低密度奇偶校验（LDPC）码改进译码算法研究
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摘要
低密度奇偶校验（LDPC）码具有描述简单、译码复杂度低、可以并行实现、使用灵活、错误平台低等优点，在实际系统中得到了广泛的应用。随着LDPC码的潜力不断被挖掘，可以预见其将在宽带无线通信、卫星通信、以太网传输等方面有更为广泛的应用前景。因此，近年来LDPC码成为纠错码领域的研究热点。
LDPC码的研究重点之一是LDPC码的构造。LDPC码的构造涉及减少码组中短环的数量来提升码的性能，因此需要对其二分图的环数进行检测。研究的另一个重点是译码算法的改进。改进可以从两个方面展开，一是在计算复杂度基本不变的情况下，提升译码性能；二是在译码性能基本不变的情况下，降低计算复杂度。
本文主要围绕LDPC码的环数检测算法改进以及译码算法改进展开研究，研究的目的是提出更快的环数检测算法以及性能更优的译码算法。本文的主要学术贡献在于：
(1)提出了三种改进的修正加权位翻转译码算法，分别为组合修正加权位翻转（CM-WBF）译码算法，混合修正加权位翻转（MM-WBF）译码算法，可靠性修正加权位翻转（RA-WBF）译码算法。
CM-WBF译码算法是针对FG-LDPC码提出的。该算法利用现有加权位翻转译码算法在译码过程中定位错误比特顺序不同的特点，选取两种现有的加权位翻转译码算法进行组合构造而成。与现有的M-WBF、LC-WBF及RR-WBF等译码算法相比，CM-WBF译码算法在总体计算复杂度相差不大的情况下，译码性能提升了0.15~0.3dB，平均译码时间缩短近一半。
MM-WBF译码算法主要是针对非正则LDPC码提出的。该算法在CM-WBF译码算法的基础上进行了改进，提出主算法和辅助算法概念，并制定两者译码优先策略，从而实现性能的提升。与IM-WBF、LC-WBF及RR-WBF算法相比，MM-WBF译码算法在总体计算复杂度相差不大的情况下，译码性能提升了0.3~2.0dB，平均译码时间缩短近一半。此外，仿真结果表明，MM-WBF译码算法对正则LDPC码同样能够有效提升译码性能。
RA-WBF译码算法提出在解码过程中动态调整接收码字部分比特的可靠性，以便于更加准确地定位、纠正错误比特，从而得到更佳的性能。该算法能够应用到现有的几种修正加权位翻转译码算法中，有效提升现有算法的译码性能，具有广泛应用意义。
研究成果在Journal of Computational Information Systems (JCIS)期刊、2012International Conference on Wireless Communications and Signal Processing (WCSP’2012)会议及2013International Conference on Computational Intelligence and Security (CIS’2013)会议上发表。
(2)提出了三种改进的置信传播译码算法，分别为振荡梳洗置信传播（OS-BP）译码算法、Tchebyshev-Padé逼近置信传播译码算法（TPA-BP）、有理函数逼近置信传播译码算法（RFA-BP）。
OS-BP译码算法以Shuffled BP算法为基础进行改进，进一步优化了校验节点与信息节点之间的消息传递方式。在总体计算复杂度与Shuffled BP算法基本相同的情况下，有效提升译码性能0.15dB以上，平均迭代次数为Shuffled BP算法的70%。此外，为了解决硬件实现中时延较大的问题，我们进一步提出分组振荡梳洗置信传播（GOS-BP）译码算法；GOS-BP算法通过分组处理，能够解决硬件实现中译码时延较大的问题。
TPA-BP译码算法和RFA-BP译码算法针对标准对数似然比置信传播（LLR BP）译码算法计算复杂度较高的缺点，分别通过Tchebyshev-Padé多项式和有理函数多项式逼近置信传播消息传递公式，在几乎没有任何译码性能损失的情况下，大大降低了计算复杂度。并且由于其通用性，能够广泛地应用在各类BP译码算法中。
研究成果在Journal of Computational Information Systems (JCIS)期刊上发表。
(3)提出了一种快速检测LDPC码环数的新算法。该算法根据LDPC码校验矩阵对应Tanner图的特点，基于Dijkstra算法进行构造。该算法不需依次检测环长为4、6、8等长度的环数，而是通过一次检测得到所有环数，从而实现了LDPC码环数的快速检测。与传统的环分析检测算法相比，新算法极大降低了计算复杂度，提升了检测速度；仿真表明，快速检测环数的新算法比环分析检测算法减少检测时间1~2个数量级。
研究成果在《计算机应用》期刊上发表。
Low-density parity-check (LDPC) codes have been widely used in the real systembecause of its outstanding metrics, such as simple description, low decoding complexity,possible parallel implementation, flexible to be used, low error floor, etc. With the potential ofLDPC codes being continuously excavated, the codes are expected to have broaderapplications in broadband wireless communications, satellite communications, Ethernettransmission, etc. That is why LDPC codes have been a hot research field in recent years.
We focus on the construction of LDPC codes. In code construction, it is necessary toreduce the number of short cycles to achieve better code performance. Therefore we need todetect the number of cycles in the Tanner graph of the corresponding LDPC code. Anotherfocus of our research is to improve decoding performance. The improvement could be carriedout in two ways. One is to improve the decoding performance and speed of convergencewithout increasing the total computational complexity. The other is to reduce the totalcomputational complexity without performance loss.
In this dissertation, we focus on the improved algorithms for counting the number ofcycles and decoding for LDPC codes. The target of this study is to propose a faster algorithmfor counting the number of cycles and better decoding algorithms. The major contributions ofthis dissertation are as follows:
(1) Three new modified weighted bit-flipping (WBF) decoding algorithms have beenproposed to achieve better performance. They are combined modified weighted bit-flipping(CM-WBF) decoding algorithm, mixed modified weighted bit-flipping (MM-WBF) decodingalgorithm and reliability adjustment weighted bit-flipping (RA-WBF) decoding algorithm.
CM-WBF is proposed for finite geometry low-density parity-check (FG-LDPC) codes. Itis constructed by combining two WBF algorithms and makes use of the feature that two WBFalgorithms often select different error bits in decoding process. Simulation results show thatby carefully choosing the two WBF algorithms to be combined, the new algorithm can offer abetter trade-off among performance, complexity, and convergence compared with M-WBF,LC-WBF and RR-WBF algorithms for FG-LDPC codes. With almost the same complexity,CM-WBF performs about0.15dB~0.3dB better and reduces the average convergence numberof iterations by half compared with other WBF algorithms.
MM-WBF is proposed mainly for irregular LDPC codes. Based on CM-WBF algorithm,MM-WBF innovatively proposes the concepts of the main and assistant algorithms, anddevelops priority strategies of the two algorithms to achieve performance improvements. Simulation results show that with almost the same computational complexity, MM-WBFperforms about0.3dB~2.0dB better and reduces the average convergence number of iterationsby half compared with RR-WBF, IM-WBF and LC-WBF algorithms. Moreover, MM-WBFachieves similar improvement for regular LDPC codes.
RA-WBF is an innovative WBF algorithm. It locates and corrects the error bits moreaccurately to obtain better performance by adjusting the reliability of some bits of a receivedcodeword in the decoding process. This idea of RA-WBF algorithm can be easily extended toexisting WBF decoding algorithms and improve the decoding performance effectively.
The research productions are published in Journal of Computational InformationSystems (JCIS),2012International Conference on Wireless Communications and SignalProcessing (WCSP’2012) and2013International Conference on Computational Intelligenceand Security (CIS’2013).
(2) Three new belief propagation decoding algorithms have been proposed to achievebetter performance or lower complexity. They are Oscillating Shuffled BP (OS-BP),Tchebyshev-Padé approximation BP (TPA-BP) and rational function approximation BP(RFA-BP) decoding algorithms.
OS-BP is based on shuffled BP decoding algorithm and optimizes the messagetransmission route from check nodes to bit nodes. It improves performance by0.15dB andreduces the average number of iterations for convergence by about30%without increasingthe total complexity compared with the shuffled BP algorithm. Furthermore, to solve theproblem of relatively large delay in hardware implementation, we propose group oscillatingshuffled BP (GOS-BP) decoding algorithm, which solves the problem by the groupingprocess.
TPA-BP and RFA-BP decoding algorithm are proposed to reduce the high computationalcomplexity in the standard LLR BP algorithm. TPA-BP and RFA-BP use Tchebyshev-Padéapproximation polynomial and rational functions approximation polynomial respectively toreplace tanh(x) function to reduce complexity. Compared with the standard LLR BP decodingalgorithm, the proposed algorithms significantly reduce computational complexity withoutnoticeable performance loss. These approximation ideas can be easily extended to othervariations of BP decoding.
The research productions are published in Journal of Computational InformationSystems (JCIS), etc.
(3) A new fast algorithm for counting the number of cycles in LDPC codes is proposedin this dissertation. The existing typical counting algorithm called cycle shape analysis counting algorithm counts the number of cycles sequentially, such as4-cycle first, then6-cycle, next8-cycle, etc., which is not efficient. The proposed algorithm, which is based onDijkstra algorithm in graph theory and the nature of the Tanner graph of LDPC codes, countsthe number of cycles once detected. The new algorithm greatly improves counting efficiency.Simulation results show that the new algorithm reduces counting time by1to2orders ofmagnitude compared with the typical counting algorithm.
The research productions are published in Journal of Computer Applications.
LDPC码的研究重点之一是LDPC码的构造。LDPC码的构造涉及减少码组中短环的数量来提升码的性能，因此需要对其二分图的环数进行检测。研究的另一个重点是译码算法的改进。改进可以从两个方面展开，一是在计算复杂度基本不变的情况下，提升译码性能；二是在译码性能基本不变的情况下，降低计算复杂度。
本文主要围绕LDPC码的环数检测算法改进以及译码算法改进展开研究，研究的目的是提出更快的环数检测算法以及性能更优的译码算法。本文的主要学术贡献在于：
(1)提出了三种改进的修正加权位翻转译码算法，分别为组合修正加权位翻转（CM-WBF）译码算法，混合修正加权位翻转（MM-WBF）译码算法，可靠性修正加权位翻转（RA-WBF）译码算法。
CM-WBF译码算法是针对FG-LDPC码提出的。该算法利用现有加权位翻转译码算法在译码过程中定位错误比特顺序不同的特点，选取两种现有的加权位翻转译码算法进行组合构造而成。与现有的M-WBF、LC-WBF及RR-WBF等译码算法相比，CM-WBF译码算法在总体计算复杂度相差不大的情况下，译码性能提升了0.15~0.3dB，平均译码时间缩短近一半。
MM-WBF译码算法主要是针对非正则LDPC码提出的。该算法在CM-WBF译码算法的基础上进行了改进，提出主算法和辅助算法概念，并制定两者译码优先策略，从而实现性能的提升。与IM-WBF、LC-WBF及RR-WBF算法相比，MM-WBF译码算法在总体计算复杂度相差不大的情况下，译码性能提升了0.3~2.0dB，平均译码时间缩短近一半。此外，仿真结果表明，MM-WBF译码算法对正则LDPC码同样能够有效提升译码性能。
RA-WBF译码算法提出在解码过程中动态调整接收码字部分比特的可靠性，以便于更加准确地定位、纠正错误比特，从而得到更佳的性能。该算法能够应用到现有的几种修正加权位翻转译码算法中，有效提升现有算法的译码性能，具有广泛应用意义。
研究成果在Journal of Computational Information Systems (JCIS)期刊、2012International Conference on Wireless Communications and Signal Processing (WCSP’2012)会议及2013International Conference on Computational Intelligence and Security (CIS’2013)会议上发表。
(2)提出了三种改进的置信传播译码算法，分别为振荡梳洗置信传播（OS-BP）译码算法、Tchebyshev-Padé逼近置信传播译码算法（TPA-BP）、有理函数逼近置信传播译码算法（RFA-BP）。
OS-BP译码算法以Shuffled BP算法为基础进行改进，进一步优化了校验节点与信息节点之间的消息传递方式。在总体计算复杂度与Shuffled BP算法基本相同的情况下，有效提升译码性能0.15dB以上，平均迭代次数为Shuffled BP算法的70%。此外，为了解决硬件实现中时延较大的问题，我们进一步提出分组振荡梳洗置信传播（GOS-BP）译码算法；GOS-BP算法通过分组处理，能够解决硬件实现中译码时延较大的问题。
TPA-BP译码算法和RFA-BP译码算法针对标准对数似然比置信传播（LLR BP）译码算法计算复杂度较高的缺点，分别通过Tchebyshev-Padé多项式和有理函数多项式逼近置信传播消息传递公式，在几乎没有任何译码性能损失的情况下，大大降低了计算复杂度。并且由于其通用性，能够广泛地应用在各类BP译码算法中。
研究成果在Journal of Computational Information Systems (JCIS)期刊上发表。
(3)提出了一种快速检测LDPC码环数的新算法。该算法根据LDPC码校验矩阵对应Tanner图的特点，基于Dijkstra算法进行构造。该算法不需依次检测环长为4、6、8等长度的环数，而是通过一次检测得到所有环数，从而实现了LDPC码环数的快速检测。与传统的环分析检测算法相比，新算法极大降低了计算复杂度，提升了检测速度；仿真表明，快速检测环数的新算法比环分析检测算法减少检测时间1~2个数量级。
研究成果在《计算机应用》期刊上发表。
Low-density parity-check (LDPC) codes have been widely used in the real systembecause of its outstanding metrics, such as simple description, low decoding complexity,possible parallel implementation, flexible to be used, low error floor, etc. With the potential ofLDPC codes being continuously excavated, the codes are expected to have broaderapplications in broadband wireless communications, satellite communications, Ethernettransmission, etc. That is why LDPC codes have been a hot research field in recent years.
We focus on the construction of LDPC codes. In code construction, it is necessary toreduce the number of short cycles to achieve better code performance. Therefore we need todetect the number of cycles in the Tanner graph of the corresponding LDPC code. Anotherfocus of our research is to improve decoding performance. The improvement could be carriedout in two ways. One is to improve the decoding performance and speed of convergencewithout increasing the total computational complexity. The other is to reduce the totalcomputational complexity without performance loss.
In this dissertation, we focus on the improved algorithms for counting the number ofcycles and decoding for LDPC codes. The target of this study is to propose a faster algorithmfor counting the number of cycles and better decoding algorithms. The major contributions ofthis dissertation are as follows:
(1) Three new modified weighted bit-flipping (WBF) decoding algorithms have beenproposed to achieve better performance. They are combined modified weighted bit-flipping(CM-WBF) decoding algorithm, mixed modified weighted bit-flipping (MM-WBF) decodingalgorithm and reliability adjustment weighted bit-flipping (RA-WBF) decoding algorithm.
CM-WBF is proposed for finite geometry low-density parity-check (FG-LDPC) codes. Itis constructed by combining two WBF algorithms and makes use of the feature that two WBFalgorithms often select different error bits in decoding process. Simulation results show thatby carefully choosing the two WBF algorithms to be combined, the new algorithm can offer abetter trade-off among performance, complexity, and convergence compared with M-WBF,LC-WBF and RR-WBF algorithms for FG-LDPC codes. With almost the same complexity,CM-WBF performs about0.15dB~0.3dB better and reduces the average convergence numberof iterations by half compared with other WBF algorithms.
MM-WBF is proposed mainly for irregular LDPC codes. Based on CM-WBF algorithm,MM-WBF innovatively proposes the concepts of the main and assistant algorithms, anddevelops priority strategies of the two algorithms to achieve performance improvements. Simulation results show that with almost the same computational complexity, MM-WBFperforms about0.3dB~2.0dB better and reduces the average convergence number of iterationsby half compared with RR-WBF, IM-WBF and LC-WBF algorithms. Moreover, MM-WBFachieves similar improvement for regular LDPC codes.
RA-WBF is an innovative WBF algorithm. It locates and corrects the error bits moreaccurately to obtain better performance by adjusting the reliability of some bits of a receivedcodeword in the decoding process. This idea of RA-WBF algorithm can be easily extended toexisting WBF decoding algorithms and improve the decoding performance effectively.
The research productions are published in Journal of Computational InformationSystems (JCIS),2012International Conference on Wireless Communications and SignalProcessing (WCSP’2012) and2013International Conference on Computational Intelligenceand Security (CIS’2013).
(2) Three new belief propagation decoding algorithms have been proposed to achievebetter performance or lower complexity. They are Oscillating Shuffled BP (OS-BP),Tchebyshev-Padé approximation BP (TPA-BP) and rational function approximation BP(RFA-BP) decoding algorithms.
OS-BP is based on shuffled BP decoding algorithm and optimizes the messagetransmission route from check nodes to bit nodes. It improves performance by0.15dB andreduces the average number of iterations for convergence by about30%without increasingthe total complexity compared with the shuffled BP algorithm. Furthermore, to solve theproblem of relatively large delay in hardware implementation, we propose group oscillatingshuffled BP (GOS-BP) decoding algorithm, which solves the problem by the groupingprocess.
TPA-BP and RFA-BP decoding algorithm are proposed to reduce the high computationalcomplexity in the standard LLR BP algorithm. TPA-BP and RFA-BP use Tchebyshev-Padéapproximation polynomial and rational functions approximation polynomial respectively toreplace tanh(x) function to reduce complexity. Compared with the standard LLR BP decodingalgorithm, the proposed algorithms significantly reduce computational complexity withoutnoticeable performance loss. These approximation ideas can be easily extended to othervariations of BP decoding.
The research productions are published in Journal of Computational InformationSystems (JCIS), etc.
(3) A new fast algorithm for counting the number of cycles in LDPC codes is proposedin this dissertation. The existing typical counting algorithm called cycle shape analysis counting algorithm counts the number of cycles sequentially, such as4-cycle first, then6-cycle, next8-cycle, etc., which is not efficient. The proposed algorithm, which is based onDijkstra algorithm in graph theory and the nature of the Tanner graph of LDPC codes, countsthe number of cycles once detected. The new algorithm greatly improves counting efficiency.Simulation results show that the new algorithm reduces counting time by1to2orders ofmagnitude compared with the typical counting algorithm.
The research productions are published in Journal of Computer Applications.
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