Progressive Bit-Flipping Decoding of Polar Codes: A Critical-Set Based Tree Search Approach
08 Oct 2018-IEEE Access (Institute of Electrical and Electronics Engineers (IEEE))-Vol. 6, pp 57738-57750
TL;DR: Numerical results show that the proposed ET-bit-flipping decoders can provide almost the same BLER performance as the state-of-the-art cyclic redundancy check-aided SC list decmoders, with an average computational complexity and decoding latency similar to that of the SC decoder at medium to a high SNR regime.
Abstract: In successive cancellation (SC) polar decoding, an incorrect estimate of any prior unfrozen bit may bring about severe error propagation in the following decoding, and thus it is desirable to find out and correct an error as early as possible. In this paper, we investigate a progressive bit-flipping decoder which corrects at most $L$ - independent errors in SC decoding. In particular, we first study the distribution of the first error position in SC decoding, and a critical set which with high probability includes the bit where the first error occurs regardless of the channel realizations is proposed. Second, a progressive bit-flipping decoding algorithm is proposed based on a search tree, which is established with a modified critical set in a progressive manner. The maximum level of the search tree is shown to coincide well with the number of independent errors that could be corrected. On this basis, the lower bound on BLER performance of a progressive bit-flipping decoder which corrects at most $L$ errors is derived, and we show the bound can be tightly achieved by the proposed algorithm for some $L$ . Moreover, an early-terminated bit-flipping (ET-Bit-Flipping) decoder is proposed to reduce the computational complexity and decoding latency of the original progressive bit-flipping scheme. Finally, numerical results show that the proposed ET-bit-flipping decoders can provide almost the same BLER performance as the state-of-the-art cyclic redundancy check-aided SC list decoders, with an average computational complexity and decoding latency similar to that of the SC decoder at medium to a high SNR regime.
Citations
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TL;DR: 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.
Abstract: 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.
38 citations
Cites background or methods from "Progressive Bit-Flipping Decoding o..."
...To reduce the complexity of the SCL decoder, in recent works [1], [2], [7], [8], it is shown that the SC bit-flip (SCF) decoder is able to yield the same BLER as the CA-SCL decoder, and the complexity of SCF decoding approaches that of the SC decoding in high signal-to-noise ratio (SNR) region....
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...For more details about CS, readers may refer to [1] and [2]....
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...CS-ω is a truncated version of the modified critical set (MCS) [1], [2] in the sense that CS-ω consists of the most error-prone elements in MCS (the details are in Section IV)....
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...In addition, based on the observations in [1] and [2], we further explain why critical set is powerful in identifying unreliable bits....
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...However, the MCS includes massive (approximately |CS|) error-prone indices that are required to be tested by the bit-flip decoding [1], [2]....
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TL;DR: All the simplifications and approximations are shown to have minimal impact on the error-correction performance, and the reported Fast-DSCF decoder is the only SCF-based architecture that can correct multiple errors.
Abstract: SC-Flip (SCF) is a low-complexity polar code decoding algorithm with improved performance, and is an alternative to high-complexity (CRC)-aided SC-List (CA-SCL) decoding. However, the performance improvement of SCF is limited since it can correct up to only one channel error ( $\omega =1$ ). Dynamic SCF (DSCF) algorithm tackles this problem by tackling multiple errors ( $\omega \geq 1$ ), but it requires logarithmic and exponential computations, which make it infeasible for practical applications. In this work, we propose simplifications and approximations to make DSCF practically feasible. First, we reduce the transcendental computations of DSCF decoding to a constant approximation. Then, we show how to incorporate special node decoding techniques into DSCF algorithm, creating the Fast-DSCF decoding. Next, we reduce the search span within the special nodes to further reduce the computational complexity. Following, we describe a hardware architecture for the Fast-DSCF decoder, in which we introduce additional simplifications such as metric normalization and sorter length reduction. All the simplifications and approximations are shown to have minimal impact on the error-correction performance, and the reported Fast-DSCF decoder is the only SCF-based architecture that can correct multiple errors. The Fast-DSCF decoders synthesized using TSMC 65 nm CMOS technology can achieve a 1.25, 1.06 and 0.93 Gbps throughput for $\omega \in \lbrace 1,2,3\rbrace$ , respectively. Compared to the state-of-the-art fast CA-SCL decoders with equivalent FER performance, the proposed decoders are up to $5.8\times$ more area-efficient. Finally, observations at energy dissipation indicate that the Fast-DSCF is more energy-efficient than its CA-SCL-based counterparts.
22 citations
Cites background or methods from "Progressive Bit-Flipping Decoding o..."
...Based on (27), the FER for a Rate-1 node under SC decoding can be derived using its top-node LLRs rather than its leaf node LLRs, by exploiting the fact that there are no frozen (parity) bits involved [28]....
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...has been carried out in [28] but it is limited to ω = 0 and δ = 1, which cannot be used for higher order FER approximations....
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TL;DR: This work proposes new SC-Flip decoders that take advantage of both the CRC and distributed parity checks (PCs) to detect, identify and flip erroneously decoded non-frozen bits.
Abstract: When polar codes are decoded by the successive cancellation (SC) decoding algorithm, erroneously decoded non-frozen bits are caused by either channel noise or error propagation. SC-Flip algorithms aim to improve error performance by first identifying erroneous hard decisions due to channel noise and then flipping them during the decoding process to reduce error propagation. In existing SC-Flip algorithms, cyclic redundancy check (CRC) is used to check the decoded codeword to detect incorrect hard decisions. Differing from this detection approach based on CRC, we propose new SC-Flip decoders that take advantage of both the CRC and distributed parity checks (PCs) to detect, identify and flip erroneously decoded non-frozen bits. The proposed decoders terminate SC decoding early when a distributed PC is not satisfied. In addition, we propose a new metric to help locate the incorrect hard decisions. Finally, simulation results demonstrate that our SC-Flip decoders achieve better performance complexity tradeoffs than prior flipping algorithms, and approach the performance and complexity of the SC-Oracle algorithms.
11 citations
TL;DR: Simulation results in additive white Gaussian noise channel demonstrated that the segmented decoding patterns can efficiently improve the error-correction performance of the flip-based scheme at different rates of polar codes while keeping a lower complexity.
Abstract: As a new flavor for the successive cancellation decoding of polar codes, bit-flipping technology can be used to improve the frame error rate performance of polar codes at moderate code lengths. In this paper, a generalized segmented scheme of bit-flipping technology is taken into full consideration, in which several segmented decoding patterns are described. First, the basic ones considering single-error-correcting and multiple-error-correcting are described, where a fully protected CRC is used. Second, with a constructed critical set for the segmented scheme, the improved version of the above basic patterns is fully discussed. Unfrozen bits corresponding to the critical set called critical information bits are divided into several parts and protected by cyclic redundancy check codes. Third, for a better performance of the segmented scheme, a segmented strategy indicating the positions of segments is effectively designed by analyzing the effect of positions of cyclic redundancy check and the error probability of each bit. At last, the frame error rate performance and the average computational complexity of the segmented decoding patterns are analyzed at matching average computational complexity or under equivalent frame error rate for different rates of polar codes. By proper design, the simulation results in additive white Gaussian noise channel demonstrated that the segmented decoding patterns can efficiently improve the error-correction performance of the flip-based scheme at different rates of polar codes while keeping a lower complexity.
10 citations
TL;DR: An improved multi-successive cancellation bit flipping (M-SCFlip) decoding algorithm is proposed to execute the bit flipping operation after CRC check-in each segment to solve the error propagation problem of successive cancellation decoding for polar codes.
Abstract: A multi-cyclic redundancy check (Multi-CRC) polar code construction algorithm is proposed in this paper to solve the error propagation problem of successive cancellation decoding for polar codes. In this algorithm, the information sequence is optimized into several segments to allow decoding errors to be corrected in time, minimizing the impact of error propagation. An improved multi-successive cancellation bit flipping (M-SCFlip) decoding algorithm is proposed to execute the bit flipping operation after CRC check-in each segment. In the low-SNR region, the proposed new multi-CRC polar code with successive cancellation list (SCL) decoding has a slight frame-error rate (FER) degradation compared with the original CRC polar code. With the M-SCFlip decoding algorithm developed in this paper, it achieves a better FER performance compared with the CRC polar code with successive cancellation (SC) and SCL ( $L\!=\!2$ ) decoding algorithms. In addition, it has a lower decoding delay and requires a lower memory space. For example, at a FER of 10−4 with the same code length and effective code rate, the proposed multi-CRC polar code with M-CFlip decoding achieves a 1.19 dB and 0.79 dB gains over existing CRC polar codes with the SC and SCL ( $L\!=\!2$ ) decoding algorithms, respectively.
7 citations
References
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Book•
01 Jan 1983
25,017 citations
"Progressive Bit-Flipping Decoding o..." refers methods in this paper
...The block length of polar codes is N = 1024, and K = 512 information bits is concatenated with a r = 24 CRC bits using generator polynomial g(D) = D24 + D23 + D6 + D5 + D + 1 (see [20]), i....
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TL;DR: The paper proves that, given any B-DMC W with I(W) > 0 and any target rate R< I( W) there exists a sequence of polar codes {Cfrn;nges1} such that Cfrn has block-length N=2n, rate ges R, and probability of block error under successive cancellation decoding bounded as Pe(N,R) les O(N-1/4) independently of the code rate.
Abstract: A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity I(W) of any given binary-input discrete memoryless channel (B-DMC) W. The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability. Channel polarization refers to the fact that it is possible to synthesize, out of N independent copies of a given B-DMC W, a second set of N binary-input channels {WN(i)1 les i les N} such that, as N becomes large, the fraction of indices i for which I(WN(i)) is near 1 approaches I(W) and the fraction for which I(WN(i)) is near 0 approaches 1-I(W). The polarized channels {WN(i)} are well-conditioned for channel coding: one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining. Codes constructed on the basis of this idea are called polar codes. The paper proves that, given any B-DMC W with I(W) > 0 and any target rate R< I(W) there exists a sequence of polar codes {Cfrn;nges1} such that Cfrn has block-length N=2n , rate ges R, and probability of block error under successive cancellation decoding bounded as Pe(N,R) les O(N-1/4) independently of the code rate. This performance is achievable by encoders and decoders with complexity O(N logN) for each.
3,554 citations
"Progressive Bit-Flipping Decoding o..." refers background or methods in this paper
...Next, invoking the expressions (A[1],A[2]) = (B[1],C[1]) × G2 and (A[3],A[4]) = (B[2],C[2]) × G2, the polar codeword is obtained as c(4)1 = (A[1],A[2],A[3],A[4]) = (1, 1, 0, 0)....
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...On this basis, the constituent code at node B is obtained by (B[1],B[2]) = (D[1],E[1]) × G2, which gives B[1] = B[2] = 0....
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...Proof: According to [2], the likelihood ratio (LR) of ui is defined as...
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...Similarly, we have C[1] = 1 and C[2] = 0....
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...INTRODUCTION Polar codes, which can be proven to achieve the capacity of any symmetric binary-input discrete memoryless channel (B-DMC) as the block-length goes to infinity [2], have straightforward construction method and efficient encoding and decoding algorithm....
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03 Oct 2011
TL;DR: It appears that the proposed list decoder bridges the gap between successive-cancellation and maximum-likelihood decoding of polar codes, and devise an efficient, numerically stable, implementation taking only O(L · n log n) time and O( L · n) space.
Abstract: We describe a successive-cancellation list decoder for polar codes, which is a generalization of the classic successive-cancellation decoder of Arikan. In the proposed list decoder, up to L decoding paths are considered concurrently at each decoding stage. Simulation results show that the resulting performance is very close to that of a maximum-likelihood decoder, even for moderate values of L. Thus it appears that the proposed list decoder bridges the gap between successive-cancellation and maximum-likelihood decoding of polar codes. The specific list-decoding algorithm that achieves this performance doubles the number of decoding paths at each decoding step, and then uses a pruning procedure to discard all but the L “best” paths. In order to implement this algorithm, we introduce a natural pruning criterion that can be easily evaluated. Nevertheless, straightforward implementation still requires O(L · n2) time, which is in stark contrast with the O(n log n) complexity of the original successive-cancellation decoder. We utilize the structure of polar codes to overcome this problem. Specifically, we devise an efficient, numerically stable, implementation taking only O(L · n log n) time and O(L · n) space.
1,338 citations
"Progressive Bit-Flipping Decoding o..." refers background in this paper
...Next, invoking the expressions (A[1],A[2]) = (B[1],C[1]) × G2 and (A[3],A[4]) = (B[2],C[2]) × G2, the polar codeword is obtained as c(4)1 = (A[1],A[2],A[3],A[4]) = (1, 1, 0, 0)....
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...To improve the performance of polar codes with finite block-lengths, a successive cancellation list (SCL) decoder which considers L decoding paths concurrently at each decoding stage was investigated in [3] and [4]....
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TL;DR: Simulations show that the resulting performance is very close to that of maximum-likelihood decoding, even for moderate values of L, and it is shown that such a genie can be easily implemented using simple CRC precoding.
Abstract: We describe a successive-cancellation list decoder for polar codes, which is a generalization of the classic successive-cancellation decoder of Arikan. In the proposed list decoder, $L$ decoding paths are considered concurrently at each decoding stage, where $L$ is an integer parameter. At the end of the decoding process, the most likely among the $L$ paths is selected as the single codeword at the decoder output. Simulations show that the resulting performance is very close to that of maximum-likelihood decoding, even for moderate values of $L$ . Alternatively, if a genie is allowed to pick the transmitted codeword from the list, the results are comparable with the performance of current state-of-the-art LDPC codes. We show that such a genie can be easily implemented using simple CRC precoding. The specific list-decoding algorithm that achieves this performance doubles the number of decoding paths for each information bit, and then uses a pruning procedure to discard all but the $L$ most likely paths. However, straightforward implementation of this algorithm requires $\Omega (L n^{2})$ time, which is in stark contrast with the $O(n \log n)$ complexity of the original successive-cancellation decoder. In this paper, we utilize the structure of polar codes along with certain algorithmic transformations in order to overcome this problem: we devise an efficient, numerically stable, implementation of the proposed list decoder that takes only $O(L n \log n)$ time and $O(L n)$ space.
1,263 citations
TL;DR: By using the Gaussian approximation for message densities under density evolution, the sum-product decoding algorithm can be visualize and the optimization of degree distributions can be understood and done graphically using the visualization.
Abstract: Density evolution is an algorithm for computing the capacity of low-density parity-check (LDPC) codes under message-passing decoding. For memoryless binary-input continuous-output additive white Gaussian noise (AWGN) channels and sum-product decoders, we use a Gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm. We convert the infinite-dimensional problem of iteratively calculating message densities, which is needed to find the exact threshold, to a one-dimensional problem of updating the means of the Gaussian densities. This simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better, but also makes it easier to design good irregular LDPC codes for AWGN channels. For various regular LDPC codes we have examined, thresholds can be estimated within 0.1 dB of the exact value. For rates between 0.5 and 0.9, codes designed using the Gaussian approximation perform within 0.02 dB of the best performing codes found so far by using density evolution when the maximum variable degree is 10. We show that by using the Gaussian approximation, we can visualize the sum-product decoding algorithm. We also show that the optimization of degree distributions can be understood and done graphically using the visualization.
1,204 citations
"Progressive Bit-Flipping Decoding o..." refers background or methods in this paper
...As indicated in [17], provided that the symmetry condition is always satisfied, the whole LLRs involved in SC decoding can be seen as Gaussian random variables whose variance values are twice of the mean values....
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...GA was introduced in [17] and used for the construction of polar codes in [13]....
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