Performance Of Polar Codes Research Articles

Performance of Adaptive Bit-Interleaved Polar Coded Modulation in FSOC System

This paper proposes an adaptive bit-interleaved polar coded modulation (A-BIPCM) method based on minimum logarithmic upper bound weight (MLUW). It is designed to reduce the fading effects and long string of bit error interference caused by atmospheric turbulence in free-space optical communications (FSOC). To assess the effectiveness of this method across turbulent channels of varying intensities, we conducted an evaluation of the bit error rate (BER) performance of polar codes in turbulent channels. The results demonstrate significant performance improvements provided by the A-BIPCM method compared to conventional polar code encoding and decoding. Specifically, under weak, moderate, and strong turbulence conditions, the A-BIPCM method achieves performance gains of 0.96 dB, 1.66 dB, and 1.35 dB, respectively. Additionally, an experimental verification platform for FSOC employing intensity modulation direct detection (IM/DD) with an atmospheric turbulence simulation channel, is established in this work. When the optical power of the detector is −16 dBm, the traditional polar code encoding and decoding performance at BER = 2.36 × 10−5, whereas the A-BIPCM scheme exhibits a significantly higher performance at BER = 2.11 × 10−6. The BER has been improved by representing an order of magnitude.

Sparse graph neural network aided efficient decoder for polar codes under bursty interference

In this paper, a sparse graph neural network-aided (SGNN-aided) decoder is proposed for improving the decoding performance of polar codes under bursty interference. Firstly, a sparse factor graph is constructed using the encoding characteristic to achieve high-throughput polar decoding. To further improve the decoding performance, a residual gated bipartite graph neural network is designed for updating embedding vectors of heterogeneous nodes based on a bidirectional message passing neural network. This framework exploits gated recurrent units and residual blocks to address the gradient disappearance in deep graph recurrent neural networks. Finally, predictions are generated by feeding the embedding vectors into a readout module. Simulation results are shown that the proposed decoder is more robust than the existing ones in the presence of bursty interference and exhibits high universality.

A Note on the Mixing Factor of Polar Codes.

Over binary-input memoryless symmetric (BMS) channels, the performance of polar codes under successive cancellation list (SCL) decoding can approach maximum likelihood (ML) algorithm when the list size L is greater than or equal to 2MF, where MF, known as mixing factor of code, represents the number of information bits before the last frozen bit. Recently, Yao et al. showed the upper bound of the mixing factor of decreasing monomial codes with length n=2m and rate R≤12 when m is an odd number; moreover, this bound is reachable. Herein, we obtain an achievable upper bound in the case of an even number. Further, we propose a new decoding hard-decision rule beyond the last frozen bit of polar codes under BMS channels.

New Structure of Channel Coding: Serial Concatenation of Polar Codes

In this paper, we introduce a new coding and decoding structure for enhancing the reliability and performance of polar codes, specifically at low error rates. We achieve this by concatenating two polar codes in series to create robust error-correcting codes. The primary objective here is to optimize the behavior of individual elementary codes within polar codes. In this structure, we incorporate interleaving, a technique that rearranges bits to maximize the separation between originally neighboring symbols. This rearrangement is instrumental in converting error clusters into distributed errors across the entire sequence. To evaluate their performance, we proposed to model a communication system with seven components: an information source, a channel encoder, a modulator, a channel, a demodulator, a channel decoder, and a destination. This work focuses on evaluating the bit error rate (BER) of codes for different block lengths and code rates. Next, we compare the bit error rate (BER) performance between our proposed method and polar codes.

Performance of polar codes with low complexity cyclic redundancy check aided bit‐flipping for successive cancellation list decoding over power line channels

SummaryPower lines have complex transmission characteristics and noise interference when used as information transmission channel, thus limiting their data‐carrying rate and reliability. Impulsive noise is the most dominant factor degrading the performance of communication systems over power lines. To combat the impulsive noise, polar codes are introduced into the orthogonal frequency division multiplexing (OFDM)‐based power line communication (PLC) system. Because the successive cancelation list bit‐flip (SCLF) decoding algorithm of polar codes has high decoding complexity, a low complexity segmented cyclic redundancy check (CRC)‐aided SCLF decoding is proposed to improve the decoding performance. The decoding process can be terminated early with segmented CRC, which can reduce the bit‐flip decoding complexity significantly with short and medium block lengths. Experimental results demonstrate that for various configurations, the proposed decoder outperforms the traditional CRC‐aided SCL decoder in terms of decoding performance and complexity, and the suggested polar codes‐based OFDM‐based PLC system can reduce the bit error rate of the system in the presence of impulsive noise.

Research on different channel coding methods for 5G communication

The implementation of channel coding techniques has the potential to yield coding gains, augment the capacity of the communication system, and significantly enhance the performance of the communication system. The quest for a code possessing a channel capacity surpassing the Shannon limit has been a longstanding pursuit among researchers. The utilization of Polar codes, a coding technique proposed by E. Arikan in 2007, was initially implemented. Theoretical demonstrations have shown that the Shannon limit can be achieved for binary, discrete, memoryless channels through the utilization of a code with a straightforward algorithm. This has resulted in notable progress within the realm of channel coding. As a new coding technology, polar codes have attracted extensive attention in the wireless communication field and become one of the most attractive research hotspots in the coding field. This paper systematically expounds polar codes, analyzes the coding and decoding principles of polar codes, and compares them with Turbo codes and LDPC codes. In this study, polar code coding and decoding are simulated and implemented, and the impact of coding block length, coding rate, and iteration time variations on the performance of polar codes is explored. Through simulation and comparison, the performance, benefits, and drawbacks of these three codes are compared with the polar codes, turbo codes, and LDPC codes.

Polar-Coded Transmission over 7.8-km Terrestrial Free-Space Optical Links

Free-space optical (FSO) communications can offer high-capacity transmission owing to the properties of the laser beams. However, performance degradation caused by atmospheric turbulence is an urgent issue. Recently, the application of polar codes, which can provide capacity-achieving error-correcting performance with low computational cost for decoding, to FSO communications has been studied. However, long-distance and real-field experiments have not been conducted in these studies. To the best of our knowledge, this study is the first to present the experimental results of polar-coded transmission over 7.8-km FSO links. Using experimental data, we investigated the performance of polar codes over atmospheric channels, including their superiority to regular low-density parity-check codes. We expect that our results will offer a path toward the application of polar codes in high-speed optical communication networks including satellites.

Concatenated turbo polar-convolutional codes based on soft cancellation algorithm

This paper presents the Parallel and Serial Concatenated Turbo Polar-Convolutional Codes (CTPCCs) to improve the performance of polar codes in the finite code length regime. Both proposed concatenated codes use the Systematic Polar Code (SPC) and Recursive Systematic Convolutional (RSC) code as constituent codes. The iterative decoding algorithms of the proposed systems consist of a Soft Cancellation (SCAN) decoder with a single internal iteration for the SPC and SOVA or LogMap decoder for the RSC code. The simulation results show that the performance of the Parallel Concatenated Turbo Polar-Convolutional Code (PCTPCC) at low signal-to-noise ratios (SNRs) is better than that of the Serial Concatenated Turbo Polar-Convolutional Code (SCTPCC) and also better than the performance of other Turbo Polar Codes (TPCs). At BER=10−2, the proposed PCTPCC with SOVA and LogMap algorithms has about 0.3 dB and 0.25 dB performance gains over the SCTPCC models, respectively. In the high SNR regions, the performance of the SCTPCC with the LogMap algorithm significantly outperforms that of other models. At BER=10−6, the proposed SCTPCC models with SOVA and LogMap algorithms outperform the PCTPCC model by gains of 0.3 dB and 0.25 dB, respectively. The PCTPCC model with the SOVA decoder is superior to other models in terms of complexity.

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.

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.

A Weighted Sum Based Construction of PAC Codes

The polarization-adjusted convolutional (PAC) codes concatenate an outer convolutional transform with an inner polar transform to improve the error-correction performance of polar codes. In the short-to-medium codeword length regime, it can approach the normal approximation (NA) bound. However, its optimal rate-profile remains unknown. This letter proposes a new construction for PAC codes based on a weighted sum (WS) metric, which considers both the cutoff rates and the utilizations of the polarized subchannels. In comparison with the existing Reed-Muller (RM) rate-profiles, it can yield a flexible rate. By adjusting the design signal-to-noise ratio (SNR), an appropriate tradeoff between the decoding performance and complexity can be achieved. Simulation results show the designed codes can outperform the ones that are designed by the existing techniques.

Adaptive List Flip Decoder for Polar Codes with High-Order Error Correction Capability and a Simplified Flip Metric.

Designing an efficient decoder is an effective way to improve the performance of polar codes with limited code length. List flip decoders have received attention due to their good performance trade-off between list decoders and flip decoders. In particular, the newly proposed dynamic successive cancellation list flip (D-SCLF) decoder employs a new flip metric to effectively correct high-order errors and thus enhances the performance potential of present list flip decoders. However, this flip metric introduces extra exponential and logarithmic operations, and the number of these operations rises exponentially with the increase in the order of error correction and the number of information bits, which then limits its application value. Therefore, we designed an adaptive list flip (ALF) decoder with a new heuristic simplified flip metric, which replaces these extra nonlinear operations in the original flip metric with linear operations. Simulation results show that the simplified flip metric does not reduce the performance of the D-SCLF decoder. Moreover, based on the in-depth theoretical analyses of the combination of the adaptive list and the list flip decoders, the ALF decoder adopts the adaptive list to further reduce the average complexity.

THE EFFECT OF DIFFERENT DECODING TECHNIQUES WITH GAUSSIAN APPROXIMATION ON THE PERFORMANCE OF POLAR CODES

In the last ten years, polar code research has piqued the interest of firms and researchers, particularly in the communication industry. Polar codes have been utilised as a coding method for the fifth-generation wireless standard (5G). However, the polar decoder does not adequately correct errors in successive cancellation (SC) decoding when dealing with short- to intermediate-length codes. However, SC decoding can correct errors more efficiently by using sequential cancellation list (SCL) decoding. The main drawback of SCL is its higher cost due to computational complexity and throughput. The present research investigates the effect of Gaussian approximation (GA) and different decoding approaches on the performance of polar codes. First, SC and SCL decoders are developed utilising amplitude shift keying modulation; a decoder using GA is then integrated. According to simulation data, the SCL, both with and without GA, exhibits a better block error rate (BLER) than SC. The maximum difference between the SCL decoder and SC decoder is 0.6 dB at BLER=0.1 for N=2048. Furthermore, at BLER=5.6 x 10-6, the SCL decoder with GA performs better than the SC decoder for block lengths, N=1024, with a maximum difference of 2.72 dB. When the polar decoder with GA is utilised, enhancements are observed in polar code performance for various list sizes and block lengths, although time complexity is increased.

Performance Improvement of Polar Codes via UEP Product Coding

Aiming to improve the error correction performance of polar codes, researchers have proposed employing the product coding structure involving RS codes of different rates in the horizontal direction and the short polar codewords along the vertical direction. However, there is no efficient algorithm optimizing the rate allocation of RS codes. In order to address this problem, this paper provides an analytical formulation by maximizing the number of correctly decoded information bits. The proposed rate allocation formulation takes the channel statistics into consideration, and we further find that the number of different RS code rates could be limited to a small value. By doing so, the complexities of both the rate allocation optimization and the iterative product decoding could be reduced. Simulation results demonstrate the superiority of RS-polar product codes with the proposed rate allocation method over both additive white Gaussian noise channels and Gilbert–Elliott channels. When the inner polar codes of rate 1/2 are utilized, the optimized RS-polar product codes of low-to-medium rates significantly outperform the successive cancellation decoding of long polar codes, in the regime of a frame error rate ≥10−3.

Performance of Polar Codes over Generalized Correlated Fading Channels

Polar coding, as introduced by Arikan in 2009, is an error correcting code scheme that uses the polarization technique to obtain almost noise-free channels that are suitable for transmission. Polar codes are being widely studied due to their applications on the 5th Generation (5G) of mobile communication systems. In 5G communications, the wireless channel is subject to fading that can significantly degrade the quality of the transmitted signals. In this context, the generalized fading models are useful due to their capacity to model a variety of channel conditions ranging from moderate to severe fading by adjusting some parameters. In this work, the authors investigate the performance of polar codes considering the transmission over generalized fading channels. Additionally, the effect of channel correlation is analyzed and a random interleaver is proposed to mitigate the negative impact of the correlation on the system performance. For the system with generalized fading modeled by eta-mu distribution and Cyclic Redundancy Check (CRC) Aided Successive Cancellation List (CA-SCL) decoding, the scenario with correlated fading channel and interleaving shows a performance improvement compared to the system without interleaving. In this case, a gain of approximately 3.5 dB is obtained considering a Bit Error Rate (BER) of 10^{-2}. Considering the alpha-mu (alpha=2 and mu=1) and the kappa-mu&nbsp; (kappa=mu=2) fading channels, &nbsp;the uncorrelated systems outperform the correlated ones. Among them, the interleaver improves the performance of the systems. For a BER of 10^{-2}, as an example, the use of fading correlated channels and interleaver shows a performance improvement of 12 dB and 0.5 dB for the alpha-mu and kappa-mu fading channels, respectively.

On Sequential Decoding Metric Function of Polarization-Adjusted Convolutional (PAC) Codes

In this paper, we present a sequential decoding metric function, which leads to significantly improved computational complexity while maintaining the superiority of polarization-adjusted convolutional (PAC) codes’ error-correction performance. With the proposed metric function, the PAC codes’ decoding computational complexity is comparable to the computational complexity of sequential decoding of conventional convolutional codes (CCs). Moreover, simulation results show an improvement in the error-correction performance of low rate PAC codes when using the proposed metric function. Simulation results also show that using the proposed metric, the upper bound on the PAC codes’ computational complexity has a Pareto distribution. To reduce the worst-case latency of PAC sequential decoder, we limit the number of searches performed by sequential decoder. The results show that for PAC codes of length 128, search-limited sequential decoding can achieve an error-correction performance close to the error-correction performance of polar codes with successive cancellation list decoding with list size 64 and CRC length 11 with considerably less computational complexity.

List Decoding of Arıkan's PAC Codes.

Polar coding gives rise to the first explicit family of codes that provably achieve capacity with efficient encoding and decoding for a wide range of channels. However, its performance at short blocklengths under standard successive cancellation decoding is far from optimal. A well-known way to improve the performance of polar codes at short blocklengths is CRC precoding followed by successive-cancellation list decoding. This approach, along with various refinements thereof, has largely remained the state of the art in polar coding since it was introduced in 2011. Recently, Arıkan presented a new polar coding scheme, which he called polarization-adjusted convolutional (PAC) codes. At short blocklengths, such codes offer a dramatic improvement in performance as compared to CRC-aided list decoding of conventional polar codes. PAC codes are based primarily upon the following main ideas: replacing CRC codes with convolutional precoding (under appropriate rate profiling) and replacing list decoding by sequential decoding. One of our primary goals in this paper is to answer the following question: is sequential decoding essential for the superior performance of PAC codes? We show that similar performance can be achieved using list decoding when the list size L is moderately large (say, ). List decoding has distinct advantages over sequential decoding in certain scenarios, such as low-SNR regimes or situations where the worst-case complexity/latency is the primary constraint. Another objective is to provide some insights into the remarkable performance of PAC codes. We first observe that both sequential decoding and list decoding of PAC codes closely match ML decoding thereof. We then estimate the number of low weight codewords in PAC codes, and use these estimates to approximate the union bound on their performance. These results indicate that PAC codes are superior to both polar codes and Reed–Muller codes. We also consider random time-varying convolutional precoding for PAC codes, and observe that this scheme achieves the same superior performance with constraint length as low as .

Error correction in data storage systems using polar codes

This paper investigates error correction in two-dimensional (2-D) intersymbol interference (ISI) channels using polar codes. 2-D channels offer high data storage capacity compared to traditional one-dimensional (1-D) channels but suffer from greater ISI effects. Error correction codes can be used to mitigate the effects of ISI in 2-D data storage systems and improve their performance and reliability. Polar codes achieve the symmetric capacity for the class of binary-input discrete memoryless channels (B-DMCs). A post-processing technique is proposed to improve the performance of polar codes in the 2-D ISI channel. The simplified successive cancellation (SSC) polar decoder is useful for data storage systems owing to its good waterfall region performance, low error floors and good throughput rates and is therefore adopted for the study. Simulation results indicate that the proposed technique has an error correction performance comparable to some existing decoders and has better performance than other decoding techniques.

Bidirekciono stek dekodovanje polarnih kodova

Introduction/purpose: The paper introduces a reduced latency stack decoding algorithm of polar codes, inspired by the bidirectional stack decoding of convolutional codes and based on the folding technique. Methods: The stack decoding algorithm (also known as stack search) that is useful for decoding tree codes, the list decoding technique introduced by Peter Elias and the folding technique for polar codes which is used to reduce the latency of the decoding algorithm. The simulation was done using the Monte Carlo procedure. Results: A new polar code decoding algorithm, suitable for parallel implementation, is developed and the simulation results are presented. Conclusions: Polar codes are a class of capacity achieving codes that have been adopted as the main coding scheme for control channels in 5G New Radio. The main decoding algorithm for polar codes is the successive cancellation decoder. This algorithm performs well at large blocklengths with a low complexity, but has very low reliability at short and medium blocklengths. Several decoding algorithms have been proposed in order to improve the error correcting performance of polar codes. The successive cancellation list decoder, in conjunction with a cyclic redundancy check, provides very good error-correction performance, but at the cost of a high implementation complexity. The successive cancellation stack decoder provides similar error-correction performance at a lower complexity. Future machine-type and ultra reliable low latency communication applications require high-speed low latency decoding algorithms with good error correcting performance. In this paper, we propose a novel decoding algorithm, inspired by the bidirectional stack decoding of classical convolutional codes, with reduced latency that achieves similar performance as the classical successive cancellation list and successive cancellation stack decoding algorithms. The results are presented analytically and verified by simulation.

Polar Coded Diversity on Block Fading Channels Via Polar Spectrum

Due to the advantage of capacity-achieving, polar codes have been extended to the block fading channel whereas most constructions involve complex iterative-calculation. In this paper, we establish a systematic framework to analyze the error performance of polar codes in the case of block mapping and random mapping. For both the mappings, by introducing the new concept, named split polar spectrum, we derive the upper bound on the error probability of polarized channel which explicitly reveals the relationship between the diversity order L and the block-wise weight distribution of the codeword. For the special case L=2 in the block mapping, we design the enumeration algorithm to calculate the exact split polar spectrum based on the general MacWilliams identities. For arbitrary diversity order in the random mapping, with the help of uniform interleaving, we derive the approximate split polar spectrum by combining the polar spectrum and the probability of fading pattern for a specific weight. Furthermore, we propose the design criteria to construct polar codes over the block fading channel. The full diversity criterion is the primary target so as to achieve the diversity gain and the product distance criterion requires to maximize the product of the block-wise Hamming distance whereby obtain the coding gain. Guided by these design criteria, the construction metric, named polarized diversity weight (PDW) is proposed to design the polar codes in both mappings. Such a simple metric can construct polar codes with similar or better performance over those based on traditional methods in block fading channel.