Binary Input Additive White Gaussian Noise Channels Research Articles

Analysis and Design of QC-LDPC Coded BICM Ensembles Based on RCA Density Evolution

This paper presents density evolution (DE) techniques based on reciprocal channel approximation (RCA) for general bit-interleaved coded modulation (BICM) systems. The original RCA-based DE (RCA-DE) technique is implemented under the assumption of binary-input additive white Gaussian noise channels (BI-AWGNCs). In BICM systems, however, <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-ary modulation schemes are generally used, and they can be regarded as to be a parallel transmission of <inline-formula> <tex-math notation="LaTeX">$\log _{2} M$ </tex-math></inline-formula> bits. Thus, <inline-formula> <tex-math notation="LaTeX">$\log _{2} M$ </tex-math></inline-formula> separate bit-level channels, which are not Gaussian equivalent, need to be considered. In order to extend the conventional RCA-DE technique to BICM systems, we first establish a model of protograph BICM ensembles. Two methods are developed based on the bit error rate (BER) and the BICM capacity in order to find a corresponding BI-AWGNC that models each of the bit-level channels. Using these methods, we implement the protograph RCA-DE technique for BICM systems and show that it achieves an accurate estimation. As a practical application of the proposed RCA-DE method, we design bit interleavers in order to achieve better BICM performance in the 3GPP New Radio (NR) LDPC coding system. Numerical results show that performance gain of up to 0.3 dB is consistently achieved over a wide range of parameter values.

A Multi-Objective Optimization Framework for URLLC With Decoding Complexity Constraints

Stringent constraints on both reliability and latency must be guaranteed in ultra-reliable low-latency communication (URLLC). To fulfill these constraints with computationally constrained receivers, such as low-budget IoT receivers, optimal transmission parameters need to be studied in detail. In this paper, we introduce a multi-objective optimization framework for the optimal design of URLLC in the presence of decoding complexity constraints. We consider transmission of short-blocklength codewords that are encoded with linear block encoders, transmitted over a binary-input AWGN channel, and finally decoded with order-statistics (OS) decoder. We investigate the optimal selection of a transmission rate and power pair, while satisfying the constraints. For this purpose, a multi-objective optimization problem (MOOP) is formulated. Based on the empirical model that accurately quantifies the trade-off between the performance of an OS decoder and its computational complexity, the MOOP is solved and the Pareto boundary is derived. In order to assess the overall performance among several Pareto-optimal transmission pairs, two scalarization methods are investigated. To exemplify the importance of the MOOP, a case study on a battery-powered communication system is provided. It is shown that, compared to the classical fixed rate-power transmissions, the MOOP provides the optimum usage of the battery and increases the energy efficiency of the communication system while maintaining the constraints.

Latency and Reliability Trade-Off With Computational Complexity Constraints: OS Decoders and Generalizations

In this article, we study the problem of latency and reliability trade-off in ultra-reliable low-latency communication (URLLC) in the presence of decoding complexity constraints. We consider linear block encoded codewords transmitted over a binary-input AWGN channel and decoded with order-statistic (OS) decoder. We first investigate the performance of OS decoders as a function of decoding complexity and propose an empirical model that accurately quantifies the corresponding trade-off. Next, a consistent way to compute the aggregate latency for complexity constrained receivers is presented, where the latency due to decoding is also included. It is shown that, with strict latency requirements, decoding latency cannot be neglected in complexity constrained receivers. Next, based on the proposed model, several optimization problems, relevant to the design of URLLC systems, are introduced and solved. It is shown that the decoding time has a drastic effect on the design of URLLC systems when constraints on decoding complexity are considered. Finally, it is also illustrated that the proposed model can closely describe the performance versus complexity trade-off for other candidate coding solutions for URLLC such as tail-biting convolutional codes, polar codes, and low-density parity-check codes.

Increasing Throughput in Wireless Communications by Grouping Similar Quality Bits

This letter proposes and studies a technique for grouping the bits transmitted through a wireless channel into codewords according to their quality (SNR). It proves that splitting the bits into multiple codewords of different rates provides a higher throughput than mixing heterogeneous quality bits into fixed-rate codewords. The letter first analyzes the pros and cons of different mappings of bits and codewords to the available time, frequency, and modulation resources. Then it describes the proposed scheme for a 16-QAM modulation and illustrates its benefits through simulations. Finally, it provides a mathematical proof of its superiority in a binary-input parallel AWGN channel with finite length error correcting codes. The proposed scheme can be applied to any communications channel using error correcting codes (ECC), but it is of particular interest for millimeter-wave (mmWave) wireless communications, where the channel quality is closely monitored and high order modulations are used over wide bandwidths. The simulations suggest that modest gains in throughput can be obtained with negligible additional complexity.

REF Codes: Intermediate Performance Oriented Fountain Codes With Feedback

In this article, a novel class of fountain codes with feedback, called relative-entropy-based fountain (REF) codes, is proposed. The transmitter of REF codes adapts the degrees of encoded symbols to make the degree distribution at the receiver close to the robust soliton distribution, where the distance between two distributions is measured by relative entropy. The proposed REF codes are shown to achieve excellent intermediate performance over binary erasure channels (BECs), and binary-input additive white Gaussian noise channels (BI-AWGNCs) for both unicast, and multicast scenarios. For multicast, a non-uniform input symbol selection scheme is proposed to enhance the performance of REF codes. Furthermore, since the feedback is imprecise under noisy channels, the concept “belief” is introduced to improve the reliability of REF codes. Theoretical analysis is performed for the proposed REF codes, with an upper bound, and an approximate lower bound of the intermediate performance of REF codes over BECs derived. Both theoretical analysis, and simulations show that the proposed REF codes outperform the state-of-the-art fountain codes with feedback, in terms of the intermediate performance, with low overhead.

Efficient Successive Cancellation Stack Decoder for Polar Codes

As an improved version of successive cancellation (SC) polar decoder, an SC stack (SCS) decoder has been proposed for performance improvement. However, the existing SCS polar decoder suffers a lot from high time complexity at low signal-to-noise ratio (SNR) region and space complexity compared with the SC decoder. To this end, two improved decoders are proposed to reduce time and space complexity in both low and high SNR regions. The first one is the segmented cyclic redundancy check (CRC)-aided SCS (SCA-SCS) decoder, which is based on segmented parity checkers. The second one is the adaptive SCS (ASCS) decoder, which has the flexibility of stack depth and searching width. Furthermore, a channel condition estimator is proposed to select appropriate decision criteria for different SNR scenarios. Results have shown that for the polar code of length 1024 and rate 1/2, two improved SCS decoders can perform better than the traditional SCS decoder. The proposed SCA-SCS decoder and the ASCS decoder can achieve 10.8% and 11.42% time complexity reduction and 31.68% and 60.85% space complexity reduction on average over binary-input additive white Gaussian noise channels (BI-AWGNCs), respectively. Efficient parallel hardware architecture of the SCS polar decoder is first proposed and implemented with 90- and 65-nm technologies. Results have verified its advantages over the state of the art (SOA).

Construction of Polar Codes in Rayleigh Fading Channel

In this letter, we investigate the effective construction method of the polar codes in the Rayleigh fading channel (RFC) based on the Gaussian approximation (GA) algorithm. For the RFC, two algorithms are proposed to find the equivalent binary input AWGN channel (BIAWGNC). The first one transforms the given RFC to an equivalent BIAWGNC with the same average mutual information, and then GA algorithm is performed to construct the polar codes. For the second one, the given RFC is handled as an equivalent BIAWGNC with an analytical expression in terms of the Kullback–Leibler divergence so as to further reduce the complexity of the first algorithm. Finally, the simulation results show the effectiveness of the proposed methods.

Expanding window fountain codes with intermediate feedback over BIAWGN channels

Expanding window fountain (EWF) codes use a windowing technique to achieve unequal error protection (UEP) and unequal recovery time. UEP affords significant performance improvement of the more important bits (MIB), but at the cost of performance deterioration of the less important bits (LIB), resulting in poor intermediate performance of the EWF coding. In this study, the authors propose the utilisation of intermediate feedback in EWF coding to terminate the encoding of the MIB after their recovery, with the purpose of enhancing the intermediate performance of EWF codes over binary input additive white Gaussian noise (BIAWGN) channels. The effectiveness of the proposed coding scheme is analysed and evaluated by using extrinsic information transfer functions to investigate the theoretical performance and obtain the closed-form formula of the bit error rate (BER) of the LIB after the intermediate feedback. The results of the theoretical analysis and simulation of the scheme using additive white Gaussian noises with different signal-to-noise ratios reveal an improvement in the BER performance of the LIB compared to the case of conventional EWF coding, and also show that the scheme significantly enhances the intermediate performance of the EWF codes and improves the transmission efficiency.

Delay-Universal Channel Coding With Feedback

In this paper, the design of error-correcting or channel codes for delay-universal/anytime communication is shown while considering systems with and without a feedback link. We construct practical and low complexity anytime channel codes based on spatially-coupled repeat-accumulate (SC-RA) codes. Performance and density evolution analysis are shown for the binary erasure channel (BEC) and the binary input additive white Gaussian noise (BIAWGN) channel. We observe that the erasure/error floors exist even at low decoding delay in the following cases: 1) when the code rate is close to the Shannon capacity; and/or 2) when the code parameters are chosen to target a high decaying rate of erasure/error probability. To mitigate erasure/error floors, we present feedback algorithms for BEC and BIAWGN channels. We show that the proposed feedback strategies can greatly enhance the performance of anytime SC-RA codes. Numerical results also show that the feedback strategies significantly reduce the decoder complexity. The proposed feedback approach is applied to an aircraft tracking application to track/calculate/estimate the state information of the aircraft. Based on comparisons of the results obtained from the traditional block and anytime coding scenarios, it is observed that the latter significantly outperforms the former in terms of tracking performance.

Design of LDPC Codes: A Survey and New Results

This survey paper provides fundamentals in the design of LDPC codes. To provide a target for the code designer, we first summarize the EXIT chart technique for determining(near-)optimal degree distributions for LDPC code ensembles. We also demonstrate the simplicity of representing codes by protographs and how this naturally leads to quasi-cyclic LDPC codes. The EXIT chart technique is then extended to the special case of protograph-based LDPC codes. Next, we present several design approaches for LDPC codes which incorporate one or more accumulators, including quasi-cyclic accumulatorbased codes. The second half the paper then surveys severalalgebraic LDPC code design techniques. First, codes based on finite geometries are discussed and then codes whose designs are based on Reed-Solomon codes are covered. The algebraic designs lead to cyclic, quasi-cyclic, and structured codes. The masking technique for converting regular quasi-cyclic LDPC codes to irregular codes is also presented. Some of these results and codes have not been presented elsewhere. The paper focuses on the binary-input AWGN channel (BI-AWGNC). However, as discussed in the paper, good BI-AWGNC codes tend to be universally good across many channels. Alternatively, the reader may treat this paper as a starting point for extensions to more advanced channels. The paper concludes with a brief discussion of open problems.

Construction of rate-compatible (RC) low-density parity-check (LDPC) convolutional codes based on RC-LDPC block codes

In this paper, a family of rate-compatible (RC) low-density parity-check (LDPC) convolutional codes can be obtained from RC-LDPC block codes by graph extension method. The resulted RC-LDPC convolutional codes, which are derived by permuting the matrices of the corresponding RC-LDPC block codes, are systematic and have maximum encoding memory. Simulation results show that the proposed RC-LDPC convolutional codes with belief propagation (BP) decoding collectively offer a steady improvement on performance compared with the block counterparts over the binary-input additive white Gaussian noise channels (BI-AWGNCs).

Design of Spatially Coupled LDPC Codes Over GF $(q)$ for Windowed Decoding

In this paper we consider the generalization of binary spatially coupled low-density parity-check (SC-LDPC) codes to finite fields GF$(q)$, $q\geq 2$, and develop design rules for $q$-ary SC-LDPC code ensembles based on their iterative belief propagation (BP) decoding thresholds, with particular emphasis on low-latency windowed decoding (WD). We consider transmission over both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (BIAWGNC) and present results for a variety of $(J,K)$-regular SC-LDPC code ensembles constructed over GF$(q)$ using protographs. Thresholds are calculated using protograph versions of $q$-ary density evolution (for the BEC) and $q$-ary extrinsic information transfer analysis (for the BIAWGNC). We show that WD of $q$-ary SC-LDPC codes provides significant threshold gains compared to corresponding (uncoupled) $q$-ary LDPC block code (LDPC-BC) ensembles when the window size $W$ is large enough and that these gains increase as the finite field size $q=2^m$ increases. Moreover, we demonstrate that the new design rules provide WD thresholds that are close to capacity, even when both $m$ and $W$ are relatively small (thereby reducing decoding complexity and latency). The analysis further shows that, compared to standard flooding-schedule decoding, WD of $q$-ary SC-LDPC code ensembles results in significant reductions in both decoding complexity and decoding latency, and that these reductions increase as $m$ increases. For applications with a near-threshold performance requirement and a constraint on decoding latency, we show that using $q$-ary SC-LDPC code ensembles, with moderate $q>2$, instead of their binary counterparts results in reduced decoding complexity.

Optimization Design and Asymptotic Analysis of Systematic Luby Transform Codes Over BIAWGN Channels

In this paper, we study systematic Luby transform (SLT) codes over binary input additive white Gaussian noise (BIAWGN) channels. To improve the bit-error-rate (BER) performance of SLT codes on the BIAWGN channel, this paper presents a novel optimization design of degree distributions. First, we apply the Gaussian approximation (GA) to analyze the asymptotic BER performance of SLT codes and calculate overhead thresholds for successful decoding. Second, we derive an approximate closed-form expression for lower bound on BER by applying the GA and further simplify the expression in low and high signal-to-noise ratio regimes, respectively. Third, we adopt the conventional linear programming (CLP) constrained by the GA to optimize the degree distribution of SLT codes. The objective of CLP is to minimize the average degree of the degree distribution that may result in bad error floor. To improve the error floor caused by CLP, we put forth a novel optimization model constrained by the lower bound on BER, and using the model we can design the degree distribution more flexibly with any desired BER. Simulation results show that the proposed degree distribution can provide better BER performance of SLT codes over the BIAWGN channel.

Construction of regular rate-compatible LDPC convolutional codes

In this paper, we propose a new method to derive a family of regular rate-compatible low-density parity-check (RC-LDPC) convolutional codes from RC-LDPC block codes. In the RC-LDPC convolutional family, each extended sub-matrix of each extended code is obtained by choosing specified elements from two fixed matrices Hk E1 and Hk E2 , which are derived by modifying the extended matrices H E1 and H E2 of a systematic RC-LDPC block code. The proposed method which is based on graph extension simplifies the design, and prevent the defects caused by the puncturing method. It can be used to generate both regular and irregular RC-LDPC convolutional codes. All resulted codes in the family are systematic which simplify the encoder structure and have maximum encoding memories which ensure the property. Simulation results show the family collectively offer a steady improvement in performance with code compatibility over binary-input additive white Gaussian noise channel (BI-AW-GNC).

Randomly Punctured LDPC Codes

In this paper, we present a random puncturing analysis of low-density parity-check (LDPC) code ensembles. We derive a simple analytic expression for the iterative belief propagation (BP) decoding threshold of a randomly punctured LDPC code ensemble on the binary erasure channel (BEC) and show that, with respect to the BP threshold, the strength and suitability of an LDPC code ensemble for random puncturing is completely determined by a single constant that depends only on the rate and the BP threshold of the mother code ensemble. We then provide an efficient way to accurately predict BP thresholds of randomly punctured LDPC code ensembles on the binary-input additive white Gaussian noise channel (BI-AWGNC), given only the BP threshold of the mother code ensemble on the BEC and the design rate, and we show how the prediction can be improved with knowledge of the BI-AWGNC threshold. We also perform an asymptotic minimum distance analysis of randomly punctured code ensembles and present simulation results that confirm the robust decoding performance promised by the asymptotic results. Protograph-based LDPC block code and spatially coupled LDPC code ensembles are used throughout as examples to demonstrate the results.

Anytime Characteristics of Protograph-Based LDPC Convolutional Codes

Anytime transmission has been shown to be an effective means of stabilizing an unstable system over noisy channels. A crucial issue in implementing anytime transmission strategies is the design of anytime codes that offer good finite-length performance and asymptotic anytime properties. This paper introduces an efficient anytime code derived from protograph-based low-density parity-check convolutional codes known as extended S-LDPCC (XS-LDPCC) codes. The asymptotic anytime properties of XS-LDPCC codes over both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise (BIAWGN) channel are demonstrated through density evolution. By means of certain observations verified by simulations, we provide accurate estimates of the delay exponents of XS-LDPCC codes over the BEC and the BIAWGN channel. Through simulations, the finite-length performance of XS-LDPCC codes and the effects of different code design parameters are quantified.

유한한 길이에서 성능이 향상된 BP 극 복호기

In this paper, we discuss the belief propagation decoding algorithm for polar codes. The performance of Polar codes for shorter lengths is not satisfactory. Motivated by this, we propose a novel technique to improve its performance at short lengths. We showed that the probability of messages passed along the factor graph of polar codes, can be increased by multiplying the current message of nodes with their previous message. This is like a feedback path in which the present signal is updated by multiplying with its previous signal. Thus the experimental results show that performance of belief propagation polar decoder can be improved using this proposed technique. Simulation results in binary-input additive white Gaussian noise channel (BI-AWGNC) show that the proposed belief propagation polar decoder can provide significant gain of 2 dB over the original belief propagation polar decoder with code rate 0.5 and code length 128 at the bit error rate (BER) of .

Protograph-Based Raptor-Like LDPC Codes

This paper proposes protograph-based Raptor-like (PBRL) codes as a class of rate-compatible low-density parity-check codes for binary-input AWGN channels. As with the Raptor codes, exclusive-OR operations on precoded bits produce additional parity bits providing extensive rate compatibility. Unlike Raptor codes, each additional parity bit in the protograph is explicitly designed to optimize the density evolution threshold. During the lifting process, approximate cycle extrinsic message degree (ACE) and circulant progressive edge growth (CPEG) constraints are used to avoid undesirable graphical structures. Some density-evolution performance is sacrificed to obtain lower error floors, particularly at short block-lengths. Simulation results are shown for information block sizes of k = 1032 and 16 384. For a target frame 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 each rate, the k = 1032 and 16 384 code families perform within 1 dB and 0.4 dB of both the Gallager bound and the normal approximation, respectively. The 16 384 code family outperforms the best known standardized code family, namely, the AR4JA codes. The PBRL codes also outperform DVB-S2 codes that have the advantages of longer blocklengths and outer BCH codes. Performance is similar to RC code families designed by Nguyen et al. that do not constrain codes to have the PBRL structure and involve simulation in the optimization process at each rate.

A Design of Physical-Layer Raptor Codes for Wide SNR Ranges

It is known that the Raptor code can be constructed to achieve a rate close to the capacity of the binary-input AWGN (BIAWGN) channel with a fixed signal-to-noise ratio (SNR). In this letter, we propose a constructive approach of designing the Raptor code over the BIAWGN channel such that the achieved rates are close to the associated capacities for wide SNR ranges.

Ripple Design of LT Codes for BIAWGN Channels

This paper presents a novel framework, which enables a design of rateless codes for binary input additive white Gaussian noise (BIAWGN) channels, using the ripple-based approach known from the works for the binary erasure channel (BEC). We reveal that several aspects of the analytical results from the BEC also hold in BIAWGN channels. The presented framework is applied in a code design example, which shows promising results compared to existing work. In particular it shows a great robustness towards variations in the signal-to-noise power ratio (SNR), contrary to existing codes.