智能天线波束赋形算法研究
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摘要
智能天线技术是未来无线移动通信技术发展的主要方向之一。如何消除同信道干扰(CCI)、多址干扰(MAI)、以及多径衰落的影响成为人们在提高无线移动通信系统性能时考虑的主要因素。智能天线利用数字信号处理技术，采用先进的自适应阵列处理技术，产生空间定向波束，使波束主瓣对准用户信号到达方向，旁瓣或零陷对准干扰信号方向，删除或抑制干扰信号，从而提高期望信号的接收信噪比，提高系统容量。智能天线是解决频率资源匮乏的有效途径。近年来，智能天线技术成为移动通信领域中的研究热点之一。
论文在分析前人对智能天线技术、特别是智能天线波束赋形算法研究的基础上，深入研究了智能天线波束赋形算法的两个方面：一是针对智能天线的非理想情况，即存在阵元几何误差、角度估计误差、传播介质的不一致性等，研究采用鲁棒性波束赋形算法以提高波束赋形性能；二是针对复杂的电磁环境，即存在强干扰的情况下，研究采用智能天线零陷技术消除强干扰的算法。在此基础上，论文提出了一种综合智能天线零陷技术和动态信道分配技术的方案来消除由于基站间时隙分配不同造成的各种干扰。
首先，针对智能天线应用中的非理想情况，论文提出了一种基于多波束的鲁棒性波束赋形算法。算法先利用阵列导向矢量扰动的先验信息和接收阵列数据的后验信息，构造了一个新型的代价函数；再通过最速下降法，获得鲁棒性波束赋形权值。论文通过理论推导证明了该算法的收敛性，又经过仿真验证了算法的良好性能。
然后，论文深入研究了各种传统零陷算法，提出了以下零陷算法：
1．传统幅值扰动算法不能够形成深度零陷，且形成零陷个数较少；论文提出了一种在阵元关于阵列中心对称的线阵中进行幅值扰动形成零陷的算法，仿真证明，该算法性能远好于传统的幅值扰动算法。在此基础上，将各阵元对零陷的贡献大小进行分析，发
Smart antennas technique is a primary development direction of future wireless communication systems. How to cancel co-channel interference, multi-access interference, and combat the multi-paths fading effects, are major factors needed to be considered, when people attempt to enhance the communication system. Smart antennas system, which introduces advanced adaptive array processing, produces space beams whose main beam directs at the direction of desired user, null steering beams directing at the direction of interference, thus suppressing or canceling the interfering signals, therefore, it can improve the signal-to-interference ratio and system capacity. Recently, the smart antennas technique is one of the hot research topics in mobile communication area as a solution to the frequency lack.
Based on the other persons' research on smart antennas algorithms, especially on beamforming techniques, two aspects of the smart antennas beamforming algorithms are thoroughly investigated in this paper. Firstly, to combat the array non-ideal geometry error, DOA estimation error, not exact calibration of antennas, multi-beams based robust beamforming algorithms are studied to improve the beamforming performance. Secondly, aiming at the complex electromagnetic environments when existing strong interference, null steering algorithms are exploited to cancel the strong interference. Based on this, a proposal synthesizing null steering technique and dynamic channel allocation is presented to cancel interference resulted from inconsistent timeslot allocation between base stations.
论文在分析前人对智能天线技术、特别是智能天线波束赋形算法研究的基础上，深入研究了智能天线波束赋形算法的两个方面：一是针对智能天线的非理想情况，即存在阵元几何误差、角度估计误差、传播介质的不一致性等，研究采用鲁棒性波束赋形算法以提高波束赋形性能；二是针对复杂的电磁环境，即存在强干扰的情况下，研究采用智能天线零陷技术消除强干扰的算法。在此基础上，论文提出了一种综合智能天线零陷技术和动态信道分配技术的方案来消除由于基站间时隙分配不同造成的各种干扰。
首先，针对智能天线应用中的非理想情况，论文提出了一种基于多波束的鲁棒性波束赋形算法。算法先利用阵列导向矢量扰动的先验信息和接收阵列数据的后验信息，构造了一个新型的代价函数；再通过最速下降法，获得鲁棒性波束赋形权值。论文通过理论推导证明了该算法的收敛性，又经过仿真验证了算法的良好性能。
然后，论文深入研究了各种传统零陷算法，提出了以下零陷算法：
1．传统幅值扰动算法不能够形成深度零陷，且形成零陷个数较少；论文提出了一种在阵元关于阵列中心对称的线阵中进行幅值扰动形成零陷的算法，仿真证明，该算法性能远好于传统的幅值扰动算法。在此基础上，将各阵元对零陷的贡献大小进行分析，发
Smart antennas technique is a primary development direction of future wireless communication systems. How to cancel co-channel interference, multi-access interference, and combat the multi-paths fading effects, are major factors needed to be considered, when people attempt to enhance the communication system. Smart antennas system, which introduces advanced adaptive array processing, produces space beams whose main beam directs at the direction of desired user, null steering beams directing at the direction of interference, thus suppressing or canceling the interfering signals, therefore, it can improve the signal-to-interference ratio and system capacity. Recently, the smart antennas technique is one of the hot research topics in mobile communication area as a solution to the frequency lack.
Based on the other persons' research on smart antennas algorithms, especially on beamforming techniques, two aspects of the smart antennas beamforming algorithms are thoroughly investigated in this paper. Firstly, to combat the array non-ideal geometry error, DOA estimation error, not exact calibration of antennas, multi-beams based robust beamforming algorithms are studied to improve the beamforming performance. Secondly, aiming at the complex electromagnetic environments when existing strong interference, null steering algorithms are exploited to cancel the strong interference. Based on this, a proposal synthesizing null steering technique and dynamic channel allocation is presented to cancel interference resulted from inconsistent timeslot allocation between base stations.
引文
[1] Liang Yuzhu, Zou Qiyue, and Meng Hwa Er, "A new approach to robust beamforming against generalized phase errors", Emerging Technologies: Frontiers of Mobile and Wireless Communication, 2004. Proceedings of the IEEE 6th Circuits and Systems Symposium on Volume 2, 31 May-2 June 2004, pp.775 - 778.
[2] Jisung Oh, Seung-Jean Kim, and Kan-Lin Hsiung, "A Computationally Efficient Method for Robust Minimum Variance Beamforming", Vehicular Technology Conference, 2005. VTC 2005-Spring. 2005 IEEE 61st Volume 2, 30 May-1 June 2005, pp. 1162-1165.
[3] B.D Carlson, "Covariance matrix estimation errors and diagonal loading in adaptive arrays" , IEEE Transactions Aerospace Electronic Systems, vol. 24, no. 4, July 1988, pp. 397-401.
[4] Kogon, S., "Eigenvectors, diagonal loading and white noise gain constraints for robust adaptive beamforming". Signals, Systems and Computers, 2003. Conference Record of the Thirty-Seventh Asilomar Conference on Volume 2, 9-12 Nov. 2003, pp.1853 - 1857.
[5] J. Li, P. Stoica, and Z. Wang, "On robust Capon beamforming and diagonal loading" , IEEE Transactions Signal Processing, vol. 51, no. 7, July 2003, pp. 1702-1715.
[6] Gu, J, "Robust beamforming based on variable loading", Electronics Letters Volume 41, Issue 2, 20 Jan. 2005, pp.55 - 56.
[7] R. Lorenz and S. Boyd, "Robust minimum variance beamforming", Signal Processing, IEEE Transactions on [see also Acoustics, Speech, and Signal Processing], IEEE Transactions on Volume 53, Issue 5, May 2005, pp.1684 -1696.
[8] S. Vorobyov, A. Gershman, and Z. Luo, "Robust adaptive beamforming using worst-case performance optimization: A solution to the signal mismatch problem" , IEEE Transactions Signal Processing, vol. 51, no. 2, February 2003, pp. 313—324.
[9] M. H. Er and A. Cantoni, "Derivative constraints for broad-band element space antenna array processors", IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, Dec. 1983, pp. 1378-1393.
[10]R. T. Compton, "The effect of random steering vector errors in the Applebaum
??adaptive array", IEEE Trans. Aerosp. Electron. Syst., vol.AES-18, Sept. 1982, pp. 392-400.
[11]A. M. Vural, "Effects of perturbation on the performance of optimum adaptive arrays" , IEEE Trans. Aerosp. Electron. Syst., vol. AES-15, Feb. 1979, pp.76 - 87.
[12]Viberg, M., and Swindlehurst, A., "Analysis of the combined effects infinite samples and model errors on array processing performance", IEEE Trans. Signal Process., 42, (11), 1994, pp. 3073-3083
[13] N. Jablon, "Adaptive beamforming with the generalized sidelobe canceller in the presence of array imperfections", IEEE Transactions Antennas Propagation, vol. 34, no. 8, August 1986, pp. 996-1012.
[14]J. T. Mayhan, "Area coverage adaptive nulling from geosynchronous satellites: Phased arrays versus multiple-beam antennas", IEEE Trans. Antennas Propag., vol. AP-34, Mar. 1986, pp. 410-419.
[15]K.-B. Yu, "Adaptive beamforming for satellite communication with selective earth coverage and jammer nulling capability", IEEE Trans. Signal Process., vol. 44, Dec. 1996, pp. 3162-3166.
[16]J.-H. Lee and T.-F. Hsu, "Adaptive beamforming with multiple-beam constraints in the presence of coherent jammers", Signal Process., vol. 80, Nov. 2000, pp. 2475-2480.
[17]Yashchyshyn, Y. and Starszuk, G. "Investigation of a simple four-element steering antenna array" , Microwaves, Antennas and Propagation, IEE Proceedings Volume 152, Issue 2, 8, April 2005, pp.92 - 96.
[18]H. Steyskal, "Simple method for pattern nulling by phase perturbation", IEEE Trans. Antennas Propagation, vol. AP-31, 1983, pp. 163-166.
[19] Shore, R. "A unified treatment of nulling in linear array patterns with minimized weight perturbations", Antennas and Propagation Society International Symposium, Vol. 20, 1982, pp.703 - 706.
[20]T. Ismail and M. Dawoud, "Null steering in phased arrays by controlling the elements positions", IEEE Trans. Antennas Propagation, vol. 39, 1991, pp. 1561-1566.
[21]Hejres, Jasem A., "Null steering in phased arrays by controlling the positions of selected elements", IEEE Transactions on Antennas and Propagation, Vol 52, 2004, pp 2891-2895.
[22]Ali AKDAGLI and Kerim G, "Pattern Nulling by Phase and Position Perturbationswith the Use of a Modified Tabu Search Algorithm", Turk J Electric Engineering, VOL.10, NO.3, 2002, pp. 449-458.
[23] R.A. Shore, "Nulling at Symmetric Pattern Location with Phase-Only Weight Control", IEEE Trans. Antennas and Propagation, Vol. AP-32, 1984, pp. 530-533.
[24] Haupt, R.L. and Johnson, J.M, "Dynamic phase-only array beam control using a genetic algorithm", Proceedings of the First NASA/DoD Workshop,on 19-21 1999, pp.217-224.[1] Quentin Spencer, Michael Rice, Brian Jeffs, and Michael Jensen, "A Statistical Model for Angle of Arrival in Indoor Multipath Propagation", Vehicular Technology Conference, 1997 IEEE 47th Volume 3, 4-7 May 1997, pp.1415-1419.
[2] Klaus Ingemann Pedersen, Preben E. Mogensen, and Bernard H. Fleury, "A Stochastic Model of the Temporal and Azimuthal Dispersion Seen at the Base Station in Outdoor Propagation Environments", Vehicular Technology, IEEE Transactions on Volume 49, Issue 2, March 2000, pp.437-447.
[3] Albert Algans, Klaus Ingemann Pedersen, and Preben Elgaard Mogensen, "Experimental Analysis of the Joint Statistical Properties of Azimuth Spread, Delay Spread, and Shadow Fading", Selected Areas in Communications, IEEE Journal on Volume 20, Issue 3, April 2002, pp.523-531.[1] Zhu Liang Yu, Qiyue Zou and Meng Hwa Er, "A new approach to robust beamforming against generalized phase errors", Emerging Technologies: Frontiers of Mobile and Wireless Communication, 2004. Proceedings of the IEEE 6th Circuits and Systems Symposium on Volume 2, 31 May-2 June 2004, pp.775-778.
[2] R. T. Compton, Jr., "The effect of random steering vector errors in the Applebaum adaptive array", IEEE Trans. Aerosp. Electron. Syst., vol.AES-18, Sept. 1982, pp. 392-400.
[3] A. M. Vural, "Effects of perturbation on the performance of optimum adaptive arrays", IEEE Trans. Aerosp. Electron. Syst., vol. AES-15, Feb. 1979, pp.76-87.
[4] Viberg, M., and Swindlehurst, A., "Analysis of the combined effects offinite samples and model errors on array processing performance", IEEETrans. Signal Process. 42, (11), 1994, pp. 3073-3083.
[5] Gu, J, "Robust beamforming based on variable loading", Volume 41, Issue 2, 20 Jan. 2005, pp.55-56.
[6] N. Jablon, "Adaptive beamforming with the generalized sidelobe canceller in the presence of array imperfections", IEEE Transactions Antennas Propagation, vol. 34, no. 8, August 1986, pp. 996-1012.
[7] Blair D. Carlson, "Covariance Matrix Estimation Errors and Diagonal Loading in Adaptive Arrays", IEEE Trans. Aerospace and Electronic Systems vol.24, no.4,1988, pp397-401.
[8] Kogon, S., "Eigenvectors, diagonal loading and white noise gain constraints for robust adaptive beamforming", Signals, Systems and Computers, 2003. Conference Record of the Thirty-Seventh Asilomar Conference on Volume 2, 9-12 Nov. 2003, pp.1853-1857.
[9] R. Lorenz and S. Boyd, "Robust minimum variance beamforming", Signal Processing, IEEE Transactions on [see also Acoustics, Speech, and Signal Processing], IEEE Transactions on Volume 53, Issue 5, May 2005, pp.1684-1696.
[10] M. H. Er and A. Cantoni, "Derivative constraints for broad-band element space antenna array processors", IEEE Trans. Acoust., Speech, Signal Processing, vol.??ASSP-31, Dec. 1983, pp. 1378-1393.
[11] S. Vorobyov, A. Gershman, and Z. Luo, "Robust adaptive beamforming using worst-case performance optimization: A solution to the signal mismatch problem", IEEE Transactions Signal Processing, vol. 51, no. 2, pp. 313-324, February 2003.
[12] J. Li, P. Stoica, and Z. Wang, "On robust Capon beamforming and diagonal loading", IEEE Transactions Signal Processing, vol. 51, no. 7, pp. 1702-1715, July 2003.
[13] J. T. Mayhan, "Area coverage adaptive nulling from geosynchronous satellites: Phased arrays versus multiple-beam antennas", IEEE Trans. Antennas Propag., vol. AP-34, Mar. 1986, pp. 410-419.
[14] K.-B. Yu, "Adaptive beamforming for satellite communication with selective earth coverage and jammer nulling capability", IEEE Trans. Signal Process., vol. 44, Dec. 1996, pp. 3162-3166.
[15] J.-H. Lee and T.-E Hsu, "Adaptive beamforming with multiple-beam constraints in the presence of coherent jammers", Signal Process., vol. 80, Nov. 2000, pp. 2475-2480.[1] T. B. Vu, "Method of null steering without using phase shifters", Proc. Inst. Elect. Eng., pt. H, vol. 131, Aug. 1984, pp. 242-246.
[2] T. B. VU, "Null Steering by Controlling Current Amplitude Only", Antennas and Propagation Society International Symposium,Volume 22, Jun 1984, pp.811-814.
[3] T. B. VU, "Pattern Nulling in Difference Pattem by Amplitude Control", Antennas and Propagation, IEEE Transactions on, Volume 33, Issue 6, Jun 1985, pp.669-671.
[4] Hosny M. Ibrahim, "Null Steering by Real-Weight Control-A method of Decoupling the Weights", Vol. AP-39,1991, pp.1648-1650.
[5] W.P. Liao and F.L. Chu, "Array Pattem Synthesis with Null Steering Using Genetic Algorithms by Controlling Only the Current Amplitudes", Int. J. Electronics, Vol. 86, 1999, pp. 445-457.
[6] Shore, R. "A unified treatment of nulling in linear array pattems with minimized weight perturbations", Antennas and Propagation Society Intemational Symposium, 1982, Vol. 20, pp.703-706.
[7] H. Steyskal, "Simple method for pattem nulling by phase perturbation", IEEE Trans. Antennas Propagat., vol. AP-31, Jan. 1983, pp. 163-166.
[8] Hejres, Jasem A. "Null steering in phased arrays by controlling the positions of selected elements", IEEE Transactions on Antennas and Propagation, v52, n11, November, 2004, pp. 2891-2895.
[9] Haupt, R.L.; Johnson, J.M, "Dynamic phase-only array beam control using a genetic algorithm", Evolvable Hardware, 1999. Proceedings of the First NASA/DoD Workshop on 19-21 July 1999, pp.217-224.
[10] Haupt, R.L.; Haupt, S.E., "Phase-only adaptive nulling with a genetic algorithm", Aerospace Conference, 1997. Proceedings., IEEE Volume 3, 1-8 Feb. 1997 pp.151-160.
[11] T. Ismail and M. Dawoud, "Null steering in phased arrays by controlling the elements positions", IEEE Trans. Antennas Propagat., vol. 39, Nov. 1991, pp. 1561-1566.
[12] Ali AKDAGLI, Kerim G, "Pattern Nulling by Phase and Position Perturbations with the Use of a Modified Tabu Search Algorithm", Turk J Elec Engin, 2002, VOL.10,??NO.3, pp. 449-458.
[13] R.A. Shore, "Nulling at Symmetric Pattern Location with Phase-Only Weight Control", IEEE Trans. Antennas and Propagation, Vol. AP-32, 1984, pp. 530-533.
[14] Castella, F.R., Kuttler, J.R., "Optimised array antenna nulling with phase-only control", Radar and Signal Processing, IEE Proceedings F Volume 138, Issue 3, June 1991, pp.241-246.[1] Hongbo Yan, "Parallel Interference Cancellation for Uplink Multirate Overlay CDMA Channels", Communications, IEEE Transactions on Volume 53, Issue 1, Jan. 2005, pp.152-161.
[2] Y. C. Yoon, R. Kohno, and H. Imai, "A spread-spectrum multiaccess system with cochannel interference cancellation for multipath fading channels", Selected Areas in Communications, IEEE Journal on Volume 11, Issue 7, Sep. 1993, pp.1067-1075.
[3] R. Buehrer, N. Correal, and B. Woerner, "A comparison of multiuser receivers for cellular CDMA", Global Telecommunications Conference, 1996. The Key to Global Prosperity Volume 3, 18-22, 1996, pp.1571-1577.
[4] Alle-Jan van der Veen, "Spatial Filtering Of RF Interference In Radio Astronomy Using A Reference Antenna", Acoustics, Speech, and Signal Processing, Volume 2, May 2004, pp.17-21.
[2] Jisung Oh, Seung-Jean Kim, and Kan-Lin Hsiung, "A Computationally Efficient Method for Robust Minimum Variance Beamforming", Vehicular Technology Conference, 2005. VTC 2005-Spring. 2005 IEEE 61st Volume 2, 30 May-1 June 2005, pp. 1162-1165.
[3] B.D Carlson, "Covariance matrix estimation errors and diagonal loading in adaptive arrays" , IEEE Transactions Aerospace Electronic Systems, vol. 24, no. 4, July 1988, pp. 397-401.
[4] Kogon, S., "Eigenvectors, diagonal loading and white noise gain constraints for robust adaptive beamforming". Signals, Systems and Computers, 2003. Conference Record of the Thirty-Seventh Asilomar Conference on Volume 2, 9-12 Nov. 2003, pp.1853 - 1857.
[5] J. Li, P. Stoica, and Z. Wang, "On robust Capon beamforming and diagonal loading" , IEEE Transactions Signal Processing, vol. 51, no. 7, July 2003, pp. 1702-1715.
[6] Gu, J, "Robust beamforming based on variable loading", Electronics Letters Volume 41, Issue 2, 20 Jan. 2005, pp.55 - 56.
[7] R. Lorenz and S. Boyd, "Robust minimum variance beamforming", Signal Processing, IEEE Transactions on [see also Acoustics, Speech, and Signal Processing], IEEE Transactions on Volume 53, Issue 5, May 2005, pp.1684 -1696.
[8] S. Vorobyov, A. Gershman, and Z. Luo, "Robust adaptive beamforming using worst-case performance optimization: A solution to the signal mismatch problem" , IEEE Transactions Signal Processing, vol. 51, no. 2, February 2003, pp. 313—324.
[9] M. H. Er and A. Cantoni, "Derivative constraints for broad-band element space antenna array processors", IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, Dec. 1983, pp. 1378-1393.
[10]R. T. Compton, "The effect of random steering vector errors in the Applebaum
??adaptive array", IEEE Trans. Aerosp. Electron. Syst., vol.AES-18, Sept. 1982, pp. 392-400.
[11]A. M. Vural, "Effects of perturbation on the performance of optimum adaptive arrays" , IEEE Trans. Aerosp. Electron. Syst., vol. AES-15, Feb. 1979, pp.76 - 87.
[12]Viberg, M., and Swindlehurst, A., "Analysis of the combined effects infinite samples and model errors on array processing performance", IEEE Trans. Signal Process., 42, (11), 1994, pp. 3073-3083
[13] N. Jablon, "Adaptive beamforming with the generalized sidelobe canceller in the presence of array imperfections", IEEE Transactions Antennas Propagation, vol. 34, no. 8, August 1986, pp. 996-1012.
[14]J. T. Mayhan, "Area coverage adaptive nulling from geosynchronous satellites: Phased arrays versus multiple-beam antennas", IEEE Trans. Antennas Propag., vol. AP-34, Mar. 1986, pp. 410-419.
[15]K.-B. Yu, "Adaptive beamforming for satellite communication with selective earth coverage and jammer nulling capability", IEEE Trans. Signal Process., vol. 44, Dec. 1996, pp. 3162-3166.
[16]J.-H. Lee and T.-F. Hsu, "Adaptive beamforming with multiple-beam constraints in the presence of coherent jammers", Signal Process., vol. 80, Nov. 2000, pp. 2475-2480.
[17]Yashchyshyn, Y. and Starszuk, G. "Investigation of a simple four-element steering antenna array" , Microwaves, Antennas and Propagation, IEE Proceedings Volume 152, Issue 2, 8, April 2005, pp.92 - 96.
[18]H. Steyskal, "Simple method for pattern nulling by phase perturbation", IEEE Trans. Antennas Propagation, vol. AP-31, 1983, pp. 163-166.
[19] Shore, R. "A unified treatment of nulling in linear array patterns with minimized weight perturbations", Antennas and Propagation Society International Symposium, Vol. 20, 1982, pp.703 - 706.
[20]T. Ismail and M. Dawoud, "Null steering in phased arrays by controlling the elements positions", IEEE Trans. Antennas Propagation, vol. 39, 1991, pp. 1561-1566.
[21]Hejres, Jasem A., "Null steering in phased arrays by controlling the positions of selected elements", IEEE Transactions on Antennas and Propagation, Vol 52, 2004, pp 2891-2895.
[22]Ali AKDAGLI and Kerim G, "Pattern Nulling by Phase and Position Perturbationswith the Use of a Modified Tabu Search Algorithm", Turk J Electric Engineering, VOL.10, NO.3, 2002, pp. 449-458.
[23] R.A. Shore, "Nulling at Symmetric Pattern Location with Phase-Only Weight Control", IEEE Trans. Antennas and Propagation, Vol. AP-32, 1984, pp. 530-533.
[24] Haupt, R.L. and Johnson, J.M, "Dynamic phase-only array beam control using a genetic algorithm", Proceedings of the First NASA/DoD Workshop,on 19-21 1999, pp.217-224.[1] Quentin Spencer, Michael Rice, Brian Jeffs, and Michael Jensen, "A Statistical Model for Angle of Arrival in Indoor Multipath Propagation", Vehicular Technology Conference, 1997 IEEE 47th Volume 3, 4-7 May 1997, pp.1415-1419.
[2] Klaus Ingemann Pedersen, Preben E. Mogensen, and Bernard H. Fleury, "A Stochastic Model of the Temporal and Azimuthal Dispersion Seen at the Base Station in Outdoor Propagation Environments", Vehicular Technology, IEEE Transactions on Volume 49, Issue 2, March 2000, pp.437-447.
[3] Albert Algans, Klaus Ingemann Pedersen, and Preben Elgaard Mogensen, "Experimental Analysis of the Joint Statistical Properties of Azimuth Spread, Delay Spread, and Shadow Fading", Selected Areas in Communications, IEEE Journal on Volume 20, Issue 3, April 2002, pp.523-531.[1] Zhu Liang Yu, Qiyue Zou and Meng Hwa Er, "A new approach to robust beamforming against generalized phase errors", Emerging Technologies: Frontiers of Mobile and Wireless Communication, 2004. Proceedings of the IEEE 6th Circuits and Systems Symposium on Volume 2, 31 May-2 June 2004, pp.775-778.
[2] R. T. Compton, Jr., "The effect of random steering vector errors in the Applebaum adaptive array", IEEE Trans. Aerosp. Electron. Syst., vol.AES-18, Sept. 1982, pp. 392-400.
[3] A. M. Vural, "Effects of perturbation on the performance of optimum adaptive arrays", IEEE Trans. Aerosp. Electron. Syst., vol. AES-15, Feb. 1979, pp.76-87.
[4] Viberg, M., and Swindlehurst, A., "Analysis of the combined effects offinite samples and model errors on array processing performance", IEEETrans. Signal Process. 42, (11), 1994, pp. 3073-3083.
[5] Gu, J, "Robust beamforming based on variable loading", Volume 41, Issue 2, 20 Jan. 2005, pp.55-56.
[6] N. Jablon, "Adaptive beamforming with the generalized sidelobe canceller in the presence of array imperfections", IEEE Transactions Antennas Propagation, vol. 34, no. 8, August 1986, pp. 996-1012.
[7] Blair D. Carlson, "Covariance Matrix Estimation Errors and Diagonal Loading in Adaptive Arrays", IEEE Trans. Aerospace and Electronic Systems vol.24, no.4,1988, pp397-401.
[8] Kogon, S., "Eigenvectors, diagonal loading and white noise gain constraints for robust adaptive beamforming", Signals, Systems and Computers, 2003. Conference Record of the Thirty-Seventh Asilomar Conference on Volume 2, 9-12 Nov. 2003, pp.1853-1857.
[9] R. Lorenz and S. Boyd, "Robust minimum variance beamforming", Signal Processing, IEEE Transactions on [see also Acoustics, Speech, and Signal Processing], IEEE Transactions on Volume 53, Issue 5, May 2005, pp.1684-1696.
[10] M. H. Er and A. Cantoni, "Derivative constraints for broad-band element space antenna array processors", IEEE Trans. Acoust., Speech, Signal Processing, vol.??ASSP-31, Dec. 1983, pp. 1378-1393.
[11] S. Vorobyov, A. Gershman, and Z. Luo, "Robust adaptive beamforming using worst-case performance optimization: A solution to the signal mismatch problem", IEEE Transactions Signal Processing, vol. 51, no. 2, pp. 313-324, February 2003.
[12] J. Li, P. Stoica, and Z. Wang, "On robust Capon beamforming and diagonal loading", IEEE Transactions Signal Processing, vol. 51, no. 7, pp. 1702-1715, July 2003.
[13] J. T. Mayhan, "Area coverage adaptive nulling from geosynchronous satellites: Phased arrays versus multiple-beam antennas", IEEE Trans. Antennas Propag., vol. AP-34, Mar. 1986, pp. 410-419.
[14] K.-B. Yu, "Adaptive beamforming for satellite communication with selective earth coverage and jammer nulling capability", IEEE Trans. Signal Process., vol. 44, Dec. 1996, pp. 3162-3166.
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