基于有限几何LDPC编码的研究及其FPGA实现
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摘要
LDPC码由Gallager于1962年首次提出,只是限于当时的技术发展水平这方面的研究在接下来的30多年里沉寂了,直到上世纪九十年代中期,Mackay, Neal等人重新发现LDPC码与Turbo码一样也是一种能够逼近香农限的非常好码,过去几年LDPC码有了很大发展。
本文以欧氏几何空间、循环码、线性分组码为理论背景对LDPC码做了研究,并设计了DVB-S2标准中LDPC编码在硬件FPGA匕的实现方案。主要工作如下：
1.研究了LDPC码奇偶校验矩阵的构造方法,基于置信度传播的译码算法,一般编码算法和快速编码算法。重点研究了欧氏空间LDPC码和基于奇偶校验矩阵编码的算法。
2.设计了构造第一类二维循环(2,0,s)阶EG-LDPC码的的具体实现步骤；提出一种直接从奇偶校验矩阵经过简单计算即可获得循环EG-LDPC码的生成多项式的方法。
3.设计了构造第一类多维准循环(m,0,s)阶EG-LDPC码的的具体实现方法,提出基于准循环矩阵实现q(q>2)进制LDPC编码的思路,并给出了硬件编码电路。
4.针对LDPC码通用编码算法的复杂度与码长的平方成正比的问题重点研究了LDPC码的快速编码方法。根据DVB-S2标准中给出的LDPC码的奇偶校验矩阵为非规则校验矩阵,此矩阵可以看成由一个稀疏矩阵和一个双对角线矩阵组成,符合利用校验矩阵直接进行编码的条件,并且编码复杂度为线性。给出了在FPGA上实现的编码器的方案,针对设计中的乘法运算提出了累加的改进方法,节约了存储空间,利用硬件描述语言完成了编码器设计。
The Low Density Parity-Check (LDPC) code was first proposed by Gallager in 1962, of which the research had been halted due to constraint of the technology development in the following more than 30 years. In the 1990s, Mackay and Neal found that LDPC is one kind of good codes as well as turbo and near to Shannon limit. Based on this discovery, the research about LDPC has grown and got a big development in the past several years.
Based upon Euclidean Geometry, Cyclic codes and linear block codes scheme, this dissertation is trying to research on LDPC and presenting a proposal, for how to implement the LDPC encoder on hardware according to DVB-S2 standard. The details are as below:
1. The structure method of LDPC parity check matrix has been studied in detail, as well as belief-propagation decoding algorithms, general encoding and fast encoding algorithms. Euclidean geometry LDPC and encoding algorithms based on the parity check matrix are mainly researched.
2. A new method for constructing the first class 2-dimention (2,0,s) order EG-LDPC is presented; and a new method for getting generated polynomial from parity check matrix of cyclic EG-LDPC by simple calculation is proposed.
3. A method for constructing the first class multi-dimension (m,0 s) order cyclic EG-LDPC is presented; and an implementation of non binary LDPC encoding from QC-LDPC is proposed, with the hardware encoder circuit introduced.
4. On the question that the complexity and the quadratic of code length are proportional by general algorithms, the fast encoding method of LDPC is mainly studied. An encoder is presented on FPGA with hardware language according to DVB-S2 standard. The LDPC parity check matrix in DVB-S2 standard is irregular and is composed of one sparse matrix and one double diagonal matrix and has condition to be encoded directly in linear time. By corrective method of accumulation to replace multiply operation, the proposal gives a method to save memory space.
本文以欧氏几何空间、循环码、线性分组码为理论背景对LDPC码做了研究,并设计了DVB-S2标准中LDPC编码在硬件FPGA匕的实现方案。主要工作如下：
1.研究了LDPC码奇偶校验矩阵的构造方法,基于置信度传播的译码算法,一般编码算法和快速编码算法。重点研究了欧氏空间LDPC码和基于奇偶校验矩阵编码的算法。
2.设计了构造第一类二维循环(2,0,s)阶EG-LDPC码的的具体实现步骤；提出一种直接从奇偶校验矩阵经过简单计算即可获得循环EG-LDPC码的生成多项式的方法。
3.设计了构造第一类多维准循环(m,0,s)阶EG-LDPC码的的具体实现方法,提出基于准循环矩阵实现q(q>2)进制LDPC编码的思路,并给出了硬件编码电路。
4.针对LDPC码通用编码算法的复杂度与码长的平方成正比的问题重点研究了LDPC码的快速编码方法。根据DVB-S2标准中给出的LDPC码的奇偶校验矩阵为非规则校验矩阵,此矩阵可以看成由一个稀疏矩阵和一个双对角线矩阵组成,符合利用校验矩阵直接进行编码的条件,并且编码复杂度为线性。给出了在FPGA上实现的编码器的方案,针对设计中的乘法运算提出了累加的改进方法,节约了存储空间,利用硬件描述语言完成了编码器设计。
The Low Density Parity-Check (LDPC) code was first proposed by Gallager in 1962, of which the research had been halted due to constraint of the technology development in the following more than 30 years. In the 1990s, Mackay and Neal found that LDPC is one kind of good codes as well as turbo and near to Shannon limit. Based on this discovery, the research about LDPC has grown and got a big development in the past several years.
Based upon Euclidean Geometry, Cyclic codes and linear block codes scheme, this dissertation is trying to research on LDPC and presenting a proposal, for how to implement the LDPC encoder on hardware according to DVB-S2 standard. The details are as below:
1. The structure method of LDPC parity check matrix has been studied in detail, as well as belief-propagation decoding algorithms, general encoding and fast encoding algorithms. Euclidean geometry LDPC and encoding algorithms based on the parity check matrix are mainly researched.
2. A new method for constructing the first class 2-dimention (2,0,s) order EG-LDPC is presented; and a new method for getting generated polynomial from parity check matrix of cyclic EG-LDPC by simple calculation is proposed.
3. A method for constructing the first class multi-dimension (m,0 s) order cyclic EG-LDPC is presented; and an implementation of non binary LDPC encoding from QC-LDPC is proposed, with the hardware encoder circuit introduced.
4. On the question that the complexity and the quadratic of code length are proportional by general algorithms, the fast encoding method of LDPC is mainly studied. An encoder is presented on FPGA with hardware language according to DVB-S2 standard. The LDPC parity check matrix in DVB-S2 standard is irregular and is composed of one sparse matrix and one double diagonal matrix and has condition to be encoded directly in linear time. By corrective method of accumulation to replace multiply operation, the proposal gives a method to save memory space.
引文
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[34]M.G. Luby, M. Mitzenmacher, and M.A. Shokrollahi. Analysis of random processes via and-or tree evaluation. In Proc.9th Annu. ACM-SIAM Symp. Discrete Algorithms,1998. pp.364-373.
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[36]T.J. Richardson and R.L. Urbanke. The capacity of low-density parity-check codes under message-passing decoding. IEEE Trans. Inf. Theory,47(2):Feb. 2001.599-618.
[37]H.-A. Loeliger, F. Lustenberger, M. Helfenstein etc. "Probability propagation and decoding inanalog VLSI," Proc.1998, IEEE Intl. Symp. Inform. Theory (Cambridge, MA), Aug.1998 pp.146.
[38]Research and Development for Space Data System Standards, Low Density Parity Check Codes for Use in Near-Earth and Deep Space Applications, Experimental specification CCSDS 131.1-0-1, August 2006
[39]中华人民共和国广播电影电视行业标准-移动多媒体广播第1部分：广播信道帧结构、信道编码和调制.GY/T 220.1-2006
[40]IEEE 802.16e/D 10-2005.Part 16:Air interface for fixed and mobile broadband wireless accesssystems-Amendment for physical and medium access control layers for combined fixed and mobile operation in licensed bands.August 9,2005
[41]Luby, M. Mitzenmacher, M.A. Shokrollahi, and D.A. Spielman. Improved low-density parity-check codes using irregular graphs and belief propagation. In Proc. of 1998 IEEE ISIT, Cambridge, MA, pp.117, Aug.1998.
[42]M.G.Luby,M.Mitzenmacher,M.A.Shokrollah.Improved low-density Parity-check Codes UsingIrregular Graphs,Information Theory. IEEE Transactions on,2001,vol.47:585-598.
[43]. M.G.Luby,M.Mitzenmacher,M.A.Shokrollah,D.A.Spielman. Analysis of Low density Parity-check Codes and Improved Designs Using Irregular Graghs.in Proceeding of the 30th Annual Symposium on Theory of computing,1998,:249-252.
[44]D.J.C.MacKay.Good Error-Correcting Codes based on Very Sparse Matrices.IEEE TransInform Theory,1996,45(3):399-431.
[45]Abhiram Prabhakar, Krashna Narayanan. Pseuorandom Construction of Low-density Parity-check Using Linear Congruential Sequences. IEEE Transactions On Communications, Vol.50,No 9,sep.2002.
[46]Jorge Campello,Dharmendra S Modha and Sridhar Ragagopalan. Designing LDPC Cods Using Bit-filling. ICC 2001,No.1,June 2001.
[47]Jorge Campello,Dharmendra S Modha.Extended Bit-filling and LDPC Code Design. Globecom 2001,No.1,Nov 2001.
[48]E J Wendon, Jr,"Euclidean Geometry Cyclic Codes," Proc. Symp. Combinatorial Math., University of North Carolina,Chapel Hill, April 1967.
[49]T Kasami, S Lin, and W W Peterson,"Polynomial Codes," IEEE Trans. Inform. Theory,14:1968.807-14.
[50]E J Weldon, Jr, "Some Results on Majority-Logic Decoding," Chap.8 in Error-correcting Codes,Hmann, ed., John Wiley, New York,1968.
[51]J Rosenthal P.O.Vontobel. Constructions of LDPC codes use Ramanujan Graphs and ideas from Margulis. Proceedings of 38th Allerton Conference on Communication and compitong, October,2000.
[52]Lin S and Costello D J. Error control coding: fundamentals and applications. New Jersey: Prentice-Hall Inc.,1993:184-257.
[53]T. J. Richardson and R. L. Urbanke, "Efficient Encoding of Low-Density Parity-Check Codes", IEEE Transactions on Information Theory, Vol 47, No.2, Feb.2001.
[54]Michael Yang, William E. Ryan, Yan Li. Design of Efficiently Encodable Moderate-Length High-Rate Irregular LDPC Codes. IEEE Transactions on Communication.2004,52(4):566-571.
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[2]C E Shannon,"A Mathenmatical Theory of communication," Bell Syst. Tech.J., pp.379-423 (part 1); July 1948, pp.623-656(part 2).
[3]王育民,梁传甲.信息与编码理论.西北电讯工程学院出版社,1986,pp.1-10.
[4]Shu Lin,Daniel J.Costello.Jr.差错控制编码机械工业出版社,2007.6,pp350-360.
[5]J.G. Proakis. Digital Communications,3rd ed. New York: McGraw-Hill,1995, pp. 613-646.
[6]王新梅,肖国镇.纠错码原理与方法.西安电子科技大学出版社,1991.pp.100-120.
[7]T.M. Cover and J.A. Thomas, Elements of Information Theory. New York, Wiley 1991.
[8]R. E. Blahut, Principles and Practice of Information Theory. Reading, MA: Addison-wesley,1987.
[9]C. Berrou and A. Glavieux. Near optimum error correcting coding and decoding: turbo-codes. IEEE Trans. Comm.,44(10):Oct.1996,1261-1271.
[10]C. Berrou, A. Glavieux, and P. Thitimajshima. Near Shannon limit error- correct ing coding and decoding: turbo-codes. In Proc. of ICC'93, May 1993 pp.1064-1070.
[11]R.G. Gallager. Low density parity-check codes. Cambridge, MA:MIT Press,196 3.
[12]R.G. Gallager. Low density parity-check codes. IRE Trans. Inf. Theory,8(1):Jan. 1962 pp 21-28.
[13]G. D. Forney, Jr, "Codes on graphs:New and wiews," In Proc.2 International Symposium on Turbo Codes and Related Topics, Brest France, Sept.1988 pp. 9-16.
[14]G. D. Forney, Jr, "Codes on graphs:normal realizations," IEEE Trans. Inform. Theory, vol47, no2, Feb.2001 pp.520-548.
[15]R. J. McEliece, "Are Turbo-like codes effective on nonstandard channels" IEEE Inform. Theory Society Newsletter, vol.51, no.4, Dec.2001.
[16]D.J.C. MacKay. Good error correcting codes based on very sparse matrices. IEEE Trans. Inf. Theory,45(2):1999. pp 399-431.
[17]N. Wiberg. Codes and decoding on general graphs. Ph.D. dissertation. Linkoping University, Linkoping, Sweden,1996.
[18]D.J.C. MacKay. Turbo codes are low-density parity-check codes. [Online], available: http://www.cs.toronto.edu/-mackay/abstracts/-turbo-ldpc-html,1998.
[19]N. Wiberg, H.-A. Loeliger, and R. Koetter. Codes and iterative decoding on general graphs. European Trans. Telecomm.,6 1995. pp 513-525.
[20]R.M. Tanner. A recursive approach to low complexity codes. IEEE Trans. Inf. Theory, Sept.1981.27:533-547.
[21]D.J.C. MacKay and R.M. Neal. Near Shannon limit performance of low-density parity check codes. Electron. Lett., Aug.1996.32:1645-1646.
[22]M.C. Davey. Error-correction using low-density parity-check codes. Ph.D. dissertation, University Cambridge, Cambridge, UK, Dec.1999.
[23]M.C. Davey and D.J.C. Mackay. Low density parity check codes GF(q). IEEE Comm. Letters,2(6):June 1998. pp 165-167.
[24]D. A. Spielman, Computationally efficient error-correcting codes and holographic proofs. Ph. D dissertation. MIT,1995
[25]D.A. Spielman. Linear-time encodeable and decodable error-correcting codes. IEEE Trans. Inf. Theory,42(6):Nov.1996.1723-1731.
[26]M. Sipser and D.A. Spielman. Expander codes. IEEE Trans. Inf. Theory,42(6): Nov.1996.1710-1722.
[27]D.A. Spielman. "Constructing error-correcting codes from expander graphs," Available at http://www-math.mit.edu/-spielman.
[28]M.G. Luby, M. Mitzenmacher, M.A. Shokrollahi, D.A.Spielman, and V. Stemann. Practical loss-resilient codes. In Proc.29th Annu. Symp. Theory of Computing,,1997. pp.150-159.
[29]J. W. Byers, M. Luby, M. Mitzenmacher, and A. Rege. A digital fountain approach to reliable distribution of bulk data. [Online], available at http://www.icsi. berkeley. edu/-luby/,1998.
[30]S.-Y. Chung, G.D. Forney, Jr., T.J. Richardson, and R. Urbanke. On the design of low-density parity-check codes within 0.0045dB of the Shannon limit. IEEE Comm. Letters,5(2):Feb.2001.58-60.
[31]M.G. Luby, M. Mitzenmacher, M.A. Shokrollahi, and D.A. Spielman. Analysis of low-density codes and improved designs using irregular graphs. In Proc.30th Annu. Symp. Theory of Computing,1998. pp.249-258.
[32]T.J. Richardson, M.A. Shokrollahi, and R.L. Urbanke. Design of capacity-approaching irregular low-density parity-check codes. IEEE Trans. Inf. Theory,47(2):Feb.2001.619-637.
[33]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 Trans. Inf. Theory,47(2):Feb.2001.657-670.
[34]M.G. Luby, M. Mitzenmacher, and M.A. Shokrollahi. Analysis of random processes via and-or tree evaluation. In Proc.9th Annu. ACM-SIAM Symp. Discrete Algorithms,1998. pp.364-373.
[35]G.D. Forney, Jr. Codes on graphs:normal realizations. IEEE Trans. Inf. Theory, 47(2):2001.520-548.
[36]T.J. Richardson and R.L. Urbanke. The capacity of low-density parity-check codes under message-passing decoding. IEEE Trans. Inf. Theory,47(2):Feb. 2001.599-618.
[37]H.-A. Loeliger, F. Lustenberger, M. Helfenstein etc. "Probability propagation and decoding inanalog VLSI," Proc.1998, IEEE Intl. Symp. Inform. Theory (Cambridge, MA), Aug.1998 pp.146.
[38]Research and Development for Space Data System Standards, Low Density Parity Check Codes for Use in Near-Earth and Deep Space Applications, Experimental specification CCSDS 131.1-0-1, August 2006
[39]中华人民共和国广播电影电视行业标准-移动多媒体广播第1部分：广播信道帧结构、信道编码和调制.GY/T 220.1-2006
[40]IEEE 802.16e/D 10-2005.Part 16:Air interface for fixed and mobile broadband wireless accesssystems-Amendment for physical and medium access control layers for combined fixed and mobile operation in licensed bands.August 9,2005
[41]Luby, M. Mitzenmacher, M.A. Shokrollahi, and D.A. Spielman. Improved low-density parity-check codes using irregular graphs and belief propagation. In Proc. of 1998 IEEE ISIT, Cambridge, MA, pp.117, Aug.1998.
[42]M.G.Luby,M.Mitzenmacher,M.A.Shokrollah.Improved low-density Parity-check Codes UsingIrregular Graphs,Information Theory. IEEE Transactions on,2001,vol.47:585-598.
[43]. M.G.Luby,M.Mitzenmacher,M.A.Shokrollah,D.A.Spielman. Analysis of Low density Parity-check Codes and Improved Designs Using Irregular Graghs.in Proceeding of the 30th Annual Symposium on Theory of computing,1998,:249-252.
[44]D.J.C.MacKay.Good Error-Correcting Codes based on Very Sparse Matrices.IEEE TransInform Theory,1996,45(3):399-431.
[45]Abhiram Prabhakar, Krashna Narayanan. Pseuorandom Construction of Low-density Parity-check Using Linear Congruential Sequences. IEEE Transactions On Communications, Vol.50,No 9,sep.2002.
[46]Jorge Campello,Dharmendra S Modha and Sridhar Ragagopalan. Designing LDPC Cods Using Bit-filling. ICC 2001,No.1,June 2001.
[47]Jorge Campello,Dharmendra S Modha.Extended Bit-filling and LDPC Code Design. Globecom 2001,No.1,Nov 2001.
[48]E J Wendon, Jr,"Euclidean Geometry Cyclic Codes," Proc. Symp. Combinatorial Math., University of North Carolina,Chapel Hill, April 1967.
[49]T Kasami, S Lin, and W W Peterson,"Polynomial Codes," IEEE Trans. Inform. Theory,14:1968.807-14.
[50]E J Weldon, Jr, "Some Results on Majority-Logic Decoding," Chap.8 in Error-correcting Codes,Hmann, ed., John Wiley, New York,1968.
[51]J Rosenthal P.O.Vontobel. Constructions of LDPC codes use Ramanujan Graphs and ideas from Margulis. Proceedings of 38th Allerton Conference on Communication and compitong, October,2000.
[52]Lin S and Costello D J. Error control coding: fundamentals and applications. New Jersey: Prentice-Hall Inc.,1993:184-257.
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