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arXiv:1309.3173 · IET Communications

Low complexity list successive cancellation decoding of polar codes

Congzhe Cao, Zesong Fei, Jinhong Yuan, Jingming Kuang

Published Sep 12, 2014• 26 citations
arXivHTMLPDF

Abstract

The authors propose a low complexity list successive cancellation (LCLSC) decoding algorithm, where the advantages of the successive cancellation (SC) decoding and the list successive cancellation (LSC) decoding are both considered. In the proposed decoding, SC decoding instead of LSC decoding is implemented when all information bits from bad subchannels are received reliably. While the reliability of each information bit is estimated by its likelihood ratio (LR), the bit channel quality is measured via its Bhattacharyya parameter. To achieve this goal, the authors introduce two thresholds: LR threshold and Bhattacharyya parameter threshold. Also, the methods to determine them are both elaborated. The numerical results suggest that the complexity of LCLSC decoding is much lower than LSC decoding and can be close to that of SC decoding, while the error performance is almost equal to that of LSC decoding. Especially, when the code rate is in low region, the advantage of our decoding is more obvious.

DOI
10.1049/iet-com.2014.0227

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The authors propose a low complexity list successive cancellation (LCLSC) decoding algorithm, where the advantages of the successive cancellation (SC) decoding and the list successive cancellation (LSC) decoding are both considered. In the proposed decoding, SC decoding instead of LSC decoding is implemented when all information bits from bad subchannels are received reliably. While the reliability of each information bit is estimated by its likelihood ratio (LR), the bit channel quality is measured via its Bhattacharyya parameter. To achieve this goal, the authors introduce two thresholds: LR threshold and Bhattacharyya parameter threshold. Also, the methods to determine them are both elaborated. The numerical results suggest that the complexity of LCLSC decoding is much lower than LSC decoding and can be close to that of SC decoding, while the error performance is almost equal to that of LSC decoding. Especially, when the code rate is in low region, the advantage of our decoding is more obvious.
Successive cancellation (SC) decoding algorithm and successive cancellation list (SCL) decoding algorithm are two common decoding methods for polar codes. The SC decoding algorithm has low complexity, but in its decoding process, the bits with small channel indices are used to assist the decoding of the bits with large channel indices. Therefore, the bit being decoded depends on the bits that have already been decoded which causes error propagation. On the basis of SC decoding, the SCL decoding algorithm overcomes the problem of error propagation by widening the path search width and retaining multiple paths at the same time in the decoding process. However, the complexity of SCL decoding is higher than that of SC decoding. Without changing the performance of the SCL decoding algorithm, we propose a low-complexity SCL decoding algorithm based on path metric (PM). After the last frozen bit is decoded, the path with the smallest PM is selected, and the subsequent nodes adopt SC decoding. Through this combination of SCL decoding and SC decoding, the complexity of the SCL decoding algorithm can be effectively reduced while the SCL decoding performance remains unchanged, especially in the case of high code rates.
A two-part successive syndrome-check decoding of polar codes is proposed with the first part successively refining the received codeword and the second part checking its syndrome. A new formulation of the successive-cancellation (SC) decoding algorithm is presented that allows for successively refining the received codeword by comparing the log-likelihood ratio value of a frozen bit with its predefined value. The syndrome of the refined received codeword is then checked for possible errors. In case there are no errors, the decoding process is terminated. Otherwise, the decoder continues to refine the received code-word. The proposed method is extended to the case of SC list (SCL) decoding by terminating the decoding process when the syndrome of the best candidate in the list indicates no errors. Simulation results show that the proposed method reduces the time-complexity of SC and SCL decoders and their fast variants, especially at high signal-to-noise ratios.
Representing continuous values by streams of random binary bits, stochastic decoding has shown advantages in both hardware efficiency and fault tolerance, therefore has been widely adopted by iterative decoding of error correction codes such as low-density parity-check (LDPC) codes and so on. Recently, polar codes, the first codes that can provably achieve the capacity of symmetric binary-input discrete memoryless channels (B-DMCs), have drawn a lot of attentions from both academia and industry. Although, polar codes with list successive cancellation (SC) decoding can outperform several best-known LDPC codes even within high error-rate regions, the linearly increasing hardware complexity makes its efficient implementation difficult. To this end, the stochastic list SC polar decoding algorithm is proposed in this paper to provide a good tradeoff between performance and complexity. In order to increase the decoding performance of stochastic list SC polar decoder, doubling probability approach is presented. The corresponding hardware architecture is also given. The approximate doubling approach is employed to facilitate the efficient implementation. Implementation results have shown that the proposed stochastic list SC polar decoder can achieve a good trade-off between performance and complexity.
Polar codes are the first provably capacity-achieving novel channel codes. Successive cancellation list (SCL) decoding algorithm has a performance close to maximum-likelihood decoding when code length is finite, though the complexity is high. To reduce the complexity of SCL, a partial path expansion with segmented check and pruning SCL (PPE-SCP-SCL) algorithm is proposed in this paper. Based on the successive decoding structure, if the decoding result of a bit is judged to be reliable enough, then a hard decision is made directly and no path expansion is necessary. Moreover, several parity-check points are introduced to perform segmented parity-check during the decoding process, and if the check fails, the current path is directly abandoned. The rationality of the operations is theoretically verified, and numerical simulations also verify that the proposed algorithm can significantly reduce the complexity of decoding with trivial loss of error performance.
Successive cancellation list (SCL) decoding algorithm for polar codes is one of the best decoding algorithms due to its excellent decoding performance. The decoding complexity of conventional SCL decoding algorithm is O(LN log N). The decoding complexity will increase as the size of the list increases. To solve this problem, we introduce an improved SCL decoding algorithm based on the sequential probability ratio test (SPRT). The proposed algorithm use double thresholds to decode the polar codes instead of single threshold decision used by the conventional SCL algorithm. Numerical results have shown that for (1024, 512) polar code, the proposed SPRT-SCL decoder can achieve a reduction in decoding complexity, and keep the similar decoding performance compared with conventional SCL decoder.
A method for efficiently successive cancellation (SC) decoding of polar codes with high-dimensional linear binary kernels (HDLBK) is presented and analyzed. We devise a lll-expressions method which can obtain simplified recursive formulas of SC decoder in likelihood ratio form for arbitrary linear binary kernels to reduce the complexity of corresponding SC decoder. By considering the bit-channel transition probabilities WG(⋅)(⋅∣0)W_{G}^{(\cdot)}(\cdot|0)WG(⋅)​(⋅∣0) and WG(⋅)(⋅∣1)W_{G}^{(\cdot)}(\cdot|1)WG(⋅)​(⋅∣1) separately, a WWW-expressions method is proposed to further reduce the complexity of HDLBK based SC decoder. For a m×mm\times mm×m binary kernel, the complexity of straightforward SC decoder is O(2mNlog⁡N)O(2^{m}N\log N)O(2mNlogN). With WWW-expressions, we reduce the complexity of straightforward SC decoder to O(m2Nlog⁡N)O(m^{2}N\log N)O(m2NlogN) when m≤16m\leq 16m≤16. Simulation results show that 16×1616\times1616×16 kernel polar codes offer significant advantages in terms of error performances compared with 2×22\times22×2 kernel polar codes under SC and list SC decoders.