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IEEE Access

Low Complexity Early Stopping Belief Propagation Decoder for Polar Codes

Chungsu Lee, Chansoo Park, Sungyeol Back, Wangrok Oh

Published Jan 1, 2024• 4 citations
Publisher

Abstract

Belief propagation is one of low latency decoding algorithms for polar codes but it requires relatively high decoding complexity due to its inherent iterative decoding nature. To remedy the problem, early stopping belief propagation algorithm was proposed. Unfortunately, it requires additional complexity to test stopping criterion at each end of decoder iterations. In this paper, we propose a low complexity early stopping belief propagation decoder for polar codes which requires lower complexities not only for stopping criterion test but also for decoding by reducing the number of message updates within the decoder without any performance loss.

DOI
10.1109/access.2024.3402662

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Belief propagation is one of low latency decoding algorithms for polar codes but it requires relatively high decoding complexity due to its inherent iterative decoding nature. To remedy the problem, early stopping belief propagation algorithm was proposed. Unfortunately, it requires additional complexity to test stopping criterion at each end of decoder iterations. In this paper, we propose a low complexity early stopping belief propagation decoder for polar codes which requires lower complexities not only for stopping criterion test but also for decoding by reducing the number of message updates within the decoder without any performance loss.
Belief propagation (BP) algorithm of polar codes suffers from very high computational complexity when the iteration number becomes larger. In this paper, an early stopping criterion is proposed to avoid unnecessary iterations efficiently. The proposed only needs to detect whether the proportion of unchanged soft-output messages in a detection space exceeds a threshold. The proposed iteration can achieve a desired trade-off between complexity and decoding performance. Simulation results have shown that when the maximum iteration number is 40, the flooding BP (FO-BP) decoder with the proposed criterion can reduce 82.17% of iterations at 3.5dB compared with the FO-BP decoder without early stopping criterion. The proposed criterion costs Nl32 additions and Nl32 comparisons, where N indicates the code length, which effectively reduces the computational complexity.
In order to reduce the decoding latency, a new early stopping criterion is proposed for belief propagation (BP) decoding. A kind of special processing elements (PEs) of BP decoder called frozen and information PE (FIPE) is selected to predict whether decoding is successful or not. Statistics indicate that FIPE can be considered reliable when the frozen bit is decoded successfully. The proposed criterion is based on the fact that the number of reliable FIPE increase along with iterations and the variation trend is approximate to the ratio of correct information bits. In the term of hardware complexity, the proposed method has a linear correlation with the number of FIPE. This criterion consumes only ‘xor’ and ‘or’ gates to check stopping condition. Simulation results show that the proposed criterion achieves lower latency than Worst Information Bits (WIB) and Frozen Bit Error Rate (FBER) without BLER degradation compared with fixed and G-matrix. Compared with WIB, FBER, Best Frozen Bits (BFB) criterion, the proposed criterion has the lowest hardware complexity.
The belief propagation algorithm is desirable for a polar code based decoder, because of the potentially low latency and the ability of integration in digital signal processing units or other multi-core processor systems to parallelize the computations. Although belief propagation polar code decoder algorithms have the ability for a highly parallelized imple-mentation, the algorithms require many iterations to achieve a comparable frame error rate and bit error rate with respect to a successive cancellation polar code algorithm. The iterative nature of the belief propagation algorithms also result in a higher computational complexity, i.e. O(IN(2log_{2} N-1)) compared to the computational complexity O(Nlog_{2}N) of the successive cancellation decoder algorithm. In this paper we propose several simplifications for a simplified belief propagation algorithm for polar code decoders, where the arithmetic complexity of the nodes is reduced. The proposed belief propagation algorithm shows preliminary results of a net reduction of the arithmetic complexity of ≈ 13%. This reduction is a result of the reduced number of arithmetic operations, i.e., additions, compares, and multiplications, without a lost in error-correcting performance.
The belief propagation algorithm is one of the preferred algorithms for a polar code based decoder that can potentially achieve the lowest latency. Although belief propagation polar code algorithms have the ability for a highly parallelized implementation, they require more iterations to achieve a comparable frame error rate and bit error rate compared to the successive cancellation polar code algorithm. The iterative nature of the belief propagation algorithms also results in a higher computational complexity, i.e. O(N(2log2N-1)) compared to the computational complexity O(Nlog2N) of the successive cancellation decoder algorithm. In this paper we propose several software implementations of belief propagation polar code decoders using the computational complexity reduced belief propagation algorithm and the enhanced loop-weakened belief propagation algorithm, to increase the number of decoded codes per second. Compared to a baseline belief propagation implementation, our proposed radix-2 and radix-4 implementations increase the throughput by approximately 15.4% and 22.04% respectively. This gain is the result of a reduction of arithmetic operations, i.e., additions, compares, and multiplications, which is obtained without a loss in error-correcting performance.
The error correction performance of Belief Propagation (BP) decoding for polar codes is satisfactory compared with the Successive Cancellation (SC) decoding. Nevertheless, it has to complete a fixed number of iterations, which results in high computational complexity. This necessitates an intelligent identification of successful BP decoding for early termination of the decoding process to avoid unnecessary iterations and minimize the computational complexity of BP decoding. This paper proposes a hybrid technique that combines the “parity-check” with the “G-matrix” to reduce the computational complexity of BP decoder for polar codes. The proposed hybrid technique takes advantage of the parity-check to intelligently identify the valid codeword at an early stage and terminate the BP decoding process, which minimizes the overhead of the G-matrix and reduces the computational complexity of BP decoding. We explore a detailed mechanism incorporating the parity bits as outer code and prove that the proposed hybrid technique minimizes the computational complexity while preserving the BP error correction performance. Moreover, mathematical formulation for the proposed hybrid technique that minimizes the computation cost of the G-matrix is elaborated. The performance of the proposed hybrid technique is validated by comparing it with the state-of-the-art early stopping criteria for BP decoding. Simulation results show that the proposed hybrid technique reduces the iterations by about 90% of BP decoding in a high Signal-to-Noise Ratio (SNR) (i.e., 3.5 ∼ 4 ​dB), and approaches the error correction performance of G-matrix and conventional BP decoder for polar codes.
In this paper, a belief propagation (BP) decoding algorithm with early termination is proposed for polar code. The early termination is accomplished using the syndrome generated by the parity-check matrix, which can be transferred from the generator matrix. With the assistance of the early termination, the BP decoder can provide not only a better bit error rate performance but also significantly reduce the decoding complexity. For (1024, 512) polar codes, the average numbers of iterations can be reduced from 80 to 23 when Eb/N0=3 dB. The reduction in the computation complexity is about 70%.