基于RMT的MIMO信道分析研究
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摘要
MIMO技术可以在不增加带宽的前提下大幅度的提高系统容量,因此成为近年来移动通信的研究热点之一,而信道容量的大小直接影响MIMO系统的通信速率,因而成为被广泛研究的一个重要对象。
无线信道根据其特征主要分为平坦性快衰落信道、频率选择性快衰落信道、平坦性慢衰落信道和频率选择性慢衰落信道四类,由于实际的通信信道是时变的,且具有一定的随机性,因此在研究MIMO系统时通常讨论两种统计意义上的信道容量：信道为快衰落时讨论中断容量,为慢衰落时讨论各态历经容量。本文主要研究在空域相关条件下,MIMO通信系统在平坦性慢衰落信道和频率选择性慢衰落信道中的各态历经容量的问题,利用Wishart矩阵的联合特征值概率密度函数给出了在上述两种衰落下半相关信道的各态历经容量的可计算公式,并利用Wishart矩阵分布的性质和詹森不等式,推导出了在全相关信道下各态历经容量的上下限表达式,弥补了已有文献中只考虑发送端或接收端一侧相关的不足。
本文利用Matlab对上述理论结果进行了数值计算,计算结果表明：利用Wishart矩阵特征值的概率密度函数计算所得出的半相关各态历经容量值与利用蒙特卡罗方法进行1000次实验近似得出的各态历经容量值非常接近。随着接收端和发送端天线数目的增大,蒙特卡罗方法的运算时间大大增加,而利用本文信道容量公式可以快速计算出各态历经容量值,且与蒙特卡罗方法具有近似的计算精度。随着信噪比不断增大,本文给出的全相关信道容量的上下限值逐渐接近于各态历经容量值,当信噪比足够大时,信道容量的下限值近似等于各态历经容量值。另外,本文根据数值计算结果进一步讨论了影响空域相关性的三个重要参数(天线间的距离、来波角度和角度扩展)与各态历经容量之间的关系。
MIMO technology is paid more and more attention because it can improve capacity without increasing the bandwidth, and the capacity is one of the very important indexes which is used to measure a MIMO communication system performance and had a lot of in the past paper. But since the actual channel state varies over the time and has some randomicity, so the research about MIMO system mainly based on ergodic capacity and outage capacity which are of statistical significance.
According to the characteristics of wireless channel, it is classified as flat fading channel and frequency selective fading channel or fast fading channel and slow fading channel. This thesis mainly studied the ergodic capacity of the slow fading channel under the spatial correlation conditions. The calculation formula of the ergodic capacity based on the probability density function of Wishart matrix is derived. Moreover, according the characteristic of Wishart matrix's distribution and the Jansen inequality, the upper and lower limit of expressions of the channel capacity about the full spatial correlation channel are derived.
The results obtained by Matlab numeric calculation show that, the value calculated by the formula derived in this thesis is very closed to the value derived by Monte Carlo method. Along with the numbers of receiver and the transmitter antenna increasing, the Monte Carlo method's operand increased greatly, but the method in this thesis gets the results which have approximate calculation accuracy compared with the Monte Carlo method quickly. Moreover, with the Signal-to-Noise Ratio (SNR) increasing, the curve of the correlated fading channel on the lower limit is closer to the curve of the channel capacity. And discussing the interrelation between the channel capacity and the three important parameters (antenna distance, incoming wave angle and angle expansion) based on the results of Matlab numeric calculation.
无线信道根据其特征主要分为平坦性快衰落信道、频率选择性快衰落信道、平坦性慢衰落信道和频率选择性慢衰落信道四类,由于实际的通信信道是时变的,且具有一定的随机性,因此在研究MIMO系统时通常讨论两种统计意义上的信道容量：信道为快衰落时讨论中断容量,为慢衰落时讨论各态历经容量。本文主要研究在空域相关条件下,MIMO通信系统在平坦性慢衰落信道和频率选择性慢衰落信道中的各态历经容量的问题,利用Wishart矩阵的联合特征值概率密度函数给出了在上述两种衰落下半相关信道的各态历经容量的可计算公式,并利用Wishart矩阵分布的性质和詹森不等式,推导出了在全相关信道下各态历经容量的上下限表达式,弥补了已有文献中只考虑发送端或接收端一侧相关的不足。
本文利用Matlab对上述理论结果进行了数值计算,计算结果表明：利用Wishart矩阵特征值的概率密度函数计算所得出的半相关各态历经容量值与利用蒙特卡罗方法进行1000次实验近似得出的各态历经容量值非常接近。随着接收端和发送端天线数目的增大,蒙特卡罗方法的运算时间大大增加,而利用本文信道容量公式可以快速计算出各态历经容量值,且与蒙特卡罗方法具有近似的计算精度。随着信噪比不断增大,本文给出的全相关信道容量的上下限值逐渐接近于各态历经容量值,当信噪比足够大时,信道容量的下限值近似等于各态历经容量值。另外,本文根据数值计算结果进一步讨论了影响空域相关性的三个重要参数(天线间的距离、来波角度和角度扩展)与各态历经容量之间的关系。
MIMO technology is paid more and more attention because it can improve capacity without increasing the bandwidth, and the capacity is one of the very important indexes which is used to measure a MIMO communication system performance and had a lot of in the past paper. But since the actual channel state varies over the time and has some randomicity, so the research about MIMO system mainly based on ergodic capacity and outage capacity which are of statistical significance.
According to the characteristics of wireless channel, it is classified as flat fading channel and frequency selective fading channel or fast fading channel and slow fading channel. This thesis mainly studied the ergodic capacity of the slow fading channel under the spatial correlation conditions. The calculation formula of the ergodic capacity based on the probability density function of Wishart matrix is derived. Moreover, according the characteristic of Wishart matrix's distribution and the Jansen inequality, the upper and lower limit of expressions of the channel capacity about the full spatial correlation channel are derived.
The results obtained by Matlab numeric calculation show that, the value calculated by the formula derived in this thesis is very closed to the value derived by Monte Carlo method. Along with the numbers of receiver and the transmitter antenna increasing, the Monte Carlo method's operand increased greatly, but the method in this thesis gets the results which have approximate calculation accuracy compared with the Monte Carlo method quickly. Moreover, with the Signal-to-Noise Ratio (SNR) increasing, the curve of the correlated fading channel on the lower limit is closer to the curve of the channel capacity. And discussing the interrelation between the channel capacity and the three important parameters (antenna distance, incoming wave angle and angle expansion) based on the results of Matlab numeric calculation.
引文
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[2]Bolcskei H, Gesbert D, Paulraj A J. On the capacity of wireless systems employing OFDM-based spatial multiplexing[J]. IEEE Trans. on Comm.,2002,50(1):225-234.
[3]Chiani M, Win M Z, Zanella A. On the capacity of spatially correlated MIMO rayleigh-fading channels[J]. IEEE Trans. Infor. Theory,2003,49(10):2363-2371.
[4]Chuah C, Tse D M C, Kahn J, Valenzuela R. Capacity scaling in MIMO wireless systems under correlated fading[J], IEEE Trans. Inform. Theory,2002,48(3):637-650.
[5]Foschini G J, Gans M J. On limits of wireless communications in a fading environment when using multiple antennas[J]. Wireless Personal Communications,1998,6(3):311-335.
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[8]Jakes W C. Microwave mobile communication[M]. New Jersey, USA:John Wiley & Sons, Inc.,1974.
[9]Johansson K. Shape fluctuations and random matrices[J]. Comm. Math Phys,2000,20(9): 437-476.
[10]Kafedziski V. Capacity of frequency selective fading channels[J]. IEEE ISIT,1997,13(5): 335-339.
[11]Khatri C G On the moments of traces of two matrices in three situations for complex multivariate normal populations [J]. The Indian J. Statistical,1970,32(6):65-80.
[12]Lee W C Y. Effects on correlation between two mobile radio base station antennas[J]. IEEE Trans. Comm.,1973,21(11):1214-1224.
[13]李水祥.在相关信道下MIMO-OFDM信道容量的分析[J].微电子技术,2006,239(24)：132-136.
[14]李玮.MIMO多天线技术研究[D].西安：西安电子科技大学硕士学位论文,2008.
[15]李忻,聂在平.MIMO信道中衰落信号的空域相关性评估[J].电子学报,2004,32(12)：1949-1953.
[16]林云,何丰.MIMO技术原理及应用[M].北京：人民邮电出版社,2010.
[17]Mehtam L. Random matrices (third edition) [M]. London:Academic Press,2006.
[18]宁波MIMO信道模型及衰落信号的空域相关性分析[D].北京：北京邮电大学硕士学 位论文,2007.
[19]庞继勇MIMO相关信道容量研究[D].西安：西安电子科技大学硕士学位论文,2006.
[20]Pedersen K I, Mogensen P E, Fieury B H. Spatial channel characteristics in outdoor environments and their impact on BS antenna system performance[C]. Ottawa:IEEE Vehi. Tech. Conf.,1998,719-723.
[21]Pohl V, Jungnickel V, Haustein T, Helmolt C V. Antenna spacing in MIMO indoor channels[J]. IEEE Trans. Comm.,2002,45(3):749-753.
[22]彭林.第三代移动通信技术[M].北京：电子工业出版社,2001.
[23]Proakis J K. Digital communications (third edition) [M]. New York:Mc Graw-Hill,1995.
[24]Ratnarajah T, Vaillancourt R, Alvo M. Complex random matrices and rayleigh channel capacity[J]. Communications in Information and Systems,2003,3:119-138.
[25]Robb J M. Aspects of multivariate statistical theory[M]. New York:John Wiley,1982.
[26]Salz J, Winters J H. Effect of fading correlation on adaptive arrays in digital mobile radio [J]. IEEE Trans. Veh. Techno.,1994,43(4):1049-1057.
[27]Shannon C E. A mathematical theory of communication [J]. The Bell System Technical Journal,1948,27(7):379-423.
[28]Shin H D, Win M Z, Lee J H, Chiani M. On the capacity of doubly correlated MIMO channels[J]. IEEE Trans. on Wire. Comm.,2006,5(8):2253-2264.
[29]Shiu D S, Foschini G. J, Gans M J, et al. Fading correlation and its effect on the capacity of multielement antenna systems[J]. IEEE Trans. Comm.,2000,48(3):502-513.
[30]Std. IEEE 802.11n draftl.0[S]. IEEE-SA Standard Broad,2005.
[31]Telatar I E. Capacity of multi-antenna gaussian channels[J]. European Transactions on Telecommunications,1999,10(6):585-595.
[32]Tracy C A, Widom H. On orthogonal and symplectic matrix ensembles[J]. Communications in Mathematical Physics,1996,177(3):727-754.
[33]Tulino M A, Verdu S. Random matrix theory and wireless communications[M]. USA:Now Publisher Inc.,2004.
[34]王君,朱世华,王磊.频率选择性衰落环境中MIMO系统信道容量研究[J].电波科学学报,2006,21(2)：209-214.
[35]吴晓乐.信道不准确估计条件下MIMO系统的信道容量分析[D].重庆：重庆邮电大学硕士学位论文,2008.
[36]Yu K, Bengtsson M, Ottersten B, et al. Second order statistics of NLOS indoor MIMO channels based on 5.2 GHz measurements [J]. IEEE Global Telecomm.,2001,5(1): 156-160.
[37]张华炜.无线MIMO信道建模与信道容量研究[D].西安：西安电子科技大学硕士学位论文,2009.
[2]Bolcskei H, Gesbert D, Paulraj A J. On the capacity of wireless systems employing OFDM-based spatial multiplexing[J]. IEEE Trans. on Comm.,2002,50(1):225-234.
[3]Chiani M, Win M Z, Zanella A. On the capacity of spatially correlated MIMO rayleigh-fading channels[J]. IEEE Trans. Infor. Theory,2003,49(10):2363-2371.
[4]Chuah C, Tse D M C, Kahn J, Valenzuela R. Capacity scaling in MIMO wireless systems under correlated fading[J], IEEE Trans. Inform. Theory,2002,48(3):637-650.
[5]Foschini G J, Gans M J. On limits of wireless communications in a fading environment when using multiple antennas[J]. Wireless Personal Communications,1998,6(3):311-335.
[6]Goodman N R. The distribution of the determinant of a complex Wishart distributed matrix[J]. Ann. Math. Stat.,1963,34(1):178-180.
[7]Horn R A, Johnson C R. Matrix analysis[M]. New York:Cambridge Press,1985.
[8]Jakes W C. Microwave mobile communication[M]. New Jersey, USA:John Wiley & Sons, Inc.,1974.
[9]Johansson K. Shape fluctuations and random matrices[J]. Comm. Math Phys,2000,20(9): 437-476.
[10]Kafedziski V. Capacity of frequency selective fading channels[J]. IEEE ISIT,1997,13(5): 335-339.
[11]Khatri C G On the moments of traces of two matrices in three situations for complex multivariate normal populations [J]. The Indian J. Statistical,1970,32(6):65-80.
[12]Lee W C Y. Effects on correlation between two mobile radio base station antennas[J]. IEEE Trans. Comm.,1973,21(11):1214-1224.
[13]李水祥.在相关信道下MIMO-OFDM信道容量的分析[J].微电子技术,2006,239(24)：132-136.
[14]李玮.MIMO多天线技术研究[D].西安：西安电子科技大学硕士学位论文,2008.
[15]李忻,聂在平.MIMO信道中衰落信号的空域相关性评估[J].电子学报,2004,32(12)：1949-1953.
[16]林云,何丰.MIMO技术原理及应用[M].北京：人民邮电出版社,2010.
[17]Mehtam L. Random matrices (third edition) [M]. London:Academic Press,2006.
[18]宁波MIMO信道模型及衰落信号的空域相关性分析[D].北京：北京邮电大学硕士学 位论文,2007.
[19]庞继勇MIMO相关信道容量研究[D].西安：西安电子科技大学硕士学位论文,2006.
[20]Pedersen K I, Mogensen P E, Fieury B H. Spatial channel characteristics in outdoor environments and their impact on BS antenna system performance[C]. Ottawa:IEEE Vehi. Tech. Conf.,1998,719-723.
[21]Pohl V, Jungnickel V, Haustein T, Helmolt C V. Antenna spacing in MIMO indoor channels[J]. IEEE Trans. Comm.,2002,45(3):749-753.
[22]彭林.第三代移动通信技术[M].北京：电子工业出版社,2001.
[23]Proakis J K. Digital communications (third edition) [M]. New York:Mc Graw-Hill,1995.
[24]Ratnarajah T, Vaillancourt R, Alvo M. Complex random matrices and rayleigh channel capacity[J]. Communications in Information and Systems,2003,3:119-138.
[25]Robb J M. Aspects of multivariate statistical theory[M]. New York:John Wiley,1982.
[26]Salz J, Winters J H. Effect of fading correlation on adaptive arrays in digital mobile radio [J]. IEEE Trans. Veh. Techno.,1994,43(4):1049-1057.
[27]Shannon C E. A mathematical theory of communication [J]. The Bell System Technical Journal,1948,27(7):379-423.
[28]Shin H D, Win M Z, Lee J H, Chiani M. On the capacity of doubly correlated MIMO channels[J]. IEEE Trans. on Wire. Comm.,2006,5(8):2253-2264.
[29]Shiu D S, Foschini G. J, Gans M J, et al. Fading correlation and its effect on the capacity of multielement antenna systems[J]. IEEE Trans. Comm.,2000,48(3):502-513.
[30]Std. IEEE 802.11n draftl.0[S]. IEEE-SA Standard Broad,2005.
[31]Telatar I E. Capacity of multi-antenna gaussian channels[J]. European Transactions on Telecommunications,1999,10(6):585-595.
[32]Tracy C A, Widom H. On orthogonal and symplectic matrix ensembles[J]. Communications in Mathematical Physics,1996,177(3):727-754.
[33]Tulino M A, Verdu S. Random matrix theory and wireless communications[M]. USA:Now Publisher Inc.,2004.
[34]王君,朱世华,王磊.频率选择性衰落环境中MIMO系统信道容量研究[J].电波科学学报,2006,21(2)：209-214.
[35]吴晓乐.信道不准确估计条件下MIMO系统的信道容量分析[D].重庆：重庆邮电大学硕士学位论文,2008.
[36]Yu K, Bengtsson M, Ottersten B, et al. Second order statistics of NLOS indoor MIMO channels based on 5.2 GHz measurements [J]. IEEE Global Telecomm.,2001,5(1): 156-160.
[37]张华炜.无线MIMO信道建模与信道容量研究[D].西安：西安电子科技大学硕士学位论文,2009.