Successive Cancellation Research Articles

Fast Polarization-Adjusted Convolutional (PAC) Software Decoders: Algorithm and Implementation

In this paper, a novel high throughput software polarization-adjusted convolutional (PAC) decoder based on the fast successive cancellation (FSC) decoding algorithm is first proposed. The FSC decoding algorithm is obtained by expanding the fast list decoding algorithm to suit parallel designs. To improve the parallel processing capabilities of the proposed decoder, we use SIMD and multiple cores techniques to decode multiple codewords in parallel, thus effectively parallel mapping the FSC decoding algorithm to processors. The proposed decoder is sufficiently general to be implemented on X86 and embedded processors (NEON, SSE, AVX256, and AVX512 instructions). Experimentations show that the proposed software PAC decoder can achieve 370 Mbps under the multithreading mode.

The implementation of Shannon-limited polar codes-based information reconciliation for quantum key distribution

Quantum key distribution (QKD) gives a way to generate unconditionally secure keys for two remote users, Alice and Bob. Information reconciliation (IR), which can correct the errors caused by the imperfections of the QKD systems, is a critical component in QKD. Due to the high-security requirements and large volumes of data processing, robustness and efficiency are two main factors that must be considered for the implementation of IR. The polar codes-based IR has several potential advantages, such as capacity to reach Shannon-limit, high IR efficiency, and low computational complexity. Although CPU-based IR is always implemented in most of the previous works, it is not the optimal implementation in terms of performance and power dissipation. To the best of our knowledge, there is still no work to build a special-purpose hardware module for polar codes-based IR. In this paper, a dedicated design of hardware accelerator is first proposed for polar codes-based IR, in which the block-checked successive cancellation list (SCL) algorithm is used to verify the consistency of the sifted keys, and the overall failure probability of IR is significantly reduced. The proposed design is constructed into a partially-unrolled parallel architecture to accelerate the core decoder as well as balance the resource utilization. Furthermore, the hardware implementation is completed based on Xilinx Zynq UltraScale+ XCZU5EV MPSoC platform and achieves an IR throughput of 15 Mbps with a block length of 212, while less than of the amount of on-chip resources are used in other previous designs of SCL decoder. The proposed design can provide a real-time, low-cost solution for IR in QKD systems, and enhance the performance of QKD.

Joint Method of Moments (JMoM) and Successive Moment Cancellation (SMC) Multiuser Time Synchronization for ZP-OFDM-Based Waveforms Applicable to Joint Communication and Sensing

It has been recently shown that zero padding (ZP)-orthogonal frequency-division multiplexing (OFDM) is a promising candidate for 6G wireless systems requiring joint communication and sensing. In this paper, we consider a multiuser uplink scenario where users are separated in power domain, i.e., non-orthogonal multiple access (NOMA), and use ZP-OFDM signals. The uplink transmission is grant-free and users are allowed to transmit asynchronously. In this setup, we address the problem of time synchronization by estimating the timing offset (TO) of all the users. We propose two non-data-aided (NDA) estimators, i.e., the joint method of moment (JMoM) and the successive moment cancellation (SMC), that employ the periodicity of the second order moment (SoM) of the received samples for TO estimation. Moreover, the coding assisted (CA) version of the proposed estimators, i.e., CA-JMoM and CA-SMC, are developed for the case of short observation samples. We also extend the proposed estimators to multiuser multiple-input multiple-output (MIMO) systems. The effectiveness of the proposed estimators is evaluated in terms of lock-in probability under various practical scenarios. Simulation results show that the JMoM estimator can reach the lock-in probability of one for the moderate range of Eb/N0 values. While existing NDA TO estimators in the literature either offer low lock-in probability, high computational complexity that prevents them from being employed in MIMO systems, or are designed for single-user scenarios, the proposed estimators in this paper address all of these issues.

On the improvements of successive cancellation Creeper decoding for polar codes

In this paper, an improvement for successive cancellation (SC) decoding is presented for polar codes. This improvement is based on applying Creeper approach that was used to decode convolutional codes. The proposed decoder combines a stack and a Fano approaches, allowing to significantly decrease complexity of Successive Cancellation Fano decoder at the cost of a slight increase in space complexity. We also show how to combine the SC-Creeper decoding with a bit-flipping approach to further improve the performance of SC-Creeper due to introducing additional decoding attempts.Simulation results show that the proposed SC-Creeper and SC-Creeper Flip decoders with carefully selected parameters have the same performance as both CRC-aided Successive Cancellation List decoder with list size 8 and Fano decoder, while having much smaller complexity.

Application research of polar coded OFDM underwater acoustic communications

To further accelerate the practical process of polar codes in underwater acoustic communication (UWC) and improve the reliability and efficiency of underwater data transmission, we first propose a polar code construction method suitable for the underwater acoustic channel, which drastically reduces the complexity and meets the need for practical UWC. Then, we establish a practical polar coded UWC scheme by devising a two-step procedure based on the Orthogonal Frequency Division Multiplexing technique. Moreover, through simulation, the scheme's performances with the unimproved and the proposed polar code construction method are analyzed. Moreover, its performances with four different polar decoders are also given. The performance comparison with the LDPC coded UWC system is studied as well. Furthermore, lake-trial results under two different channel conditions demonstrate that the proposed construction method and the scheme effectively guarantee the reliability of data transmission, achieving error-free transmission at a transmission distance of 1718 m with the signal-to-noise-ratio (SNR) of 21 dB, and a transmission distance of 749 m with the SNR of 16.5 dB, better than the LDPC coded system with same conditions. The semi-experimental results show that the proposed polar coded system under the Cyclic Redundancy Check-Aided Successive Cancellation List decoder outperforms the LDPC coded system by approximately 2.1–3.7 dB.

Hardware Efficient Successive-Cancellation Polar Decoders Using Approximate Computing

Benefiting for its capacity-achieving property, the polar codes have attracted wide attention in various forward error correction (FEC) codes. Although Successive-Cancellation (SC) decoders have low hardware complexity, their limited thoughput leads to low hardware efficiency. In this paper, two novel processing elements (PE) units for SC decoders are proposed by the reconstruction of key functions. Approximate comparator and adder-subtractor are developed in the proposed PEs to further enhance the hardware efficiency of the SC decoders with neglectable bit error rate (BER) performance degradation. The designs are implemented and evaluated on 28nm CMOS technology. Simulation results indicate that the area efficiency of two designs are 17.758r <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {Gb/s/mm^{2}}$ </tex-math></inline-formula> and 14.416r <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {Gb/s/mm^{2}}$ </tex-math></inline-formula> with 1024-bit code length and the code rate r, which are 26.7% and 9.7% better than the state-of-the-art SC implementations. Moreover, the low complexity Fast Simplified SC (Fast-SSC) decoder implemented with proposed approximate PEs has 60.3% area efficiency improvement than the prior Fast-SSC decoders with equivalent BER performance.

List-serial pipelined hardware architecture for SCL decoding of polar codes

For polar codes, the performance of successive cancellation list (SCL) decoding is capable of approaching that of maximum likelihood decoding. However, the existing hardware architectures for the SCL decoding suffer from high hardware complexity due to calculating L decoding paths simultaneously, which are unfriendly to the devices with limited logical resources, such as field programmable gate arrays (FPGAs). In this paper, we propose a list-serial pipelined hardware architecture with low complexity for the SCL decoding, where the serial calculation and the pipelined operation are elegantly combined to strike a balance between the complexity and the latency. Moreover, we employ only one successive cancellation (SC) decoder core without L×L crossbars, and reduce the number of inputs of the metric sorter from 2L to L + 2. Finally, the FPGA implementations show that the hardware resource consumption is significantly reduced with negligible decoding performance loss.

Performance analysis of Polar Codes in a Visible Light Communication System

One of the vital entity of any communication systems is Channel coding, which leads to the design of high performance codes for future wireless systems having low complexity encoder and decoder design. These systems will have a requirement of operating in highly reliable conditions, maximum throughput, and low and high code rates and to work with short and long information messages. Polar codes are one of the promising error correcting channel codes that can be used in these situations to obtain maximum throughput and coding gain in a communication systems because of their capacity approaching performance and finds interests in Satellite communication and 4G/5G services. These codes uses the concept of channel polarization to be constructed. The work proposed in this paper focusses on the Bit Error Rate evaluation and analysis of the Polar codes using traditional approach and Deep learning approach. The feedforward deep learning networks using different activation functions were used for the Deep learning approach. The Successive Cancellation algorithm and its variant using List decoding were used as traditional decoding methodology. The results obtained using Deep learning approach were satisfactory and was matching as per the traditional decoding.

A fast computing decoder for polar codes with a neural network

This paper proposes high-speed computing decoders for polar codes based on a neural network. To compensate for the performance gap with the successive cancellation decoder, we propose applying the recurrent neural network to the BP decoder with a modified factor graph to reduce the computational complexity of the decoder without any performance degradation. The results of the performance simulation conducted in this paper reveal that the proposed decoder requires substantially less computational complexity than the conventional decoders to achieve the same bit error rate performance.

Optimized Higher-Order Polarization Weight Incremental Selective Decoding and Forwarding in Cooperative Satellite Sensor Networks

An optimized higher-order polarization weight (HPW) incremental selective decoding and forwarding (ISDF) scheme is proposed for cooperative transmissions in satellite sensor networks (SSNs). First, an HPW method is designed with subchannel power allocation in encoding to enhance the antiinterference. Second, a cooperative decision in the destination is used to avoid error propagation for better outage probability (OP) and bit error rate (BER) performance. It switches between direct and cooperative transmissions alternatively by channel state information (CSI). Finally, the HPW coding information, cyclic redundancy codes (CRCs), and four special frame structures of polar subcodes are jointly fast successive cancellation list (FSCL) decoded to improve efficiency and BER performance. Finally, we apply the optimized polarization encoding and decoding to the relay system to form a new ISDF relay system for better coding efficiency, BER, and OP performance. Simulation results show that the proposed scheme obtains approximately 0.8 and 1 dB performance gains at OP of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${{10}^{-{2}}}$ </tex-math></inline-formula> and BER of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${{10}^{-{6}}}$ </tex-math></inline-formula> , respectively, when compared with those of the current ISDF scheme. Therefore, the cooperative scheme possesses efficient decoding and cooperation, which endows it a promising candidate relay scheme in SSNs.

A golden decade of polar codes: From basic principle to 5G applications

After the pursuit of seventy years, the invention of polar codes indicates that we have found the first capacity-achieving coding with low complexity construction and decoding, which is the great breakthrough of the coding theory in the past two decades. In this survey, we retrospect the history of polar codes and summarize the advancement in the past ten years. First, the primary principle of channel polarization is investigated such that the basic construction, coding method and the classic successive cancellation (SC) decoding are reviewed. Second, in order to improve the performance of the finite code length, we introduce the guiding principle and conclude five design criteria for the construction, design and implementation of the polar code in the practical communication system based on the exemplar schemes in the literature. Especially, we explain the design principle behind the concatenated coding and rate matching of polar codes in 5G wireless system. Furthermore, the improved SC decoding algorithms, such as SC list (SCL) decoding and SC stack (SCS) decoding etc., are investigated and compared. Finally, the research prospects of polar codes for the future 6G communication system are explored, including the optimization of short polar codes, coding construction in fading channels, polar coded modulation and HARQ, and the polar coded transmission, namely polar processing. Predictably, as a new coding methodology, polar codes will shine a light on communication theory and unveil a revolution in transmission technology.

Successive Cancellation Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding

A product code with single parity-check component codes can be described via the tools of a multi-kernel polar code, where the rows of the generator matrix are chosen according to the constraints imposed by the product code construction. Following this observation, successive cancellation decoding of such codes is introduced. In particular, the error probability of single parity-check product codes over binary memoryless symmetric channels under successive cancellation decoding is characterized. A bridge with the analysis of product codes introduced by Elias is also established for the binary erasure channel. Successive cancellation list decoding of single parity-check product codes is then described. For the provided example, simulations over the binary input additive white Gaussian channel show that successive cancellation list decoding outperforms belief propagation decoding applied to the code graph. Finally, the performance of the concatenation of a product code with a high-rate outer code is investigated via distance spectrum analysis. Examples of concatenations performing within 0.7 dB from the random coding union bound are provided.

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.

Simplified Metric Sorting for Multi-Bit Successive Cancellation List Decoding of Polar Codes

This letter focuses on the metric sorting step for multi-bit successive cancellation list (SCL) decoders of polar codes, which is an important part of the whole decoder. We propose a primary-secondary exchange sorting (PSES) method. In the primary sorting, most of the survivals, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L}$ </tex-math></inline-formula> paths are selected from the first minor <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2L</i> candidates. In the secondary sorting, a small number of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${N}$ </tex-math></inline-formula> paths are extracted from the other huge number of candidates to remedy the missing survivals. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${N}$ </tex-math></inline-formula> paths in the secondary stage are exchanged with the largest <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${N}$ </tex-math></inline-formula> elements in the first <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L}$ </tex-math></inline-formula> paths. Due to the minor candidates in the primary stage and the few survivals in the secondary stage, this method reduces the complexity with negligible performance loss. Additionally, a sorting network based on compare-and-swap unit (CASU) PSES (CPSES) and a sorting network based on Radix-2L PSES (RPSES) are proposed to improve secondary sorting. The evaluation results show that compared with state-of-the-art metric sorting methods, the proposed CPSES architecture significantly reduces hardware resources, and the proposed RPSES architecture significantly reduces the latency.

Intelligent Path-Selection-Aided Decoding of Polar Codes

CRC-aided successive cancellation list (CA-SCL) decoding is a powerful algorithm that dramatically improves the error performance of polar codes. Path selection is a major issue that affects the decoding latency of SCL decoders. Generally, path selection is implemented using a metric sorter, which causes its latency to increase as the list grows. In this paper, intelligent path selection (IPS) is proposed as an alternative to the traditional metric sorter. First, we found that in the path selection, only the most reliable paths need to be selected, and it is not necessary to completely sort all paths. Second, based on a neural network model, an intelligent path selection scheme is proposed, including a fully connected network construction, a threshold and a post-processing unit. Simulation results show that the proposed path-selection method can achieve comparable performance gain to the existing methods under SCL/CA-SCL decoding. Compared with the conventional methods, IPS has lower latency for medium and large list sizes. For the proposed hardware structure, IPS's time complexity is O(klog2(L)) where k is the number of hidden layers of the network and L is the list size.

Polar-code-based information reconciliation scheme with the frozen-bit erasure strategy for quantum key distribution

Information reconciliation (IR) ensures the correctness of quantum key distribution systems, by correcting the error bits existed in the sifted keys. In this article, we propose a polar codes-based IR scheme with the frozen bits erasure strategy, where an equivalent transmission of sifted keys is conducted, so that the frozen bits in the decoding procedure is erased to 0. Thus, our IR scheme can be implemented efficiently without the assumption of true random numbers. Furthermore, we implement the proposed IR scheme with the fast simplified successive cancellation list decoder and its throughput reaches to 0.88Mbps with the yield of 0.8333, where the decoder list size is 16, the block size is 1Mb and the quantum bit error rate is 0.02.

Performance Analysis of Polar Codes Over Fox’s H-Fading Channels With RHI and IQI Impairments

This study examined the error probability performance of polar codes across Fox’s <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H$ </tex-math></inline-formula> -fading (FHF) channels while taking into account in-phase and quadrature-phase imbalance (IQI), as well as residual hardware impairments (RHI). This fading model generalizes most of the commonly used fading and turbulence models where Málaga- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {M}$ </tex-math></inline-formula> turbulence is taken into account as a specific case of FHF in this paper. Specifically, an upper bound on block error probability (BLER) for the successive cancellation (SC) decoding method of polar codes is derived in terms of the Fox’s <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H$ </tex-math></inline-formula> -function. According to numerical results, by lengthening polar codes and changing the properties of fading channels, it is possible to decrease the impact of system impairments on the error probability performance.

Modified SC Decoding for BDPSK Modulated Polar Codes

In this letter, we derive a modified successive cancellation (SC) decoder for binary differential shift keying (BDPSK) modulated polar codes. The proposed decoder takes the phase differences of adjacent even-indexed BDPSK symbols as additional inputs, and the simulations show that the proposed decoder could improve the error performance of polar coded BDPSK systems.

Fast Successive-Cancellation Decoding of 5G Parity-Check Polar Codes

The successive-cancellation (SC) decoding of polar codes suffers from long latency due to its serial processing nature. The fast SC (FSC) decoding was proposed to address this issue by applying parallel decoders to various special nodes in the binary decoding tree. In this letter, we explore multiple special nodes containing parity-check bits of 5G polar codes. The characteristics of these nodes are well investigated. The corresponding fast decoders are developed, which significantly reduce the decoding latency without degrading the error-correction performance.

An improved signal detection algorithm for a mining-purposed MIMO-OFDM IoT-based system

&lt;abstract&gt; &lt;p&gt;The coal mine internet of things (IoT) communication system is used for real-time monitoring of mining production to ensure the safety and reliability of personnel and equipment in the mine. To eliminate multipath fading in the process of wireless communication in mines, multiple-output multiplexing (MIMO) and orthogonal frequency division multiplexing (OFDM) technologies are introduced. In this paper, a wireless communication system architecture of IoT in mining based on MIMO-OFDM is constructed. Aiming to solve the problems of intersymbol interference and frequency selective fading at the receiver, an improved minimum mean square error ordered successive interferences cancellation (MMSE-OSIC) signal detection algorithm is proposed. First, the signal-to-interference plus noise ratio of the received signal is calculated and the calculation results are sorted. The lowest signal-to-noise ratio is selected as the weakest signal layer. Then, the MMSE-OSIC algorithm is used to extract all of the signals, except the weakest layer. Finally, a maximum likelihood (ML) algorithm is used to traverse the whole signal domain; the signal symbol with the smallest distance from the weakest signal layer is found as the original signal of the weakest signal layer, and it is combined with the signal detected by MMSE-OSIC; then, the final signal detection result is obtained. The simulation results show that, compared with three benchmark algorithms, the proposed MMSE-OSIC algorithm has better signal detection performance under the conditions of different modulation methods and different channel numbers.&lt;/p&gt; &lt;/abstract&gt;