LDPC码及其反馈迭代均衡技术的研究
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摘要
在与BP迭代译码算法相结合的条件下具有逼近Shannon限的性能,是继turbo码后在纠错编码领域的又一重大进展。近年来,LDPC码以其优异的性能和巨大的潜在应用价值而受到编码界的极大关注,已成为目前最热门的研究领域之一。
本文在对LDPC码进行系统的分析和研究的基础上,探讨了高斯消元法和基于近似下三角矩阵的两种有效编码方式,并在加性高斯白噪声信道下,对LDPC码进行了BP译码算法性能仿真。同时,为了克服码间干扰(ISI)问题,本文主要研究了基于LDPC码的反馈迭代均衡技术。运用turbo译码的原理,在接收端的均衡器和BP译码器之间迭代地传递外信息,形成turbo均衡。首先,运用最小均方误差(MMSE)准则,将LDPC码的BP译码器分别与线性均衡器(LE)、判决反馈均衡器(DFE)相结合,形成非自适应的均衡接收机。其次,将最小均方(LMS)算法与判决反馈均衡器(DFE)相结合,提出了基于LDPC码的自适应均衡技术。最后,在自适应均衡的基础上,将常数模算法(CMA)与判决反馈均衡器(DFE)相结合,提出了基于LDPC码的盲均衡技术。
本文对这几种不同的均衡技术,进行了性能仿真和比较研究。结果表明,基于以上turbo均衡原理的接收端能够很好地提高系统的误码率性能。同等条件下,非自适应均衡误码率优于自适应均衡,收敛速度快,但算法相对复杂,且需要已知的信道特性,因而实用价值不大。而在信道特性未知的情况下,盲均衡技术性能并不比自适应均衡差,并且不需要训练序列,提高了频谱利用率,因此在移动通信中有着很好的应用前景。
The linear block code is called a binary low-density parity-check code if it is constructed based on a sparse parity-check matrix. Since it has been proved that the performance of LDPC codes is extremely close to the Shannon limits when combined with iterative BP algorithm, the discovery of LDPC codes is a great progress in channel coding field after turbo codes. In recent years, LDPC codes has drawn the world-wide attentions in channel coding community due to its great performance and potential worth in application.
On the basis of comprehensive studies of the performance of LDPC codes, this thesis discusses two effective encoding methods, Gauss elimination and the one based on approximate lower triangular matrix. As for BP decoding algorithm, simulation results are obtained over an AWGN channel. Furthermore, in order to overcome the problem of inter-symbol interference (ISI), we study the equalization techniques with feedback iterations suitable for LDPC codes. Applying the principles of turbo decoding, extrinsic information is exchanged between the decoder and the equalizer at the receiving end iteratively to form turbo equalization. First, by applying MMSE criteria, we combine the BP decoder of LDPC codes with linear equalizer (LE) and decision feedback equalizer (DFE) respectively to form the non-adaptive equalization receiver. Second, we present LDPC/LMS-DFE algorithm of adaptive equalization. Finally, we focus on the performance of LDPC/CMA-DFE algorithm of blind equalization.
For all three kinds of equalization techniques, we perform simulation and comparative studies. The results show that the LDPC receiver based on turbo equalization can effectively improve system performance. Compared to adaptive equalization, the non-adaptive equalization offers better BER performance and faster convergence speed. But the computational complexity and the request of given channel characteristics seriously limit its applications in practice. When the channel characteristics are unknown, blind equalization has similar performance to adaptive equalization. Because it doesn’t need training sequence, it can use the frequency band more efficiently, which makes it quite promising for the future applications in mobile communications.
本文在对LDPC码进行系统的分析和研究的基础上,探讨了高斯消元法和基于近似下三角矩阵的两种有效编码方式,并在加性高斯白噪声信道下,对LDPC码进行了BP译码算法性能仿真。同时,为了克服码间干扰(ISI)问题,本文主要研究了基于LDPC码的反馈迭代均衡技术。运用turbo译码的原理,在接收端的均衡器和BP译码器之间迭代地传递外信息,形成turbo均衡。首先,运用最小均方误差(MMSE)准则,将LDPC码的BP译码器分别与线性均衡器(LE)、判决反馈均衡器(DFE)相结合,形成非自适应的均衡接收机。其次,将最小均方(LMS)算法与判决反馈均衡器(DFE)相结合,提出了基于LDPC码的自适应均衡技术。最后,在自适应均衡的基础上,将常数模算法(CMA)与判决反馈均衡器(DFE)相结合,提出了基于LDPC码的盲均衡技术。
本文对这几种不同的均衡技术,进行了性能仿真和比较研究。结果表明,基于以上turbo均衡原理的接收端能够很好地提高系统的误码率性能。同等条件下,非自适应均衡误码率优于自适应均衡,收敛速度快,但算法相对复杂,且需要已知的信道特性,因而实用价值不大。而在信道特性未知的情况下,盲均衡技术性能并不比自适应均衡差,并且不需要训练序列,提高了频谱利用率,因此在移动通信中有着很好的应用前景。
The linear block code is called a binary low-density parity-check code if it is constructed based on a sparse parity-check matrix. Since it has been proved that the performance of LDPC codes is extremely close to the Shannon limits when combined with iterative BP algorithm, the discovery of LDPC codes is a great progress in channel coding field after turbo codes. In recent years, LDPC codes has drawn the world-wide attentions in channel coding community due to its great performance and potential worth in application.
On the basis of comprehensive studies of the performance of LDPC codes, this thesis discusses two effective encoding methods, Gauss elimination and the one based on approximate lower triangular matrix. As for BP decoding algorithm, simulation results are obtained over an AWGN channel. Furthermore, in order to overcome the problem of inter-symbol interference (ISI), we study the equalization techniques with feedback iterations suitable for LDPC codes. Applying the principles of turbo decoding, extrinsic information is exchanged between the decoder and the equalizer at the receiving end iteratively to form turbo equalization. First, by applying MMSE criteria, we combine the BP decoder of LDPC codes with linear equalizer (LE) and decision feedback equalizer (DFE) respectively to form the non-adaptive equalization receiver. Second, we present LDPC/LMS-DFE algorithm of adaptive equalization. Finally, we focus on the performance of LDPC/CMA-DFE algorithm of blind equalization.
For all three kinds of equalization techniques, we perform simulation and comparative studies. The results show that the LDPC receiver based on turbo equalization can effectively improve system performance. Compared to adaptive equalization, the non-adaptive equalization offers better BER performance and faster convergence speed. But the computational complexity and the request of given channel characteristics seriously limit its applications in practice. When the channel characteristics are unknown, blind equalization has similar performance to adaptive equalization. Because it doesn’t need training sequence, it can use the frequency band more efficiently, which makes it quite promising for the future applications in mobile communications.
引文
[1] Shannon, C.E:“A mathematical theory of communication”, Bell Syst. Tech .J., 27, pp.379-423, 623-656 (1948)
[2]王新梅,肖国镇,“纠错码-原理与方法”(修订版),西安电子科技大学出版社,2002
[3] R.G.Gallager,“Low-Density Parity-Check Codes”, IRE Transaction on Information Theory, 1962
[4] D.J.C.Mackay and R.M.Neal,“Near Shannon Limit Performance of Low Density Parity Check Codes”, Electric Lett., vol.32, pp1645-1646, Aug.1996
[5] Thomas J.Richardson and Rudiger L.Urbanke,“Capacity of Low-Density Parity-Check Codes Under Message-Passing Decoding”, IEEE Transaction on Information Theory, vol.47, No.2, Feb.2001
[6] Thomas J. Richardson and Rudiger L. Urbanke,“Efficient Encoding of Low-Density Parity-Check Codes”, IEEE Transactions on Information Theory, vol.47, No.2, Feb.2001
[7] Thomas J. Richardson, M.Amin Shokrollahi,“Design of Capacity-Approaching Irregular Low-Density Parity-Check Codes”, IEEE Transactions on Information Theory, vol.47, No.2, Feb.2001
[8] Zang Di, et al.“Low-Density Parity-Check Codes Used for Compressed Image Transmission over Noise Channel”, from Internet,2003, http://image.chonbuk.ac.kr/kwak/datal/2002-4.pdf
[9] Mittelholzer T, Dholakia A, Eleftheriou E.Reduced,“Complexity Decoding of Low Density Parity Check Codes for Generalized Partial Response Channels”, IEEE Transactions on Magnetics, vol.37, No.2, pp.721-727, Mar.2001
[10] E.Yeo, P.Pakzad, B.Nikolic, and V. Anantharam,“VLSI Architectures for Iterative Decoders in Magnetic Recording Channels”, IEEE Trans. Magnetics, vol.37, No.2, pp.748-755, Mar.2001
[11] 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, vol.19, No.5, May 2001, pp.918-923
[12] Eleftheriou E et al.,“Low-Density Parity-Check Codes for DSL Transmission”, Temporary Document BI-095, ITU-T SS, Study Group 15/4, Goa, India, 23-27 Oct, 2000
[13]“Performance of LDPC Coded Modulation for ADSL bits under latency constraints”, Temporary Document RN-073, Study Group 15/4, Red Bank, New Jersey, 21-25 May 2001
[14] Mohammad M. Mansour and Naresh R. Shanbhag,“A 1.6Gbit/sec 2048-bit Programmable and Code-Rate Tunable LDPC Decoder Chip”,3rd International Symposium on Turbo Codes an Related Topics-Brest-France-2003, pp.137-140
[15] M.Luby, M.Mitzenmacher, M.A.Shokrollahi, D.A.Spielman, and V.Stemann,“Practical Loss-Resilient Codes”, Proc.29th Symp. On Theory of Computing, pp.150-159, 1997
[16] D.J.C.Mackay, S.T.Wilson and M.C.Davey,“Comparison of Constructions of Irregular Gallager Codes”, IEEE Trans. Communications, vol.10, pp.1449-1454, Oct,1999
[17]卢开澄,卢华明,“图论及其应用”,北京:清华大学出版社,1995
[18] Marc P.C.Fossorier,“Iterative Reliability-Based Decoding of Low-Density Parity Check Codes”, IEEE Journal on Selected Areas in Communications, vol.19, No.5, May 2001
[19] M.Sipser, D.A.Spielman,“Expander Codes”, IEEE Transactions on Information Theory, vol.42, No.6, pp.1710-1722, Nov.1996
[20] Sae-Young Chung, Richardson T.J and Urbanke R.L.,“Analysis of Sum-Product Decoding of Low-Density Parity-Check Codes Using a Gaussian Approximation”, IEEE Transactions on Information Theory, vol.47, No.2, pp.657-670, Feb.2001
[21] Marc P.C.Fossorier,“Reduced Complexity Iterative Decoding of Low-Density Parity-Check Codes Based on Belief Propagation”, IEEE Transactions on Communications, vol.47, No.5, pp.673-680, May 1999
[22] H.Pishro-Nik and F.Fekri,“Improved Decoding Algorithms for Low-Density Parity-Check Codes”, 3rd International Symposium on Turbo Codes and Related Topics-Brest-France-2003, pp.117-120
[23] B.Vasic, K.Pedagani and M.Ivkovic,“High-Rate Girth-Eight Low-Density Parity-Check Codes on Rectangular Integer Lattices”, IEEE Trans. On Communications, 2004, vol.52, no.8
[24] Haotian Zhang and Moura.J.M.F,“The Design of Structured Regular LDPC Codes with Large Girth”, Golbal Telecommunications Conference, IEEE, 2003, vol.7:4022-4027
[25] I.Djurdjevic, S.Liu and K.A.Ghaffar,“Graph-Theoretic Construction of LDPC Codes”, IEEE Communications Letters, 2003, vol.7, no.4
[26] C.Berrou, A.Glavieux,“Near Optimum Error Correcting Coding and Decoding: TurboCodes”, IEEE Transactions on Communicacations,1996, vol.44, no.10
[27] Michael Tuchler, Ralf Koetter, and Andrew C.Singer,“Turbo Equalization: Principles and New Results”, IEEE Transactions on Communications, 2002, vol.50, no.5, pp.754-766
[28] S.Ariyavisitakul and Y.Li,“Joint Coding and Decision Feedback Equalization for Broadband Wireless Channels”, IEEE Journal on Selected Areas in Communications, 1998, vol.16:1670-1678
[29] K.Zhou, J.G.Proakis,“Decision Feedback Equalization of Time-dispersive Channels with Coded Modulation”, IEEE Trans.Commun, 1990, vol.38:18-24
[30] Michael Tuchler, Ralf Koetter, and Andrew C.Singer,“Minimum Mean Squared Error Equalization Using A Priori Infromation”, IEEE Trans. on Signal Processin, 2002, vol.50, no.3
[31] John G.Proakis,“Digital Communications (Fourth Edition)”,电子工业出版社,2003.1,pp.431-465, pp.477-490
[32] Tanner R.,“A Recursive Approach to Low Complexity Codes”, IEEE Trans. on Inform. Theory, 1981, 27(5):533-547
[33]丁玉美,阔永红,高新波编著,数字信号处理——时域离散随机信号处理,西安电子科技大学出版社,2002
[34] Sato.Y,“A Method for Self-recovering Equalization”, IEEE Trans. On Comm., 1975, COM-23(6), pp.679-682
[35] Dominique N.Gogard,“Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems”, IEEE Transactions on Communications, 1980, vol.28, no.11, pp.1867-1875
[36] J.R.Treichler,“A New Approach to Multipath Correction of Constant Modulus Signals”, IEEE Trans. On Acoustics, speech, and Signal Processing, 1983, vol. assp-31(2)
[37] C. B. Papadias A. Paulraj,“Decision-Feedback Equalization and Identification of Linear Channels Using Blind Algorithms of the Bussgang Type”, IEEE Proceedings of ASILOMAR-29, 1996, pp.335-340
[38] Halzinakos.D, Nikias.C.L,“Blind Decision Feedback Equalization Structures Based on Adaptive Cumuland Techniques”, ICC’89, 1989, pp.1278-1282
[39] Halzinakos.D, Nikias.C.L,“Blind Equalization Using a Tricepstrun-based Algorithm”, IEEE Trans.com, 1991
[40] S.K.Jha, J.J.Soraghan, T.S.Durrani,“Equalization Using Neural Networks”, The First CNN of IEEE, 1987, pp.356-380
[41] L.Tong, D.Liu and H.Zeng,“On Blind Decision Feedback Equalization”, In Proc.26th Asilomar Conference on Signal, System and Computer, 1996, 305-309
[42] L.Tong, D.Liu,“Bind Predictive Decision Feedback Equalization via the constant modulus algorithm”, Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, 1997, 3901-3904
[43] Li Ping, W.K.Leung,“Decoding Low Density Parity Check Codes with Finite Quantization Bits”, IEEE Communications Letters, 2000, vol.4, no.2
[2]王新梅,肖国镇,“纠错码-原理与方法”(修订版),西安电子科技大学出版社,2002
[3] R.G.Gallager,“Low-Density Parity-Check Codes”, IRE Transaction on Information Theory, 1962
[4] D.J.C.Mackay and R.M.Neal,“Near Shannon Limit Performance of Low Density Parity Check Codes”, Electric Lett., vol.32, pp1645-1646, Aug.1996
[5] Thomas J.Richardson and Rudiger L.Urbanke,“Capacity of Low-Density Parity-Check Codes Under Message-Passing Decoding”, IEEE Transaction on Information Theory, vol.47, No.2, Feb.2001
[6] Thomas J. Richardson and Rudiger L. Urbanke,“Efficient Encoding of Low-Density Parity-Check Codes”, IEEE Transactions on Information Theory, vol.47, No.2, Feb.2001
[7] Thomas J. Richardson, M.Amin Shokrollahi,“Design of Capacity-Approaching Irregular Low-Density Parity-Check Codes”, IEEE Transactions on Information Theory, vol.47, No.2, Feb.2001
[8] Zang Di, et al.“Low-Density Parity-Check Codes Used for Compressed Image Transmission over Noise Channel”, from Internet,2003, http://image.chonbuk.ac.kr/kwak/datal/2002-4.pdf
[9] Mittelholzer T, Dholakia A, Eleftheriou E.Reduced,“Complexity Decoding of Low Density Parity Check Codes for Generalized Partial Response Channels”, IEEE Transactions on Magnetics, vol.37, No.2, pp.721-727, Mar.2001
[10] E.Yeo, P.Pakzad, B.Nikolic, and V. Anantharam,“VLSI Architectures for Iterative Decoders in Magnetic Recording Channels”, IEEE Trans. Magnetics, vol.37, No.2, pp.748-755, Mar.2001
[11] 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, vol.19, No.5, May 2001, pp.918-923
[12] Eleftheriou E et al.,“Low-Density Parity-Check Codes for DSL Transmission”, Temporary Document BI-095, ITU-T SS, Study Group 15/4, Goa, India, 23-27 Oct, 2000
[13]“Performance of LDPC Coded Modulation for ADSL bits under latency constraints”, Temporary Document RN-073, Study Group 15/4, Red Bank, New Jersey, 21-25 May 2001
[14] Mohammad M. Mansour and Naresh R. Shanbhag,“A 1.6Gbit/sec 2048-bit Programmable and Code-Rate Tunable LDPC Decoder Chip”,3rd International Symposium on Turbo Codes an Related Topics-Brest-France-2003, pp.137-140
[15] M.Luby, M.Mitzenmacher, M.A.Shokrollahi, D.A.Spielman, and V.Stemann,“Practical Loss-Resilient Codes”, Proc.29th Symp. On Theory of Computing, pp.150-159, 1997
[16] D.J.C.Mackay, S.T.Wilson and M.C.Davey,“Comparison of Constructions of Irregular Gallager Codes”, IEEE Trans. Communications, vol.10, pp.1449-1454, Oct,1999
[17]卢开澄,卢华明,“图论及其应用”,北京:清华大学出版社,1995
[18] Marc P.C.Fossorier,“Iterative Reliability-Based Decoding of Low-Density Parity Check Codes”, IEEE Journal on Selected Areas in Communications, vol.19, No.5, May 2001
[19] M.Sipser, D.A.Spielman,“Expander Codes”, IEEE Transactions on Information Theory, vol.42, No.6, pp.1710-1722, Nov.1996
[20] Sae-Young Chung, Richardson T.J and Urbanke R.L.,“Analysis of Sum-Product Decoding of Low-Density Parity-Check Codes Using a Gaussian Approximation”, IEEE Transactions on Information Theory, vol.47, No.2, pp.657-670, Feb.2001
[21] Marc P.C.Fossorier,“Reduced Complexity Iterative Decoding of Low-Density Parity-Check Codes Based on Belief Propagation”, IEEE Transactions on Communications, vol.47, No.5, pp.673-680, May 1999
[22] H.Pishro-Nik and F.Fekri,“Improved Decoding Algorithms for Low-Density Parity-Check Codes”, 3rd International Symposium on Turbo Codes and Related Topics-Brest-France-2003, pp.117-120
[23] B.Vasic, K.Pedagani and M.Ivkovic,“High-Rate Girth-Eight Low-Density Parity-Check Codes on Rectangular Integer Lattices”, IEEE Trans. On Communications, 2004, vol.52, no.8
[24] Haotian Zhang and Moura.J.M.F,“The Design of Structured Regular LDPC Codes with Large Girth”, Golbal Telecommunications Conference, IEEE, 2003, vol.7:4022-4027
[25] I.Djurdjevic, S.Liu and K.A.Ghaffar,“Graph-Theoretic Construction of LDPC Codes”, IEEE Communications Letters, 2003, vol.7, no.4
[26] C.Berrou, A.Glavieux,“Near Optimum Error Correcting Coding and Decoding: TurboCodes”, IEEE Transactions on Communicacations,1996, vol.44, no.10
[27] Michael Tuchler, Ralf Koetter, and Andrew C.Singer,“Turbo Equalization: Principles and New Results”, IEEE Transactions on Communications, 2002, vol.50, no.5, pp.754-766
[28] S.Ariyavisitakul and Y.Li,“Joint Coding and Decision Feedback Equalization for Broadband Wireless Channels”, IEEE Journal on Selected Areas in Communications, 1998, vol.16:1670-1678
[29] K.Zhou, J.G.Proakis,“Decision Feedback Equalization of Time-dispersive Channels with Coded Modulation”, IEEE Trans.Commun, 1990, vol.38:18-24
[30] Michael Tuchler, Ralf Koetter, and Andrew C.Singer,“Minimum Mean Squared Error Equalization Using A Priori Infromation”, IEEE Trans. on Signal Processin, 2002, vol.50, no.3
[31] John G.Proakis,“Digital Communications (Fourth Edition)”,电子工业出版社,2003.1,pp.431-465, pp.477-490
[32] Tanner R.,“A Recursive Approach to Low Complexity Codes”, IEEE Trans. on Inform. Theory, 1981, 27(5):533-547
[33]丁玉美,阔永红,高新波编著,数字信号处理——时域离散随机信号处理,西安电子科技大学出版社,2002
[34] Sato.Y,“A Method for Self-recovering Equalization”, IEEE Trans. On Comm., 1975, COM-23(6), pp.679-682
[35] Dominique N.Gogard,“Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems”, IEEE Transactions on Communications, 1980, vol.28, no.11, pp.1867-1875
[36] J.R.Treichler,“A New Approach to Multipath Correction of Constant Modulus Signals”, IEEE Trans. On Acoustics, speech, and Signal Processing, 1983, vol. assp-31(2)
[37] C. B. Papadias A. Paulraj,“Decision-Feedback Equalization and Identification of Linear Channels Using Blind Algorithms of the Bussgang Type”, IEEE Proceedings of ASILOMAR-29, 1996, pp.335-340
[38] Halzinakos.D, Nikias.C.L,“Blind Decision Feedback Equalization Structures Based on Adaptive Cumuland Techniques”, ICC’89, 1989, pp.1278-1282
[39] Halzinakos.D, Nikias.C.L,“Blind Equalization Using a Tricepstrun-based Algorithm”, IEEE Trans.com, 1991
[40] S.K.Jha, J.J.Soraghan, T.S.Durrani,“Equalization Using Neural Networks”, The First CNN of IEEE, 1987, pp.356-380
[41] L.Tong, D.Liu and H.Zeng,“On Blind Decision Feedback Equalization”, In Proc.26th Asilomar Conference on Signal, System and Computer, 1996, 305-309
[42] L.Tong, D.Liu,“Bind Predictive Decision Feedback Equalization via the constant modulus algorithm”, Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, 1997, 3901-3904
[43] Li Ping, W.K.Leung,“Decoding Low Density Parity Check Codes with Finite Quantization Bits”, IEEE Communications Letters, 2000, vol.4, no.2