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IEEE Symposium on Computers & Informatics (ISCI)

Performance Analysis of a CRC-aided BP Decoder in an AWGN Channel

Hamizan Suhaimi, Roslina Mohamad, Ezmin Abdullah, Nurain Izzati Shuhaimi

Published Oct 14, 2023• 1 citation

Abstract

Polar codes have emerged as a powerful error correction technique for reliable data transmission in various communication systems. The addition of cyclic redundancy check (CRC) can significantly enhance the reliability of polar codes by providing an additional layer of error detection and correction. Without CRC, a system will lack a mechanism that detects and corrects decoding errors. By appending CRC bits to the polar-encoded data, the receiver with a belief propagation (BP) decoder can perform CRC to verify the correctness of the received information, enabling error detection beyond the capabilities of polar codes alone. This paper mainly focuses on incorporating CRC codes into polar codes and evaluating performance after the codes were transmitted into an additive white Gaussian channel. The CRC-aided BP decoder was simulated using MATLAB. The block length, N, was set to 256 and 1024 bits, and CRC varied between 12 and 16 bits for each N. The decoder system was tested with different numbers of iterations (50, 100, 200, and 500) to verify the performance in terms of bit error rate and frame error rate. Overall, this research provides a comprehensive analysis of the utilization of CRC for polar codes, considering different CRC sizes, numbers of iterations for BP decoding, and various block lengths. The average number of iterations of 200 and increasing CRC shows the lowest error rate readings throughout an increase in noise. The use of 12 and 16 bits of CRC further enhances the decoder’s error detection and correction capability.

DOI: 10.1109/isci58771.2023.10391888

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Polar codes have emerged as a powerful error correction technique for reliable data transmission in various communication systems. The addition of cyclic redundancy check (CRC) can significantly enhance the reliability of polar codes by providing an additional layer of error detection and correction. Without CRC, a system will lack a mechanism that detects and corrects decoding errors. By appending CRC bits to the polar-encoded data, the receiver with a belief propagation (BP) decoder can perform CRC to verify the correctness of the received information, enabling error detection beyond the capabilities of polar codes alone. This paper mainly focuses on incorporating CRC codes into polar codes and evaluating performance after the codes were transmitted into an additive white Gaussian channel. The CRC-aided BP decoder was simulated using MATLAB. The block length, N, was set to 256 and 1024 bits, and CRC varied between 12 and 16 bits for each N. The decoder system was tested with different numbers of iterations (50, 100, 200, and 500) to verify the performance in terms of bit error rate and frame error rate. Overall, this research provides a comprehensive analysis of the utilization of CRC for polar codes, considering different CRC sizes, numbers of iterations for BP decoding, and various block lengths. The average number of iterations of 200 and increasing CRC shows the lowest error rate readings throughout an increase in noise. The use of 12 and 16 bits of CRC further enhances the decoder’s error detection and correction capability.

Polar codes are the first class of error correcting codes that provably achieve the channel capacity at infinite code length. They were selected for use in the fifth generation of cellular mobile communications (5G). In practical scenarios such as 5G, a cyclic redundancy check (CRC) is concatenated with polar codes to improve their finite length performance. This is mostly beneficial for sequential successive-cancellation list decoders. However, for parallel iterative belief propagation (BP) decoders, CRC is only used as an early stopping criterion with incremental error-correction performance improvement. In this paper, we first propose a CRC-polar BP (CPBP) decoder by exchanging the extrinsic information between the factor graph of the polar code and that of the CRC. We then propose a neural CPBP (NCPBP) algorithm which improves the CPBP decoder by introducing trainable normalizing weights on the concatenated factor graph. Our results on a 5G polar code of length 128 show that at the frame error rate of 10-5and with a maximum of 30 iterations, the error-correction performance of CPBP and NCPBP are approximately 0.25 dB and 0.5 dB better than that of the conventional CRC-aided BP decoder, respectively, while introducing almost no latency overhead.

Abstract In this letter, a modified message passing algorithm to prevent short cycles in the factor graph is proposed and combined with a relaxation technique to improve the performance of cyclic redundancy check aided belief propagation decoding for polar codes. Numerical results show that the proposed scheme can achieve a 0.7 dB coding gain, for the code rate of 0.5 at the block error rate of 10 −5 with a comparable average number of iterations over the pure belief propagation decoding scheme when the code block length is 1024.

Belief propagation (BP) has attracted increasing attention due to its high parallelism. In this letter, a correction method is proposed to improve the block error rate (BLER) performance of the BP decoder for polar codes so that it is comparable to that of the successive cancellation list (SCL) decoder. Compared with BP flipping (BPF) decoding, the proposed decoder focuses on prior knowledge of the code bits in the factor graph. Two strategies have been proposed to construct the correction set. The simulation results show that the BPC decoder exhibits a BLER similar to that of BPF decoder while reducing the average latency and complexity by half. In addition, the BLER of the BPC decoder can approach that of the cyclic redundancy check (CRC)-aided SCL (CA-SCL) decoder in regions with low and medium signal-to-noise ratios (SNRs).

Novel successive cancellation list (SCL) decoding schemes of Polar codes by exploiting the error correction (EC) capability of the cyclic redundancy check (CRC) codes are proposed in this paper. Different from the schemes in the literature that only considered the error detection capability of CRC, the proposed schemes aim to improve the performance of CRC-aided Polar decoding to the most by benefiting from the potential EC capability of CRC. Specifically, we first propose a basic decoding scheme, named CRC error-correction-aided successive cancellation list (CRC-EC-SCL) decoding, by introducing the look-up-table-based CRC EC method into the Polar SCL decoding process. Then, based on the basic scheme, we further propose the segmented-CRC-EC-SCL decoding scheme. By segmented-CRC, the information block is divided into multiple segments and each segment is attached with a shorter CRC sequence. By conducting the CRC-EC-aided coding for each of the segments, multiple erroneous bits can be corrected early and error propagation can, thus, be efficiently suppressed. The simulation results show that both of the proposed decoding schemes have a significant performance gain over the classic CRC-EC-SCL decoding scheme. Especially for the segmented-CRC-EC-SCL, the performance gain at a frame error rate of 10−4is about 0.4 dB. Moreover, the average computation complexity of the segmented-CRC-EC-SCL is close to the classic scheme.

Permutation decoding gained recent interest as it can exploit the symmetries of a code in a parallel fashion. Moreover, it has been shown that by viewing permuted polar codes as polar subcodes, the set of usable permutations in permutation decoding can be increased. We extend this idea to pre-transformed polar codes, such as cyclic redundancy check (CRC)-aided polar codes, which previously could not be decoded using permutations due to their lack of automorphisms. Using belief propagation (BP)-based subdecoders, we showcase a performance close to CRC-aided SCL (CA-SCL) decoding. The proposed algorithm outperforms the previously best performing iterative CRC-aided belief propagation list (CA-BPL) decoder both in error-rate performance and decoding latency.

Although iterative decoding of polar codes has recently made huge progress based on the idea of permuted factor graphs, it still suffers from a non-negligible performance degradation when compared to state-of-the-art CRC-aided successive cancellation list (CA-SCL) decoding. In this work, we show that iterative decoding of polar codes based on the belief propagation list (BPL) algorithm can approach the error-rate performance of CA-SCL decoding and, thus, can be efficiently used for decoding the standardized 5G polar codes. Rather than only utilizing the cyclic redundancy check (CRC) as a stopping condition (i.e., for error-detection), we also aim to benefit from the error-correction capabilities of the outer CRC code. For this, we develop two distinct soft-decision CRC decoding algorithms: a Bahl-Cocke-Jelinek-Raviv (BCJR)-based approach and a sum product algorithm (SPA)-based approach. Further, an optimized selection of permuted factor graphs is analyzed and shown to reduce the decoding complexity significantly. Finally, we benchmark the proposed CRC-aided belief propagation list (CA-BPL) decoding to state-of-the-art 5G polar codes under CA-SCL decoding and, thereby, showcase an error-rate performance not just close to the CA-SCL but also close to the maximum likelihood (ML) bound as estimated by ordered statistic decoding (OSD).