多天线MIMO预编码传输技术的研究
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摘要
多输入多输出技术(MIMO,Multiple Input Multiple Output)是一种收发两端采用多天线传输的一种技术,相对于传统的单发单收系统来说,具有更高的系统容量等优良特性,因此成为未来高速率传输的移动通信中最具竞争力的技术之一。随着通信体制的发展与硬件设备的更新,系统可以通过一定的开销使发送端获得用于传输的信道信息,如:在频分复用(FDD)系统中,发送端可以通过接收端反馈而获得信道状态信息;在时分复用(TDD)系统中,发送端通过下行信道与上行信道所具有的互易性而获得信道状态信息。在发送端进行预编码设计可以带来以下好处,一方面,发端进行预编码可以有效的避免误码扩散;另一方面,系统进行预编码也可以大大的降低接收机的复杂度。这样,如何设计MIMO预编码来提高系统的频谱效率或者获得更佳的误码性能成为现在通信领域中研究的热点问题。
在分别对单用户MIMO预编码系统与多用户MIMO预编码系统进行了简要的介绍之后,我们提出了一种新颖的适用于单用户MIMO-THP(Tomlinson-Harashima precoding)系统的功率分配算法。在传统的单用户MIMO-THP系统中,利用矩阵分解将MIMO系统划分为多个独立并行的子信道。这些并行子信道之间的信道衰落差异不同,带来的性能损失也不相同。系统总体的性能又是由性能最差的子信道决定,因此,我们首先引入功率分配矩阵,在不改变发送总功率的情况下,来降低子信道之间的性能差异,提升系统总体性能。其次,在计算功率分配矩阵的过程中,我们考虑了THP算法中求模操作带来的预编码损失。这样计算的发送功率更加精确,性能明显优于传统的MIMO-THP系统,提高了系统的可靠性。最终,通过用指数函数对互补误差函数进行近似,得到了发送功率分配矩阵的闭式解。
接下来,在对实际应用环境进行考虑的基础上,我们又提出了应用在更广泛的移动通信下行广播链路中的多用户MIMO-THP收发系统联合设计算法。在本算法中,首先给出了迫零(ZF)准则和最小均方误差(MMSE)准则下用户信干噪比(SINR)的统一表达式。根据这个统一的SINR表达式,我们得到了总功率约束下的最小误比特率算法。
其次,将预编码损失因素考虑进来,系统性能获得了一定的提升,最后分析排序算法对性能的影响,最大化所有用户的信噪比(SNR)。综合以上处理,提出的新算法相对于传统的THP算法带来较大的性能提升。随后对所提出的算法进行了计算机仿真实验,仿真结果证明本算法具有很好的性能。
The multiple-input multiple-output (MIMO) system has been considered as one of the most important techniques owing to the significantly high capacity improvement compared with the traditional single-input single-output (SISO) wireless channel. With the development of the telecommunication system, the Channel State Information (CSI) at the transmitter is possible to be obtained with the cost of all system. In some cases, if the communication environment is slowly time varying, the CSI at the transmitter is possible to be obtained by feedback in frequency-division duplex (FDD) systems or the reciprocal principle in time-division duplex (TDD) systems. When CSI is achieved at the transmitter, the downlink precoding algorithms can be designed with these advantages: From one hand, the receiver error propagation can be avoided. From the other hand, the complexity of user mobile stations can be greatly reduced. As a result, how to realize the precoding algorithms in MIMO systems are hot-spot in communication researching area.
In this paper, we bring a Transmit Power Allocation (TPA) matrix into the conventional MIMO-THP system for point-to-point transmission. In tradition MIMO-THP system, the MIMO channel is divided into multiple parallel sub-channels via matrix factorization. The performances of sub-channels are different. The average system performance is dominated by the worst sub-channel. As a result, we first introduce the TPA matrix to the conventional MIMO-THP system and an effective TPA method that minimizes the bit-error rate (BER) performance under sum-power constraint is proposed. Second, this method is capable of achieving further performance improvement because the precoding loss factor is considered in the derivations. When employed to an uncoded QAM, a tightly approximation expression of the complementary cumulative distribution function (Q-function) is introduced. Finally, a closed-form expression of TPA scheme is derived.
After considering the outdoor multi-user transmission scenario, a THP structure for the downlink multi-user MIMO-THP system is proposed. During the derivation, unified expressions of the user signal to interference-plus-noise ratio (SINR) under ZF and MMSE criterions are achieved from the THP structure. According to the unified expressions, a closed-form solution of TPA matrix is determined with the minimum bit error rate (BER) criterion under sum-power constraint. It is referred to as the minimum BER-THP (MBER-THP) algorithm. Furthermore, an improved MBER-THP (IMBER-THP) algorithm is proposed under the novel structure, which invokes the precise precoding loss factor for performance improvement. Finally, the IMBER-THP algorithm is applicable to the ordered MIMO-THP system for maximizing the SNR of all users and achieves better performance than conventional ordered MIMO-THP systems.
在分别对单用户MIMO预编码系统与多用户MIMO预编码系统进行了简要的介绍之后,我们提出了一种新颖的适用于单用户MIMO-THP(Tomlinson-Harashima precoding)系统的功率分配算法。在传统的单用户MIMO-THP系统中,利用矩阵分解将MIMO系统划分为多个独立并行的子信道。这些并行子信道之间的信道衰落差异不同,带来的性能损失也不相同。系统总体的性能又是由性能最差的子信道决定,因此,我们首先引入功率分配矩阵,在不改变发送总功率的情况下,来降低子信道之间的性能差异,提升系统总体性能。其次,在计算功率分配矩阵的过程中,我们考虑了THP算法中求模操作带来的预编码损失。这样计算的发送功率更加精确,性能明显优于传统的MIMO-THP系统,提高了系统的可靠性。最终,通过用指数函数对互补误差函数进行近似,得到了发送功率分配矩阵的闭式解。
接下来,在对实际应用环境进行考虑的基础上,我们又提出了应用在更广泛的移动通信下行广播链路中的多用户MIMO-THP收发系统联合设计算法。在本算法中,首先给出了迫零(ZF)准则和最小均方误差(MMSE)准则下用户信干噪比(SINR)的统一表达式。根据这个统一的SINR表达式,我们得到了总功率约束下的最小误比特率算法。
其次,将预编码损失因素考虑进来,系统性能获得了一定的提升,最后分析排序算法对性能的影响,最大化所有用户的信噪比(SNR)。综合以上处理,提出的新算法相对于传统的THP算法带来较大的性能提升。随后对所提出的算法进行了计算机仿真实验,仿真结果证明本算法具有很好的性能。
The multiple-input multiple-output (MIMO) system has been considered as one of the most important techniques owing to the significantly high capacity improvement compared with the traditional single-input single-output (SISO) wireless channel. With the development of the telecommunication system, the Channel State Information (CSI) at the transmitter is possible to be obtained with the cost of all system. In some cases, if the communication environment is slowly time varying, the CSI at the transmitter is possible to be obtained by feedback in frequency-division duplex (FDD) systems or the reciprocal principle in time-division duplex (TDD) systems. When CSI is achieved at the transmitter, the downlink precoding algorithms can be designed with these advantages: From one hand, the receiver error propagation can be avoided. From the other hand, the complexity of user mobile stations can be greatly reduced. As a result, how to realize the precoding algorithms in MIMO systems are hot-spot in communication researching area.
In this paper, we bring a Transmit Power Allocation (TPA) matrix into the conventional MIMO-THP system for point-to-point transmission. In tradition MIMO-THP system, the MIMO channel is divided into multiple parallel sub-channels via matrix factorization. The performances of sub-channels are different. The average system performance is dominated by the worst sub-channel. As a result, we first introduce the TPA matrix to the conventional MIMO-THP system and an effective TPA method that minimizes the bit-error rate (BER) performance under sum-power constraint is proposed. Second, this method is capable of achieving further performance improvement because the precoding loss factor is considered in the derivations. When employed to an uncoded QAM, a tightly approximation expression of the complementary cumulative distribution function (Q-function) is introduced. Finally, a closed-form expression of TPA scheme is derived.
After considering the outdoor multi-user transmission scenario, a THP structure for the downlink multi-user MIMO-THP system is proposed. During the derivation, unified expressions of the user signal to interference-plus-noise ratio (SINR) under ZF and MMSE criterions are achieved from the THP structure. According to the unified expressions, a closed-form solution of TPA matrix is determined with the minimum bit error rate (BER) criterion under sum-power constraint. It is referred to as the minimum BER-THP (MBER-THP) algorithm. Furthermore, an improved MBER-THP (IMBER-THP) algorithm is proposed under the novel structure, which invokes the precise precoding loss factor for performance improvement. Finally, the IMBER-THP algorithm is applicable to the ordered MIMO-THP system for maximizing the SNR of all users and achieves better performance than conventional ordered MIMO-THP systems.
引文
[1]陈明,尤肖虎,“B3G移动通信系统的研究框架,”中兴通讯技术,pp.27-31,2006
[2]E.Telatar.Capacity of multi-antenna Gaussian channels.European Transactions on Telecommunications,pp.585-596,Nov.1999.
[3]G.J.Foschini and M.J.Gans,"On limits of wireless communications in a fading environment when using multiple antennas," Wireless Personal Communications,vol.6,pp.311-335,March 1998.
[4]S.M.Alamouti.A simple transmit diversity technique for wireless communications.IEEE J.Select.Areas.Commum.,1998,16:1451-1458.
[5]G.J.Foschini,"Layered space-time architecture for wireless communication in a fading environment when using multielement antennas," Bell Labs Tech.J.,pp.41-59,Autumn 1996.
[6]3GPP TS 36.211 V8.2.0,2008-03.
[7]陈旭彬,杨大成,“多用户MIMO在移动通信系统中的应用”
[8]Q.H.Spencer,C.B.Peel,A.L.Swindlehurst,and M.Haardt,"An introduction to the multi-user MIMO downlink," IEEE Communications Magazine,vol.42,no.10,pp.60-67,October 2004.
[9]David J.Love,Robert W.Heath and Thomas Strohmer,"Grassmannian Beamforming for Multiple-Input Multiple-Output Wireless Systems," IEEE Trans.Inf.Theory,vol.49,pp.2735-2747,October.2003.
[10]David J.Love and Robert W.Heath,"Limited Feedback Unitary Precoding for Orthogonal Space-Time Block Codes." IEEE Trans.Sigal Precoding,vol.53,No.1,pp.64-73,JAN.2005.
[11]David J.Love and Robert W.Heath,"Limited Feedback Unitary Precoding for Spatial Multiplexing Systems," IEEE Trans.Inf.Theory,pp.2967-2976,August 2005.
[12]R1-072936,"Way Forward on Householder and DFT based codebooks",RAN1#49bis,AT&T.
[13]R1-072935,"LTE MIMO:Flexible Codebook Design for Improved Performance",RAN1 #49bis,AT&T.
[14]R1-073100,"Way Forward for MU-MIMO Design",RAN1 #49b,Samsung.
[15]H.Sampath and A.Paulraj,"Joint transmit and receive optimization for high data rate wireless communications using multiple antennas," in Proc.Asilomar Conf.Signals,Systems and Computers,vol.1,1999.
[16]N.Khaled,S.Thoen,L.Deneire,and H.De Man,"Spatial-mode selection for the joint transmit receive mmse design over flat-fading mimo channels," in Proc.IEEE Workshop Signal Process.Adv.Wireless Commun.,Jun.2003.
[17]R.F.H.Fischer,C.Windpassinger,A.Lampe,and J.B.Huber,"Space-Time Transmission using Tomlinson-Harashima Precoding," In Proc.of 4.ITG Conf.on Source and Channel Coding,pp.139-147,January 2002.
[18]M.Tomlinson,"New automatic equaliser employing modulo arithmetic,"Electronics Letters,pp.138-139,March 1971.
[19]H.Harashima and H.Miyakawa,"Matched-transmission technique for channels with intersymbol interference," IEEE Trans.Commun.,vol.COM-20,pp.774-780,Aug.1972.
[20]Y.Jiang,J.Li and W.W.Hager,"Joint Transceiver Design for MIMO Communications Using Geometric Mean Decomposition," IEEE Trans.Signal Processing,vol.53,no.10,pp.3791-3803,Oct.2005.
[21]Spencer Q H,Swindlehurst A L and Haardt,"Zero - forcing methods for downlink spatial multiplexing in multiuser MIMO channels," IEEE Trans.Signal Processing,VOL.52,No.2,pp.461-471,2007.
[22]Choi L.U.,Murch R.D.,"A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach," IEEE Trans.Wireless Commun.,vol.3,No.1,pp.20-24,2004
[23]C.Windpassinger,R.F.H.Fisher,T.Vencel,and J.B.Huber,"Precoding in multiantenna and multiuser communications," IEEE Trans.Wireless Commun.,vol.3,no.4,pp.1305-1316,July 2004.
[24]Liu J,Krzymien W A.Improved Tomlinson-Harashima precoding for the downlink of multiple antenna multi-user systems.Proceedings of IEEE Wireless Communications and Networking Conference(WCNC),New Orleans,LA,USA,March 13-17,2005,(1):466- 472
[25]J.Liu and W.A.Krzymie(?),"Signal detection and pre-processing algorithms in layered space-time multiple input multiple output antenna systems," In Proc.Wireless'04,Calgary,Canada,July 2004,pp.9-21.
[26]Kusume K,Joham M,Utschick W,et al.Cholesky Factorization With Symmetric Permutation Applied to Detecting and Precoding Spatially Multiplexed Data Streams.IEEE Transaction on Signal Processing,2007,6(55):3089-3103
[27]M.Joham,J.Brehmer,and W.Utschick,"MMSE approaches to multiuser spatio-temporal Tomlinson-Harashima precoding," in Proc.5~(th) ITG Conf.Source and Channel Coding,Erlangen,Germany,Jan.2004,pp.387-394.
[28]G.J.Foschini,G.D.Golden,R.A.Valenzuela,and P.W.Wolniansky,"Simplified Processing for High Spectral Efficiency Wireless Communication Employing Multi-Element Arrays," IEEE Journal on Selected Areas in Communications,vol.17,no.11,pp.1841-1852,November 1999.
[29]Shaowei Lin ,Winston W. L. Ho and Ying-Chang Liang, "Block-Diagonal Geometric Mean Decomposition (BD-GMD) for Multiuser MIMO Broadcast Channels" 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications PIMRC'06.
[30]P. W. Wolniansky, G J. Foschini, G D. Golden, and R. A. Valenzuela, "V-BLAST:An architecture for realizing very high data rates over the rich-scattering wireless channel," presented at the ISSSE-98, Pise, Italy, Sept. 1998. invited paper.
[31]G Ginis and J. M. Cioffi, "On the relation between V-BLAST and the GDFE,"IEEE Commun. Lett., pp. 364-366, Sept. 2001.
[32]Bizaki, H.K. and Falahati, A, "Power Loading by Minimizing the Average Symbol Error Rate on MIMO THP Systems," In Proc. of 9. ICACT. Vol. 2, pp.1323-1326, Feb. 2007.
[33]P. Edward; L. Ying-Chang, "Power and Modulo Loss Tradeoff for Tomlinson-Harashima Precoding Applied to Geometric Mean Decomposition based MIMO Systems," In Proc. of 18th. PIMRC, pp. 1-5, Sep. 2007.
[34] Yun X, Lv T J, GONG P, et al. Approximate Minimum BER Power Allocation for MIMO-THP System. IEEE International Conference on Wireless Communications and Mobile Computing Conference 2008 (IWCMC 08): Aug.6-8,2008, Crete Island, GREECE: 2008, 888-892
[35] Fischer R F H. Precoding and Signal Shaping for Digital Transmission. New York USA: John Wiley & Sons, 2002.
[36] M. Payar'o, A. P'erez-Neira, and M. Lagunas, "Achievable rates for generalized spatial Tomlinson-Harashima precoding in MIMO systems," in Proc. IEEE Vehic.Tech. Conf. Fall, Sept. 2004.
[37] S. H. Nam, Oh-S. Shin, K. B. Lee, "Transmit power allocation for a modified V-BLAST system," IEEE Trans. Commun., vol. 52, pp. 1074 -1079, July 2004.
[38]N. Wang, S. D. Blostein, "Minimum BER power allocation for MIMO spatial multiplexing systems," IEEE Trans. Commun., vol. 55, pp. 180 -187, Jan. 2007.
[39] Richard D. Wesel, and John M. Cioffi, "Achievable Rates for Tomlinson-Harashima Precoding" IEEE Trans, on Inf. Theory, Vol.44, No.2,March 1998
[40] Vishwanath, S, Jindal N, Goldsmith A. Duality, Achievable Rates, and Sum-Rate Capacity of Gaussian MIMO Broadcast Channels. IEEE Transaction on Information Theory, 2003,10(49): 2658-2668.
[41]S. Zhou and G. B. Giannakis, "Adaptive modulation for multi-antenna transmissions with channel mean feedback," in Proc. IEEE ICC, Anchorage, AK,vol. 4, pp. 2281-2285, May 2003.
[42]J. G. Proakis "Digital communications (4th edition)," New York,NY:McGraw-Hill,2001.
[2]E.Telatar.Capacity of multi-antenna Gaussian channels.European Transactions on Telecommunications,pp.585-596,Nov.1999.
[3]G.J.Foschini and M.J.Gans,"On limits of wireless communications in a fading environment when using multiple antennas," Wireless Personal Communications,vol.6,pp.311-335,March 1998.
[4]S.M.Alamouti.A simple transmit diversity technique for wireless communications.IEEE J.Select.Areas.Commum.,1998,16:1451-1458.
[5]G.J.Foschini,"Layered space-time architecture for wireless communication in a fading environment when using multielement antennas," Bell Labs Tech.J.,pp.41-59,Autumn 1996.
[6]3GPP TS 36.211 V8.2.0,2008-03.
[7]陈旭彬,杨大成,“多用户MIMO在移动通信系统中的应用”
[8]Q.H.Spencer,C.B.Peel,A.L.Swindlehurst,and M.Haardt,"An introduction to the multi-user MIMO downlink," IEEE Communications Magazine,vol.42,no.10,pp.60-67,October 2004.
[9]David J.Love,Robert W.Heath and Thomas Strohmer,"Grassmannian Beamforming for Multiple-Input Multiple-Output Wireless Systems," IEEE Trans.Inf.Theory,vol.49,pp.2735-2747,October.2003.
[10]David J.Love and Robert W.Heath,"Limited Feedback Unitary Precoding for Orthogonal Space-Time Block Codes." IEEE Trans.Sigal Precoding,vol.53,No.1,pp.64-73,JAN.2005.
[11]David J.Love and Robert W.Heath,"Limited Feedback Unitary Precoding for Spatial Multiplexing Systems," IEEE Trans.Inf.Theory,pp.2967-2976,August 2005.
[12]R1-072936,"Way Forward on Householder and DFT based codebooks",RAN1#49bis,AT&T.
[13]R1-072935,"LTE MIMO:Flexible Codebook Design for Improved Performance",RAN1 #49bis,AT&T.
[14]R1-073100,"Way Forward for MU-MIMO Design",RAN1 #49b,Samsung.
[15]H.Sampath and A.Paulraj,"Joint transmit and receive optimization for high data rate wireless communications using multiple antennas," in Proc.Asilomar Conf.Signals,Systems and Computers,vol.1,1999.
[16]N.Khaled,S.Thoen,L.Deneire,and H.De Man,"Spatial-mode selection for the joint transmit receive mmse design over flat-fading mimo channels," in Proc.IEEE Workshop Signal Process.Adv.Wireless Commun.,Jun.2003.
[17]R.F.H.Fischer,C.Windpassinger,A.Lampe,and J.B.Huber,"Space-Time Transmission using Tomlinson-Harashima Precoding," In Proc.of 4.ITG Conf.on Source and Channel Coding,pp.139-147,January 2002.
[18]M.Tomlinson,"New automatic equaliser employing modulo arithmetic,"Electronics Letters,pp.138-139,March 1971.
[19]H.Harashima and H.Miyakawa,"Matched-transmission technique for channels with intersymbol interference," IEEE Trans.Commun.,vol.COM-20,pp.774-780,Aug.1972.
[20]Y.Jiang,J.Li and W.W.Hager,"Joint Transceiver Design for MIMO Communications Using Geometric Mean Decomposition," IEEE Trans.Signal Processing,vol.53,no.10,pp.3791-3803,Oct.2005.
[21]Spencer Q H,Swindlehurst A L and Haardt,"Zero - forcing methods for downlink spatial multiplexing in multiuser MIMO channels," IEEE Trans.Signal Processing,VOL.52,No.2,pp.461-471,2007.
[22]Choi L.U.,Murch R.D.,"A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach," IEEE Trans.Wireless Commun.,vol.3,No.1,pp.20-24,2004
[23]C.Windpassinger,R.F.H.Fisher,T.Vencel,and J.B.Huber,"Precoding in multiantenna and multiuser communications," IEEE Trans.Wireless Commun.,vol.3,no.4,pp.1305-1316,July 2004.
[24]Liu J,Krzymien W A.Improved Tomlinson-Harashima precoding for the downlink of multiple antenna multi-user systems.Proceedings of IEEE Wireless Communications and Networking Conference(WCNC),New Orleans,LA,USA,March 13-17,2005,(1):466- 472
[25]J.Liu and W.A.Krzymie(?),"Signal detection and pre-processing algorithms in layered space-time multiple input multiple output antenna systems," In Proc.Wireless'04,Calgary,Canada,July 2004,pp.9-21.
[26]Kusume K,Joham M,Utschick W,et al.Cholesky Factorization With Symmetric Permutation Applied to Detecting and Precoding Spatially Multiplexed Data Streams.IEEE Transaction on Signal Processing,2007,6(55):3089-3103
[27]M.Joham,J.Brehmer,and W.Utschick,"MMSE approaches to multiuser spatio-temporal Tomlinson-Harashima precoding," in Proc.5~(th) ITG Conf.Source and Channel Coding,Erlangen,Germany,Jan.2004,pp.387-394.
[28]G.J.Foschini,G.D.Golden,R.A.Valenzuela,and P.W.Wolniansky,"Simplified Processing for High Spectral Efficiency Wireless Communication Employing Multi-Element Arrays," IEEE Journal on Selected Areas in Communications,vol.17,no.11,pp.1841-1852,November 1999.
[29]Shaowei Lin ,Winston W. L. Ho and Ying-Chang Liang, "Block-Diagonal Geometric Mean Decomposition (BD-GMD) for Multiuser MIMO Broadcast Channels" 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications PIMRC'06.
[30]P. W. Wolniansky, G J. Foschini, G D. Golden, and R. A. Valenzuela, "V-BLAST:An architecture for realizing very high data rates over the rich-scattering wireless channel," presented at the ISSSE-98, Pise, Italy, Sept. 1998. invited paper.
[31]G Ginis and J. M. Cioffi, "On the relation between V-BLAST and the GDFE,"IEEE Commun. Lett., pp. 364-366, Sept. 2001.
[32]Bizaki, H.K. and Falahati, A, "Power Loading by Minimizing the Average Symbol Error Rate on MIMO THP Systems," In Proc. of 9. ICACT. Vol. 2, pp.1323-1326, Feb. 2007.
[33]P. Edward; L. Ying-Chang, "Power and Modulo Loss Tradeoff for Tomlinson-Harashima Precoding Applied to Geometric Mean Decomposition based MIMO Systems," In Proc. of 18th. PIMRC, pp. 1-5, Sep. 2007.
[34] Yun X, Lv T J, GONG P, et al. Approximate Minimum BER Power Allocation for MIMO-THP System. IEEE International Conference on Wireless Communications and Mobile Computing Conference 2008 (IWCMC 08): Aug.6-8,2008, Crete Island, GREECE: 2008, 888-892
[35] Fischer R F H. Precoding and Signal Shaping for Digital Transmission. New York USA: John Wiley & Sons, 2002.
[36] M. Payar'o, A. P'erez-Neira, and M. Lagunas, "Achievable rates for generalized spatial Tomlinson-Harashima precoding in MIMO systems," in Proc. IEEE Vehic.Tech. Conf. Fall, Sept. 2004.
[37] S. H. Nam, Oh-S. Shin, K. B. Lee, "Transmit power allocation for a modified V-BLAST system," IEEE Trans. Commun., vol. 52, pp. 1074 -1079, July 2004.
[38]N. Wang, S. D. Blostein, "Minimum BER power allocation for MIMO spatial multiplexing systems," IEEE Trans. Commun., vol. 55, pp. 180 -187, Jan. 2007.
[39] Richard D. Wesel, and John M. Cioffi, "Achievable Rates for Tomlinson-Harashima Precoding" IEEE Trans, on Inf. Theory, Vol.44, No.2,March 1998
[40] Vishwanath, S, Jindal N, Goldsmith A. Duality, Achievable Rates, and Sum-Rate Capacity of Gaussian MIMO Broadcast Channels. IEEE Transaction on Information Theory, 2003,10(49): 2658-2668.
[41]S. Zhou and G. B. Giannakis, "Adaptive modulation for multi-antenna transmissions with channel mean feedback," in Proc. IEEE ICC, Anchorage, AK,vol. 4, pp. 2281-2285, May 2003.
[42]J. G. Proakis "Digital communications (4th edition)," New York,NY:McGraw-Hill,2001.