LTE系统上行链路信道估计算法研究
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摘要
LTE是3GPP组织近几年启动的面向新一代通信系统的大规模新技术研发项目,以正交频分复用、单载波频域均衡等为核心技术。信道估计在宽带通信系统中具有重要的实际意义,专门针对LTE系统进行信道估计算法的研究也十分必要。
本文以LTE上行链路为背景,研究了适用于LTE系统的信道估计方法。介绍了最小二乘(LS)和线性最小均方误差(LMMSE)两个基本准则,并通过仿真比较了它们的性能。仿真表明,LMMSE可以获得较好的性能,但是复杂度高。通过把LS估计的结果在时域进行截断,可以明显提高估计的准确度。
LTE系统中使用的是块导频,而块导频结构在抗多普勒方面具有先天的不足。本文使用多项式拟合与内插的方法来增强系统在高多普勒情况下的性能。分别在三种信道(多普勒)环境和四个信噪比等十二种组合的情况下,仿真了各种拟合参数的结果,按照低复杂度、高性能的原则确定了不同条件下的最佳拟合参数,形成了自适应的多项式拟合与内插策略。仿真表明,高阶拟合仅适合在高多普勒和高信噪比的条件下使用。该策略可以在性能和复杂度之间获得良好的平衡。
如果利用信道的时域相关性,通过对信道的初始频域估计进行时域滤波,可以进一步提高信道估计的准确度。本文对时域维纳滤波进行了仿真,并考察了其抗多普勒的性能。根据归一化最小均方(NLMS)滤波器和递归最小二乘(RLS)滤波器的原理,推导了自适应信道估计的过程。通过仿真,分析了滤波器的各个参数对NLMS算法及RLS算法的影响,并分析了二者的收敛特性。仿真表明,RLS算法比NLMS算法具有更强的抗多普勒性能;多普勒和滤波器长度对二者的收敛速度没有明显影响。
The Long Term Evolution (LTE) is a new research and development project launched by 3rd Generation Partnership Project (3GPP) organization for the next generation communication systems. The LTE systems takes orthogonal frequency division multiplexing and single-carrier frequency domain equalization as core technology. Channel estimation is of great significance in broadband communication systems, and it is necessary to study the channel estimation algorithm specially for LTE systems.
This paper studies channel estimation method applicable for LTE uplink. We introduce two basic criterion, namely Least Square (LS) and Linear Minimum Mean Square Error (LMMSE), and the simulations show that LMMSE can achieve better performance than LS, but is more complex. The estimation accuracy can be greatly improved by truncating the LS results in time domain, according to the length of channel.
It’s block pilots that will be used in LTE systems, and block pilots has inherent deficiencies of resisting Doppler frequency shift. In this paper, we use polynomial fit and interpolation method to enhance the performance of systems under high Doppler spread. By simulation, we give the MSE results for different fit parameters, under twelve kinds of channel environment, which is made up with three sets of channel parameters and four SNRs. Under the principle of lower complexity and higher performance, we choose the best fit parameters for each channel environment, forming an adaptive polynomial fit and interpolation strategy. The simulation shows that high order fit applies only for high Doppler spread and high SNR. This strategy can make a better balance between performance and complexity.
The estimation accuracy can be further improved by exploring the time domain correlation of the channel, that is, passing the initial estimation results in frequency domain through a time domain filter. We analyze the Doppler-resisting performance of time domain Wiener filter. According to the principle of normalized least mean square (NLMS) filter and recursive least square (RLS) filter, the method of adaptive channel estimation is derived. By simulation, we analyze the impact of different filter parameters on NLMS and RLS algorithm, and the convergence properties of the two. Simulation shows that RLS achieves better Doppler-resisting performance than NLMS, and the Doppler spread and the length of filter has no obvious influence on the speed of convergence.
本文以LTE上行链路为背景,研究了适用于LTE系统的信道估计方法。介绍了最小二乘(LS)和线性最小均方误差(LMMSE)两个基本准则,并通过仿真比较了它们的性能。仿真表明,LMMSE可以获得较好的性能,但是复杂度高。通过把LS估计的结果在时域进行截断,可以明显提高估计的准确度。
LTE系统中使用的是块导频,而块导频结构在抗多普勒方面具有先天的不足。本文使用多项式拟合与内插的方法来增强系统在高多普勒情况下的性能。分别在三种信道(多普勒)环境和四个信噪比等十二种组合的情况下,仿真了各种拟合参数的结果,按照低复杂度、高性能的原则确定了不同条件下的最佳拟合参数,形成了自适应的多项式拟合与内插策略。仿真表明,高阶拟合仅适合在高多普勒和高信噪比的条件下使用。该策略可以在性能和复杂度之间获得良好的平衡。
如果利用信道的时域相关性,通过对信道的初始频域估计进行时域滤波,可以进一步提高信道估计的准确度。本文对时域维纳滤波进行了仿真,并考察了其抗多普勒的性能。根据归一化最小均方(NLMS)滤波器和递归最小二乘(RLS)滤波器的原理,推导了自适应信道估计的过程。通过仿真,分析了滤波器的各个参数对NLMS算法及RLS算法的影响,并分析了二者的收敛特性。仿真表明,RLS算法比NLMS算法具有更强的抗多普勒性能;多普勒和滤波器长度对二者的收敛速度没有明显影响。
The Long Term Evolution (LTE) is a new research and development project launched by 3rd Generation Partnership Project (3GPP) organization for the next generation communication systems. The LTE systems takes orthogonal frequency division multiplexing and single-carrier frequency domain equalization as core technology. Channel estimation is of great significance in broadband communication systems, and it is necessary to study the channel estimation algorithm specially for LTE systems.
This paper studies channel estimation method applicable for LTE uplink. We introduce two basic criterion, namely Least Square (LS) and Linear Minimum Mean Square Error (LMMSE), and the simulations show that LMMSE can achieve better performance than LS, but is more complex. The estimation accuracy can be greatly improved by truncating the LS results in time domain, according to the length of channel.
It’s block pilots that will be used in LTE systems, and block pilots has inherent deficiencies of resisting Doppler frequency shift. In this paper, we use polynomial fit and interpolation method to enhance the performance of systems under high Doppler spread. By simulation, we give the MSE results for different fit parameters, under twelve kinds of channel environment, which is made up with three sets of channel parameters and four SNRs. Under the principle of lower complexity and higher performance, we choose the best fit parameters for each channel environment, forming an adaptive polynomial fit and interpolation strategy. The simulation shows that high order fit applies only for high Doppler spread and high SNR. This strategy can make a better balance between performance and complexity.
The estimation accuracy can be further improved by exploring the time domain correlation of the channel, that is, passing the initial estimation results in frequency domain through a time domain filter. We analyze the Doppler-resisting performance of time domain Wiener filter. According to the principle of normalized least mean square (NLMS) filter and recursive least square (RLS) filter, the method of adaptive channel estimation is derived. By simulation, we analyze the impact of different filter parameters on NLMS and RLS algorithm, and the convergence properties of the two. Simulation shows that RLS achieves better Doppler-resisting performance than NLMS, and the Doppler spread and the length of filter has no obvious influence on the speed of convergence.
引文
[1] Coon, J., S. Armour, M. Beach, et al. Adaptive frequency-domain equalization for single-carrier MIMO systems[C]. in IEEE International Conference on Communications. 2004: 2487-2491.
[2] Falconer, D., S.L. Ariyavisitakul, A. Benyamin-Seeyar, et al., Frequency domain equalization for single-carrier broadband wireless systems[J]. IEEE Communications Magazine, 2002. 40(4): 58-66.
[3] Myung, H.G., J. Lim, and D.J. Goodman, Single Carrier FDMA for Uplink Wireless Transmission[J]. IEEE Vehicular Technology Magazine, 2006. 1(3): 30-38.
[4] Hwang, T., C.Y. Yang, G. Wu, et al., OFDM and Its Wireless Applications: A Survey[J]. IEEE Transactions on Vehicular Technology, 2009. 58(4): 1673-1694.
[5] Hoeher, P., S. Kaiser, and P. Robertson. Two-dimensional pilot-symbol-aided channel estimation by Wiener filtering[C]. in IEEE International Conference on Acoustics, Speech, and Signal Processing. 1997: 1845-1848.
[6] Li, Y., Pilot-symbol-aided channel estimation for OFDM in wireless systems[J]. IEEE Transactions on Vehicular Technology, 2000. 49(4): 1207-1215.
[7] Morelli, M. and U. Mengali, A comparison of pilot-aided channel estimation methods for OFDM systems[J]. IEEE Transactions on Signal Processing, 2001. 49(12): 3065-3073.
[8] Beek, J.J., O. edfors, M. Sandell, et al. On channel estimation in OFDM systems[C]. in IEEE Vehicular Technology Conference. 1995: 815-819.
[9] Edfors, O., M. Sandell, J.J. van de Beek, et al., OFDM channel estimation by singular value decomposition[J]. IEEE Transactions on Communications, 1998. 46(7): 931-939.
[10] Hoeher, P. and F. Tufvesson. Channel estimation with superimposed pilot sequence[C]. in Globecom'99: Seamless Interconnection for Universal Services. 1999: 2162-2166.
[11] Tugnait, J.K. and W.L. Luo, On channel estimation using superimposed training and first-order statistics[J]. IEEE Communications Letters, 2003. 7(9): 413-415.
[12] Zhou, G.T., M. Viberg, and T. McKelvey, A first-order statistical method for channel estimation[J]. IEEE Signal Processing Letters, 2003. 10(3): 57-60.
[13] Negi, R. and J. Cioffi, Pilot tone selection for channel estimation in a mobile OFDM system[J]. IEEE Transactions on Consumer Electronics, 1998. 44(3): 1122-1128.
[14] Cavers, J.K., An analysis of pilot assisted modulation for Rayleigh channels[J]. IEEE Transactions on Vehicular Technology, 1991. 40(4): 686-693.
[15] Adireddy, S. and L. Tong, Optimal placement of training for frequency selective block-fading channels[J]. IEEE Transactions on Information Theory,2002. 48(44): 2338-2353.
[16] Ohno, S. and G.B. Giannakis, Capacity maximizing MMSE-optimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channels[J]. IEEE Transactions on Information Theory, 2004. 50(9): 2138-2145.
[17] Ma, X., G.B. Giannakis, and S. Ohno, Optimal training for block transmissions over doubly selective wireless fading channels,[J]. IEEE Transactions on Signal Processing, 2003. 51(5): 1351-1366.
[18] Park, S.Y., Y.G. Kim, C.G. Kang, et al., Iterative receiver with joint detection and channel estimation for OFDM system with multiple receiver antennas in mobile radio channels[J]. IEEE Global Telecommunications Conference, 2001. 5: 3085-3089.
[19] ten Brink, S., F. Sanzi, and J. Speidel, Two-dimensional iterative APP channel estimation and decoding for OFDM systems[J]. IEEE Global Telecommunications Conference, 2000. 2: 741-745.
[20] Mignone, V. and A. Morello, A novel demodulation scheme for fixed and mobile receivers[J]. IEEE Transactions on Communications, 1996. 44(9): 1141-1151.
[21] Jarbot, L. Combined decoding and channel estimation of OFDM systems in mobile radio networks[C]. in IEEE 47th Vehicular Technology Conference Proceedings. 1997: 1601-1604.
[22] Sun, Q., D.C. Cox, H.C. Huang, et al., Estimation of continuous flat fadingMIMO channels[J]. IEEE Transactions on Wireless Communications, 2002. 1(4): 549-553.
[23] Al-Naffouri, T.Y., A. Bahai, and A. Paulraj. An EM-based OFDM receiver for time-variant channels[C]. in IEEE Proceedings on Global Telecommunication Conference. 2002: 589-593.
[24] Li, Y., L.J. Cimini, and N.R. Sollenberger, Robust channel estimation for OFDM systems with rapid dispersive fading channels[J]. IEEE Transactions on Communications, 1998. 46(7): 902-915.
[25] Dinis, R., C.T. Lam, and D. Falconer, Joint frequency-domain equalization and channel estimation using superimposed pilots[J]. IEEE Wireless Communications & Networking Conference, 2008: 447-452.
[26] Lam, C.T., D.D. Falconer, F. Danilo-Lemoine, et al. Channel Estimation for SC-FDE Systems Using Frequency Domain Multiplexed Pilots[C]. in IEEE 64th Vehicular Technology Conference, Vols 1-6. 2006: 1438-1442.
[27] Karakaya, B., H. Arslan, and H.A. Cirpan. Channel estimation for LTE uplink in high Doppler spread[C]. in IEEE Wireless Communications & Networking Conference, Vols 1-7. 2008: 1126-1130.
[28] Zhou, M., B. Jiang, W. Zhong, et al. Efficient Channel Estimation for LTE Uplink[C]. in Wireless Communications & Signal Processing. 2009. Nanjing 1-5.
[29] Jeong, B.J. and H.K. Chung. Pilot structures for the uplink Single Carrier FDMA transmission systems[C]. in IEEE 67th Vehicular Technology Conference-Spring, Vols 1-7. 2008: 2552-2556.
[30] Ogawa, Y., T. Takata, T. Iwai, et al. Pilot Signal Generation Scheme using Frequency Dependent Cyclic Shift Sequence for Inter-cell Interference Mitigation[C]. in IEEE Radio and Wireless Symposium. 2009: 407-410.
[31] Hoeher, P. TCM on frequency-selective land-mobile fading channels[C]. in 5th Tirrenia International Workshop. 1991. Tirrenia, Italy: 317-328.
[32] Schafhuber, D., G. Matz, F. Hlawatsch, et al. MMSE estimation of time-varying channels for DVB-T systems with strong co-channel interference[C]. in Proceedings of EUSIPCO. 2002. Toulouse, France: 25-28.
[33] Schafhuber, D. and M. Rupp. Adaptive identification and tracking of doubly selective fading channels for wireless MIMO-OFDM systems[C]. in IEEE SPAWC. 2003. Rome, Italy.
[34] Schafhuber, D., G. Matz, and F. Hlawatsch. Adaptive Wiener filters for time-varying channel estimation in wireless OFDM systems[C]. in IEEE International Conference on Acoustics, Speech, and Signal Processing. 2003: 688-691.
[35]白宾锋,蔡跃明和徐友云, OFDM系统中一种维纳LMS信道跟踪算法[J].电子与信息学报, 2005. 27(11): 1699-1703.
[36] Hou, X.L., S.B. Li, C.C. Yin, et al., Two-dimensional recursive least square adaptive channel estimation for OFDM systems[J]. 2005 International Conference on Wireless Communications, Networking and Mobile Computing Proceedings, Vols 1 and 2, 2005: 232-236.
[37] Roman, T., M. Enescu, and V. Koivunen. Time-Domain Method for Tracking Dispersive Channels in MIMO-OFDM Systems[C]. in IEEE Conference of Acoustics, Speech and Signal Process. 2003: 393-396.
[38] Jeruchim, M.C., P. Balaban, and K.S. Shanmugan, Simulation of Communication Systems[M]. 2 ed. 2000, New York: Kluwer Academic/Plenum.
[39] Weinstein, S.B. and P.M. Ebert, Data Transmission by Frequency-Division Multiplexing Using the Discrete Fourier Transform[J]. IEEE Transactions on Communication Technology, 1971. Com-19(5): 628-634.
[40] Peled, A. and A. Ruiz. Frequency domain data transmission using reduced computational complexity algorithms[C]. in IEEE International Conference of Acoustics, Speech and Signal Processing. 1980. Denver, CO.: 964-967.
[41] 3GPP, Physical Channels and Modulation (36.211 V9.1.0). 2010, Technical Specification Group Radio Access Network.
[42] Chu, D.C., Polyphase codes with good periodic correlation properties[J]. IEEE Transactions on Information Theory, 1972. IT-18: 531-532.
[43] Sarwate, D.V., Bounds on crosscorrelation and autocorrelation of sequences[J]. IEEE Transactions on Information Theory, 1979. IT-25: 720-724.
[44] Beyme, S. and C. Leung, Efficient computation of DFT of Zadoff-Chu sequences[J]. Electronics letters, 2009. 45: 461-463.
[45] Popovic, B.M., Efficient DFT of Zadoff-Chu sequences[J]. Electronics letters, 2010. 46: 502-503.
[46] Papoulis, A. and S.U. Pillai, Probability, Random Variables and Stochastic Processes[M]. 3 ed. 2002: Mcgraw Hill Higher Education.
[47] 3GPP, User Equipment (UE) radio transmission and reception (36.101 V10.0.0). 2010, Technical Specification Group Radio Access Network.
[48] Haykin, S., Adaptive Filter Theory[M]. 4 ed. 2001: Prentice Hall.
[2] Falconer, D., S.L. Ariyavisitakul, A. Benyamin-Seeyar, et al., Frequency domain equalization for single-carrier broadband wireless systems[J]. IEEE Communications Magazine, 2002. 40(4): 58-66.
[3] Myung, H.G., J. Lim, and D.J. Goodman, Single Carrier FDMA for Uplink Wireless Transmission[J]. IEEE Vehicular Technology Magazine, 2006. 1(3): 30-38.
[4] Hwang, T., C.Y. Yang, G. Wu, et al., OFDM and Its Wireless Applications: A Survey[J]. IEEE Transactions on Vehicular Technology, 2009. 58(4): 1673-1694.
[5] Hoeher, P., S. Kaiser, and P. Robertson. Two-dimensional pilot-symbol-aided channel estimation by Wiener filtering[C]. in IEEE International Conference on Acoustics, Speech, and Signal Processing. 1997: 1845-1848.
[6] Li, Y., Pilot-symbol-aided channel estimation for OFDM in wireless systems[J]. IEEE Transactions on Vehicular Technology, 2000. 49(4): 1207-1215.
[7] Morelli, M. and U. Mengali, A comparison of pilot-aided channel estimation methods for OFDM systems[J]. IEEE Transactions on Signal Processing, 2001. 49(12): 3065-3073.
[8] Beek, J.J., O. edfors, M. Sandell, et al. On channel estimation in OFDM systems[C]. in IEEE Vehicular Technology Conference. 1995: 815-819.
[9] Edfors, O., M. Sandell, J.J. van de Beek, et al., OFDM channel estimation by singular value decomposition[J]. IEEE Transactions on Communications, 1998. 46(7): 931-939.
[10] Hoeher, P. and F. Tufvesson. Channel estimation with superimposed pilot sequence[C]. in Globecom'99: Seamless Interconnection for Universal Services. 1999: 2162-2166.
[11] Tugnait, J.K. and W.L. Luo, On channel estimation using superimposed training and first-order statistics[J]. IEEE Communications Letters, 2003. 7(9): 413-415.
[12] Zhou, G.T., M. Viberg, and T. McKelvey, A first-order statistical method for channel estimation[J]. IEEE Signal Processing Letters, 2003. 10(3): 57-60.
[13] Negi, R. and J. Cioffi, Pilot tone selection for channel estimation in a mobile OFDM system[J]. IEEE Transactions on Consumer Electronics, 1998. 44(3): 1122-1128.
[14] Cavers, J.K., An analysis of pilot assisted modulation for Rayleigh channels[J]. IEEE Transactions on Vehicular Technology, 1991. 40(4): 686-693.
[15] Adireddy, S. and L. Tong, Optimal placement of training for frequency selective block-fading channels[J]. IEEE Transactions on Information Theory,2002. 48(44): 2338-2353.
[16] Ohno, S. and G.B. Giannakis, Capacity maximizing MMSE-optimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channels[J]. IEEE Transactions on Information Theory, 2004. 50(9): 2138-2145.
[17] Ma, X., G.B. Giannakis, and S. Ohno, Optimal training for block transmissions over doubly selective wireless fading channels,[J]. IEEE Transactions on Signal Processing, 2003. 51(5): 1351-1366.
[18] Park, S.Y., Y.G. Kim, C.G. Kang, et al., Iterative receiver with joint detection and channel estimation for OFDM system with multiple receiver antennas in mobile radio channels[J]. IEEE Global Telecommunications Conference, 2001. 5: 3085-3089.
[19] ten Brink, S., F. Sanzi, and J. Speidel, Two-dimensional iterative APP channel estimation and decoding for OFDM systems[J]. IEEE Global Telecommunications Conference, 2000. 2: 741-745.
[20] Mignone, V. and A. Morello, A novel demodulation scheme for fixed and mobile receivers[J]. IEEE Transactions on Communications, 1996. 44(9): 1141-1151.
[21] Jarbot, L. Combined decoding and channel estimation of OFDM systems in mobile radio networks[C]. in IEEE 47th Vehicular Technology Conference Proceedings. 1997: 1601-1604.
[22] Sun, Q., D.C. Cox, H.C. Huang, et al., Estimation of continuous flat fadingMIMO channels[J]. IEEE Transactions on Wireless Communications, 2002. 1(4): 549-553.
[23] Al-Naffouri, T.Y., A. Bahai, and A. Paulraj. An EM-based OFDM receiver for time-variant channels[C]. in IEEE Proceedings on Global Telecommunication Conference. 2002: 589-593.
[24] Li, Y., L.J. Cimini, and N.R. Sollenberger, Robust channel estimation for OFDM systems with rapid dispersive fading channels[J]. IEEE Transactions on Communications, 1998. 46(7): 902-915.
[25] Dinis, R., C.T. Lam, and D. Falconer, Joint frequency-domain equalization and channel estimation using superimposed pilots[J]. IEEE Wireless Communications & Networking Conference, 2008: 447-452.
[26] Lam, C.T., D.D. Falconer, F. Danilo-Lemoine, et al. Channel Estimation for SC-FDE Systems Using Frequency Domain Multiplexed Pilots[C]. in IEEE 64th Vehicular Technology Conference, Vols 1-6. 2006: 1438-1442.
[27] Karakaya, B., H. Arslan, and H.A. Cirpan. Channel estimation for LTE uplink in high Doppler spread[C]. in IEEE Wireless Communications & Networking Conference, Vols 1-7. 2008: 1126-1130.
[28] Zhou, M., B. Jiang, W. Zhong, et al. Efficient Channel Estimation for LTE Uplink[C]. in Wireless Communications & Signal Processing. 2009. Nanjing 1-5.
[29] Jeong, B.J. and H.K. Chung. Pilot structures for the uplink Single Carrier FDMA transmission systems[C]. in IEEE 67th Vehicular Technology Conference-Spring, Vols 1-7. 2008: 2552-2556.
[30] Ogawa, Y., T. Takata, T. Iwai, et al. Pilot Signal Generation Scheme using Frequency Dependent Cyclic Shift Sequence for Inter-cell Interference Mitigation[C]. in IEEE Radio and Wireless Symposium. 2009: 407-410.
[31] Hoeher, P. TCM on frequency-selective land-mobile fading channels[C]. in 5th Tirrenia International Workshop. 1991. Tirrenia, Italy: 317-328.
[32] Schafhuber, D., G. Matz, F. Hlawatsch, et al. MMSE estimation of time-varying channels for DVB-T systems with strong co-channel interference[C]. in Proceedings of EUSIPCO. 2002. Toulouse, France: 25-28.
[33] Schafhuber, D. and M. Rupp. Adaptive identification and tracking of doubly selective fading channels for wireless MIMO-OFDM systems[C]. in IEEE SPAWC. 2003. Rome, Italy.
[34] Schafhuber, D., G. Matz, and F. Hlawatsch. Adaptive Wiener filters for time-varying channel estimation in wireless OFDM systems[C]. in IEEE International Conference on Acoustics, Speech, and Signal Processing. 2003: 688-691.
[35]白宾锋,蔡跃明和徐友云, OFDM系统中一种维纳LMS信道跟踪算法[J].电子与信息学报, 2005. 27(11): 1699-1703.
[36] Hou, X.L., S.B. Li, C.C. Yin, et al., Two-dimensional recursive least square adaptive channel estimation for OFDM systems[J]. 2005 International Conference on Wireless Communications, Networking and Mobile Computing Proceedings, Vols 1 and 2, 2005: 232-236.
[37] Roman, T., M. Enescu, and V. Koivunen. Time-Domain Method for Tracking Dispersive Channels in MIMO-OFDM Systems[C]. in IEEE Conference of Acoustics, Speech and Signal Process. 2003: 393-396.
[38] Jeruchim, M.C., P. Balaban, and K.S. Shanmugan, Simulation of Communication Systems[M]. 2 ed. 2000, New York: Kluwer Academic/Plenum.
[39] Weinstein, S.B. and P.M. Ebert, Data Transmission by Frequency-Division Multiplexing Using the Discrete Fourier Transform[J]. IEEE Transactions on Communication Technology, 1971. Com-19(5): 628-634.
[40] Peled, A. and A. Ruiz. Frequency domain data transmission using reduced computational complexity algorithms[C]. in IEEE International Conference of Acoustics, Speech and Signal Processing. 1980. Denver, CO.: 964-967.
[41] 3GPP, Physical Channels and Modulation (36.211 V9.1.0). 2010, Technical Specification Group Radio Access Network.
[42] Chu, D.C., Polyphase codes with good periodic correlation properties[J]. IEEE Transactions on Information Theory, 1972. IT-18: 531-532.
[43] Sarwate, D.V., Bounds on crosscorrelation and autocorrelation of sequences[J]. IEEE Transactions on Information Theory, 1979. IT-25: 720-724.
[44] Beyme, S. and C. Leung, Efficient computation of DFT of Zadoff-Chu sequences[J]. Electronics letters, 2009. 45: 461-463.
[45] Popovic, B.M., Efficient DFT of Zadoff-Chu sequences[J]. Electronics letters, 2010. 46: 502-503.
[46] Papoulis, A. and S.U. Pillai, Probability, Random Variables and Stochastic Processes[M]. 3 ed. 2002: Mcgraw Hill Higher Education.
[47] 3GPP, User Equipment (UE) radio transmission and reception (36.101 V10.0.0). 2010, Technical Specification Group Radio Access Network.
[48] Haykin, S., Adaptive Filter Theory[M]. 4 ed. 2001: Prentice Hall.