准循环LDPC码的译码研究
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摘要
低密度奇偶校验码(LDPC)码是由Gallager在1962年首先提出的一种纠错码,后在90年代初被重新认知为一类具有接近香农限特性的好码。准循环LDPC码是一类结构化的LDPC码,它的译码性能和错误平底特性接近随机构造的LDPC码,而且编码算法实现简单,被认为是一种实用化的LDPC码。本论文主要对准循环LDPC码的译码算法和译码器结构进行了较深入的研究。
论文分析了准循环LDPC码的基本原理,包括准循环LDPC码的基本概念、构造方法、以及编译码算法。在编码算法里讨论了传统的编码算法以及准循环LDPC码的快速编码算法。译码算法方面分析了BP译码算法、最小和译码算法、归一化BP-Based译码算法和偏移BP-Based译码算法。
论文研究了准循环LDPC码编码仿真的实现方法;分析了BP译码算法和BP-Based算法的实现方法,并分别给出了校验节点和变量节点上迭代计算的仿真流程;对BP算法、最小和算法、归一化BP-Based算法和偏移BP-Based算法的硬件实现复杂度进行了讨论。
论文设计了准循环LDPC码的仿真系统及其仿真流程,建立了AWGN信道、Rayleigh衰落信道和PLC脉冲信道的信道模型;分析了AWGN信道下码长、码结构和最大迭代次数等因素对译码性能的影响,研究了BP算法及其改进算法在AWGN信道下的译码性能差异;分析了准循环LDPC码在非相关Rayleigh信道和PLC脉冲信道上的译码性能。
论文研究了基于置换阵的BP译码算法,给出了该算法的译码流程图;在流程图基础上,研究了一个基于交换单元的准循环LDPC码的译码器结构并给出了译码工作的具体步骤;最后研究了交换单元的电路结构和选路方法,设计了一种基于Banyan结构的交换单元,用于实现迭代信息的快速交换。
Low Density Parity-Check (LDPC) code was first discovered by Gallager in the early 1960s and was rediscovered and considered as a class of Shannon-limit-approaching code in the late 1990s. Quasi Cyclic (QC) LDPC code, which is a kind of constructed LDPC code, possess a similar decoding performance and an error floor character with random constructed LDPC code. Besides, its encoding algorithm can be easily realized. So QC LDPC code is also considered as a kind of practical LDPC code. Advanced researches on decoding algorithm and decoder structure of QC LDPC code are the focus of this thesis.
Firstly, the fundamental principles of QC LDPC code including QC LDPC code’s basic conception, construction and both the encoding and decoding algorithm are analyzed. Of encoding algorithm, a conventional algorithm and an efficient algorithm suitable for QC LDPC code are discussed. As far as the decoding algorithm concerned, Belief Propagation decoding algorithm,Min-Sum decoding algorithm, Normalized BP-Based decoding algorithm and Offset BP-Based decoding algorithm are analyzed respectively.
Secondly, the simulation method of encoding algorithm for QC LDPC code is studied; the realizable scheme of BP algorithm and BP-Based algorithm are analyzed, presenting the simulation flow chart of iterative computation on check nodes and variable nodes; finally the implement complexity of hardware for BP algorithm,Min-Sum algorithm, Normalized BP-Based algorithm and Offset BP-Based algorithm are discussed.
Besides, the simulation system and flow for QC LDPC code are designed; the channel model of AWGN channel, Rayleigh channel and power line communication pulse channel are proposed; the decoding performance under different code lengths, code structures and maximal iterative times in AWGN channel model are analyzed; the difference of decoding performance under BP algorithm and its improving algorithms are studied; at last the decoding performance in unrelated Rayleigh channel and PLC pulse channel are investigated.
Finally, the BP decoding algorithm based on permutation matrix is studied and its decoding flow chart is provided; a decoding structure based on exchange unit is designed and the working process of the structure is offered; the circuit structure and routing way of the exchange unit are studied and a exchange unit based on Banyan configuration is designed to realize the exchange of iterative message.
论文分析了准循环LDPC码的基本原理,包括准循环LDPC码的基本概念、构造方法、以及编译码算法。在编码算法里讨论了传统的编码算法以及准循环LDPC码的快速编码算法。译码算法方面分析了BP译码算法、最小和译码算法、归一化BP-Based译码算法和偏移BP-Based译码算法。
论文研究了准循环LDPC码编码仿真的实现方法;分析了BP译码算法和BP-Based算法的实现方法,并分别给出了校验节点和变量节点上迭代计算的仿真流程;对BP算法、最小和算法、归一化BP-Based算法和偏移BP-Based算法的硬件实现复杂度进行了讨论。
论文设计了准循环LDPC码的仿真系统及其仿真流程,建立了AWGN信道、Rayleigh衰落信道和PLC脉冲信道的信道模型;分析了AWGN信道下码长、码结构和最大迭代次数等因素对译码性能的影响,研究了BP算法及其改进算法在AWGN信道下的译码性能差异;分析了准循环LDPC码在非相关Rayleigh信道和PLC脉冲信道上的译码性能。
论文研究了基于置换阵的BP译码算法,给出了该算法的译码流程图;在流程图基础上,研究了一个基于交换单元的准循环LDPC码的译码器结构并给出了译码工作的具体步骤;最后研究了交换单元的电路结构和选路方法,设计了一种基于Banyan结构的交换单元,用于实现迭代信息的快速交换。
Low Density Parity-Check (LDPC) code was first discovered by Gallager in the early 1960s and was rediscovered and considered as a class of Shannon-limit-approaching code in the late 1990s. Quasi Cyclic (QC) LDPC code, which is a kind of constructed LDPC code, possess a similar decoding performance and an error floor character with random constructed LDPC code. Besides, its encoding algorithm can be easily realized. So QC LDPC code is also considered as a kind of practical LDPC code. Advanced researches on decoding algorithm and decoder structure of QC LDPC code are the focus of this thesis.
Firstly, the fundamental principles of QC LDPC code including QC LDPC code’s basic conception, construction and both the encoding and decoding algorithm are analyzed. Of encoding algorithm, a conventional algorithm and an efficient algorithm suitable for QC LDPC code are discussed. As far as the decoding algorithm concerned, Belief Propagation decoding algorithm,Min-Sum decoding algorithm, Normalized BP-Based decoding algorithm and Offset BP-Based decoding algorithm are analyzed respectively.
Secondly, the simulation method of encoding algorithm for QC LDPC code is studied; the realizable scheme of BP algorithm and BP-Based algorithm are analyzed, presenting the simulation flow chart of iterative computation on check nodes and variable nodes; finally the implement complexity of hardware for BP algorithm,Min-Sum algorithm, Normalized BP-Based algorithm and Offset BP-Based algorithm are discussed.
Besides, the simulation system and flow for QC LDPC code are designed; the channel model of AWGN channel, Rayleigh channel and power line communication pulse channel are proposed; the decoding performance under different code lengths, code structures and maximal iterative times in AWGN channel model are analyzed; the difference of decoding performance under BP algorithm and its improving algorithms are studied; at last the decoding performance in unrelated Rayleigh channel and PLC pulse channel are investigated.
Finally, the BP decoding algorithm based on permutation matrix is studied and its decoding flow chart is provided; a decoding structure based on exchange unit is designed and the working process of the structure is offered; the circuit structure and routing way of the exchange unit are studied and a exchange unit based on Banyan configuration is designed to realize the exchange of iterative message.
引文
1 G. Ungerboeck. Channel Coding with Multilevel/phase Signals. IEEE Transactions on Information Theory. 1982,28(1):55~67
2 C. Berrou, A. Glavieux, P. Thitimajshima. Near Shannon Limit Error Correcting Coding and Decoding: Turbo-codes. ICC’93, Switzerland, 1993:1064~1070
3 R. G. Gallager. Low-Density Parity-Check Codes. IEEE Transactions on Information Theory. 1962, 8(3):21~28
4 S. Y. Chung, G. D. Forney, T. J. Richardson, and R. Urbanke. On the design of low-density parity-check codes within 0.0045 db of the Shannon limit. IEEE Communictions Letters. 2001, 2(5):58~60
5 Y. Kou, S. Lin, M. P. C. Fossorier. Low-density parity-check codes based on finite geometries: A rediscovery and new results. IEEE Transactions on Information Theory. 2001, 47(7): 2711~2736
6 Sarah. J. Johnson, Steven R. Weller. Construction of Low-Density Parity-Check Codes from Kirkman Triple systems. GLOBECOM 2001-IEEE Global Telecommunications Conference. 2001, 11(1):970~974
7 M sipser, D.A.Spielman. Expander Codes. IEEE Transactions on Information Theory. 1996, 42 (6): 1710~1722
8 Rosenthal, P. O. Vontobel. Constructions of regular and irregular LDPC codes using Ramanujan graphs and ideas from Margulis. Information Theory, 2001. Proceedings. 2001 IEEE International Symposium. 2001, 4(6):24~29
9 R Michael Tanner. A Recursive Approach to Low Complexity Codes. IEEE Transactions on Information Theory. 1981,27 (5): 533~547
10 T. J. Richardson, R. Urbanke. Efficient encoding of low-density parity-check codes. IEEE Transactions on Information Theory. 2001, 47(2): 638~656
11 Marc P. C. Fossorier, Miodrag Mihaljevic, Hideki Imai. Reduced Complexity Iterative Decoding of Low-Density Parity Check Codes Based on Belief Propagation. IEEE Transaction on Communications. 1999,47(5): 673~680
12 Jinghu Chen, Marc P. C. Fossorier. Decoding Low-Density Parity-Check codes with Normalized APP-Based Algorithm. Proc. IEEE Globecom, SanAntonio, TX, 2001:1026~1030
13 Jinghu Chen, Marc P. C. Fossorier. Near Optimum Universal Belief Propagation Based Decoding of Low-Density Parity Check Codes, IEEE Transaction on Communications. 2002,50(3): 406~414
14 Jinghu Chen, Marc P. C. Fossorier, Density Evolution for Two Improved BP-Based Decoding Algorithms of LDPC Codes. IEEE Communications Letter. 2002,6(5):208~210
15 Yu Chen He, et al. Fast Decoding of LDPC using Quantization. IEEE Electronics Letter. 2002,48(4): 603~611
16 Yongyi Mao, Amir H. Banihashemi. Decoding Low-Density Parity-Check Codes with Probabilistic Scheduling. IEEE Communications Letter, 2001,5(10): 1788~1793
17 Yongyi Mao, Amir H. Banihashemi. A New Schedule for Decoding Low Density Parity-Check Codes. GLOBECOM 2001-IEEE Global Telecommunications Conference. 2001,11(1):1007~1010
18 Achilles Anastasopoulos. A Comparison between the Sum-Product and the Min-Sum Iterative Detection Algorithms Based on Density Evolution. GLOBECOM 2001-IEEE Global Telecommunications Conference. 2001,11(1):1021~1024
19 XiaoYu Hu, Evangelos Eleftheriou. Efficient Implementation of the Sum-Product Algorithm for Decoding LDPC codes. GLOBECOM 2001-IEEE Global Telecommunications Conference. 2001,11(1):1036~1041
20 Jilei Hou, Paul H.Siegel and Laurence B.Milstein, Performance Analysis and Code Optimization of Low-Density Parity-Check Codes on Rayleigh Fading Channels. IEEE Journal Selected Areas in Communications. 2001,19(5):924~934
21 Nedeljko Varnica and Aleksandar Kavcic, Optimized LDPC Codes for Partial Response Channels. IEEE Communications Letters. 1995,7(1):168~170,
22 Hongwei Song, Jingfeng Liu and B.V.K.Vijaya. Low-Density Parity-Check (LDPC) Codes for Partial Response Equalized CAD-MO Recording. IEEE Trans. Magn. 2001,37(6):676~681
23 T. J. Richardson and R. L. Urbanke. The Capacity of Low-density Parity-Check Codes Under Message-passing Decoding. IEEE Transaction onInformation Theory. 2001,47(3):599~618
24 T. J. Richardson, M. A. Shokrollahi, and R. L.Urbanke. Design of Capacity approaching Irregular Low-density Parity-check Codes. IEEE Transactions on Information Theory. 2001, 47(9):619~637
25 S.Y. Chung, T. J. Richardson, and R. L. Urbanke. Analysis of Sum-product Decoding of Low-density Parity-check Codes using a Gaussian Approximation. IEEE Transactions on Information Theory. 2001,47(8): 657~670
26 Sae-Young Chung, G. David Forney Jr., Thomas J. Richardson and Rtidiger Urbanke. On the Design of Low-density Parity-check Codes within 0.0045 dB of the Shannon Limit. IEEE Communications Letters. 2001,5(2): 58~60
27 Sae-Young Chung. On the Construction of Some Capacity-approaching Coding Schemes. PH.D thesis, MIT, Sep.2000
28 Flarion Technologies Inc, Vector-Low-Density Parity-Check Coding Solution Data Sheet[EB/OL], http://www.flarion.com
29 H. A. Loeliger. Probability Propagation and Decoding in Analog VLSI. IEEE Transactions on Information Theory. 2001, 47(2):837~843
30 T. Zhang, K. K. Parhi. VLSI Implemntation-Oriented (3,K)-Regular Low-Density Parity-Check Codes[EB/OL]. from Internet. {tzhang. parhi}@ece. Umn. Edu.
31 Tejas Bhatt, Krishna Narayanan and Nasser Kehtarnavaz. Fixed Point DSP Implementation of Low-Density Parity-Check Codes. Proceedings of the 9th IEEE DSP Workshop, Hunt, TX, 2000:921~932
32 Zang Di, et al. Low-Density Parity-Check Codes used for Compressed Image Transmission over Noisy Channel. From Internet, http://image.chonbuk.ac.kr/ kwak/ data1/2002-4.pdf,Apr.2003
33 Low-density parity-check codes for DSL transmission. Temporary Document BI-095, Study Group 15/4, Goa, India, 23~27 Oct 2000.
34 Hongxin Song, Richard M. Todd and J. R. Cruz. Applications of Low-Density Parity-Check Codes to Magnetic Recording Channels. IEEE Journal on Selected Areas in Communications. 2001,19(5): 918~923
35 Ka Leong Lo. Layered Space Time Structures with Low-Density Parity-Check and Convolution Codes. Spread Spectrum Techniques and Applications,2002 IEEE Seventh International Symposium. 2002,2:308~312
36 R. M. Tanner. Arecursive Approach to Low Complexity Codes. IEEE Transactions on Information Theory. 1981, 11(27):533~547
37 M. Fossorier. Quasi-cyclic Low Density Parity Check Codes. Proc IEEE Int Symp Information Theory. 2003,11(4):150~161
38 Seho Myung, Kyeongcheol Yang. Quasi-cyclic LDPC Codes for Fast Encoding. IEEE Transactions on Information Theory. 2005,51(8):2894~2901
39 Marc P.C.Fossorier. Quasi-cyclic Low-density Parity-check Codes form Circulant Permutation Matrices. IEEE Transactions on Information Theory. 2004,50(8):1788~1793
40 A.Sridharan, D.J.Costello Jr, D.Sridhara, T.Fuja, R.M.Tanner. A Construction for Low-density Parity Check Convolution Codes based on Quasi-cyclic Block codes. Proc. IEEE Int. Symp. Information Theory. 2002:481~453
41 J. Fan. Array codes as LDPC codes. Proc. 2nd Int. Symp. Turbo Codes, 2000:543~546.
42 Eleftheriou E. and ?l?er S. Low density parity-check codes for digital subscriber lines. Proc., ICC’2002, New York, NY, 2002:1752~1757
43 E.Eleftheriou, S.Olcer. Low-density Parity-check Codes for Multilevel Modulation. Proc. IEEE Int. Symp. Information Theory. 2002:442.
44 Hao Zhong, Tong Zhang. Block-LDPC: A Practical LDPC Coding System Design Approach. IEEE Transactions on Information Theory. April 2005, 52(4):766~775
45 Li Zongwang, Chen Lei, Zeng Lingqi, Lin Shu. Efficient Encoding of Quasi-Cyclic Low-Density Parity-Check Codes. IEEE Transactions on Communications, 2006, 54(1):71~81
46 Y.Chen, K.K.Parhi. High Throughput Overlapped Message Passing for Low-density Parity Check Codes. Proc.IEEE/ACM GLSVLSI. 2003:245~248
47 N.Wiberg. Codes and Decoding on General Graphs. Ph.D.dessertation Univ.Linkoping, Dept. Elect.Eng.,Linkoping, Sweden,1996
48 F.R.Kschischang, B.J.Frey, H.-A.Loeliger. Factor Graphs and the sum-product algorithm. IEEE Transactions on Information Theory. 2001,47(2): 498~519
49 D. J. C. Mackay. Good error-correcting codes based on very sparse matrices.IEEE Transactions on Information Theory. 1999, 4 (5):3~431
50 Jinghu Chen, Ajay Dholakia, Evangelos Eleftheriou, Marc P. C. Fossorier and Xiao-Yu Hu.Reduced-Complexity Decoding of LDPC Codes.IEEE IEEE Transactions on Communications. 2001, 53(8):1288~1299
51 Motohiko Isaka, Marc Fossorier and Hideki Imai. Iterative Reliability-based Decoding of Turbo-like Codes. ICC’2002. 2002, 25(1): 1495~500.
52高锋,董亚波.低压电力线载波通信中信号传输特性分析.电力系统自动化. 2000, (7): 36~40
53赵云峰,汪晓岩,徐立忠等.低压电力线噪声分析与建模.电力系统通信. 2003, (1):31~34.
54 D. Middleton. Statistical-physical model of electromagnetic interference. IEEE. Trans. Electromagn. Compat. EMC-19. 1977,8(3): 106~126
55 D. Umehara, H. Yamaguchi, and Y. Morihiro. Turbo decoding over impulsive noise channel. Proc. 2004 ISPLC Conf. 2004,4(2):51~56
56 ?l?er S. Decoder architecture for array-code-based LDPC codes. Global Telecommunications Conference, GLOBECOM '03, San Francisco, 2003: 2046~2050
57 LEE T H, LIU S J. Banyan Network Nonblocking With Respect to Cyclic Shift. Electronics Letters, 1991, 27(16): 1474~1476
2 C. Berrou, A. Glavieux, P. Thitimajshima. Near Shannon Limit Error Correcting Coding and Decoding: Turbo-codes. ICC’93, Switzerland, 1993:1064~1070
3 R. G. Gallager. Low-Density Parity-Check Codes. IEEE Transactions on Information Theory. 1962, 8(3):21~28
4 S. Y. Chung, G. D. Forney, T. J. Richardson, and R. Urbanke. On the design of low-density parity-check codes within 0.0045 db of the Shannon limit. IEEE Communictions Letters. 2001, 2(5):58~60
5 Y. Kou, S. Lin, M. P. C. Fossorier. Low-density parity-check codes based on finite geometries: A rediscovery and new results. IEEE Transactions on Information Theory. 2001, 47(7): 2711~2736
6 Sarah. J. Johnson, Steven R. Weller. Construction of Low-Density Parity-Check Codes from Kirkman Triple systems. GLOBECOM 2001-IEEE Global Telecommunications Conference. 2001, 11(1):970~974
7 M sipser, D.A.Spielman. Expander Codes. IEEE Transactions on Information Theory. 1996, 42 (6): 1710~1722
8 Rosenthal, P. O. Vontobel. Constructions of regular and irregular LDPC codes using Ramanujan graphs and ideas from Margulis. Information Theory, 2001. Proceedings. 2001 IEEE International Symposium. 2001, 4(6):24~29
9 R Michael Tanner. A Recursive Approach to Low Complexity Codes. IEEE Transactions on Information Theory. 1981,27 (5): 533~547
10 T. J. Richardson, R. Urbanke. Efficient encoding of low-density parity-check codes. IEEE Transactions on Information Theory. 2001, 47(2): 638~656
11 Marc P. C. Fossorier, Miodrag Mihaljevic, Hideki Imai. Reduced Complexity Iterative Decoding of Low-Density Parity Check Codes Based on Belief Propagation. IEEE Transaction on Communications. 1999,47(5): 673~680
12 Jinghu Chen, Marc P. C. Fossorier. Decoding Low-Density Parity-Check codes with Normalized APP-Based Algorithm. Proc. IEEE Globecom, SanAntonio, TX, 2001:1026~1030
13 Jinghu Chen, Marc P. C. Fossorier. Near Optimum Universal Belief Propagation Based Decoding of Low-Density Parity Check Codes, IEEE Transaction on Communications. 2002,50(3): 406~414
14 Jinghu Chen, Marc P. C. Fossorier, Density Evolution for Two Improved BP-Based Decoding Algorithms of LDPC Codes. IEEE Communications Letter. 2002,6(5):208~210
15 Yu Chen He, et al. Fast Decoding of LDPC using Quantization. IEEE Electronics Letter. 2002,48(4): 603~611
16 Yongyi Mao, Amir H. Banihashemi. Decoding Low-Density Parity-Check Codes with Probabilistic Scheduling. IEEE Communications Letter, 2001,5(10): 1788~1793
17 Yongyi Mao, Amir H. Banihashemi. A New Schedule for Decoding Low Density Parity-Check Codes. GLOBECOM 2001-IEEE Global Telecommunications Conference. 2001,11(1):1007~1010
18 Achilles Anastasopoulos. A Comparison between the Sum-Product and the Min-Sum Iterative Detection Algorithms Based on Density Evolution. GLOBECOM 2001-IEEE Global Telecommunications Conference. 2001,11(1):1021~1024
19 XiaoYu Hu, Evangelos Eleftheriou. Efficient Implementation of the Sum-Product Algorithm for Decoding LDPC codes. GLOBECOM 2001-IEEE Global Telecommunications Conference. 2001,11(1):1036~1041
20 Jilei Hou, Paul H.Siegel and Laurence B.Milstein, Performance Analysis and Code Optimization of Low-Density Parity-Check Codes on Rayleigh Fading Channels. IEEE Journal Selected Areas in Communications. 2001,19(5):924~934
21 Nedeljko Varnica and Aleksandar Kavcic, Optimized LDPC Codes for Partial Response Channels. IEEE Communications Letters. 1995,7(1):168~170,
22 Hongwei Song, Jingfeng Liu and B.V.K.Vijaya. Low-Density Parity-Check (LDPC) Codes for Partial Response Equalized CAD-MO Recording. IEEE Trans. Magn. 2001,37(6):676~681
23 T. J. Richardson and R. L. Urbanke. The Capacity of Low-density Parity-Check Codes Under Message-passing Decoding. IEEE Transaction onInformation Theory. 2001,47(3):599~618
24 T. J. Richardson, M. A. Shokrollahi, and R. L.Urbanke. Design of Capacity approaching Irregular Low-density Parity-check Codes. IEEE Transactions on Information Theory. 2001, 47(9):619~637
25 S.Y. Chung, T. J. Richardson, and R. L. Urbanke. Analysis of Sum-product Decoding of Low-density Parity-check Codes using a Gaussian Approximation. IEEE Transactions on Information Theory. 2001,47(8): 657~670
26 Sae-Young Chung, G. David Forney Jr., Thomas J. Richardson and Rtidiger Urbanke. On the Design of Low-density Parity-check Codes within 0.0045 dB of the Shannon Limit. IEEE Communications Letters. 2001,5(2): 58~60
27 Sae-Young Chung. On the Construction of Some Capacity-approaching Coding Schemes. PH.D thesis, MIT, Sep.2000
28 Flarion Technologies Inc, Vector-Low-Density Parity-Check Coding Solution Data Sheet[EB/OL], http://www.flarion.com
29 H. A. Loeliger. Probability Propagation and Decoding in Analog VLSI. IEEE Transactions on Information Theory. 2001, 47(2):837~843
30 T. Zhang, K. K. Parhi. VLSI Implemntation-Oriented (3,K)-Regular Low-Density Parity-Check Codes[EB/OL]. from Internet. {tzhang. parhi}@ece. Umn. Edu.
31 Tejas Bhatt, Krishna Narayanan and Nasser Kehtarnavaz. Fixed Point DSP Implementation of Low-Density Parity-Check Codes. Proceedings of the 9th IEEE DSP Workshop, Hunt, TX, 2000:921~932
32 Zang Di, et al. Low-Density Parity-Check Codes used for Compressed Image Transmission over Noisy Channel. From Internet, http://image.chonbuk.ac.kr/ kwak/ data1/2002-4.pdf,Apr.2003
33 Low-density parity-check codes for DSL transmission. Temporary Document BI-095, Study Group 15/4, Goa, India, 23~27 Oct 2000.
34 Hongxin Song, Richard M. Todd and J. R. Cruz. Applications of Low-Density Parity-Check Codes to Magnetic Recording Channels. IEEE Journal on Selected Areas in Communications. 2001,19(5): 918~923
35 Ka Leong Lo. Layered Space Time Structures with Low-Density Parity-Check and Convolution Codes. Spread Spectrum Techniques and Applications,2002 IEEE Seventh International Symposium. 2002,2:308~312
36 R. M. Tanner. Arecursive Approach to Low Complexity Codes. IEEE Transactions on Information Theory. 1981, 11(27):533~547
37 M. Fossorier. Quasi-cyclic Low Density Parity Check Codes. Proc IEEE Int Symp Information Theory. 2003,11(4):150~161
38 Seho Myung, Kyeongcheol Yang. Quasi-cyclic LDPC Codes for Fast Encoding. IEEE Transactions on Information Theory. 2005,51(8):2894~2901
39 Marc P.C.Fossorier. Quasi-cyclic Low-density Parity-check Codes form Circulant Permutation Matrices. IEEE Transactions on Information Theory. 2004,50(8):1788~1793
40 A.Sridharan, D.J.Costello Jr, D.Sridhara, T.Fuja, R.M.Tanner. A Construction for Low-density Parity Check Convolution Codes based on Quasi-cyclic Block codes. Proc. IEEE Int. Symp. Information Theory. 2002:481~453
41 J. Fan. Array codes as LDPC codes. Proc. 2nd Int. Symp. Turbo Codes, 2000:543~546.
42 Eleftheriou E. and ?l?er S. Low density parity-check codes for digital subscriber lines. Proc., ICC’2002, New York, NY, 2002:1752~1757
43 E.Eleftheriou, S.Olcer. Low-density Parity-check Codes for Multilevel Modulation. Proc. IEEE Int. Symp. Information Theory. 2002:442.
44 Hao Zhong, Tong Zhang. Block-LDPC: A Practical LDPC Coding System Design Approach. IEEE Transactions on Information Theory. April 2005, 52(4):766~775
45 Li Zongwang, Chen Lei, Zeng Lingqi, Lin Shu. Efficient Encoding of Quasi-Cyclic Low-Density Parity-Check Codes. IEEE Transactions on Communications, 2006, 54(1):71~81
46 Y.Chen, K.K.Parhi. High Throughput Overlapped Message Passing for Low-density Parity Check Codes. Proc.IEEE/ACM GLSVLSI. 2003:245~248
47 N.Wiberg. Codes and Decoding on General Graphs. Ph.D.dessertation Univ.Linkoping, Dept. Elect.Eng.,Linkoping, Sweden,1996
48 F.R.Kschischang, B.J.Frey, H.-A.Loeliger. Factor Graphs and the sum-product algorithm. IEEE Transactions on Information Theory. 2001,47(2): 498~519
49 D. J. C. Mackay. Good error-correcting codes based on very sparse matrices.IEEE Transactions on Information Theory. 1999, 4 (5):3~431
50 Jinghu Chen, Ajay Dholakia, Evangelos Eleftheriou, Marc P. C. Fossorier and Xiao-Yu Hu.Reduced-Complexity Decoding of LDPC Codes.IEEE IEEE Transactions on Communications. 2001, 53(8):1288~1299
51 Motohiko Isaka, Marc Fossorier and Hideki Imai. Iterative Reliability-based Decoding of Turbo-like Codes. ICC’2002. 2002, 25(1): 1495~500.
52高锋,董亚波.低压电力线载波通信中信号传输特性分析.电力系统自动化. 2000, (7): 36~40
53赵云峰,汪晓岩,徐立忠等.低压电力线噪声分析与建模.电力系统通信. 2003, (1):31~34.
54 D. Middleton. Statistical-physical model of electromagnetic interference. IEEE. Trans. Electromagn. Compat. EMC-19. 1977,8(3): 106~126
55 D. Umehara, H. Yamaguchi, and Y. Morihiro. Turbo decoding over impulsive noise channel. Proc. 2004 ISPLC Conf. 2004,4(2):51~56
56 ?l?er S. Decoder architecture for array-code-based LDPC codes. Global Telecommunications Conference, GLOBECOM '03, San Francisco, 2003: 2046~2050
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