In the construction of polar code, the selection of frozen bits affects the error-correcting performance significantly. Several calculation-based algorithms have been proposed for general binary-input discrete memoryless channels (B-DMCs) like the additive white Gaussian noise (AWGN) channel. In this paper, a method for frozen bits selection based on Monte Carlo simulation and belief propagation (BP) decoding is proposed. The information bits are selected out one by one incrementally. The numerical results show that the proposed algorithm can effectively improve the bit error rate (BER) and frame error rate (FER) performance compared with the conventional selection method, especially in high signal noise ratio (SNR) region. Moreover, the algorithm can be used to construct polar codes with any rate through a complete iteration.
In the construction of polar code, the selection of frozen bits affects the error-correcting performance significantly. Several calculation-based algorithms have been proposed for general binary-input discrete memoryless channels (B-DMCs) like the additive white Gaussian noise (AWGN) channel. In this paper, a method for frozen bits selection based on Monte Carlo simulation and belief propagation (BP) decoding is proposed. The information bits are selected out one by one incrementally. The numerical results show that the proposed algorithm can effectively improve the bit error rate (BER) and frame error rate (FER) performance compared with the conventional selection method, especially in high signal noise ratio (SNR) region. Moreover, the algorithm can be used to construct polar codes with any rate through a complete iteration.
In this paper, we focus on the low-complexity construction method of polar codes with belief propagation (BP) decoder over additive white Gaussian noise (AWGN) channels. Firstly, the zero-noise BP decoding analysis method is proposed to determine the reliability of each bit channel via recursively calculating log-likelihood ratios (LLRs) of nodes from right to left in the factor graph. The final LLR can comprehensively reflect the structural characteristics of each bit channel. Then, the zero-noise LLR criterion based construction is introduced to select the frozen bit indices. Specifically, the lower the LLR value, the higher the priority of the corresponding bit selected as the frozen bit. Due to the assumed noise-free environment, the proposed zero-noise LLR criterion based construction is channel-independent. Finally, the simulation results show that, the zero-noise LLR criterion based construction can sort out the critical bit channels to load the frozen bits. When compared with other construction methods for BP decoder, such as the LLR-based bit-swapping construction and the simulation-based bit-swapping construction, the proposed zero-noise LLR criterion based construction not only has the best error correction ability over medium to high signal-to-noise ratio (SNR) region, but also exhibits the lowest construction complexity.
Polar codes have been recently proposed as the first low complexity class of codes that can provably achieve the capacity of symmetric binary-input memoryless channels. Here, we study the bit error rate performance of finite-length polar codes under Belief Propagation (BP) decoding. We analyze the stopping sets of polar codes and the size of the minimal stopping set, called “stopping distance”. Stopping sets, as they contribute to the decoding failure, play an important role in bit error rate and error floor performance of the code. We show that the stopping distance for binary polar codes, if carefully designed, grows as O(√N) where N is the code-length. We provide bit error rate (BER) simulations for polar codes over binary erasure and gaussian channels, showing no sign of error floor down to the BERs of 10-11. Our simulations asserts that while finite-length polar codes do not perform as good as LDPC codes in terms of bit error rate, they show superior error floor performance. Motivated by good error floor performance, we introduce a modified version of BP decoding employing a guessing algorithm to improve the BER performance of polar codes. Our simulations for this guessing algorithm show two orders of magnitude improvement over simple BP decoding for the binary erasure channel (BEC), and up to 0.3 dB improvement for the gaussian channel at BERs of 10-6.
Belief Propagation(BP) decoders of Polar Codes can run in parallel and have a remarkable throughput achievement at a cost of performance degradation, when compared with other decoding algorithms of polar codes. The existing improved algorithms still have a gap to be optimized in performance, complexity or latency. This paper proposes a modified BP flipping algorithm by introducing random penalty items into the reliability metrics of bits to correct un-converged errors under the type of noise with Gaussian distributions considered. Simulation results illustrate that the proposed penalizing method can exhibit significant performance gains over the original BP and other three existing BP flipping decoding algorithms with different blocks in the additive white Gaussian noise channel(AWGN) under a lower decoding latency. This shows the proposed scheme identifies the error-prone bits more efficiently.
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 have become the channel coding scheme for control channel of enhanced mobile broadband in the 5G communication systems. Belief propagation (BP) decoding of polar codes has advantages of low decoding latency and high parallelism but achieves worse bit error ratio (BER) performance compared with the successive cancellation list (SCL) decoding scheme. In this paper, an improved BP list (IBPL) decoding algorithm is proposed with comparable BER performance to SCL algoritm. Firstly, the optimal permuted factor graph is analyzed for polar codes, which improves the performance of the BP decoder without path extension. Furthermore, based on the optimal graph, the bit metric and decoding path metric are proposed to extend and prune the decoding path. The proposed IBPL decoder is focused on not only the permutation of polar codes but also the reliabilities of decoded codewords during each iteration of BP decoding, which has a more accurate decoding path list. The simulation results show that the proposed IBPL decoder improves the BER performance compared with the original BP decoder significantly, and can approach the performance of the SCL decoder at low signal to noise ratio regions.
The bit-flip method has been successfully applied to the successive cancellation (SC) decoder to improve the block error rate (BLER) performance for polar codes in the finite code length region. However, due to the sequential decoding, the SC decoder inherently suffers from longer decoding latency than that of the belief propagation (BP) decoder with efficient early stopping criterion. It is natural to ask how to perform bit-flip in a polar BP decoder. In this paper, bit-flip is introduced into the BP decoder for polar codes. The idea of critical set (CS), that is, originally proposed by Zhang et al. for identifying unreliable bits in a SC bit-flip decoder, is extended to the BP decoder here. After revealing the relationship between CS and the incorrect BP decoding results, critical set with order ω (CS-ω) is constructed to identify unreliable bit decisions in polar BP decoding. The simulation results demonstrate that compared with the conventional BP decoder, the BLER of the proposed bit-flip decoder can achieve significant signal-to-noise ratio (SNR) gain which is comparable to that of a cyclic redundancy check-aided SC list decoder with a moderate list size. In addition, the decoding latency of the proposed BP bit-flip decoder is only slightly higher than that of the conventional BP decoder in the medium and high SNR regions.