MIMO通信系统中有限反馈技术研究
|
摘要
随着移动通信系统的不断演进,除传统话音业务之外的移动多媒体业务将逐渐成为未来移动通信系统中的主流业务?对于移动多媒体业务的支持,需要通信系统能够提供极高的数据速率,因此能够极大提高无线系统频谱效率的多天线技术将成为未来高速移动通信系统中的重要技术?理论证明,在点对点的多天线无线传输模型中,当发送端确知信道信息时,利用由多天线提供的额外自由度,系统可以获取可观的空分复用增益,极大提高无线通信系统的数据速率?尤其在多用户多天线系统中,除了可以获得空分增益外,由于较多用户的存在,通过多用户调度,系统可以获取多用户分集增益,使系统获得相比点对点系统更高的吞吐率?以上高速率多天线传输的实现无不极大的依赖于系统对于用户信道信息的获取,但是在实际系统中,反馈完善的信道信息会带来反馈信道无法承受的负担,因此基于有限反馈的多天线系统及其反馈策略就成为了研究的热点?
本文聚焦应用于多天线系统中的有限反馈技术?首先,介绍有限反馈技术的发展方向?有限反馈的基本原则?以及现有理论分析的讨论热点,针对不同系统详细介绍不同的反馈策略,对比评价采用不同反馈策略的系统性能;其次,考虑非理想信道信息,包括信道估计误差,信道时延等非理想因素,分析其对于基于码本的多用户有限反馈系统的性能影响,针对非理想信道估计,推导得到使基于码本的多用户有限反馈系统保持固定复用增益时的信道估计性能要求,以及所需反馈比特数变化规律,针对反馈信道时延,以单用户有限反馈系统为例,分别在高信噪比场景下和低信噪比场景下分析推导由于反馈信道延时导致的吞吐率损失;再次,研究现有的LTE系统的码本设计,提出优化的码本设计,利用相关性对于码本进行重排,使系统在反馈信道存在差错的情况下,依靠码本本身的结构,减小由于用户信道和预编码矩阵失配造成的吞吐率和误码率损失?最后,在总结以上研究成果的同时,展望了有限反馈技术在未来移动通信系统中的研究前景?
Accompanied by the persistent evolution of the mobile communication system, the mobile media services will gradually become the main-stream services besides the traditional voice service. In order to support the promising services, even higher data rate is urgently required, therefore multiple-input multiple-output technology, aka MIMO, which can greatly improve the spectrum efficiency of the wireless communication systems will, without a doubt ,be the killer application in future mobile communication systems. Theoretically, considerable spatial-multiplexing gain, which will highly enhance the system performance, can be extracted by taking advantage of the extra spatial freedom which is offered by the application of MIMO technology, when perfect channel state information is available at the transmitter in a point-to-point wireless MIMO transmission model. In multiuser systems, because of the multiuser diversity, even higher data rate, compared to the point-to-point system, can be achieved by proper multiuser scheduling that extracts the multiuser diversity gain. However, all of the high rate transmissions mentioned above are totally based on perfect channel state information acquisition at the transmitter. For practical systems, it is unrealistic that the receiver feedbacks the full channel information, which will be regarded as a heavy burden to the system, through the feedback channel with finite data rate. That is why limited feedback based MIMO systems as well as the feedback scheme has become the key point of the research related.
In this paper limited feedback scheme in MIMO systems is discussed. Firstly, we introduce the principle, trend, as well as the main topic of limited feedback technology. In terms of different limited feedback schemes used in different systems, an comparison and evaluation is also included in the introduction. Secondly, imperfect channel state information, including channel estimation error and channel delay, is considered. In terms of channel estimation error, in order to guarantee the multiplexing gain of the codebook based multiuser limited feedback systems, the basic requirement of channel estimation as well as the law of feedback bits is analyzed and derived. Taking single-user MIMO system as an example, we analyze the throughput loss caused by feedback delay. An upper bound of throughput loss is presented in high SNR scenario, so as the formulation of the throughput loss in low SNR scenario. Thirdly, we investigate the codebook design in LTE system, and propose an new method for codebook optimization, that will reduce the system performance penalty due to the feedback error. Finally, based on the result we have got, an perspective of limited feedback in future mobile communication is given.
本文聚焦应用于多天线系统中的有限反馈技术?首先,介绍有限反馈技术的发展方向?有限反馈的基本原则?以及现有理论分析的讨论热点,针对不同系统详细介绍不同的反馈策略,对比评价采用不同反馈策略的系统性能;其次,考虑非理想信道信息,包括信道估计误差,信道时延等非理想因素,分析其对于基于码本的多用户有限反馈系统的性能影响,针对非理想信道估计,推导得到使基于码本的多用户有限反馈系统保持固定复用增益时的信道估计性能要求,以及所需反馈比特数变化规律,针对反馈信道时延,以单用户有限反馈系统为例,分别在高信噪比场景下和低信噪比场景下分析推导由于反馈信道延时导致的吞吐率损失;再次,研究现有的LTE系统的码本设计,提出优化的码本设计,利用相关性对于码本进行重排,使系统在反馈信道存在差错的情况下,依靠码本本身的结构,减小由于用户信道和预编码矩阵失配造成的吞吐率和误码率损失?最后,在总结以上研究成果的同时,展望了有限反馈技术在未来移动通信系统中的研究前景?
Accompanied by the persistent evolution of the mobile communication system, the mobile media services will gradually become the main-stream services besides the traditional voice service. In order to support the promising services, even higher data rate is urgently required, therefore multiple-input multiple-output technology, aka MIMO, which can greatly improve the spectrum efficiency of the wireless communication systems will, without a doubt ,be the killer application in future mobile communication systems. Theoretically, considerable spatial-multiplexing gain, which will highly enhance the system performance, can be extracted by taking advantage of the extra spatial freedom which is offered by the application of MIMO technology, when perfect channel state information is available at the transmitter in a point-to-point wireless MIMO transmission model. In multiuser systems, because of the multiuser diversity, even higher data rate, compared to the point-to-point system, can be achieved by proper multiuser scheduling that extracts the multiuser diversity gain. However, all of the high rate transmissions mentioned above are totally based on perfect channel state information acquisition at the transmitter. For practical systems, it is unrealistic that the receiver feedbacks the full channel information, which will be regarded as a heavy burden to the system, through the feedback channel with finite data rate. That is why limited feedback based MIMO systems as well as the feedback scheme has become the key point of the research related.
In this paper limited feedback scheme in MIMO systems is discussed. Firstly, we introduce the principle, trend, as well as the main topic of limited feedback technology. In terms of different limited feedback schemes used in different systems, an comparison and evaluation is also included in the introduction. Secondly, imperfect channel state information, including channel estimation error and channel delay, is considered. In terms of channel estimation error, in order to guarantee the multiplexing gain of the codebook based multiuser limited feedback systems, the basic requirement of channel estimation as well as the law of feedback bits is analyzed and derived. Taking single-user MIMO system as an example, we analyze the throughput loss caused by feedback delay. An upper bound of throughput loss is presented in high SNR scenario, so as the formulation of the throughput loss in low SNR scenario. Thirdly, we investigate the codebook design in LTE system, and propose an new method for codebook optimization, that will reduce the system performance penalty due to the feedback error. Finally, based on the result we have got, an perspective of limited feedback in future mobile communication is given.
引文
[1] E.Telatar,“Capacity of Multi-Antenna Gaussian Channels,”Euro. Trans. Telecommun.,vol.10,pp.585-595,Nov.-Dec.1999.
[2] G. J. Foschini,“Layered Space-Time Architecture for Wireless Communication in a Fading Environment When Using Multi-Element Antennas,”in Bell Labs. Tech. J, vol. pp. 41-59, Autumn 1996.
[3] V. Tarokh, N. Seshadri, and A. R. Calderbank,“Space-Time Codes for High Data Rate Wireless Communication: Performance Criterion and Code Construction,”IEEE Trans. Info. Theory, vol. 44, pp. 744-765, Mar.1998.
[4] V. Tarokh, H. Jafarhani, and A. R. Calderbank,“Space-Time Block Coding from Orthogonal Designs,”IEEE JSAC, vol. 48, pp. 1456-1467, July 1999.
[5] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valen-zuela,“V-BLAST: An architecture for realizing very high data rates over the rich-scattering wireless channel,”Proc. URSI ISSSE, pp. 295-300,1998
[6] C. E. Shannon,“The zero error capacity of a noisy channel,”IRE Trans. Inform. Theory, vol. 2, no. 3, pp. 8–19, Sept. 1956
[7] C. E. Shannon,“Channels with side information at the transmitter,”IBM Journal Research and Dev., vol. 2, pp. 289–293, 1958
[8] J. Hayes,“Adaptive feedback communications,”IEEE Trans. Commun., vol. 16, no. 1, pp. 29–34, Feb. 1968.
[9] J. Schalkwijk,“Coding scheme for additive noise channels with feedback - Part II: Band-limited signals,”IEEE Trans. Inform. Theory, vol. 12, pp. 183–189, April 1966.
[10] J. Schalkwijk and T. Kailath,“A coding scheme for additive noise channels with feedback - Part I: No bandwidth constraint,”IEEE Trans. Inform. Theory, vol. 12, pp. 172–183, April 1966.
[11] G. L. Turin,“Signal design for sequential detection systems with feedback,”IEEE Trans. Inform. Theory, vol. 11, pp. 401–408, July 1965.
[12] G. L. Turin,“Comparison of sequential and nonsequential detection systems with uncertainty feedback,”IEEE Trans. Inform. Theory, vol. 12, pp. 5–8, Jan. 1966.
[13] A. Goldsmith, S. A. Jafar, N. Jindal, and S. Vishwanath,“Capacity limits of MIMO channels,”IEEE J. Select. Areas Commun., vol. 21, no. 5, pp. 684–702, June 2003.
[14] E. Visotsky and U. Madhow,“Space-time transmit precoding with imperfect feedback,”IEEE Trans. Inform. Theory, vol. 47, no. 6, pp. 2632–2639, Sept. 2001.
[15] M. Skoglund and G. Joongren,“On the capacity of a multiple-antenna communication link with channel side information,”IEEE J. Select. Areas Commun., vol. 21, no. 3, pp. 395–405, April 2003.
[16] R. S. Blum,“MIMO with limited feedback of channel state information,”in Proc. IEEE Int. Conf. Acoust., Speech and Sig. Proc., vol. 4, pp. 89–92, April 2003.
[17] V. Lau, Y. Liu, and T.-A. Chen,“On the design of MIMO block-fading channels with feedback-link capacity constraint,”IEEE Trans. Commun., vol. 52, no. 1, pp. 62–70, Jan. 2004.
[18] A. D. Dabbagh and D. J. Love,“Feedback rate-capacity loss tradeoff for limited feedback MIMO systems,”IEEE Trans. Inform. Theory, vol. 52, no. 5, pp. 2190–2202, May 2006.
[19] B. C. Banister and J. R. Zeidler,“Feedback assisted transmission subspace tracking for MIMO systems,”IEEE J. Select. Areas Commun., vol. 21, no. 3, pp. 452–463, April 2003.
[20] B. C. Banister and J. R. Zeidler,“A simple gradient sign algorithm for transmit antenna weight adaptation with feedback,”IEEE Trans. Signal Processing, vol. 51, no. 5, pp. 1156–1171, May 2003.
[21] J. Yang and D. B. Williams,“Transmission subspace tracking for MIMO systems with low-rate feedback,”IEEE Trans. Commun., vol. 55, no. 8, pp. 1629–1639, Aug. 2007.
[22] E. Zacarias, S. Werner, and R. Wichman,“Partial update adaptive transmit beamforming with limited feedback,”in Proc. IEEE Int. Conf. Acoust., Speech and Sig. Proc., vol. 4, May 2006.
[23] B. Mondal and R. W. Heath, Jr.,“Channel adaptive quantization for limited feedback MIMO beamforming systems,”IEEE Trans. Signal Processing, vol. 54, no. 12, pp. 4717–4729, Dec. 2006.
[24] K. Huang, B. Mondal, R. W. Heath, Jr., and J. G. Andrews,“Effect of feedback delay on multi-antenna limited feedback for temporally-correlated channels,”in Proc. IEEE Glob. Telecom. Conf., Nov. 2006.
[25] K. Huang, B. Mondal, R. W. Heath, Jr., and J. G. Andrews,“Markov models for limited feedback MIMO systems,”in Proc. IEEE Int. Conf. Acoust., Speech and Sig. Proc., vol. 4, May 2006.
[26]“Multi-antenna limited feedback for temporally-correlated channels: Feedback compression,”in Proc. IEEE Glob. Telecom. Conf., Nov. 2006.
[27] D. J. Love and R. W. Heath, Jr.,“Equal gain transmission in multiple-input multiple-output wireless systems,”IEEE Trans. Commun,vol. 51, no. 7, pp. 1102–1110, July 2003.
[28] M. K. Simon and M. Alouini, Digital Communication over Fading Channels. New York, NY: John Wiley and Sons, Inc., 2004.
[29] D. Gerlach and A. Paulraj,“Adaptive transmitting antenna arrays with feedback,”IEEE Sig. Proc. Lett., vol. 1, no. 10, pp. 150–152, Oct 1994.
[30] C. D. Frank,“Optimal transmit array weighting for DS-CDMA with channel feedback,”in Proc. Conf. Info. Sciences and Systems, March 2000.
[31] A. Hottinen, O. Tirkkonen, and R. Wichman,“Closed-loop transmit diversity techniques for multi-element transceivers,”in Proc. IEEE Veh. Technol. Conf., vol. 1, pp. 70–73, Sept. 2000.
[32] A. Hottinen and R. Wichman,“Transmit diversity using filtered feedback weights in the FDD/WCDMA system,”in Proc. Int. Zurich Seminar on Broadband Commun., Feb. 2000, pp. 15–21.
[33] S. Nagaraj and Y.-F. Huang,“Downlink transmit beamforming with selective feedback,”in Proc. IEEE Asilomar Conf. on Signals, Systems, and Comp., vol. 2, Oct.-Nov. 2000, pp. 1608–1612.
[34] S. Nagaraj and Y.-F. Huang,“Prediction error based feedback for downlink transmit beamforming,”in Proc. IEEE Int. Conf. Acoust., Speech and Sig. Proc., vol. 5, May 2001, pp. 2969–2972.
[35] N. R. Sollenberger,“Diversity and automatic link transfer for a TDMA wireless access link,”in Proc. IEEE Glob. Telecom. Conf., vol. 1, Nov.- Dec. 1993, pp. 532–536.
[36] S. Thoen, L. Van der Perre, B. Gyselinckx, and M. Engels,“Performance analysis of combined transmit-SC/receive-MRC,”IEEE Trans. Commun., vol. 49, no. 1, pp. 5–8, Jan. 2001.
[37] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell,“Efficient use of side information in mulitiple-antenna data transmission over fading channels,”IEEE J. Select. Areas Commun., vol. 16, no. 8, pp. 1423–1436, Oct. 1998.
[38] R. W. Heath, Jr. and A. Paulraj,“A simple scheme for transmit diversity using partial channel feedback,”in Proc. IEEE Asilomar Conf. on Signals, Systems, and Comp., vol. 2, Nov. 1998, pp. 1073–1078.
[39] D. J. Love, R. W. Heath Jr., and T. Strohmer,“Grassmannian beamforming for multiple-input multiple-output wireless systems,”IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2735–2747, Oct. 2003.
[40] C. R. Murthy and B. D. Rao,“Quantization methods for equal gain transmission with finite rate feedback,”IEEE Trans. Signal Processing, vol. 55, no. 1, pp. 233–245, Jan. 2007.
[41] X. Zheng, Y. Xie, J. Li, and P. Stoica,“MIMO transmit beamforming under uniform elemental power constraint,”IEEE Trans. Signal Processing, vol. 55, no. 11, pp. 5395–5406, Nov. 2007.
[42] K. K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang,“On beamforming with finite rate feedback in multiple antenna systems,”IEEE Trans. Inform. Theory, vol. 49, pp. 2562–2579, Oct. 2003.
[43] J. C. Roh and B. D. Rao,“Transmit beamforming in multiple-antenna systems with finite rate feedback: A VQ-based approach,”IEEE Trans. Inform. Theory, vol. 52, no. 3, pp. 1101–1112, March 2006.
[44] W. Santipach and M. L. Honig,“Asymptotic performance of MIMO wireless channels with limited feedback,”in Proc. IEEE Mil. Comm. Conf., vol. 1, Oct. 2003, pp. 141–146.
[45] A. F. Molisch, M. Z. Win, Y. Choi, and J. H. Winters,“Capacity of MIMO systems with antenna selection,”IEEE Trans. Wireless Comm., vol. 4, no. 4, pp. 1759–1772, July 2005.
[46] D. J. Love and R. W. Heath, Jr.,“Limited feedback unitary precoding for spatial multiplexing systems,”IEEE Trans. Inform. Theory, vol. 51, no. 8, pp. 2967–2976, Aug. 2005.
[47] A. Barg and D. Y. Nogin,“Bounds on packings of spheres in the Grassmann manifold,”IEEE Trans. Inform. Theory, vol. 48, no. 9, pp. 2450–2454, Sept. 2002.
[48] W. Dai, Y. Liu, and B. Rider,“Quantization bounds on Grassmann manifolds and applications to MIMO communications,”IEEE Trans. Inform. Theory, vol. 54, no. 3, pp. 1108-1123, March 2008.
[49] J. Choi and R. W. Heath, Jr.,“Interpolation based transmit beamforming for MIMO-OFDM with limited feedback,”IEEE Trans. Signal Processing, vol. 53, no. 11, pp. 4125–4135, Nov. 2005.
[50] T. Pande, D. J. Love, and J. V. Krogmeier,“A weighted least squares approach to precoding with pilots for MIMO-OFDM,”IEEE Trans. Signal Processing, vol. 54, no. 10, pp. 4067–4073, Oct. 2006.
[51] S. Zhou, B. Li, and P. Willett,“Recursive and trellis-based feedback reduction for MIMO-OFDM with rate-limited feedback,”IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3400–3405, Dec. 2006.
[52] L. Liu and H. Jafarkhani,“Successive transmit beamforming algorithms for multiple-antenna OFDM systems,”IEEE Trans. Wireless Comm., vol. 6, no. 4, pp. 1512–1522, April 2007.
[53] H. Zhang, Y. Li, V. Stolpman, and N. Van Waes,“A reduced CSI feedback approach for precoded MIMO-OFDM systems,”IEEE Trans. Wireless Commun., vol. 6, no. 1, pp. 55–58, Jan. 2007.
[54] N. Jindal,“MIMO broadcast channels with finite-rate feedback,”IEEE Trans. Inform. Theory, vol. 52, no. 11, pp. 5045–5060, Nov. 2006.
[55] C. Swannack, G. W. Wornell, and E. Uysal-Biyikoglu,“Efficient quantization for feedback in MIMO broadcasting systems,”in Proc. IEEE Asilomar Conf. on Signals, Systems, and Comp., Oct.-Nov. 2006, pp. 784–788.
[56] M. Sharif and B. Hassibi,“On the capacity of MIMO broadcast channels with partial side information,”IEEE Trans. Inform. Theory, vol. 51, no. 2, pp. 506–522, Feb. 2005.
[57] J. S. Kim, H. Kim, and K. B. Lee,“Limited feedback signaling for MIMO broadcast channels,”in Proc. IEEE Int. Workshop on Signal Proc. Adv. Wireless Commun., June 2005, pp. 855–859.
[58] D. Gesbert and M. Alouini,“Selective multi-user diversity,”in Proc. IEEE Int. Symp. Signal Proc. Info. Theory, Dec. 2003, pp. 162–165.
[59] R. Zakhour and D. Gesbert,“A two-stage approach to feedback design in multi-user MIMO channels with limited channel state information,”in Proc. IEEE Int. Symp. Personal, Indoor, and Mobile Radio Commun., Sept. 2007.
[60] M. Kountouris, R. de Francisco, D. Gesbert, D. Slock, and T. Salzer,“Multiuser diversity - multiplexing tradeoff in MIMO broadcast channels with limited feedback,”in Proc. IEEE Asilomar Conf. on Signals, Systems, and Comp., Oct.-Nov. 2006, pp. 364–368.
[61] W. Choi, A. Forenza, J. G. Andrews, and R. W. Heath, Jr.,“Opportunistic space-division multiple access with beam selection,”IEEE Trans. Commun., vol. 55, no. 12, pp. 2371–2380, Dec. 2007.
[62] C. Wang and R. D. Murch,“MU-MISO transmission with limited feedback,”IEEE Trans. Wireless Commun., vol. 6, no. 11, pp. 3907–3913, Nov. 2007.
[63] C.K. Au Yeung and D.J. Love,“On the Performance of Random Vector Quantization limited Feedback Beamforming in a MISO System,”IEEE Transactions on Communications, vol. 6, no. 2, pp. 458–462, February 2007.
[64] Liu Yupeng,Yang Zhe,Qiu Ling, On the Performance and Codebook Reorder in MISO Systems with Limited Feedback in the Presence of Feedback Error?,Chinese Journal of Electronics, submitted
[65]沈佳,索士强,全海洋,赵训威,胡海静,姜怡华等, 3GPP长期演进ξLTEο技术原理与系统设计?,2008年7月
[2] G. J. Foschini,“Layered Space-Time Architecture for Wireless Communication in a Fading Environment When Using Multi-Element Antennas,”in Bell Labs. Tech. J, vol. pp. 41-59, Autumn 1996.
[3] V. Tarokh, N. Seshadri, and A. R. Calderbank,“Space-Time Codes for High Data Rate Wireless Communication: Performance Criterion and Code Construction,”IEEE Trans. Info. Theory, vol. 44, pp. 744-765, Mar.1998.
[4] V. Tarokh, H. Jafarhani, and A. R. Calderbank,“Space-Time Block Coding from Orthogonal Designs,”IEEE JSAC, vol. 48, pp. 1456-1467, July 1999.
[5] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valen-zuela,“V-BLAST: An architecture for realizing very high data rates over the rich-scattering wireless channel,”Proc. URSI ISSSE, pp. 295-300,1998
[6] C. E. Shannon,“The zero error capacity of a noisy channel,”IRE Trans. Inform. Theory, vol. 2, no. 3, pp. 8–19, Sept. 1956
[7] C. E. Shannon,“Channels with side information at the transmitter,”IBM Journal Research and Dev., vol. 2, pp. 289–293, 1958
[8] J. Hayes,“Adaptive feedback communications,”IEEE Trans. Commun., vol. 16, no. 1, pp. 29–34, Feb. 1968.
[9] J. Schalkwijk,“Coding scheme for additive noise channels with feedback - Part II: Band-limited signals,”IEEE Trans. Inform. Theory, vol. 12, pp. 183–189, April 1966.
[10] J. Schalkwijk and T. Kailath,“A coding scheme for additive noise channels with feedback - Part I: No bandwidth constraint,”IEEE Trans. Inform. Theory, vol. 12, pp. 172–183, April 1966.
[11] G. L. Turin,“Signal design for sequential detection systems with feedback,”IEEE Trans. Inform. Theory, vol. 11, pp. 401–408, July 1965.
[12] G. L. Turin,“Comparison of sequential and nonsequential detection systems with uncertainty feedback,”IEEE Trans. Inform. Theory, vol. 12, pp. 5–8, Jan. 1966.
[13] A. Goldsmith, S. A. Jafar, N. Jindal, and S. Vishwanath,“Capacity limits of MIMO channels,”IEEE J. Select. Areas Commun., vol. 21, no. 5, pp. 684–702, June 2003.
[14] E. Visotsky and U. Madhow,“Space-time transmit precoding with imperfect feedback,”IEEE Trans. Inform. Theory, vol. 47, no. 6, pp. 2632–2639, Sept. 2001.
[15] M. Skoglund and G. Joongren,“On the capacity of a multiple-antenna communication link with channel side information,”IEEE J. Select. Areas Commun., vol. 21, no. 3, pp. 395–405, April 2003.
[16] R. S. Blum,“MIMO with limited feedback of channel state information,”in Proc. IEEE Int. Conf. Acoust., Speech and Sig. Proc., vol. 4, pp. 89–92, April 2003.
[17] V. Lau, Y. Liu, and T.-A. Chen,“On the design of MIMO block-fading channels with feedback-link capacity constraint,”IEEE Trans. Commun., vol. 52, no. 1, pp. 62–70, Jan. 2004.
[18] A. D. Dabbagh and D. J. Love,“Feedback rate-capacity loss tradeoff for limited feedback MIMO systems,”IEEE Trans. Inform. Theory, vol. 52, no. 5, pp. 2190–2202, May 2006.
[19] B. C. Banister and J. R. Zeidler,“Feedback assisted transmission subspace tracking for MIMO systems,”IEEE J. Select. Areas Commun., vol. 21, no. 3, pp. 452–463, April 2003.
[20] B. C. Banister and J. R. Zeidler,“A simple gradient sign algorithm for transmit antenna weight adaptation with feedback,”IEEE Trans. Signal Processing, vol. 51, no. 5, pp. 1156–1171, May 2003.
[21] J. Yang and D. B. Williams,“Transmission subspace tracking for MIMO systems with low-rate feedback,”IEEE Trans. Commun., vol. 55, no. 8, pp. 1629–1639, Aug. 2007.
[22] E. Zacarias, S. Werner, and R. Wichman,“Partial update adaptive transmit beamforming with limited feedback,”in Proc. IEEE Int. Conf. Acoust., Speech and Sig. Proc., vol. 4, May 2006.
[23] B. Mondal and R. W. Heath, Jr.,“Channel adaptive quantization for limited feedback MIMO beamforming systems,”IEEE Trans. Signal Processing, vol. 54, no. 12, pp. 4717–4729, Dec. 2006.
[24] K. Huang, B. Mondal, R. W. Heath, Jr., and J. G. Andrews,“Effect of feedback delay on multi-antenna limited feedback for temporally-correlated channels,”in Proc. IEEE Glob. Telecom. Conf., Nov. 2006.
[25] K. Huang, B. Mondal, R. W. Heath, Jr., and J. G. Andrews,“Markov models for limited feedback MIMO systems,”in Proc. IEEE Int. Conf. Acoust., Speech and Sig. Proc., vol. 4, May 2006.
[26]“Multi-antenna limited feedback for temporally-correlated channels: Feedback compression,”in Proc. IEEE Glob. Telecom. Conf., Nov. 2006.
[27] D. J. Love and R. W. Heath, Jr.,“Equal gain transmission in multiple-input multiple-output wireless systems,”IEEE Trans. Commun,vol. 51, no. 7, pp. 1102–1110, July 2003.
[28] M. K. Simon and M. Alouini, Digital Communication over Fading Channels. New York, NY: John Wiley and Sons, Inc., 2004.
[29] D. Gerlach and A. Paulraj,“Adaptive transmitting antenna arrays with feedback,”IEEE Sig. Proc. Lett., vol. 1, no. 10, pp. 150–152, Oct 1994.
[30] C. D. Frank,“Optimal transmit array weighting for DS-CDMA with channel feedback,”in Proc. Conf. Info. Sciences and Systems, March 2000.
[31] A. Hottinen, O. Tirkkonen, and R. Wichman,“Closed-loop transmit diversity techniques for multi-element transceivers,”in Proc. IEEE Veh. Technol. Conf., vol. 1, pp. 70–73, Sept. 2000.
[32] A. Hottinen and R. Wichman,“Transmit diversity using filtered feedback weights in the FDD/WCDMA system,”in Proc. Int. Zurich Seminar on Broadband Commun., Feb. 2000, pp. 15–21.
[33] S. Nagaraj and Y.-F. Huang,“Downlink transmit beamforming with selective feedback,”in Proc. IEEE Asilomar Conf. on Signals, Systems, and Comp., vol. 2, Oct.-Nov. 2000, pp. 1608–1612.
[34] S. Nagaraj and Y.-F. Huang,“Prediction error based feedback for downlink transmit beamforming,”in Proc. IEEE Int. Conf. Acoust., Speech and Sig. Proc., vol. 5, May 2001, pp. 2969–2972.
[35] N. R. Sollenberger,“Diversity and automatic link transfer for a TDMA wireless access link,”in Proc. IEEE Glob. Telecom. Conf., vol. 1, Nov.- Dec. 1993, pp. 532–536.
[36] S. Thoen, L. Van der Perre, B. Gyselinckx, and M. Engels,“Performance analysis of combined transmit-SC/receive-MRC,”IEEE Trans. Commun., vol. 49, no. 1, pp. 5–8, Jan. 2001.
[37] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell,“Efficient use of side information in mulitiple-antenna data transmission over fading channels,”IEEE J. Select. Areas Commun., vol. 16, no. 8, pp. 1423–1436, Oct. 1998.
[38] R. W. Heath, Jr. and A. Paulraj,“A simple scheme for transmit diversity using partial channel feedback,”in Proc. IEEE Asilomar Conf. on Signals, Systems, and Comp., vol. 2, Nov. 1998, pp. 1073–1078.
[39] D. J. Love, R. W. Heath Jr., and T. Strohmer,“Grassmannian beamforming for multiple-input multiple-output wireless systems,”IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2735–2747, Oct. 2003.
[40] C. R. Murthy and B. D. Rao,“Quantization methods for equal gain transmission with finite rate feedback,”IEEE Trans. Signal Processing, vol. 55, no. 1, pp. 233–245, Jan. 2007.
[41] X. Zheng, Y. Xie, J. Li, and P. Stoica,“MIMO transmit beamforming under uniform elemental power constraint,”IEEE Trans. Signal Processing, vol. 55, no. 11, pp. 5395–5406, Nov. 2007.
[42] K. K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang,“On beamforming with finite rate feedback in multiple antenna systems,”IEEE Trans. Inform. Theory, vol. 49, pp. 2562–2579, Oct. 2003.
[43] J. C. Roh and B. D. Rao,“Transmit beamforming in multiple-antenna systems with finite rate feedback: A VQ-based approach,”IEEE Trans. Inform. Theory, vol. 52, no. 3, pp. 1101–1112, March 2006.
[44] W. Santipach and M. L. Honig,“Asymptotic performance of MIMO wireless channels with limited feedback,”in Proc. IEEE Mil. Comm. Conf., vol. 1, Oct. 2003, pp. 141–146.
[45] A. F. Molisch, M. Z. Win, Y. Choi, and J. H. Winters,“Capacity of MIMO systems with antenna selection,”IEEE Trans. Wireless Comm., vol. 4, no. 4, pp. 1759–1772, July 2005.
[46] D. J. Love and R. W. Heath, Jr.,“Limited feedback unitary precoding for spatial multiplexing systems,”IEEE Trans. Inform. Theory, vol. 51, no. 8, pp. 2967–2976, Aug. 2005.
[47] A. Barg and D. Y. Nogin,“Bounds on packings of spheres in the Grassmann manifold,”IEEE Trans. Inform. Theory, vol. 48, no. 9, pp. 2450–2454, Sept. 2002.
[48] W. Dai, Y. Liu, and B. Rider,“Quantization bounds on Grassmann manifolds and applications to MIMO communications,”IEEE Trans. Inform. Theory, vol. 54, no. 3, pp. 1108-1123, March 2008.
[49] J. Choi and R. W. Heath, Jr.,“Interpolation based transmit beamforming for MIMO-OFDM with limited feedback,”IEEE Trans. Signal Processing, vol. 53, no. 11, pp. 4125–4135, Nov. 2005.
[50] T. Pande, D. J. Love, and J. V. Krogmeier,“A weighted least squares approach to precoding with pilots for MIMO-OFDM,”IEEE Trans. Signal Processing, vol. 54, no. 10, pp. 4067–4073, Oct. 2006.
[51] S. Zhou, B. Li, and P. Willett,“Recursive and trellis-based feedback reduction for MIMO-OFDM with rate-limited feedback,”IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3400–3405, Dec. 2006.
[52] L. Liu and H. Jafarkhani,“Successive transmit beamforming algorithms for multiple-antenna OFDM systems,”IEEE Trans. Wireless Comm., vol. 6, no. 4, pp. 1512–1522, April 2007.
[53] H. Zhang, Y. Li, V. Stolpman, and N. Van Waes,“A reduced CSI feedback approach for precoded MIMO-OFDM systems,”IEEE Trans. Wireless Commun., vol. 6, no. 1, pp. 55–58, Jan. 2007.
[54] N. Jindal,“MIMO broadcast channels with finite-rate feedback,”IEEE Trans. Inform. Theory, vol. 52, no. 11, pp. 5045–5060, Nov. 2006.
[55] C. Swannack, G. W. Wornell, and E. Uysal-Biyikoglu,“Efficient quantization for feedback in MIMO broadcasting systems,”in Proc. IEEE Asilomar Conf. on Signals, Systems, and Comp., Oct.-Nov. 2006, pp. 784–788.
[56] M. Sharif and B. Hassibi,“On the capacity of MIMO broadcast channels with partial side information,”IEEE Trans. Inform. Theory, vol. 51, no. 2, pp. 506–522, Feb. 2005.
[57] J. S. Kim, H. Kim, and K. B. Lee,“Limited feedback signaling for MIMO broadcast channels,”in Proc. IEEE Int. Workshop on Signal Proc. Adv. Wireless Commun., June 2005, pp. 855–859.
[58] D. Gesbert and M. Alouini,“Selective multi-user diversity,”in Proc. IEEE Int. Symp. Signal Proc. Info. Theory, Dec. 2003, pp. 162–165.
[59] R. Zakhour and D. Gesbert,“A two-stage approach to feedback design in multi-user MIMO channels with limited channel state information,”in Proc. IEEE Int. Symp. Personal, Indoor, and Mobile Radio Commun., Sept. 2007.
[60] M. Kountouris, R. de Francisco, D. Gesbert, D. Slock, and T. Salzer,“Multiuser diversity - multiplexing tradeoff in MIMO broadcast channels with limited feedback,”in Proc. IEEE Asilomar Conf. on Signals, Systems, and Comp., Oct.-Nov. 2006, pp. 364–368.
[61] W. Choi, A. Forenza, J. G. Andrews, and R. W. Heath, Jr.,“Opportunistic space-division multiple access with beam selection,”IEEE Trans. Commun., vol. 55, no. 12, pp. 2371–2380, Dec. 2007.
[62] C. Wang and R. D. Murch,“MU-MISO transmission with limited feedback,”IEEE Trans. Wireless Commun., vol. 6, no. 11, pp. 3907–3913, Nov. 2007.
[63] C.K. Au Yeung and D.J. Love,“On the Performance of Random Vector Quantization limited Feedback Beamforming in a MISO System,”IEEE Transactions on Communications, vol. 6, no. 2, pp. 458–462, February 2007.
[64] Liu Yupeng,Yang Zhe,Qiu Ling, On the Performance and Codebook Reorder in MISO Systems with Limited Feedback in the Presence of Feedback Error?,Chinese Journal of Electronics, submitted
[65]沈佳,索士强,全海洋,赵训威,胡海静,姜怡华等, 3GPP长期演进ξLTEο技术原理与系统设计?,2008年7月