空间谱估计算法研究
|
摘要
在无源定位系统中,对信号源的DOA估计是一个关键问题,它将直接关系到系统定位的准确性。DOA估计作为阵列信号处理中的一个重要研究方向,以其优异的参数估计性能、广阔的应用前景引起了人们极大的兴趣。
本文针对如何减少特征子空间类估计算法的运算量和相干信号源DOA估计这两方面的问题做了一些研究工作,主要包括以下内容:
关于特征子空间类估计算法的实时性研究:介绍了经典的MUSIC算法,详细地分析MUSIC算法的特点,在此基础上研究了波束空间的MUSIC算法和多项式求根MUSIC算法;介绍了针对阵列导向矩阵和数据协方差矩阵的实值化处理技术和将DFT变换应用于DOA估计的DFT算法;提出一种快速子空间DOA估计方法。用Matlab进行仿真实验表明:波束空间方法、求根处理技术、实值化处理和快速子空间DOA估计方法能有效降低特征子空间类算法的计算量,但实值化处理的分辨性能不高;DFT算法利用一次快拍数据估计信源DOA,因而计算量非常小,但分辨率较差。
关于相干信号源的DOA估计算法研究:介绍了空间平滑类(SS)算法的特点和解相干性能;介绍了盲估计算法,该方法对相干信号源的生成导向矢量进行频率谱估计,间接得到信号的DOA;提出一种DOA估计新方法——DSM算法,本质上,该方法仅利用了天线阵的结构特征实现DOA的估计,而不依赖采样数据空间的特征结构,算法实现限制条件少,可广泛应用于众多不同的实际环境。用Matlab进行仿真实验表明:空间平滑算法、盲估计算法和DSM算法均能有效估计相干信号源DOA;空间平滑算法和盲估计算法性能相当,且在小信噪比条件下,两者性能略优于DSM算法;DSM算法对相干和非相干信号情况进行统一处理,在独立信号源情况下,DSM算法性能略差于MUSIC方法。
本课题受到2006年航天支撑技术基金(编号20060307)和2006武汉光电国家实验室基金支持。
In passive location system, the DOA estimation is an important issue which will decide whether the object could be located or not. As an important area in array signal processing, DOA estimation draws more and more attention because of its high performance and wide usage.
This paper focuses on the two problems of how to reduce the computation load of eigen-subspace algorithm and the DOA algorithm for coherent signals. The main contents of this research are as follows.
1) The research on the real-time realization of eigen-subspace algorithm.
Based on the basic MUSIC method and its characteristic, we investigate Beam-space MUSIC method and Root-MUSIC method; give the introduction of the Unitary operation on array steer vector and data covariance matrix and the introduction of DFT method which introduces the DFT transform into the DOA estimation; propose a new fast subspace algorithm for DOA estimation. The Matlab simulation results show that the Beam-space method, the Root method, Unitary operation and fast subspace DOA method all can reduce the computation load of eigen-subspace algorithm effectively. But the performance of Unitary operation is not very good. The DFT method has a low computation complexity and a little operation time because there is only a snapshot data involved. And the performance is not very good due to the limited data.
2) The research on the DOA algorithm for coherent signals.
We introduce the characteristic and decorrelation performance of spatial smoothing(SS) method; introduce the Blind estimation method which makes a frequency spectrum estimation on the generalized steer vector of coherent signals and gets the DOA information indirectly; propose a new method—DSM method for DOA estimation. Essentially, DSM method only bases on the characteristic of sensor array and doesn’t rely on the eigen-structure of sample data subspace. And it can be widely used in different areas because of its few restriction. The Matlab simulation results show that the SS method, the Blind method and DSM method all can estimate the DOA of coherent signals effectively. The performance of SS method is the same as that of Blind method, and both slightly better than that of DSM method on the condition of low SNR. DSM method which offers the similar procedure for coherent and incoherent signals, has a slightly worse performance than MUSIC method under the situation of independent signals.
This paper is supported by national space foundation (No.20060307) and 2006 foundation of Wuhan national research lab of photoelectricity.
本文针对如何减少特征子空间类估计算法的运算量和相干信号源DOA估计这两方面的问题做了一些研究工作,主要包括以下内容:
关于特征子空间类估计算法的实时性研究:介绍了经典的MUSIC算法,详细地分析MUSIC算法的特点,在此基础上研究了波束空间的MUSIC算法和多项式求根MUSIC算法;介绍了针对阵列导向矩阵和数据协方差矩阵的实值化处理技术和将DFT变换应用于DOA估计的DFT算法;提出一种快速子空间DOA估计方法。用Matlab进行仿真实验表明:波束空间方法、求根处理技术、实值化处理和快速子空间DOA估计方法能有效降低特征子空间类算法的计算量,但实值化处理的分辨性能不高;DFT算法利用一次快拍数据估计信源DOA,因而计算量非常小,但分辨率较差。
关于相干信号源的DOA估计算法研究:介绍了空间平滑类(SS)算法的特点和解相干性能;介绍了盲估计算法,该方法对相干信号源的生成导向矢量进行频率谱估计,间接得到信号的DOA;提出一种DOA估计新方法——DSM算法,本质上,该方法仅利用了天线阵的结构特征实现DOA的估计,而不依赖采样数据空间的特征结构,算法实现限制条件少,可广泛应用于众多不同的实际环境。用Matlab进行仿真实验表明:空间平滑算法、盲估计算法和DSM算法均能有效估计相干信号源DOA;空间平滑算法和盲估计算法性能相当,且在小信噪比条件下,两者性能略优于DSM算法;DSM算法对相干和非相干信号情况进行统一处理,在独立信号源情况下,DSM算法性能略差于MUSIC方法。
本课题受到2006年航天支撑技术基金(编号20060307)和2006武汉光电国家实验室基金支持。
In passive location system, the DOA estimation is an important issue which will decide whether the object could be located or not. As an important area in array signal processing, DOA estimation draws more and more attention because of its high performance and wide usage.
This paper focuses on the two problems of how to reduce the computation load of eigen-subspace algorithm and the DOA algorithm for coherent signals. The main contents of this research are as follows.
1) The research on the real-time realization of eigen-subspace algorithm.
Based on the basic MUSIC method and its characteristic, we investigate Beam-space MUSIC method and Root-MUSIC method; give the introduction of the Unitary operation on array steer vector and data covariance matrix and the introduction of DFT method which introduces the DFT transform into the DOA estimation; propose a new fast subspace algorithm for DOA estimation. The Matlab simulation results show that the Beam-space method, the Root method, Unitary operation and fast subspace DOA method all can reduce the computation load of eigen-subspace algorithm effectively. But the performance of Unitary operation is not very good. The DFT method has a low computation complexity and a little operation time because there is only a snapshot data involved. And the performance is not very good due to the limited data.
2) The research on the DOA algorithm for coherent signals.
We introduce the characteristic and decorrelation performance of spatial smoothing(SS) method; introduce the Blind estimation method which makes a frequency spectrum estimation on the generalized steer vector of coherent signals and gets the DOA information indirectly; propose a new method—DSM method for DOA estimation. Essentially, DSM method only bases on the characteristic of sensor array and doesn’t rely on the eigen-structure of sample data subspace. And it can be widely used in different areas because of its few restriction. The Matlab simulation results show that the SS method, the Blind method and DSM method all can estimate the DOA of coherent signals effectively. The performance of SS method is the same as that of Blind method, and both slightly better than that of DSM method on the condition of low SNR. DSM method which offers the similar procedure for coherent and incoherent signals, has a slightly worse performance than MUSIC method under the situation of independent signals.
This paper is supported by national space foundation (No.20060307) and 2006 foundation of Wuhan national research lab of photoelectricity.
引文
[1] Krim H, Viberg M. Two decades of array signal processing research. IEEE Signal Processing Magzine, 1996, 13(4): 67~94.
[2] Kay S M, Marple S L. Spectrum analysis—a modern perspective. Proc. of the IEEE, 1981,11.
[3] Burg J P. Maximum entropy spectrum analysis. Pro. of the 37th meeting of the Annual Int. SEG Meeting, Oklahoma City, OK, 1967.
[4] Capon J. High-resolution frequency-wavenumber spectrum analysis. Proc. of the IEEE, 1969, 57(8): 1408~1418.
[5] Schmidt R O. Multiple emitter location and signal parameter estimation. IEEE Trans., 1986, AP-34(3): 276~280.
[6] Cadzow J A, Kim Y S, Shiue D C. General direction-of-arrival estimation: a signal subspace approach. IEEE Trans. On AES, 1989, 25(1): 31~46.
[7] Rao B D, Hari K V S. Performance analysis of Root-MUSIC. IEEE Trans. On ASSP, 1989, 37(12): 1939~1949.
[8] Kumaresan R, Tufts D W. Estimating the angles of arrival of multiple plane waves. IEEE Trans. On AES, 1983, 19(1): 134~139.
[9] Kung S Y, Arum K S, Rao D V B. State space and SVD based approximation methods for the harmonic retrieval problem. J. Opt. Soc. Amer., 1983, 73(12): 1799~1811.
[10] Roy R, Kailath T. ESPRIT—a subspace rotation approach to estimation of parameters of cissoids in noise. IEEE Trans. On ASSP, 1986, 34(10): 1340~1342.
[11] Roy R, Kailath T. ESPRIT—estimation of signal parameters via rotational invariance teachniques, IEEE Trans. on ASSP, 1989, 37(7): 984~995.
[12] Stoica P, Nehorai A. MUSIC, Maximum likehood, and Cramer-Rao bound. In Proc. ICASSP, 1988, 2296~2299.
[13] Ottersten B, Viberg M, Stoica P, Nehorai A. Exact and large sample ML techniques for parameter estimation and detection in array processing. IN Haykin, Litva, and shepherd, editors, Radar array processing, Spring-Verlag, Berlin, 1993,99~151.
[14] Cadzow J A. A high resolution direction-of-arrival algorithm for narrow-band coherent and incoherent sources. IEEE Trans. On ASSP, 1988, 36(7):965~979.
[15] Clergeot H, Tressens S, Ouamri A. Performance of high resolution frequencies estimation methods compared to the Cramer-Rao bounds. IEEE Trans. On ASSP, 1989, 37(11): 1703~1720.
[16] Akaike H. A new look at the statistical model identification. IEEE Trans. On AC, 1974, 19(6): 716~723.
[17] Schwartz G. Estimation the dimension of a model. Ann. Stat, 1987, 6: 461~464.
[18] Bishop W B, Djuric P M. Model order selection of damped sinusoids in noise by predictive densities. IEEE Trans., 1996, SP-44(3): 611~619.
[19] Shan T J, Paulray A, Kailath T. On smoothed rand profile Tests in Eigen structure Methods for Direction-of-Arrival Estimation. IEEE Trans. On ASSP, 1987, 35(10):1377~1385.
[20] Di A. Multiple sources location—a matrix decomposition approach. IEEE Trans. On ASSP, 1985, 33(4):1086~1091.
[21] Wu H T, Yang J, Chen F K. Source number estimation using transformed gerschgorin radii. IEEE Trans. On SP, 1995, 43(6):1325~1333.
[22] Chen W, Reilly J P. Detection of the Number of signals in noise with banded Covariance Matrices. IEEE Trans. On SP, 1992, 42(5): 377~380.
[23] Rao B D, Hari K V S. On spatial smoothing and weighted subspace methods. In Proc. 24th Asilomar Conf. Signals, Syst., Comput., 1990, 936~940.
[24] Rao B D, Hari K V S. Weighted state space methods/ESPRIT and spatial smoothing. ICASSP, 1991, 3317~3320.
[25]王布宏,王永良,陈辉.相干信源波达方向估计的加权空间平滑算法.通信学报, 2003, 4(24): 31~40.
[26]高世伟,保铮.利用数据矩阵分解实现对空间相干源的超分辨处理.通信学报, 1988, 9(1): 4~13.
[27] Chen Y M, Lee J H, Yeh C C. Bearing estimation without calibration for randomly perturbed arrays. IEEE Trans. On SP, 39(1): 194~197.
[28] Zoltowski M D, Mathews C P. Direction finding with circular arrays via phase mode excitation and beamspace Root-MUSIC. 1992 Antennas and Propagation Society International Symposium. 1992 AP-S, Chicago IL, USA. 1992, 7: 245~248.
[29] Moffet A T. Minimum-redundancy linear arrays. IEEE Trans. On Antennas Propagat, 1968, 16: 172~175.
[30] Chambers C, Tozer T C, Sharman K C, Durrani T S. Temporal and spatial samplinginfluence on the estimates of superimposed narrowband signals: when less can mean more. IEEE Trans. On Signal Processing, 1996, 44(12): 3085~3098.
[31] Mathews C P, Zoltowski M D. Eigenstructure techniques for 2-D angle estimation with uniform circular arrays. IEEE Trans. On Signal Processing, 1994, 42(9): 2395~2407.
[32] Friedlander B, Weiss A J. Direction finding using spatial smoothing with interpolated arrays. IEEE Transaction on Aerospace and Electronic Systems, 1992, 4(28): 574~587.
[33] Ho J C, Yang J F, Kaveh M. Parallel adaptive rooting algorithms for general frequency estimation and direction finding. IEEE Proc.–F, 1992, 139(1): 43~48.
[34] Mccormick W S, Tsui J B Y. Suboptimal real-time frequency/incident angle estimation for multiple radar pulsed. IEEE Proc.–F, 1991, 138(1): 247~254.
[35]贾永康,保铮.时-空二维信号模型下的波达方向估计方法及其性能分析.电子学报, 25(9): 69~73.
[36]孙超,杨益新.高分辨目标方位估计算法——递增阶数多参数估计的理论与实验研究.声学学报, 1999, 24(2): 210~218.
[37] Clarke I J, High discrimination target detection algorithm and estimation of parameters. In Y. T. Chan ed. Underwater Acoustic: data processing, 1989, 273~277.
[38] Shan T J, Kailath T. Adaptive beamforming for coherent signals and interference . IEEE Trans. On ASSP, 1985,33(3): 527~536.
[39] Shan T J, Wax M, Kailath T. On spatial smoothing for estimation of coherent signals. IEEE Trans. On ASSP, 1985, 33(4): 806~811.
[40] Pillai S U, Kwon B H. Performance analysis of MUSIC—type high resolution estimators for direction finding in correlated and coherent scenes. IEEE Trans. On ASSP, 1989, 37(8): 1176~1189.
[41] Viberg M, Ottersten B. Sensor array processing based on subspace fitting. IEEE Trans. On SP, 1991, 39(5): 1110~1121.
[42] Stoica P, Soderstrom T. Statistical analysis of MUSIC and subspace rotation estimates of sinusoidal frequencies. IEEE Trans. On SP, 1991, 39(8): 1836~1847.
[43]王永良,陈辉,彭应宁,万群.空间谱估计理论与算法.北京:清华大学出版社, 2004.
[44] Huarng K C, Yeh C C. A unitary transformation method for angle of arrival estimation. IEEE Trans. Acoust., Speech, Signal Processing, vol. 39, pp. 975~977,Apr. 1991.
[45] Zoltowski M D, Kautz G M, Silverstein S D. Beamspace root-MUSIC. IEEE Trans. Signal Processing, vol. 41, pp. 344~364, Jan. 1993.
[46] Marius P, Alex B. Unitary Root-MUSIC with a real-valued eigendecomposition: a theoretical and experimental performance study. IEEE Trans. On Signal Processing, vol. 48, no. 5, pp. 1306~1314, May 2000.
[47] Liang T, Kwan H K. A novel approach to fast DOA estimation of multiple spatial narrowband signals. The 2002 45th Midwest Symposium on Circuits and Systems, Tulsa, Oklahoma, 2002.
[48]陈四根,杨莘元.等距线阵信号傅里叶变换研究.哈尔滨工程大学学报, 2003, 24(4): 440~443.
[49]袁国靖,丁君,郭陈江.一种快速高分辨率的DOA估计方法——FFT-BMUSIC法.微波学报, 2005, 21(1): 10~13.
[50]曾浩,谭晓衡,杨士中,刘玲. DFT在二维DOA估计的MUSIC法中的应用.重庆大学学报, 2005, 28(8):68~71.
[51]余继周,陈定昌.一种DOA估计的快速子空间算法.现代电子技术, 2005, (28): 90~92.
[52] Yih-Min Chen. On spatial smoothing for two-dimensional direction-of-arrival estimation of coherent signals. IEEE Trans. On SP, 1997, 45(7): 1689~1696.
[53]王布宏,王永良,陈辉.一种新的相干信源DOA估计算法:加权空间平滑协方差矩阵的Toeplitz矩阵拟合.电子学报, 2003,31(9): 1394~1397.
[54] Wang Buhong, Wang Yongliang, Chen Hui. Weighted spatial smoothing for direction-of-arrival estimation of coherent signals. 2002 IEEE International Symposium on Antennas and Propagation and USNC/URSI National Radio Science Meeting, San Antonio, Texas, USA, 2002, 6: 668~671.
[55] Alireza M, Tzu C C. Signal enhancement of the spatial smoothing algorithm. IEEE Trans on SP, 1991, 39(8): 1907~1911.
[56] Jian L. Improved angular resolution for spatial smoothing techniques. IEEE Trans on Signal processing. 1992, 40(12): 3078~3081.
[57] C C Yeh, J H Lee, Y M Chen. Estimating two-dimensional angles of arrival in coherent source environment. IEEE Trans Acoust., Speech, Signal Processing, vol. 37, pp. 153~155, Jan. 1989.
[58] Yixin Yang, Chunru Wan, Chao Sun, Qing Wang. DOA estimation for coherentsources in beamspace using spatial smoothing. ICICS-PCM 2003, Singapore, 1028~1032.
[59] Ma Changlin, Peng Yingning, Tian Lisheng. Blind estimation of DOA’s of coherent sources. Proceedings of ICSP’98, pp. 327~330.
[60] Donald W, Tufts, Ramdas K. Estimation of frequencies of multiple sinusoids: Making linear prediction performance like maximum likelihood. Proceedings of IEEE, Vol. 70, No. 9, pp. 975~989, 1982.
[61]周亚强,黄甫堪.噪扰条件下数字式多基线相位干涉仪解模糊问题.通信学报, 2005, (26): 16~21.
[2] Kay S M, Marple S L. Spectrum analysis—a modern perspective. Proc. of the IEEE, 1981,11.
[3] Burg J P. Maximum entropy spectrum analysis. Pro. of the 37th meeting of the Annual Int. SEG Meeting, Oklahoma City, OK, 1967.
[4] Capon J. High-resolution frequency-wavenumber spectrum analysis. Proc. of the IEEE, 1969, 57(8): 1408~1418.
[5] Schmidt R O. Multiple emitter location and signal parameter estimation. IEEE Trans., 1986, AP-34(3): 276~280.
[6] Cadzow J A, Kim Y S, Shiue D C. General direction-of-arrival estimation: a signal subspace approach. IEEE Trans. On AES, 1989, 25(1): 31~46.
[7] Rao B D, Hari K V S. Performance analysis of Root-MUSIC. IEEE Trans. On ASSP, 1989, 37(12): 1939~1949.
[8] Kumaresan R, Tufts D W. Estimating the angles of arrival of multiple plane waves. IEEE Trans. On AES, 1983, 19(1): 134~139.
[9] Kung S Y, Arum K S, Rao D V B. State space and SVD based approximation methods for the harmonic retrieval problem. J. Opt. Soc. Amer., 1983, 73(12): 1799~1811.
[10] Roy R, Kailath T. ESPRIT—a subspace rotation approach to estimation of parameters of cissoids in noise. IEEE Trans. On ASSP, 1986, 34(10): 1340~1342.
[11] Roy R, Kailath T. ESPRIT—estimation of signal parameters via rotational invariance teachniques, IEEE Trans. on ASSP, 1989, 37(7): 984~995.
[12] Stoica P, Nehorai A. MUSIC, Maximum likehood, and Cramer-Rao bound. In Proc. ICASSP, 1988, 2296~2299.
[13] Ottersten B, Viberg M, Stoica P, Nehorai A. Exact and large sample ML techniques for parameter estimation and detection in array processing. IN Haykin, Litva, and shepherd, editors, Radar array processing, Spring-Verlag, Berlin, 1993,99~151.
[14] Cadzow J A. A high resolution direction-of-arrival algorithm for narrow-band coherent and incoherent sources. IEEE Trans. On ASSP, 1988, 36(7):965~979.
[15] Clergeot H, Tressens S, Ouamri A. Performance of high resolution frequencies estimation methods compared to the Cramer-Rao bounds. IEEE Trans. On ASSP, 1989, 37(11): 1703~1720.
[16] Akaike H. A new look at the statistical model identification. IEEE Trans. On AC, 1974, 19(6): 716~723.
[17] Schwartz G. Estimation the dimension of a model. Ann. Stat, 1987, 6: 461~464.
[18] Bishop W B, Djuric P M. Model order selection of damped sinusoids in noise by predictive densities. IEEE Trans., 1996, SP-44(3): 611~619.
[19] Shan T J, Paulray A, Kailath T. On smoothed rand profile Tests in Eigen structure Methods for Direction-of-Arrival Estimation. IEEE Trans. On ASSP, 1987, 35(10):1377~1385.
[20] Di A. Multiple sources location—a matrix decomposition approach. IEEE Trans. On ASSP, 1985, 33(4):1086~1091.
[21] Wu H T, Yang J, Chen F K. Source number estimation using transformed gerschgorin radii. IEEE Trans. On SP, 1995, 43(6):1325~1333.
[22] Chen W, Reilly J P. Detection of the Number of signals in noise with banded Covariance Matrices. IEEE Trans. On SP, 1992, 42(5): 377~380.
[23] Rao B D, Hari K V S. On spatial smoothing and weighted subspace methods. In Proc. 24th Asilomar Conf. Signals, Syst., Comput., 1990, 936~940.
[24] Rao B D, Hari K V S. Weighted state space methods/ESPRIT and spatial smoothing. ICASSP, 1991, 3317~3320.
[25]王布宏,王永良,陈辉.相干信源波达方向估计的加权空间平滑算法.通信学报, 2003, 4(24): 31~40.
[26]高世伟,保铮.利用数据矩阵分解实现对空间相干源的超分辨处理.通信学报, 1988, 9(1): 4~13.
[27] Chen Y M, Lee J H, Yeh C C. Bearing estimation without calibration for randomly perturbed arrays. IEEE Trans. On SP, 39(1): 194~197.
[28] Zoltowski M D, Mathews C P. Direction finding with circular arrays via phase mode excitation and beamspace Root-MUSIC. 1992 Antennas and Propagation Society International Symposium. 1992 AP-S, Chicago IL, USA. 1992, 7: 245~248.
[29] Moffet A T. Minimum-redundancy linear arrays. IEEE Trans. On Antennas Propagat, 1968, 16: 172~175.
[30] Chambers C, Tozer T C, Sharman K C, Durrani T S. Temporal and spatial samplinginfluence on the estimates of superimposed narrowband signals: when less can mean more. IEEE Trans. On Signal Processing, 1996, 44(12): 3085~3098.
[31] Mathews C P, Zoltowski M D. Eigenstructure techniques for 2-D angle estimation with uniform circular arrays. IEEE Trans. On Signal Processing, 1994, 42(9): 2395~2407.
[32] Friedlander B, Weiss A J. Direction finding using spatial smoothing with interpolated arrays. IEEE Transaction on Aerospace and Electronic Systems, 1992, 4(28): 574~587.
[33] Ho J C, Yang J F, Kaveh M. Parallel adaptive rooting algorithms for general frequency estimation and direction finding. IEEE Proc.–F, 1992, 139(1): 43~48.
[34] Mccormick W S, Tsui J B Y. Suboptimal real-time frequency/incident angle estimation for multiple radar pulsed. IEEE Proc.–F, 1991, 138(1): 247~254.
[35]贾永康,保铮.时-空二维信号模型下的波达方向估计方法及其性能分析.电子学报, 25(9): 69~73.
[36]孙超,杨益新.高分辨目标方位估计算法——递增阶数多参数估计的理论与实验研究.声学学报, 1999, 24(2): 210~218.
[37] Clarke I J, High discrimination target detection algorithm and estimation of parameters. In Y. T. Chan ed. Underwater Acoustic: data processing, 1989, 273~277.
[38] Shan T J, Kailath T. Adaptive beamforming for coherent signals and interference . IEEE Trans. On ASSP, 1985,33(3): 527~536.
[39] Shan T J, Wax M, Kailath T. On spatial smoothing for estimation of coherent signals. IEEE Trans. On ASSP, 1985, 33(4): 806~811.
[40] Pillai S U, Kwon B H. Performance analysis of MUSIC—type high resolution estimators for direction finding in correlated and coherent scenes. IEEE Trans. On ASSP, 1989, 37(8): 1176~1189.
[41] Viberg M, Ottersten B. Sensor array processing based on subspace fitting. IEEE Trans. On SP, 1991, 39(5): 1110~1121.
[42] Stoica P, Soderstrom T. Statistical analysis of MUSIC and subspace rotation estimates of sinusoidal frequencies. IEEE Trans. On SP, 1991, 39(8): 1836~1847.
[43]王永良,陈辉,彭应宁,万群.空间谱估计理论与算法.北京:清华大学出版社, 2004.
[44] Huarng K C, Yeh C C. A unitary transformation method for angle of arrival estimation. IEEE Trans. Acoust., Speech, Signal Processing, vol. 39, pp. 975~977,Apr. 1991.
[45] Zoltowski M D, Kautz G M, Silverstein S D. Beamspace root-MUSIC. IEEE Trans. Signal Processing, vol. 41, pp. 344~364, Jan. 1993.
[46] Marius P, Alex B. Unitary Root-MUSIC with a real-valued eigendecomposition: a theoretical and experimental performance study. IEEE Trans. On Signal Processing, vol. 48, no. 5, pp. 1306~1314, May 2000.
[47] Liang T, Kwan H K. A novel approach to fast DOA estimation of multiple spatial narrowband signals. The 2002 45th Midwest Symposium on Circuits and Systems, Tulsa, Oklahoma, 2002.
[48]陈四根,杨莘元.等距线阵信号傅里叶变换研究.哈尔滨工程大学学报, 2003, 24(4): 440~443.
[49]袁国靖,丁君,郭陈江.一种快速高分辨率的DOA估计方法——FFT-BMUSIC法.微波学报, 2005, 21(1): 10~13.
[50]曾浩,谭晓衡,杨士中,刘玲. DFT在二维DOA估计的MUSIC法中的应用.重庆大学学报, 2005, 28(8):68~71.
[51]余继周,陈定昌.一种DOA估计的快速子空间算法.现代电子技术, 2005, (28): 90~92.
[52] Yih-Min Chen. On spatial smoothing for two-dimensional direction-of-arrival estimation of coherent signals. IEEE Trans. On SP, 1997, 45(7): 1689~1696.
[53]王布宏,王永良,陈辉.一种新的相干信源DOA估计算法:加权空间平滑协方差矩阵的Toeplitz矩阵拟合.电子学报, 2003,31(9): 1394~1397.
[54] Wang Buhong, Wang Yongliang, Chen Hui. Weighted spatial smoothing for direction-of-arrival estimation of coherent signals. 2002 IEEE International Symposium on Antennas and Propagation and USNC/URSI National Radio Science Meeting, San Antonio, Texas, USA, 2002, 6: 668~671.
[55] Alireza M, Tzu C C. Signal enhancement of the spatial smoothing algorithm. IEEE Trans on SP, 1991, 39(8): 1907~1911.
[56] Jian L. Improved angular resolution for spatial smoothing techniques. IEEE Trans on Signal processing. 1992, 40(12): 3078~3081.
[57] C C Yeh, J H Lee, Y M Chen. Estimating two-dimensional angles of arrival in coherent source environment. IEEE Trans Acoust., Speech, Signal Processing, vol. 37, pp. 153~155, Jan. 1989.
[58] Yixin Yang, Chunru Wan, Chao Sun, Qing Wang. DOA estimation for coherentsources in beamspace using spatial smoothing. ICICS-PCM 2003, Singapore, 1028~1032.
[59] Ma Changlin, Peng Yingning, Tian Lisheng. Blind estimation of DOA’s of coherent sources. Proceedings of ICSP’98, pp. 327~330.
[60] Donald W, Tufts, Ramdas K. Estimation of frequencies of multiple sinusoids: Making linear prediction performance like maximum likelihood. Proceedings of IEEE, Vol. 70, No. 9, pp. 975~989, 1982.
[61]周亚强,黄甫堪.噪扰条件下数字式多基线相位干涉仪解模糊问题.通信学报, 2005, (26): 16~21.