基于非圆系数估计的宽线性波束形成算法研究
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摘要
波束形成(Beamforming)技术是阵列信号处理领域中的一个重要研究方向,其利用空间多传感器所组成传感器阵列来对空间信号进行发射或接收,广泛应用于雷达,声呐,无线通信等领域。传统的波束形成技术假设信号是圆的、平稳的,仅仅利用了观测矢量的协方差矩阵,然而在通信领域大量的人工调制信号都是非圆的、非平稳的,不仅存在协方差矩阵而且存在伪协方差矩阵。基于圆信号假设下的波束形成技术虽然可以应用于非圆信号,但是其性能并不能达到最优。近些年现代信号处理领域中同时利用观测矢量及其共轭的宽线性(WL Widely Linear)技术的不断发展促使研究者们重新审视针对非圆信号的波束形成问题。宽线性波束形成技术通过分析信号的非圆性,同时利用了观测矢量的协方差矩阵和伪协方差矩阵,相应地一系列宽线性波束形成方法也陆续出现。
本文在对宽线性波束形成技术进行概述的基础上,针对宽线性波束形成技术发展中存在的关键问题,开展基于非圆系数估计的宽线性波束形成算法研究。本研究工作的目的是在解决期望信号非圆系数估计问题的基础上,针对不同场景提出相应的宽线性波束形成算法。本论文研究工作的主要贡献与创新点归纳如下
1.针对最优宽线性波束形成算法中非圆系数难以估计的问题提出非圆系数估计算法,并得到了一个稳健的对角加载权重矢量。我们研究使用最小功率无畸变(MPDR)准则,将其转化为一个参数估计问题,通过求导得到初始估计后,分析干扰和噪声对初始估计的影响,最后得到一个修正的估计值。在得到非圆系数估计后,使用对角加载技术减小估计误差对算法性能的影响。理论与仿真实验同时证明所提算法对非圆系数有比较好的估计性能,而且能有效地应用于实际场景。
2.针对期望信号存在偏移频率的情况提出了频移宽线性波束形成算法。由于存在偏移频率的期望信号非圆系数随观测时间逐渐趋于零,共轭观测矢量与观测矢量中关于期望信号的相关信息是逐渐减少的。我们提出使用共轭循环观测矢量代替共轭观测矢量,其对应的共轭循环相关系数代替非圆系数进行宽线性波束形成。所提的算法在偏移频率为零时,等价于最优宽线性波束形成算法,在偏移频率不为零时,优于最优宽线性波束形成算法。此外针对期望信号是直线信号的情况,通过分析时间平均的非圆系数与偏移频率的关系,提出了两种估计偏移频率的算法。
3.当期望信号的偏移频率与干扰的偏移频率不同时,提出M阶频移宽线性波束形成算法。频移宽线性波束形成算法仅利用了期望信号的共轭循环相关函数,当期望信号的偏移频率与干扰的偏移频率不同时,考虑使用多个共轭循环相关观测矢量,通过增加M个约束使其干扰在对应于他们自己的循环频率上的输出为零。所提算法是频移宽线性波束形成算法的扩展版本,不增加约束的零阶算法对应着频移宽线性波束形成算法。当期望信号的偏移频率与干扰的偏移频率不同时,所提算法优于频移宽线性波束形成算法。
4.针对期望信号存在导向矢量失配的情况,引入非圆系数空间谱,提出了基于非圆系数空间谱估计的稳健宽线性波束形成算法。所提算法重构了干扰加噪声协方差矩阵和干扰加噪声伪协方差矩阵,剔除了扩展的协方差矩阵中的信号分量。在高信噪比和小快拍情况下,波束形成器输出信干噪比始终接近最优值。
Beamforming technique is one of the most important issues in the array signal processing, which utilizes the array constituted by multiple sensors to transmit or receive the space signals, and has been widely used in the area of radar, sonar, communication and etc. Conventional beamforming technique generally assumes that the signals are circular and statistically stationary and only utilizes the covariance matrix of the observation vector. However, in the area of communication system, many artificial modulation signals are noncircular and nonstationary, which only have covariance matrix but also have conjugated covariace matrix. Although the conventional beamforming technique based on the circular assumption can be used for the reception of noncircular signals, the reception performance cannot attain the optimum. Recently, in the area of modern signal processing, the development of widely linear (WL) technique attains more and more researchers to reconsider the problem of the beamforming technique for the reception of noncircular signals. By analyzing the noncircularity of noncircular signals, the WL beamforming technique simultaneously utilizes the covariance matrix and conjugated covariance matrix of the observation vector and the corresponding WL beamforming methods were successively proposed.
Based on the overview of WL beamforming technique, this paper aims at carrying out the study of the WL beamforming methods for the key issues in the development of WL beamforming technique. The target of our research is to solve the problem of noncircularity coefficient estimation and then propose corresponding WL beamforming methods under different scenarios. The main contributions and innovative points of this dissertation are listed as follows:
1. For the problem of noncircularity coefficient estimation, we propose a method to estimate the noncircularity coefficient of the desired signal, and then a robust weight vector by using the diagonal loading technique. We consider the the principle of minimum Variance Distortionless response (MVDR) and turn out this problem into a parameter estimation problem. We find out the parameter to make the output power maximal as a rough estimate of noncircularity coefficient by taking the conjugate derivative of the objective function. Then, through analyzing the effect of interferences and noise on the rough estimate of noncircularity coefficient, we obtain the final modified estimate of the noncircularity coefficient. After obtaining the modified estimate of the noncircularity coefficient, the diagonal loading technique is used to suppress the effect of the estimation error on the optimal WL MVDR beamformer.
2. For the situation that the desired signal has a non-null offset frequency, we propose a frequency-shift (FS) WL beamforming method for the reception of noncircular signals. Considering that the noncircularity rate of desired signal with a non-null offset frequency changes and approaches to0with the increment of observation time, the correlation information of the desired signal between the conjugated observation vector and observation decreases. We propose to use the conjugated cyclic observation vector and the corresponding conjugated cyclic coefficient instead of the conjugated observation vector and the noncircularity coefficient in the optimal WL beamformer. When the offset frequency of the desired signal is null. the proposed method is equivalent to the optimal WL MVDR beamformer. When the offset frequency of the desired signal is non-null.the proposed method outperforms the optimal WL MVDR beamformer. In addition, we propose two methods for the estimation of offset frequency based on relationship between the time-averaged noncircularity coefficient and offset frequency.
3. Considering the situation that the offset frequencies of the desired signal and interferences are different, we propose an Mth-order FS WL beamfoming method. The FS WL beamfomer only utilizes the conjugated coefficient of the desired signal. When the offset frequencies of the desired signal and interferences are different, the proposed method uses multiple conjugated cyclic observation vectors and suppresses the output powers of interferences on their own cyclic frequency by adding M constraints. The proposed method is the extended version of the FS WL MVDR beamformer and the0th-order FS WL MVDR beamformer corresponds to the proposed FS WL MVDR beamformer. When the offset frequencies of the desired signal and interferences are different.the proposed method outperforms the FS WL MVDR beamformer.
4. For the situation that there exists steering vector mismatch, we propose a robust WL beamforming method, in which the spatial spectrum of noncircularity coefficient is firstly introduced. The proposed method constructs the interfere-plus-noise covariance matrix (1NCM) and conjugated1NCM. Hence, the desired signal component is removed from the extended covariance matrix. Under the high signal-to-noise ratio (SNR) and small snapshots size, the output signal-to-interference-plus-noise ratio (SINR) of the proposed robust WL MVDR beamformer always approaches the optimal value.
本文在对宽线性波束形成技术进行概述的基础上,针对宽线性波束形成技术发展中存在的关键问题,开展基于非圆系数估计的宽线性波束形成算法研究。本研究工作的目的是在解决期望信号非圆系数估计问题的基础上,针对不同场景提出相应的宽线性波束形成算法。本论文研究工作的主要贡献与创新点归纳如下
1.针对最优宽线性波束形成算法中非圆系数难以估计的问题提出非圆系数估计算法,并得到了一个稳健的对角加载权重矢量。我们研究使用最小功率无畸变(MPDR)准则,将其转化为一个参数估计问题,通过求导得到初始估计后,分析干扰和噪声对初始估计的影响,最后得到一个修正的估计值。在得到非圆系数估计后,使用对角加载技术减小估计误差对算法性能的影响。理论与仿真实验同时证明所提算法对非圆系数有比较好的估计性能,而且能有效地应用于实际场景。
2.针对期望信号存在偏移频率的情况提出了频移宽线性波束形成算法。由于存在偏移频率的期望信号非圆系数随观测时间逐渐趋于零,共轭观测矢量与观测矢量中关于期望信号的相关信息是逐渐减少的。我们提出使用共轭循环观测矢量代替共轭观测矢量,其对应的共轭循环相关系数代替非圆系数进行宽线性波束形成。所提的算法在偏移频率为零时,等价于最优宽线性波束形成算法,在偏移频率不为零时,优于最优宽线性波束形成算法。此外针对期望信号是直线信号的情况,通过分析时间平均的非圆系数与偏移频率的关系,提出了两种估计偏移频率的算法。
3.当期望信号的偏移频率与干扰的偏移频率不同时,提出M阶频移宽线性波束形成算法。频移宽线性波束形成算法仅利用了期望信号的共轭循环相关函数,当期望信号的偏移频率与干扰的偏移频率不同时,考虑使用多个共轭循环相关观测矢量,通过增加M个约束使其干扰在对应于他们自己的循环频率上的输出为零。所提算法是频移宽线性波束形成算法的扩展版本,不增加约束的零阶算法对应着频移宽线性波束形成算法。当期望信号的偏移频率与干扰的偏移频率不同时,所提算法优于频移宽线性波束形成算法。
4.针对期望信号存在导向矢量失配的情况,引入非圆系数空间谱,提出了基于非圆系数空间谱估计的稳健宽线性波束形成算法。所提算法重构了干扰加噪声协方差矩阵和干扰加噪声伪协方差矩阵,剔除了扩展的协方差矩阵中的信号分量。在高信噪比和小快拍情况下,波束形成器输出信干噪比始终接近最优值。
Beamforming technique is one of the most important issues in the array signal processing, which utilizes the array constituted by multiple sensors to transmit or receive the space signals, and has been widely used in the area of radar, sonar, communication and etc. Conventional beamforming technique generally assumes that the signals are circular and statistically stationary and only utilizes the covariance matrix of the observation vector. However, in the area of communication system, many artificial modulation signals are noncircular and nonstationary, which only have covariance matrix but also have conjugated covariace matrix. Although the conventional beamforming technique based on the circular assumption can be used for the reception of noncircular signals, the reception performance cannot attain the optimum. Recently, in the area of modern signal processing, the development of widely linear (WL) technique attains more and more researchers to reconsider the problem of the beamforming technique for the reception of noncircular signals. By analyzing the noncircularity of noncircular signals, the WL beamforming technique simultaneously utilizes the covariance matrix and conjugated covariance matrix of the observation vector and the corresponding WL beamforming methods were successively proposed.
Based on the overview of WL beamforming technique, this paper aims at carrying out the study of the WL beamforming methods for the key issues in the development of WL beamforming technique. The target of our research is to solve the problem of noncircularity coefficient estimation and then propose corresponding WL beamforming methods under different scenarios. The main contributions and innovative points of this dissertation are listed as follows:
1. For the problem of noncircularity coefficient estimation, we propose a method to estimate the noncircularity coefficient of the desired signal, and then a robust weight vector by using the diagonal loading technique. We consider the the principle of minimum Variance Distortionless response (MVDR) and turn out this problem into a parameter estimation problem. We find out the parameter to make the output power maximal as a rough estimate of noncircularity coefficient by taking the conjugate derivative of the objective function. Then, through analyzing the effect of interferences and noise on the rough estimate of noncircularity coefficient, we obtain the final modified estimate of the noncircularity coefficient. After obtaining the modified estimate of the noncircularity coefficient, the diagonal loading technique is used to suppress the effect of the estimation error on the optimal WL MVDR beamformer.
2. For the situation that the desired signal has a non-null offset frequency, we propose a frequency-shift (FS) WL beamforming method for the reception of noncircular signals. Considering that the noncircularity rate of desired signal with a non-null offset frequency changes and approaches to0with the increment of observation time, the correlation information of the desired signal between the conjugated observation vector and observation decreases. We propose to use the conjugated cyclic observation vector and the corresponding conjugated cyclic coefficient instead of the conjugated observation vector and the noncircularity coefficient in the optimal WL beamformer. When the offset frequency of the desired signal is null. the proposed method is equivalent to the optimal WL MVDR beamformer. When the offset frequency of the desired signal is non-null.the proposed method outperforms the optimal WL MVDR beamformer. In addition, we propose two methods for the estimation of offset frequency based on relationship between the time-averaged noncircularity coefficient and offset frequency.
3. Considering the situation that the offset frequencies of the desired signal and interferences are different, we propose an Mth-order FS WL beamfoming method. The FS WL beamfomer only utilizes the conjugated coefficient of the desired signal. When the offset frequencies of the desired signal and interferences are different, the proposed method uses multiple conjugated cyclic observation vectors and suppresses the output powers of interferences on their own cyclic frequency by adding M constraints. The proposed method is the extended version of the FS WL MVDR beamformer and the0th-order FS WL MVDR beamformer corresponds to the proposed FS WL MVDR beamformer. When the offset frequencies of the desired signal and interferences are different.the proposed method outperforms the FS WL MVDR beamformer.
4. For the situation that there exists steering vector mismatch, we propose a robust WL beamforming method, in which the spatial spectrum of noncircularity coefficient is firstly introduced. The proposed method constructs the interfere-plus-noise covariance matrix (1NCM) and conjugated1NCM. Hence, the desired signal component is removed from the extended covariance matrix. Under the high signal-to-noise ratio (SNR) and small snapshots size, the output signal-to-interference-plus-noise ratio (SINR) of the proposed robust WL MVDR beamformer always approaches the optimal value.
引文
[1]Johnson D H, Dudgeon D E. Array signal processing:concepts and techniques[M]. Englewood Cliffs:Prentice Hall,1993.
[2]Naidu P S. Sensor array signal processing[M]. Boca Raton:CRC Press,2001.
[3]Krim H, Viberg M. Two decades of array signal processing research:the parametric approach[J]. IEEE Signal Processing Magazine,1996,13(4):67-94.
[4]Bolcskei H, Gesbert D, Papadias CB, Veen A J. Space-time wireless systems:from array processing to MIMO communications[M]. New York:Cambridge University Press,2008.
[5]Munier J, Delisle G Y. Spatial analysis in passive listening using adaptive techniques[J]. In Proceedings of IEEE.1987,75(11):1458-1471.
[6]Benesty J, Chen J, Huang Y. Microphone array signal processing[M]. Heidelberg: Springer-Verlag,2008.
[7]王永良等.空间谱估计理论与算法[M].北京：清华大学出版社,2004.
[8]Chandran S. Advances in direction-of-arrival estimation[M]. Boston:Artech House,2006.
[9]Tuncer E. Friedlander B. Classical and modern direction-of-arrival estimation[M]. Burlington: Elsevier,2009.
[10]Li J, Stoica P. Robust adaptive beamforming[M]. Hoboken:John Wiley & Sons,2006.
[11]Picinbono B. On circularity[J]. IEEE Transaction on Signal Processing,1994,42(12): 3473-3482.
[12]Veen B D V and Buckley K M. Beamforming:a versatile approach to spatial filtering[J]. IEEE Acoustic, Speech, Signal Processing Magazine,1988,5(2):4-24.
[13]Trees H L V. Detection, estimation, and modulation theory[M]. Part Ⅳ:Optimum array processing. John Wiley & Sons, New York, NY,2002.
[14]Capon J. High resolution frequency-wavenumber spectrum analysis[J]. Proceeding of IEEE, 1969,57(8):1408-1418.
[15]Schreier P J, Scharf L L. Second-order analysis of improper complex random vector and processes[J]. IEEE Transaction on Signal Processing,2003,51(3):714-725.
[16]Picinbono B. Second-order complex random vectors and normal distributions[J]. IEEE Transaction on Signal Processing,1996,44(10):2637-2640.
[17]Picinbono B. Second-order statistics of complex signals.IEE Transaction on Signal Processing[J],1997,45(2):411-420.
[18]Schreier P J, Scharf L L. Statistical Signal processing of complex-valued data:the theory of improper and noncircular signals[M]. Cambridge Univsersity Press. New York.2010.
[19]Charge P, Wang Y, Saillard J. A non-circular sources direction finding method using polynomial rooting[J]. Signal Processing,2001,81(6):1765-1770.
[20]Charge P, Wang Y, Saillard J. A root MUSIC algorithm for non-circular sources[C]. In Proc. Int. Conf. Acoust., Speech, Signal Process.,2001:2985-2988.
[21]Zoubir A, Charge P, Wang Y. Non-circular sources localization with ESPRIT[C]. The 6th Eur. Conf. Wireless Technology, Munich, Germany,2003.
[22]Haardt M, Romer F. Enhancements of unitary ESPRIT for non-circular sources[C]. In Proc. Int. Conference Acoustic Speech Signal Process, Montral, QC, Canada, May 17-21,2004, vol. II:101-104.
[23]Delmas J P. Asymptotically minimum variance second-order estimation for noncircular signals with application to DOA estimation[J]. IEEE Transaction on Signal Processing, 2004,52(5):1235-1241.
[24]Abeida H and Delmas J. MUSIC-like estimation of direction of arrival for noncircular sources[J]. IEEE Transaction on Signal Processing,2006,54(7):2678-2690.
[25]Gao F, Nallanathan A, Wang Y. Improved MUSIC under the coexistence of both circular and noncircular sources[J]. IEEE Transaction on Signal Processing,2008,56(7):3033-3038.
[27]Liu J, Huang Z, Zhou Y. Extended 2q-MUSIC algorithm for noncircular signals[J]. Signal Processing,2008,88(6):1327-1339.
[28]刘剑,于红旗,黄知涛,周一宁.用于非圆信号二维测向的扩展Root-MUSIC算法[J].信号处理,2008年第5期.
[29]冯大政,郑春弟,周祎.一种利用信号特点的实值信号特点的实值MUSIC算法[J].电波科学学报,2007,22(2)：331-335.
[30]Brown W M, Crane R B. Conjugate linear filtering[J]. IEEE Transaction on Information Theory,1969,15(4):462-465.
[31]Picinbono B and Chevalier P. Widely linear estimation with complex data[J]. IEEE Transaction on Signal Processing,1995,43(8):2030-2033.
[32]Gardner W A. Cyclic wiener filtering:theory and method[J]. IEEE Transaction on Communications,1993m 41(1):151-163.
[33]Gelli G, Paura L, Ragozini A R P. Blind widely linear multiuser detection[J]. IEEE communications letters,2000,4(6):187-189.
[34]Mandic D P, Goh V S L. Complex valued nonlinear adaptive filters:noncircularity, widely linear and neural models[M]. NewYork:Wiley,2009.
[35]Xia Y, Jelfs B, VanHulle M M, Principe J C, Mandic D P. An augmented echo state network for nonlinear adaptive filtering of complex noncircular signals[J]. IEEE Transaction Neural Network,2011,22(1):74-83.
[36]Buzzi S, Lops M, Sardellitti S. Widely linear reception strategies for layered space-time wireless communications [J], IEEE Transaction on Signal Processing,2006,54(6):2252-2262.
[37]Lampe A, Breiling M. Asymptotic analysis of widely linear MMSE multiuser detection-complex vs real modulation [C]. ITW2001, Cairns, Australia,2001:55-57.
[38]Chevalier P, Pipon F. New insights into optimal widely linear array receivers for the demodulation of BPSK, MSK, and GMSK signals corrupted by noncircular interferences-application to SAIC[J]. IEEE Transaction on Signal Processing,2006,54(3):870-883.
[39]Cacciapuoti A S, Gelli G, Paura L. Widely linear versus linear blind multiuser detection with subspace-based channel estimation:finite sample-size effects[J], IEEE Transaction on Signal Processing,2009,57(4):1426-1443.
[40]Dini D H, Mandic D P. Class of widely linear complex Kalman filters[J]. IEEE Transaction on Neural Networks and Learning,2012,23(5):775-786.
[41]Dini D H, Mandic D P, Julier S J. The widely linear complex unscented Kalman filter[J]. IEEE Transaction on Signal Processing Letter,2011,18(11):623-626.
[42]Whorter T M, Schreier P. Widely linear beamfroming[C]. In Proc.37th Asilomar Conference Signals, Systems, Computing,2003, pp.753-759.
[43]Gerstacker M H, Schober R, Lampe A. Receivers with widely linear processing for frequency-selective channels[J]. IEEE Transactions on Communications,2003,51(9): 1512-1523.
[44]Trigui H and Slock D T M. Performance bounds for cochannel interference cancellation within the current GSM standard[J]. Signal Processing,2000,80(7):1335-1346.
[45]Mattera D, Paura L, Sterle F. Widely linear decision-feedback equalizer for time-dispersive linear MIMO channels[J]. IEEE Transaction on Signal Processing,2005,53(7):2525-2536.
[46]Chevalier P, Pipon F. Optimal array receiver for synchronization of a BPSK signal corrupted by noncircular interferences[C]. In Proc. Int. Conference Acoustic Speech Signal Process., Toulouse, France,2006:Ⅳ.1061-IV.1064
[47]Schober R, Gerstacker W H, Lampe L H J. Data-aided and blind stochastic gradient algorithms for widely linear MMSE MAI suppression for DS-CDMA[J]. IEEE Transaction on Signal Processing,2004,52(3):746-756.
[48]Buzzi S and Lops M. Performance analysis for the improved linear multiuser detectors in BPSK-modulated DS-CDMA systems[J]. IEEE Transaction on Communications,2003, 51(1):37-42.
[49]Buzzi S, Lops M, Tulino A M.A new family of MMSE multiuser receivers for interference suppression in DS/CDMA systems employing BPSK modulation[J]. IEEE Transaction on Communications,2001,49(1):154-167.
[50]Buzzi S, Lops M, Tulino A M. A generalized minimum-meanoutput-energy for CDMA systems with improper MAI[J]. IEEE Transaction on Information Theory,2002,48(3): 761-767.
[51]Jeon J J, Andrews J G, Sung K M. The blind widely linear minimum output energy algorithm for DS-CDMA systems order[J]. IEEE Transaction on Signal Processing,2006, 54(5):1926-1931.
[52]Kapur A, Varanasi M K. Multiuser detection for overloaded CDMA systems[J]. IEEE Transaction on Information Theory,2003,49(7):1728-1742.
[53]Mirbagheri A, Plataniotis K N, Pasupathy S. An enhanced widely linear CDMA receiver with OQPSK modulation[J]. IEEE Transaction on Communications,2006,54(2):261-272.
[54]Sfar S, Letaief K B. Improved group multiuser detection with multiple receive antennas in the presence of improper multiaccess interference[J]. IEEE Transaction on Communications, 2005,53(4):560-563.
[55]Tulino A M, Verdu S. Asymptotic analysis of improved linear receivers for BPSK-CDMA subject to fading[J]. IEEE Journal Selected Areas Communications,2001,19(8): 1544-1555.
[56]Yoon Y C, Leib H. Maximizing SNR in improper complex noise and applications to CDMA[J]. IEEE Communications Letter,1997,1(1):5-8.
[57]Mattera D, Paura L, Sterle F. Widely linear MMSE equalizer for MIMO linear time-dispersive channel[J]. Electronics Letter,2003,39(20):1481-1482.
[58]Sterle F. Widely linear MMSE transceivers for MIMO channels[J]. IEEE Transaction on Signal Processing,2007,55(8):4258-4270.
[59]Hellings C, Utschick W. Performance gains due to improper signals in MIMO broadcast channels with widely linear transceivers[C]. In Proc. Int. Conference Acoustic Speech Signal Process., Vancouver, BC,2013:4379-4383.
[60]Proakis J G. Digital Communications[M],3rd ed. New York:McGraw-Hill,1995.
[61]Chevalier P. Filtrage d'antenne optimal pour signaux non stationnaires-Concepts, performances[C]. In Proc. GRETSI, Juan-Les-Pins, France,1995:233-236.
[62]Chevalier P. Optimal array processing for nonstationary signals[C]. In Proc. Int. Conf. Acoust., Speech, Signal Process., Atlanta, GA,1996:2868-2871.
[63]Chevalier P, Blin A. Widely linear MVDR beamformers for the reception of an unknown signal corrupted by noncircular interferences[J]. IEEE Transaction on Signal Processing, 2007,55(11):5323-5336.
[64]Chevalier P, Delmas J P, Oukaci A. Optimal widely linear MVDR beamforming for noncircular signals[C]. In Proc. Int. Conf. Acoust., Speech, Signal Process., Taipei, Taiwan, 2009:3573-3576.
[65]Chevalier P, Delmas J P, Oukaci A. Performance analysis of the optimal widely linear MVDR beamformer[C]. In Proc. Europen Signal Process. Conf., Glasgow, Scotland,2009: 587-591.
[66]Vorobyov S A. Principles of minimum variance robust adaptive beamforming design[J]. Signal Processing,2013,93(12):3264-3277.
[67]刘聪锋,刘晓军,廖桂生,稳健自适应波束形成算法综述[J].中国电子科学研究院学报,2009,4(6)：560-565.
[68]Gordara L C. Error analysis of the optimal antenna-array processor[J]. IEEE Transactionson Aerospace and Electronic Systerms,1986.22(4):395-409.
[69]刘凯,梁国龙,张光普,等.初探阵列误差对矢量阵波束形成系统的影响[J].系统仿真学报.2012.4：848-853.
[70]Vural A M, Effects of perturbation on the performance of optimum/adaptive arrays[J]. IEEE Transactions on Aerospace and Electronic Systerms,1979,15 (1):76-87.
[71]Godara L C. The effect of phase-shift errors on the performance of an antenna-array beamformer[J]. IEEE J. Ocean Engineering,1985,10(7):278-284.
[72]Kim J W, Un C K. An adaptive array robust to beam pointing error[J]. IEEE Transactions on Signal Processing,1992,40(6):1582-1584.
[73]Bell K. L, Ephraim Y, Trees H L V. A Bayesian approach to robust adaptive beamforming[J]. IEEE Transactions on Signal Processing,2000,48(2):386-398.
[74]陈红利,阵列误差校正方法及其工程实现研究[D].西安,西安电子科技大学硕士论文,2010.
[75]Lee J H. Cheng K P. Adaptive array beamforming with robust capabilities under random phase perturbations[J]. IEEE Transactions on Signal Processing,2005,53(1):365-371.
[76]Jablon N K. Adaptive beamforming with the generalized sidelobe canceller in the presence of array imperfections[J]. IEEE Transactions on Antennas and Propagation,1986,34(8): 996-1012.
[77]Wen F, Wei Q W, Fan R, etc. Robust Capon beamforming exploiting the second-order noncircularity of signals[J]..Signal Processing,2014,102(9):100-111.
[78]Schreier P J. Scharf L L. Second-order analysis of improper complex random vectors and processes[J]. IEEE Transactions on Signal Processing,2003,51(3):714-725.
[79]Schreier P J, Scharf L L, Mullis C T. Detection and estimation of improper complex random signals[J]. IEEE Transactions on Information Theory,2005,51(1):306-312.
[80]Neeser F D, Massey L. Proper complex random processes with applications to information theory[J]. IEEE Transactions on Information Theory,1993,39(4):1293-1302.
[81]Li X, Adali T. Complex-Valued Linear and Widely Linear Filtering Using MSE and Gaussian Entropy[J]. IEEE Transactions on Signal Processing,2012,60(11):5672-5684.
[82]Adali T, Peter J Schreier, Scharf L L. Complex-valued signal processing:The proper way to deal with impropriety[J]. IEEE Transactions on Signal Processing,2011,59(11): 5101-5125.
[83]Chevalier P, Delmas J P, Oukaci A. Properties, performance and practical interest of the widely linear MMSE beamformer for nonrectilinear signals[J]. Signal Processing,2014, 97(4):269-281.
[84]Schmidt R, Franks R. Multiple source DF signal processing:An experimental system[J]. IEEE Transactions on Antennas and Propagation,1986,34(3):281-290,
[85]Gabriel W. Using spectral estimation techniques in adaptive processing antenna systems[J]. IEEE Transactions on Antennas and Propagation,1986,34(3):291-300.
[86]Carlson B D. Covariance matrix estimation errors and diagonal loading in adaptive arrays [J]. IEEE Transactions on Aerospace and Electronic Systems,1988,24(4):397-401.
[87]Cox H, Zeskind R M, Owen M H. Robust adaptive beamforming[J]. IEEE Transaction on Acoustic, Speech, signal Processing,1987,35(10):1365-1376.
[88]陈晓初.自适应阵对角线加载研究[J].电子学报,1998,26(4)：29-35.
[89]沈凤麟,常春起.自适应数字波束形成中波束形成的快速收敛[J].电子学报,1996,24(7)：32-37.
[90]Bull J F, Arnao M A, Burgess L R. Hypersensitivity effects in adaptive antenna arrays[C]. IEEE Antennas and Propagation Society International Symposium,1990,1:396-399.
[91]Kim J W, Un C K. A robust adaptive array based on signal subspace approach[J], IEEE Transactions on Signal Processing,1993,41 (11):3166-3171.
[92]Chang L, Yeh C C. Performance of DMI and eigenspace-based beamformers[J]. IEEE Transactions on Antennas and Propagation,1992,40(11):1336-1347.
[93]Feldman D D, Griffiths L J. A projection approach to robust adaptive beamforming[J]. IEEE Transactions on Signal Processing,1994,42(4):867-876.
[94]Thomas J K, Scharf L L, Tufts D W. The probability of a subspace swap in the SVD[.I]. IEEE Transactions on Signal Processing,1995,43(3):730-736.
[95]Vorobyov S A, Gershman A B, Luo Z Q. Robust adaptive beamforming using worst-case performance optimization via second-order cone programming[C]. In Proc. Int. Conference Acoustic Speech Signal Process., Minneapolis,2002,3:2901-2904
[96]S. A. Vorobyov, A. B. Gershman, Z. Q. Luo. Robust Adaptive Beamforming Using Worst-Case Performance Optimization:A Solution to the Signal Mismatch Problem[J]. IEEE Transactions on Signal Processing,2003,51(2):313-324.
[97]Li J, Stoica P. On robust capon beamforming and diagonal loading[J]. IEEE Transactions on Signal Processing,2003,51(7):1702-1715.
[98]Lorenz R G, Boyd S P. Robust minimum variance beamforming[J]. IEEE Transactions on Signal Processing,2005,53(5):1684-1696.
[99]Kim S J, Mangnani A, Mutapcic A, et al. Robust Beamforming via Worst-Case SINR Maximization[]J]. IEEE Transactions on Signal Processing,2008,56(4):1539-1547.
[100]Hassanien A, Vorobyov S A, Wong K M. Robust adaptive beamforming using sequential programming:An iterative solution to the mismatch problem[J]. IEEE Signal Processing Letters,2008,15:733-736.
[101]Gu Y, Leshem A. Robust adaptive beamforming based on interference covariance matrix reconstruction and steering vector estimation [J]. IEEE Transactions on Signal Processing, 2012,60(7):3881-3885.
[102]Li J, Wei G, Ding Y. Adaptive beamforming based on covariance matrix reconstruction by exploiting interferences' cyclostationarity[J]. Signal Processing,2013,93(9):2543-2547.
[103]Wang G, Lie J P, See C S. A robust approach to optimum widely linear MVDR beamformer[C]. In Proc. Int. Conf. Acoust., Speech, Signal Process.,2012, pp.2593-2596.
[104]Douglas S C. Widely linear recursive least-squares algorithm for adaptive beamforming[C]. In Proc. Int. Conf. Acoust., Speech, Signal Process.,2009:2041-2044.
[105]Song N, Steinwandt J, Wang L, ect. Non-data-aided adaptive beamforming algorithm based on the widely linear auxiliary vector filter[C]. In Proc. Int. Conf. Acoust., Speech, Signal Process.,2011:2636-2639.
[106]Tseng C Y, Griffiths L J. A unified approach to the design of linear constraints in minimum variance adaptive beamformers[J]. IEEE Transactions on Antennas Propag.,1992,40.(12); 1533-1542.
[107]Chevalier P and Maurice A. Constrained beamforming for cyclostationary signals[C]. In Proc. Int. Conf. Acoust., Speech, Signal Process., Munich,1997:3789-3792.
[108]Zeng X, Ghrayeb A. A blind carrier frequency offset estimation scheme for OFDM systems with constant modulus signaling[J]. IEEE Transactions on Communications,2008,56(7): 1032-1037.
[109]Ciblat P, Loubaton P, Serpedin E, etc. Performance analysis of blind carrier frequency offset estimators for non-circular transmissions through frequency-selective channels[J]. IEEE Transactions on Signal Processing,2002,50(1):130-140.
[110]Beek J J V D, Sandell M. ML estimation of time and frequency offset in OFDM systems[J]. IEEE Transactions on Signal Processing,45(7):1800-1805.
[111]Morelli M, Mengali U. An improved frequency offset estimator for OFDM applications[C]. Communication Theory Mini-Conference,1999:106-109.
[112]A. B. Gershman. Robust adaptive beamforming in sensor arrays[J]. AEU International Journal of Electronics and Communications,1999,53(12):305-314.
[113]张小飞,汪飞,陈伟华.阵列信号处理的理论与应用[M].北京：国防工业出版社,2013.
[114]刘剑.非圆信号波达方向估计算法研究[D].长沙,国防科学技术大学博士论文,2007.
[115]Laurent P A. Exact and approximate construetion of digital phase modulations by superposition of amplitude modulated pulse[J]. IEEE Transactions on communications, 1986,34(2):150-160.
[116]鄢社锋,马远良.传感器阵列波束优化设计及应用[M].北京：科学出版社.2009.
[117]Albera L. Ferreol A, Como P, etc. Blind identification of overcomplete MixturEs of sources (BIOME)[J]. Linear Algebra Appl. J.,2004,391(11):3-30.
[118]Ferreol A, Albera L, Chevalier P. Fourth order blind identification of underdetermined mixtures of sources (FOBIUM)[J]. IEEE Transactions on Signal Processing,2005,53(5): 1640-1653.
[119]Stoica P, Wang Z S, Li J. Robust Capon beamforming[J]. IEEE Signal Processing Letters, 2003,10(6):172-175.
[120]Bertsekas D P. Constrained optimization and Lagrange multiplier methods[J]. Computer Science and Applied Mathematics, Boston:Academic Press,1982,1.
[121]Dahlquist G, Bjorck. Numerical methods[J]. Prentice-Hall, Englewood Cliffs,1974,504: 302-303.
[122]Cao S, Ye Z, Xu D and Xu X. A hadamard product based method for DOA estimation and gain-phase error calibration[J]. IEEE Transactions on Aerospace and Electronic Systems, 2013 49(2):1224-1233.
[2]Naidu P S. Sensor array signal processing[M]. Boca Raton:CRC Press,2001.
[3]Krim H, Viberg M. Two decades of array signal processing research:the parametric approach[J]. IEEE Signal Processing Magazine,1996,13(4):67-94.
[4]Bolcskei H, Gesbert D, Papadias CB, Veen A J. Space-time wireless systems:from array processing to MIMO communications[M]. New York:Cambridge University Press,2008.
[5]Munier J, Delisle G Y. Spatial analysis in passive listening using adaptive techniques[J]. In Proceedings of IEEE.1987,75(11):1458-1471.
[6]Benesty J, Chen J, Huang Y. Microphone array signal processing[M]. Heidelberg: Springer-Verlag,2008.
[7]王永良等.空间谱估计理论与算法[M].北京：清华大学出版社,2004.
[8]Chandran S. Advances in direction-of-arrival estimation[M]. Boston:Artech House,2006.
[9]Tuncer E. Friedlander B. Classical and modern direction-of-arrival estimation[M]. Burlington: Elsevier,2009.
[10]Li J, Stoica P. Robust adaptive beamforming[M]. Hoboken:John Wiley & Sons,2006.
[11]Picinbono B. On circularity[J]. IEEE Transaction on Signal Processing,1994,42(12): 3473-3482.
[12]Veen B D V and Buckley K M. Beamforming:a versatile approach to spatial filtering[J]. IEEE Acoustic, Speech, Signal Processing Magazine,1988,5(2):4-24.
[13]Trees H L V. Detection, estimation, and modulation theory[M]. Part Ⅳ:Optimum array processing. John Wiley & Sons, New York, NY,2002.
[14]Capon J. High resolution frequency-wavenumber spectrum analysis[J]. Proceeding of IEEE, 1969,57(8):1408-1418.
[15]Schreier P J, Scharf L L. Second-order analysis of improper complex random vector and processes[J]. IEEE Transaction on Signal Processing,2003,51(3):714-725.
[16]Picinbono B. Second-order complex random vectors and normal distributions[J]. IEEE Transaction on Signal Processing,1996,44(10):2637-2640.
[17]Picinbono B. Second-order statistics of complex signals.IEE Transaction on Signal Processing[J],1997,45(2):411-420.
[18]Schreier P J, Scharf L L. Statistical Signal processing of complex-valued data:the theory of improper and noncircular signals[M]. Cambridge Univsersity Press. New York.2010.
[19]Charge P, Wang Y, Saillard J. A non-circular sources direction finding method using polynomial rooting[J]. Signal Processing,2001,81(6):1765-1770.
[20]Charge P, Wang Y, Saillard J. A root MUSIC algorithm for non-circular sources[C]. In Proc. Int. Conf. Acoust., Speech, Signal Process.,2001:2985-2988.
[21]Zoubir A, Charge P, Wang Y. Non-circular sources localization with ESPRIT[C]. The 6th Eur. Conf. Wireless Technology, Munich, Germany,2003.
[22]Haardt M, Romer F. Enhancements of unitary ESPRIT for non-circular sources[C]. In Proc. Int. Conference Acoustic Speech Signal Process, Montral, QC, Canada, May 17-21,2004, vol. II:101-104.
[23]Delmas J P. Asymptotically minimum variance second-order estimation for noncircular signals with application to DOA estimation[J]. IEEE Transaction on Signal Processing, 2004,52(5):1235-1241.
[24]Abeida H and Delmas J. MUSIC-like estimation of direction of arrival for noncircular sources[J]. IEEE Transaction on Signal Processing,2006,54(7):2678-2690.
[25]Gao F, Nallanathan A, Wang Y. Improved MUSIC under the coexistence of both circular and noncircular sources[J]. IEEE Transaction on Signal Processing,2008,56(7):3033-3038.
[27]Liu J, Huang Z, Zhou Y. Extended 2q-MUSIC algorithm for noncircular signals[J]. Signal Processing,2008,88(6):1327-1339.
[28]刘剑,于红旗,黄知涛,周一宁.用于非圆信号二维测向的扩展Root-MUSIC算法[J].信号处理,2008年第5期.
[29]冯大政,郑春弟,周祎.一种利用信号特点的实值信号特点的实值MUSIC算法[J].电波科学学报,2007,22(2)：331-335.
[30]Brown W M, Crane R B. Conjugate linear filtering[J]. IEEE Transaction on Information Theory,1969,15(4):462-465.
[31]Picinbono B and Chevalier P. Widely linear estimation with complex data[J]. IEEE Transaction on Signal Processing,1995,43(8):2030-2033.
[32]Gardner W A. Cyclic wiener filtering:theory and method[J]. IEEE Transaction on Communications,1993m 41(1):151-163.
[33]Gelli G, Paura L, Ragozini A R P. Blind widely linear multiuser detection[J]. IEEE communications letters,2000,4(6):187-189.
[34]Mandic D P, Goh V S L. Complex valued nonlinear adaptive filters:noncircularity, widely linear and neural models[M]. NewYork:Wiley,2009.
[35]Xia Y, Jelfs B, VanHulle M M, Principe J C, Mandic D P. An augmented echo state network for nonlinear adaptive filtering of complex noncircular signals[J]. IEEE Transaction Neural Network,2011,22(1):74-83.
[36]Buzzi S, Lops M, Sardellitti S. Widely linear reception strategies for layered space-time wireless communications [J], IEEE Transaction on Signal Processing,2006,54(6):2252-2262.
[37]Lampe A, Breiling M. Asymptotic analysis of widely linear MMSE multiuser detection-complex vs real modulation [C]. ITW2001, Cairns, Australia,2001:55-57.
[38]Chevalier P, Pipon F. New insights into optimal widely linear array receivers for the demodulation of BPSK, MSK, and GMSK signals corrupted by noncircular interferences-application to SAIC[J]. IEEE Transaction on Signal Processing,2006,54(3):870-883.
[39]Cacciapuoti A S, Gelli G, Paura L. Widely linear versus linear blind multiuser detection with subspace-based channel estimation:finite sample-size effects[J], IEEE Transaction on Signal Processing,2009,57(4):1426-1443.
[40]Dini D H, Mandic D P. Class of widely linear complex Kalman filters[J]. IEEE Transaction on Neural Networks and Learning,2012,23(5):775-786.
[41]Dini D H, Mandic D P, Julier S J. The widely linear complex unscented Kalman filter[J]. IEEE Transaction on Signal Processing Letter,2011,18(11):623-626.
[42]Whorter T M, Schreier P. Widely linear beamfroming[C]. In Proc.37th Asilomar Conference Signals, Systems, Computing,2003, pp.753-759.
[43]Gerstacker M H, Schober R, Lampe A. Receivers with widely linear processing for frequency-selective channels[J]. IEEE Transactions on Communications,2003,51(9): 1512-1523.
[44]Trigui H and Slock D T M. Performance bounds for cochannel interference cancellation within the current GSM standard[J]. Signal Processing,2000,80(7):1335-1346.
[45]Mattera D, Paura L, Sterle F. Widely linear decision-feedback equalizer for time-dispersive linear MIMO channels[J]. IEEE Transaction on Signal Processing,2005,53(7):2525-2536.
[46]Chevalier P, Pipon F. Optimal array receiver for synchronization of a BPSK signal corrupted by noncircular interferences[C]. In Proc. Int. Conference Acoustic Speech Signal Process., Toulouse, France,2006:Ⅳ.1061-IV.1064
[47]Schober R, Gerstacker W H, Lampe L H J. Data-aided and blind stochastic gradient algorithms for widely linear MMSE MAI suppression for DS-CDMA[J]. IEEE Transaction on Signal Processing,2004,52(3):746-756.
[48]Buzzi S and Lops M. Performance analysis for the improved linear multiuser detectors in BPSK-modulated DS-CDMA systems[J]. IEEE Transaction on Communications,2003, 51(1):37-42.
[49]Buzzi S, Lops M, Tulino A M.A new family of MMSE multiuser receivers for interference suppression in DS/CDMA systems employing BPSK modulation[J]. IEEE Transaction on Communications,2001,49(1):154-167.
[50]Buzzi S, Lops M, Tulino A M. A generalized minimum-meanoutput-energy for CDMA systems with improper MAI[J]. IEEE Transaction on Information Theory,2002,48(3): 761-767.
[51]Jeon J J, Andrews J G, Sung K M. The blind widely linear minimum output energy algorithm for DS-CDMA systems order[J]. IEEE Transaction on Signal Processing,2006, 54(5):1926-1931.
[52]Kapur A, Varanasi M K. Multiuser detection for overloaded CDMA systems[J]. IEEE Transaction on Information Theory,2003,49(7):1728-1742.
[53]Mirbagheri A, Plataniotis K N, Pasupathy S. An enhanced widely linear CDMA receiver with OQPSK modulation[J]. IEEE Transaction on Communications,2006,54(2):261-272.
[54]Sfar S, Letaief K B. Improved group multiuser detection with multiple receive antennas in the presence of improper multiaccess interference[J]. IEEE Transaction on Communications, 2005,53(4):560-563.
[55]Tulino A M, Verdu S. Asymptotic analysis of improved linear receivers for BPSK-CDMA subject to fading[J]. IEEE Journal Selected Areas Communications,2001,19(8): 1544-1555.
[56]Yoon Y C, Leib H. Maximizing SNR in improper complex noise and applications to CDMA[J]. IEEE Communications Letter,1997,1(1):5-8.
[57]Mattera D, Paura L, Sterle F. Widely linear MMSE equalizer for MIMO linear time-dispersive channel[J]. Electronics Letter,2003,39(20):1481-1482.
[58]Sterle F. Widely linear MMSE transceivers for MIMO channels[J]. IEEE Transaction on Signal Processing,2007,55(8):4258-4270.
[59]Hellings C, Utschick W. Performance gains due to improper signals in MIMO broadcast channels with widely linear transceivers[C]. In Proc. Int. Conference Acoustic Speech Signal Process., Vancouver, BC,2013:4379-4383.
[60]Proakis J G. Digital Communications[M],3rd ed. New York:McGraw-Hill,1995.
[61]Chevalier P. Filtrage d'antenne optimal pour signaux non stationnaires-Concepts, performances[C]. In Proc. GRETSI, Juan-Les-Pins, France,1995:233-236.
[62]Chevalier P. Optimal array processing for nonstationary signals[C]. In Proc. Int. Conf. Acoust., Speech, Signal Process., Atlanta, GA,1996:2868-2871.
[63]Chevalier P, Blin A. Widely linear MVDR beamformers for the reception of an unknown signal corrupted by noncircular interferences[J]. IEEE Transaction on Signal Processing, 2007,55(11):5323-5336.
[64]Chevalier P, Delmas J P, Oukaci A. Optimal widely linear MVDR beamforming for noncircular signals[C]. In Proc. Int. Conf. Acoust., Speech, Signal Process., Taipei, Taiwan, 2009:3573-3576.
[65]Chevalier P, Delmas J P, Oukaci A. Performance analysis of the optimal widely linear MVDR beamformer[C]. In Proc. Europen Signal Process. Conf., Glasgow, Scotland,2009: 587-591.
[66]Vorobyov S A. Principles of minimum variance robust adaptive beamforming design[J]. Signal Processing,2013,93(12):3264-3277.
[67]刘聪锋,刘晓军,廖桂生,稳健自适应波束形成算法综述[J].中国电子科学研究院学报,2009,4(6)：560-565.
[68]Gordara L C. Error analysis of the optimal antenna-array processor[J]. IEEE Transactionson Aerospace and Electronic Systerms,1986.22(4):395-409.
[69]刘凯,梁国龙,张光普,等.初探阵列误差对矢量阵波束形成系统的影响[J].系统仿真学报.2012.4：848-853.
[70]Vural A M, Effects of perturbation on the performance of optimum/adaptive arrays[J]. IEEE Transactions on Aerospace and Electronic Systerms,1979,15 (1):76-87.
[71]Godara L C. The effect of phase-shift errors on the performance of an antenna-array beamformer[J]. IEEE J. Ocean Engineering,1985,10(7):278-284.
[72]Kim J W, Un C K. An adaptive array robust to beam pointing error[J]. IEEE Transactions on Signal Processing,1992,40(6):1582-1584.
[73]Bell K. L, Ephraim Y, Trees H L V. A Bayesian approach to robust adaptive beamforming[J]. IEEE Transactions on Signal Processing,2000,48(2):386-398.
[74]陈红利,阵列误差校正方法及其工程实现研究[D].西安,西安电子科技大学硕士论文,2010.
[75]Lee J H. Cheng K P. Adaptive array beamforming with robust capabilities under random phase perturbations[J]. IEEE Transactions on Signal Processing,2005,53(1):365-371.
[76]Jablon N K. Adaptive beamforming with the generalized sidelobe canceller in the presence of array imperfections[J]. IEEE Transactions on Antennas and Propagation,1986,34(8): 996-1012.
[77]Wen F, Wei Q W, Fan R, etc. Robust Capon beamforming exploiting the second-order noncircularity of signals[J]..Signal Processing,2014,102(9):100-111.
[78]Schreier P J. Scharf L L. Second-order analysis of improper complex random vectors and processes[J]. IEEE Transactions on Signal Processing,2003,51(3):714-725.
[79]Schreier P J, Scharf L L, Mullis C T. Detection and estimation of improper complex random signals[J]. IEEE Transactions on Information Theory,2005,51(1):306-312.
[80]Neeser F D, Massey L. Proper complex random processes with applications to information theory[J]. IEEE Transactions on Information Theory,1993,39(4):1293-1302.
[81]Li X, Adali T. Complex-Valued Linear and Widely Linear Filtering Using MSE and Gaussian Entropy[J]. IEEE Transactions on Signal Processing,2012,60(11):5672-5684.
[82]Adali T, Peter J Schreier, Scharf L L. Complex-valued signal processing:The proper way to deal with impropriety[J]. IEEE Transactions on Signal Processing,2011,59(11): 5101-5125.
[83]Chevalier P, Delmas J P, Oukaci A. Properties, performance and practical interest of the widely linear MMSE beamformer for nonrectilinear signals[J]. Signal Processing,2014, 97(4):269-281.
[84]Schmidt R, Franks R. Multiple source DF signal processing:An experimental system[J]. IEEE Transactions on Antennas and Propagation,1986,34(3):281-290,
[85]Gabriel W. Using spectral estimation techniques in adaptive processing antenna systems[J]. IEEE Transactions on Antennas and Propagation,1986,34(3):291-300.
[86]Carlson B D. Covariance matrix estimation errors and diagonal loading in adaptive arrays [J]. IEEE Transactions on Aerospace and Electronic Systems,1988,24(4):397-401.
[87]Cox H, Zeskind R M, Owen M H. Robust adaptive beamforming[J]. IEEE Transaction on Acoustic, Speech, signal Processing,1987,35(10):1365-1376.
[88]陈晓初.自适应阵对角线加载研究[J].电子学报,1998,26(4)：29-35.
[89]沈凤麟,常春起.自适应数字波束形成中波束形成的快速收敛[J].电子学报,1996,24(7)：32-37.
[90]Bull J F, Arnao M A, Burgess L R. Hypersensitivity effects in adaptive antenna arrays[C]. IEEE Antennas and Propagation Society International Symposium,1990,1:396-399.
[91]Kim J W, Un C K. A robust adaptive array based on signal subspace approach[J], IEEE Transactions on Signal Processing,1993,41 (11):3166-3171.
[92]Chang L, Yeh C C. Performance of DMI and eigenspace-based beamformers[J]. IEEE Transactions on Antennas and Propagation,1992,40(11):1336-1347.
[93]Feldman D D, Griffiths L J. A projection approach to robust adaptive beamforming[J]. IEEE Transactions on Signal Processing,1994,42(4):867-876.
[94]Thomas J K, Scharf L L, Tufts D W. The probability of a subspace swap in the SVD[.I]. IEEE Transactions on Signal Processing,1995,43(3):730-736.
[95]Vorobyov S A, Gershman A B, Luo Z Q. Robust adaptive beamforming using worst-case performance optimization via second-order cone programming[C]. In Proc. Int. Conference Acoustic Speech Signal Process., Minneapolis,2002,3:2901-2904
[96]S. A. Vorobyov, A. B. Gershman, Z. Q. Luo. Robust Adaptive Beamforming Using Worst-Case Performance Optimization:A Solution to the Signal Mismatch Problem[J]. IEEE Transactions on Signal Processing,2003,51(2):313-324.
[97]Li J, Stoica P. On robust capon beamforming and diagonal loading[J]. IEEE Transactions on Signal Processing,2003,51(7):1702-1715.
[98]Lorenz R G, Boyd S P. Robust minimum variance beamforming[J]. IEEE Transactions on Signal Processing,2005,53(5):1684-1696.
[99]Kim S J, Mangnani A, Mutapcic A, et al. Robust Beamforming via Worst-Case SINR Maximization[]J]. IEEE Transactions on Signal Processing,2008,56(4):1539-1547.
[100]Hassanien A, Vorobyov S A, Wong K M. Robust adaptive beamforming using sequential programming:An iterative solution to the mismatch problem[J]. IEEE Signal Processing Letters,2008,15:733-736.
[101]Gu Y, Leshem A. Robust adaptive beamforming based on interference covariance matrix reconstruction and steering vector estimation [J]. IEEE Transactions on Signal Processing, 2012,60(7):3881-3885.
[102]Li J, Wei G, Ding Y. Adaptive beamforming based on covariance matrix reconstruction by exploiting interferences' cyclostationarity[J]. Signal Processing,2013,93(9):2543-2547.
[103]Wang G, Lie J P, See C S. A robust approach to optimum widely linear MVDR beamformer[C]. In Proc. Int. Conf. Acoust., Speech, Signal Process.,2012, pp.2593-2596.
[104]Douglas S C. Widely linear recursive least-squares algorithm for adaptive beamforming[C]. In Proc. Int. Conf. Acoust., Speech, Signal Process.,2009:2041-2044.
[105]Song N, Steinwandt J, Wang L, ect. Non-data-aided adaptive beamforming algorithm based on the widely linear auxiliary vector filter[C]. In Proc. Int. Conf. Acoust., Speech, Signal Process.,2011:2636-2639.
[106]Tseng C Y, Griffiths L J. A unified approach to the design of linear constraints in minimum variance adaptive beamformers[J]. IEEE Transactions on Antennas Propag.,1992,40.(12); 1533-1542.
[107]Chevalier P and Maurice A. Constrained beamforming for cyclostationary signals[C]. In Proc. Int. Conf. Acoust., Speech, Signal Process., Munich,1997:3789-3792.
[108]Zeng X, Ghrayeb A. A blind carrier frequency offset estimation scheme for OFDM systems with constant modulus signaling[J]. IEEE Transactions on Communications,2008,56(7): 1032-1037.
[109]Ciblat P, Loubaton P, Serpedin E, etc. Performance analysis of blind carrier frequency offset estimators for non-circular transmissions through frequency-selective channels[J]. IEEE Transactions on Signal Processing,2002,50(1):130-140.
[110]Beek J J V D, Sandell M. ML estimation of time and frequency offset in OFDM systems[J]. IEEE Transactions on Signal Processing,45(7):1800-1805.
[111]Morelli M, Mengali U. An improved frequency offset estimator for OFDM applications[C]. Communication Theory Mini-Conference,1999:106-109.
[112]A. B. Gershman. Robust adaptive beamforming in sensor arrays[J]. AEU International Journal of Electronics and Communications,1999,53(12):305-314.
[113]张小飞,汪飞,陈伟华.阵列信号处理的理论与应用[M].北京：国防工业出版社,2013.
[114]刘剑.非圆信号波达方向估计算法研究[D].长沙,国防科学技术大学博士论文,2007.
[115]Laurent P A. Exact and approximate construetion of digital phase modulations by superposition of amplitude modulated pulse[J]. IEEE Transactions on communications, 1986,34(2):150-160.
[116]鄢社锋,马远良.传感器阵列波束优化设计及应用[M].北京：科学出版社.2009.
[117]Albera L. Ferreol A, Como P, etc. Blind identification of overcomplete MixturEs of sources (BIOME)[J]. Linear Algebra Appl. J.,2004,391(11):3-30.
[118]Ferreol A, Albera L, Chevalier P. Fourth order blind identification of underdetermined mixtures of sources (FOBIUM)[J]. IEEE Transactions on Signal Processing,2005,53(5): 1640-1653.
[119]Stoica P, Wang Z S, Li J. Robust Capon beamforming[J]. IEEE Signal Processing Letters, 2003,10(6):172-175.
[120]Bertsekas D P. Constrained optimization and Lagrange multiplier methods[J]. Computer Science and Applied Mathematics, Boston:Academic Press,1982,1.
[121]Dahlquist G, Bjorck. Numerical methods[J]. Prentice-Hall, Englewood Cliffs,1974,504: 302-303.
[122]Cao S, Ye Z, Xu D and Xu X. A hadamard product based method for DOA estimation and gain-phase error calibration[J]. IEEE Transactions on Aerospace and Electronic Systems, 2013 49(2):1224-1233.