Binary-input Additive White Gaussian Noise Research Articles

Dual Capacity Upper Bounds for Noisy Runlength Constrained Channels

Binary-input memoryless channels with a run length constrained input are considered. Upper bounds to the capacity of such noisy run length constrained channels are derived using the dual capacity method with Markov test distributions satisfying the Karush-Kuhn-Tucker conditions for the capacity-achieving output distribution. Simplified algebraic characterizations of the bounds are presented for the binary erasure channel and the binary symmetric channel. These upper bounds are very close to achievable rates, and improve upon previously known feedback-based bounds for a large range of channel parameters. For the binary-input additive white Gaussian noise channel, the upper bound is simplified to a small-scale numerical optimization problem. These results provide some of the simplest upper bounds for an open capacity problem that has theoretical and practical relevance.

Useful Mathematical Tools for Capacity Approaching Codes Design

Focus of this letter is the oldest class of codes that can approach the Shannon limit quite closely, i.e., low-density parity-check (LDPC) codes, and two mathematical tools that can make their design an easier job under appropriate assumptions. In particular, we present a simple algorithmic method to estimate the threshold for regular and irregular LDPC codes on memoryless binary-input continuous-output additive white Gaussian noise (AWGN) channels with sum-product decoding, and, to determine how close are the obtained thresholds to the theoretical maximum, i.e., to the Shannon limit, we give a simple and invertible expression of the AWGN channel capacity in the binary input-soft output case. For these codes, the thresholds are defined as the maximum noise level, such that an arbitrarily small bit-error probability can be achieved as the block length tends to infinity. We assume a Gaussian approximation for message densities under density evolution, a widely used simplification of the decoding algorithm.

Improved Belief Propagation Decoding Algorithm for Short Polar Codes

In this paper, we discuss the belief propagation (BP) decoding of polar codes. The performance of polar codes for short lengths is not satisfactory. Therefore, motivated by this we propose a novel technique to improve the performance of polar codes under BP decoding. In order to enhance the reliability of the variable nodes’ propagated messages in BP decoding, the messages are multiplied by previous messages of variable nodes. It is also shown that the BP decoding can be performed on polar codes if the factor graph is constructed according to the parity check matrix of polar codes. Simulation results in binary-input additive white Gaussian noise channel show that using the proposed method a gain of 1–2 dB can be achieved. We also show that the complexity of proposed decoder is same as the complexity of standard BP decoder. Furthermore, we show that the proposed decoder requires about 10–25 iterations less than the standard BP decoder.

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.

Embedded physical‐layer authentication in cognitive radio requires efficient low‐rate channel coding schemes

In this study, the author investigates the practical limitation of the recently proposed embedded cryptographic signature authentication scheme at the physical layer. By employing the log-likelihood ratio of a tag bit and its approximation, the author shows that the equivalent authentication channel observed by the secondary receiver can be viewed as a binary-input additive white Gaussian noise channel. Then, the sphere-packing lower bound can be employed to show the transmission capability for practical finite-length authentication tags. To achieve the same effective coverage area for both the primary and secondary receivers, it essentially requires efficient low-rate channel coding schemes with near sphere-packing-bound performance at the secondary receiver, which contrasts sharply with the pessimistic conclusion of Jiang et al.

Informed Fixed Scheduling for Faster Convergence of Shuffled Belief-Propagation Decoding

A novel informed fixed scheduling (IFS) scheme for shuffled belief-propagation (BP) decoding of binary low-density parity-check (LDPC) code is introduced to improve the BP decoding convergence. The IFS finds an appropriate order of variable nodes in accordance with the number of updated neighbors in the code graph, ensuring that the maximum number of latest message updates is utilized within a single iteration. This allows the utilization of most reliable message updates in a timely manner, leading to faster error-rate convergence. Simulation results show that the proposed IFS scheme improves the convergence speed of BP decoder by up to 20% for regular LDPC codes and 45% for irregular LDPC codes, without affecting the error-rate performance, at medium-to-high signal-to-noise ratio over binary-input additive white Gaussian noise channel.

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).

SC-LDPC Code Design for Half-Duplex Relay Channels

This paper studies code design for the half-duplex relay channel when the transmissions take place over binary input additive white Gaussian noise (BIAWGN) channels. Using the decode-forward relay protocol, we design the relay code based on a spatially coupled low-density parity-check (SC-LDPC) code. We show a low complexity density evolution analysis for the proposed relay code. From the density evolution results, we observe that the proposed spatially coupled relay code achieves a capacity approaching performance. We also observe that the proposed code outperforms existing optimized LDPC relay codes. Through simulation results, we evaluate the finite-length performance of the proposed code.

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).

Optimizing Transmission Lengths for Limited Feedback With Nonbinary LDPC Examples

This paper presents a general approach for optimizing the number of symbols in increments (packets of incremental redundancy) in a feedback communication system with a limited number of increments. This approach is based on a tight normal approximation on the rate for successful decoding. Applying this approach to a variety of feedback systems using nonbinary (NB) low-density parity-check (LDPC) codes shows that greater than 90% of capacity can be achieved with average blocklengths fewer than 500 transmitted bits. One result is that the performance with ten increments closely approaches the performance with an infinite number of increments. The paper focuses on binary-input additive-white Gaussian noise (BI-AWGN) channels but also demonstrates that the normal approximation works well on examples of fading channels as well as high-SNR AWGN channels that require larger QAM constellations. This paper explores both variable-length feedback codes with termination (VLFT) and the more practical variable length feedback (VLF) codes without termination that require no assumption of noiseless transmitter confirmation. For VLF, we consider both a two-phase scheme and CRC-based scheme.

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.

Parallel concatenated systematic polar codes

A parallel concatenated coding structure, which can improve the performance of polar codes in finite-length regime, is proposed. In the proposed structure, the encoder is built using two systematic polar codes (SPCs), and the decoder adopts iterative belief propagation decoding algorithm. Numerical results in the binary-input additive white Gaussian noise channel indicate that the proposed coding scheme can improve BER performance of polar codes significantly. Specially, BER of the proposed coding scheme with length 192 is almost equal to that of original SPCs with length 256, when the code rate is 1/3.

Raptor Codes in the Low SNR Regime

In this paper, we revisit the design of Raptor codes for binary input additive white Gaussian noise (BIAWGN) channels, where we are interested in very low signal to noise ratios (SNRs). A linear programming degree distribution optimization problem is defined for Raptor codes in the low SNR regime through several approximations. We also provide an exact expression for the polynomial representation of the degree distribution with infinite maximum degree in the low SNR regime, which enables us to calculate the exact value of the fractions of output nodes of small degrees. A more practical degree distribution design is also proposed for Raptor codes in the low SNR regime, where we include the rate efficiency and the decoding complexity in the optimization problem, and an upper bound on the maximum rate efficiency is derived for given design parameters. Simulation results show that the Raptor code with the designed degree distributions can approach rate efficiencies larger than 0.95 in the low SNR regime.

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.

Anytime Reliability of Spatially Coupled Codes

Anytime transmission was previously shown to be necessary and sufficient for tracking and controlling an unstable plant over a noisy channel. In this paper, we propose an efficient channel coding scheme for anytime transmission. We design this coding scheme based on spatially coupled codes and investigate its performance over binary erasure channels (BEC) and binary input additive white Gaussian noise (BIAWGN) channels. Through density evolution analysis, we asymptotically show the desired anytime properties of the proposed code. For the BEC, we derive the corresponding delay-exponent and numerically predict the finite-length performance of the proposed anytime code. We also propose adaptive window decoding techniques to ensure low complexity anytime receivers.

Decoding LDPC Codes With Locally Maximum-Likelihood Binary Messages

A new low-complexity message passing algorithm is described for decoding low-density parity-check (LDPC) codes by exchanging binary messages. The algorithm computes the local maximum-likelihood binary message (LMLBM) at each symbol node, given the combination of local channel information and partial syndrome components from adjacent parity check nodes. When channel information is quantized, the locally ML messages are pre-computed and stored in a dynamic global lookup table. The proposed algorithm uses memoryless extrinsic messages so that density evolution thresholds can be directly computed. Thresholds are obtained for regular ensembles, predicting good performance on quantized binary-input additive white Gaussian noise (biAWGN) channels.