estimation, coherent demodulation, and decoding are iteratively performed without the use of any training or pilot symbols. An expectation-maximization channel-estimation algorithm for the fading amplitude, phase, and the interference power spectral density (PSD) due to the combined interference and thermal noise is proposed for DS-CDMA systems with irregular repeat-accumulate codes. After initial estimates of the fading amplitude, phase, and interference PSD are obtained from the received symbols, subsequent values of these parameters are iteratively updated by using the soft feedback from the channel decoder. The updated estimates are combined with the received symbols and iteratively passed to the decoder. The elimination of pilot symbols simplifies the system design and allows either an enhanced information throughput, an improved bit error rate, or greater spectral efficiency. The interference-PSD estimation enables DS-CDMA systems to significantly suppress interference.

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.

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

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> 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. </para>

Outage Performance and IRA Code Design for MIMO Block Fading With Unitary Scrambling

We consider coded multiple-input-multiple-output (MIMO) systems in block-fading channels employing iterative receivers, where the MIMO channels are scrambled by unitary matrices. We show that the unitary scrambling improves the outage performance with discrete input when the outer code rate is high, i.e., R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> > 0.5. We also demonstrate that the performance improvement from scrambling can be achieved by a small number of random unitary matrices. Moreover, with unitary scrambling, the signal model can be simplified, and the performance of the MIMO demodulator can be characterized by the singular values of the channel matrix. We then consider the design of irregular repeat-accumulate (IRA) codes for unitary scrambled MIMO block-fading channels. We propose a code design method by optimizing the IRA codes for extrinsic information transfer (EXIT) average with a given effective mutual information. We also propose a pessimistic code design for some system settings to improve the frame error rate performance.

Generalized IRA Erasure Correcting Codes for Hybrid Iterative/Maximum Likelihood Decoding

The design of low-density parity-check (LDPC) codes under hybrid iterative / maximum likelihood decoding is addressed for the binary erasure channel (BEC). Specifically, we focus on generalized irregular repeat-accumulate (GeIRA) codes, which offer both efficient encoding and design flexibility. We show that properly designed GeIRA codes tightly approach the performance of an ideal maximum distance separable (MDS) code, even for short block sizes. For example, our (2048,1024) code reaches a codeword error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> at channel erasure probability isin= 0.450, where an ideal (2048,1024) MDS code would reach the same error rate at isin = 0.453.

Lowering the Error Floor of Optimized Short-Block-Length LDPC-Coded OFDM via Spreading

In this paper, short-block-length low-density parity-check (LDPC) codes are optimized and introduced, along with Walsh-Hadamard spreading, to orthogonal frequency-division multiplexing (OFDM) systems. Prior research has shown that lowering the threshold associated with LDPC codes results in higher error floors. The error floor becomes more pronounced for short-block-length LDPC codes that have been proposed for many wireless standards. In this paper, we show that the use of spreading along with code optimization and girth conditioning not only improves the performance but also lowers the error floor for short-length LDPC codes. Density evolution and differential evolution methods are used during the optimization phase of the codes for the frequency-uncorrelated Rayleigh fading channel. Performance results are obtained for extended irregular repeat-accumulate codes and progressive edge-growth algorithm generated codes. Next, the performances of LDPC-coded OFDM and spread OFDM systems in correlated Rayleigh fading channels are presented. A channel statistic, namely the number of faded bits per OFDM symbol, is theoretically derived, and it is demonstrated that the distribution of this channel statistic can be used to determine how close the correlated channel performance is going to be to that of the uncorrelated fading channel.