Binary-input Additive White Gaussian Noise Research Articles

Two-Round Selection-Based Bit Flipping Decoding Algorithm for LDPC Codes

This paper presents a novel iterative reliability-based bit flipping (BF) algorithm for decoding low-density parity-check codes. The new decoder is a single BF algorithm called two-round selection -based bit flipping. It introduces the idea of a two-round selection of the flipped bit, based successively on hard and soft received channel values. In the first stage, a set of unreliable bits is identified, and then a second selection is used, to pick out among them the bit to flip. In the second round of selection, the initial belief about received signals, contributes efficiently to selecting the best candidate bit. We demonstrate through simulations over the binary-input additive white Gaussian noise channel and the Rayleigh fading channel that the proposed algorithm exhibits better decoding performance when compared with some well-known soft decision BF algorithms. A complexity analysis of the proposal and a comparison to other BF decoders are also presented.

A New Polar Code Design Based on Reciprocal Channel Approximation

This paper revisits polar code design for a binary-input additive white Gaussian noise (BI-AWGN) channel when successive cancellation (SC) decoding is applied at the receiver. We focus on the so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">reciprocal channel approximation (RCA)</i> , which is often adopted in the design of low-density parity-check (LDPC) codes. Implementation of RCA requires the computation of the mutual information of BPSK signaling as well as a corresponding function known as the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">reciprocal channel mapping</i> , and thus we develop rigorous closed-form approximations of these that are easy to calculate numerically and also valid over a wide range of SNR. Through numerical evaluation we find that, compared to approaches based on the popular Gaussian approximation (GA) as well as the so-called improved GA (IGA), the proposed RCA approach offers better estimates of the bit error rate of polarized channels <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">with no additional computational cost</i> . As a result, polar codes designed by the proposed RCA can achieve further improvement in terms of block error rate (BLER) performance. The gain achieved by the new approach becomes significant as the codeword length increases.

An Information-Theoretic Perspective on Successive Cancellation List Decoding and Polar Code Design

This work identifies information-theoretic quantities that are closely related to the required list size on average for successive cancellation list (SCL) decoding to implement maximum-likelihood decoding over general binary memoryless symmetric (BMS) channels. It also provides upper and lower bounds for these quantities that can be computed efficiently for very long codes. For the binary erasure channel (BEC), we provide a simple method to estimate the mean accurately via density evolution. The analysis shows how to modify, e.g., Reed-Muller codes, to improve the performance when practical list sizes, e.g., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L\in {[{8, 1024}]}$ </tex-math></inline-formula> , are adopted. Exemplary constructions with block lengths <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N\in \{128,512\}$ </tex-math></inline-formula> outperform polar codes of 5G over the binary-input additive white Gaussian noise channel. It is further shown that there is a concentration around the mean of the logarithm of the required list size for sufficiently large block lengths, over discrete-output BMS channels. We provide the probability mass functions (p.m.f.s) of this logarithm, over the BEC, for a sequence of the modified RM codes with an increasing block length via simulations, which illustrate that the p.m.f.s concentrate around the estimated mean.

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).

Error Performance Prediction of Randomly Shortened and Punctured LDPC Codes

In this contribution, we show that the word error rate (WER) performance in the waterfall region of a randomly shortened and punctured low density parity check code can be accurately predicted from the WER performance of its finite-length mother code. We derive an approximate analytical expression for the mutual information (MI) required by a daughter code to achieve a given WER, based on the MI required by the mother code, which shows that the gap to the capacity of the daughter code grows the more the code is punctured or shortened. The theoretical results are confirmed by simulations (where the random shortening and puncturing pattern is either selected independently per codeword or kept the same for all codewords) for practical codes on both the binary erasure channel and the binary-input additive white Gaussian noise channel.

Performance Analysis of Practical QC-LDPC Codes: From DVB-S2 to ATSC 3.0

In this paper, a new analysis method is introduced to predict the gap between Shannon limit and successful decoding signal-to-noise ratio threshold (SNR loss) of practical quasi-cyclic low-density parity check (QC-LDPC) codes. The SNR loss of a QC-LDPC code is summed up from three aspects: 1) the base matrix; 2) the number of iterations for decoding; and 3) the finite codeword length. As an example, the SNR loss of DVB-S2 and ATSC 3.0 LDPC codes is analyzed with the proposed method. Results show that a large portion of the SNR loss of DVB-S2 LDPC codes is due to their base matrices, which is significantly improved in latest ATSC 3.0 standard. Furthermore, with the aid of the proposed analysis method, newly designed QC-LDPC codes with raptor-like structure are proposed. Bit-error rate simulations show that the proposed codes are only 0.4 dB away from Shannon limit under binary-input additive white Gaussian noise channel and only 0.65 dB away from Shannon limit under binary-input Rayleigh fading channel, which are comparable with state-of-art design.

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.

More Accurate Analysis of Sum-Product Decoding of LDPC Codes Using a Gaussian Approximation

This letter presents a more accurate mathematical analysis, with respect to the one performed in Chung et al.'s 2001 paper, of belief-propagation decoding for low-density parity-check (LDPC) codes on memoryless binary input-additive white Gaussian noise channels, when considering a Gaussian approximation (GA) for message densities under density evolution. The recurrent sequence, defined in Chung et al.'s 2001 paper, describing the message passing between variable and check nodes follows from the GA approach and involves the function φ(x), therein defined, and its inverse. The analysis of this function is here resumed and studied in depth to obtain tighter upper and lower bounds on it. Moreover, unlike the upper bound given in the above cited paper, the tighter upper bound on φ(x) is invertible. This allows a more accurate evaluation of the asymptotical performance of sum-product decoding of LDPC codes when a GA is assumed.

High efficiency continuous-variable quantum key distribution based on QC-LDPC codes

Seeking good error correcting codes to improve the efficiency of continuous-variable quantum key distribution (CVQKD) reconciliation is a concerning issue. Due to the introduction of multidimensional reconciliation, the error correcting techniques in the classical binary-input additive white Gaussian noise channel are applicable to CVQKD. In this Letter, we apply the quasi-cyclic low-density parity-check (QC-LDPC) codes, which are specified in 5G protocol, to the reconciliation process. Simulation results show that the reconciliation efficiency can reach 92.6% when the code rate is 22/68 and the signal-to-noise ratio is 0.623. Such a new error correcting code points out a new direction for the development of CVQKD.

Progressive Matrix Growth Algorithm for Constructing Rate-Compatible Length-Scalable Raptor-Like Quasi-Cyclic LDPC Codes

With the worldwide deployment and rapid development of digital television terrestrial broadcasting (DTTB) systems, the demands of diversified multimedia services from the fixed/mobile receptions increase significantly. To satisfy the various requirements of quality of service, the forward error correction codes with multiple code rates and lengths become a trend for DTTB standards. In this paper, rate-compatible length-scalable raptor-like quasi-cyclic low density parity check (RL-QC-LDPC) codes are proposed for future DTTB systems, where a wide range of code rates and information/parity block lengths can be obtained with low implementation complexity. A low-complexity construction method, called progressive matrix growth (PMG), is also proposed for such kind of RL-QC-LDPC codes. Unlike conventional construction methods, PMG performs 2-D progressive extension of both information variable nodes and check nodes with limited computational complexity and memory resources. Based on the performance prediction provided by the tool of multi-edge type density evolution, several code families are constructed with PMG as examples, and careful discussions on design parameters are given. The numerical results show that the average gaps of SNR threshold to Shannon limit of binary-input additive white Gaussian noise channel are only 0.41~0.44 dB for the code families with information-bit puncturing.

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.

Distributed Joint Source-Channel Decoding Using Systematic Polar Codes

This letter proposes a novel joint decoding scheme for correlated sources using systematic polar codes. In the proposed scheme, each source independently encodes its message into a systematic polar code word and sends the codeword via a binary-input additive white Gaussian noise channel to the common destination, where adaptive cyclic redundancy check-aided successive cancellation list (CA-SCL) decoders are employed. During the iterative decoding process, the log likelihood ratios (LLRs) fed to each adaptive CA-SCL decoder are iteratively combined with the extrinsic LLRs from other decoders. The correlation among the sources is exploited to improve the decoding performance. Simulation results show that, in comparison with independent decoding, the proposed scheme can achieve significantly improved decoding performance without increased complexity.

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.

Improved Target Acquisition Rates With Feedback Codes

This paper considers the problem of acquiring an unknown target location (among a finite number of locations) via a sequence of measurements, where each measurement consists of simultaneously probing a group of locations. The resulting observation consists of a sum of an indicator of the target's presence in the probed region, and a zero mean Gaussian noise term whose variance is a function of the measurement vector. An equivalence between the target acquisition problem and channel coding over a binary input additive white Gaussian noise (BAWGN) channel with state and feedback is established. Utilizing this information theoretic perspective, a two-stage adaptive target search strategy based on the sorted Posterior Matching channel coding strategy is proposed. Furthermore, using information theoretic converses, the fundamental limits on the target acquisition rate for adaptive and non-adaptive strategies are characterized. As a corollary to the non-asymptotic upper bound of the expected number of measurements under the proposed two-stage strategy, and to non-asymptotic lower bound of the expected number of measurements for optimal non-adaptive search strategy, a lower bound on the adaptivity gain is obtained. The adaptivity gain is further investigated in different asymptotic regimes of interest.

A Reused-Public-Path Successive Cancellation List Decoding for Polar Codes With CRC

Since the decoding of polar codes goes wrong in sometime, several decoding methods have been proposed to overcome this issue. Unfortunately, the several schemes are still suffering from high decoding complexity. To this end, we propose a reused-public-path successive-cancellation list decoder, which can re-do the decoding from ${E}$ -th level when the current decoding process fails. In addition, a reused public-path scheme is proposed. By properly designing, simulation results in binary-input additive white gaussian noise channel show that this decoding approach provides significant computational complexity reduction in the moderate and high signal-to-noise ratio regime without performance loss.