分布式多传感器阵列的信号重构方法
|
摘要
分布式多传感器阵列的传统模式为:各子阵列单独进行信号参数估计,估计得到的参数通过通信链路发送到信息融合中心进行信息处理,这种模式忽略了阵列之间的相参性。多阵列联合处理能够利用阵列之间的相参性。本文给出了分布式阵列的基本概念,阐述了几何模型和信号模型。分析了分布式阵列联合相参的理论性能,采用误差椭圆和Crame′r-Rao (CRB)下限作为性能评价方法。相参处理相对非相参处理能够获得较高的性能增益。详细分析了影响处理性能的因素:信噪比、信号DOA、信号源距离。
虚拟阵列内插方法能够将不规则的分布式阵列映射到均匀等距直线阵上,进而可以利用其Vandermonde结构。角度分区、校正方向数目、阵列的分布疏密程度都对阵列内插性能有影响,本文详细分析了这些因素,总结出其中的规律。子阵取向对阵列内插性能影响明显,与ULA取向一致的子阵会带来较高的性能。
实际实现中,阵列内插方法具有很大的实现障碍。虚拟阵列的选取具有太多的主观性,目前尚无最优虚拟阵列的设计方法。因此,如果从阵列本身的信号模型出发,从数学结构的角度去构造Vandermonde结构,思路会更为明确。流形分离方法(MST)针对分布式阵列,将导向矢量分解为采样矩阵(取决于几何形状)和范德蒙结构矢量(取决于信号场)的乘积,进而把适用于线性阵列的快速DOA估计算法推广到任意阵列,例如求根MUSIC算法。实际应用中,噪声使基于MST的算法性能下降。分析了噪声对子空间类DOA估计算法的影响,给出了最优模式数目的选取方法。仿真结果表明基于MST的DOA估计方法性能取决于信噪比和模式数。
Traditional processing involves separate parameter estimation at individual sub-arrays and data processing at the fusion center. However, coherence between dif-ferent arrays are neglected, which can be utilized by joint coherent processing ofdistributed arrays. In this paper, we define basic concepts and derive geometricaland signal model on distributed arrays. Theoretical performance of coherent andnoncoherent processing is analyzed with the help of error ellipse and Crame′RaoBound(CRB)。Parameter estimation can benefit from coherent processing. SNR,DOA, and source distance are factors that affect system performance.
Array interpolation maps irregular distributed array to a well-behavior ULA,which can benefit from its Vandermonde structure. We considered effects of sectorwidth, calibration direction number, and inter-distance of subarrays. Orientation ofsubarrays are vital to interpolation performance. When the virtual array share a similardirection with subarrays, performance are relatively better.
Actually array interpolation is an uneasy task because no philosophy on virtualarray selection is available. It is better to look for the Vandermonde structure froma mathematical view. The manifold separation technique (MST) models the steeringvector of antenna arrays with arbitrary geometry, which can be expressed as prod-uct of a sampling matrix (dependent on the antenna array only) and a Vandermondestructured vector (dependent on the wavefield only). The calibration measurementsare corrupted by noise in real-world applications, which impairs the performance ofMST-based algorithms. The effect of noisy measurements on subspace-based DOAalgorithms is analyzed. A link between the optimal number of selected modes and thestatistics of noise is established. Simulation results show satisfactory performance.
虚拟阵列内插方法能够将不规则的分布式阵列映射到均匀等距直线阵上,进而可以利用其Vandermonde结构。角度分区、校正方向数目、阵列的分布疏密程度都对阵列内插性能有影响,本文详细分析了这些因素,总结出其中的规律。子阵取向对阵列内插性能影响明显,与ULA取向一致的子阵会带来较高的性能。
实际实现中,阵列内插方法具有很大的实现障碍。虚拟阵列的选取具有太多的主观性,目前尚无最优虚拟阵列的设计方法。因此,如果从阵列本身的信号模型出发,从数学结构的角度去构造Vandermonde结构,思路会更为明确。流形分离方法(MST)针对分布式阵列,将导向矢量分解为采样矩阵(取决于几何形状)和范德蒙结构矢量(取决于信号场)的乘积,进而把适用于线性阵列的快速DOA估计算法推广到任意阵列,例如求根MUSIC算法。实际应用中,噪声使基于MST的算法性能下降。分析了噪声对子空间类DOA估计算法的影响,给出了最优模式数目的选取方法。仿真结果表明基于MST的DOA估计方法性能取决于信噪比和模式数。
Traditional processing involves separate parameter estimation at individual sub-arrays and data processing at the fusion center. However, coherence between dif-ferent arrays are neglected, which can be utilized by joint coherent processing ofdistributed arrays. In this paper, we define basic concepts and derive geometricaland signal model on distributed arrays. Theoretical performance of coherent andnoncoherent processing is analyzed with the help of error ellipse and Crame′RaoBound(CRB)。Parameter estimation can benefit from coherent processing. SNR,DOA, and source distance are factors that affect system performance.
Array interpolation maps irregular distributed array to a well-behavior ULA,which can benefit from its Vandermonde structure. We considered effects of sectorwidth, calibration direction number, and inter-distance of subarrays. Orientation ofsubarrays are vital to interpolation performance. When the virtual array share a similardirection with subarrays, performance are relatively better.
Actually array interpolation is an uneasy task because no philosophy on virtualarray selection is available. It is better to look for the Vandermonde structure froma mathematical view. The manifold separation technique (MST) models the steeringvector of antenna arrays with arbitrary geometry, which can be expressed as prod-uct of a sampling matrix (dependent on the antenna array only) and a Vandermondestructured vector (dependent on the wavefield only). The calibration measurementsare corrupted by noise in real-world applications, which impairs the performance ofMST-based algorithms. The effect of noisy measurements on subspace-based DOAalgorithms is analyzed. A link between the optimal number of selected modes and thestatistics of noise is established. Simulation results show satisfactory performance.
引文
1 R. C. Heimiller, J. E. Belyea, P. G. Tomlinson. Distributed Array Radar. IEEETransactions on Aerospace and Electronic Systems. 1983, AES-19(6):831–839
2 E. H. Attia, K. Abend. An Experimental Demonstration of a Distributed ArrayRadar. AP-S International Symposium (Digest), IEEE Antennas and PropagationSociety. London, Ont, Can, 1991, 3:1720–1723
3 R. J. Kozick, B. M. Sadler. Source Localization with Distributed Sensor Arraysand Partial Spatial Coherence. IEEE Transactions on Signal Processing. 2004,52(3):601–616
4 A. Farina. Target Tracking with Bearings-only Measurements. Signal Processing.1999, 78(1):61–78
5 J. Levine, R. Marino. Constant-speed Target Tracking via Bearings-only Mea-surements. IEEE Transactions on Aerospace and Electronic Systems. 1992,28(1):174–182
6 M. Wax, T. Kailath. Optimum Localization of Multiple Sources by Passive Ar-rays. IEEE Transactions on Acoustics, Speech, and Signal Processing. 1983,ASSP-31(5):1210–1218
7 B. D. Van Veen, K. M. Buckley. Beamforming: A Versatile Approach to SpatialFiltering. IEEE Acoustics, Speech, and Signal Processing Magazine. 1988,5(2):4–24
8 H. L. Van Trees. Optimum Array Processing, Pt. IV of Detection, Estimation andModulation Theory. Wiley-Interscience, 2002
9 P. Hyberg, M. Jansson, B. Ottersten. Array Interpolation and Bias Reduction.IEEE Transactions on Signal Processing. 2004, 52(10):2711–2720
10 P. Hyberg, M. Jansson, B. Ottersten. Array Interpolation and DOA MSE Reduc-tion. IEEE Transactions on Signal Processing. 2005, 53(12):4464–4471
11 E. Weinstein. Decentralization of the Gaussian Maximum Likelihood Estimatorand its Applications to Passive Array Processing. IEEE Transactions on Acous-tics, Speech, and Signal Processing. 1981, ASSP-29(5):945–951
12 M. Wax, T. Kailath. Decentralized Processing in Sensor Arrays. IEEE Trans-actions on Acoustics, Speech, and Signal Processing. 1985, ASSP-33(5):1123–1129
13 P. Stoica, A. Nehorai, T. Soderstrom. Decentralized Array-processing Using theMode Algorithm. Circuits Systems and Signal Processing. 1995, 14(1):17–38
14 R. J. Kozick, B. M. Sadler. Source Localization with Distributed Sensor Arraysand Partial Spatial Coherence. Proceedings of SPIE - The International Societyfor Optical Engineering. 2000, 4040:142–153
15 R. J. Kozick, B. M. Sadler. Distributed Source Localization with Multiple Sen-sor Arrays and Frequency-selective Spatial Coherence. IEEE Signal ProcessingWorkshop on Statistical Signal and Array Processing. Pennsylvania, PA, USA,2000:419–423
16 R. J. Kozick, B. M. Sadler. Tracking Moving Acoustic Sources with a Networkof Sensors. Proceedings of SPIE - The International Society for Optical Engi-neering. 2002, 4743:20–31
17 W. F. Walker, G. E. Trahey. A Fundamental Limit on Delay Estimation UsingPartially Correlated Speckle Signals. IEEE Transactions on Ultrasonics Ferro-electrics and Frequency Control. 1995, 42(2):301–308
18 H. Krim, M. Viberg. Two Decades of Array Signal Processing Research - theParametric Approach. IEEE Signal Processing Magazine. 1996, 13(4):67–94
19 B. Friedlander. Root-MUSIC Algorithm for Direction Finding with InterpolatedArrays. Signal Processing. 1993, 30(1):15–29
20 C. P. Mathews, M. D. Zoltowski. Eigenstructure Techniques for 2-D Angle Esti-mation with Uniform Circular Arrays. IEEE Transactions on Signal Processing.1994, 42(9):2395–2407
21 M. Pesavento, C. F. Mecklenbrauker, J. F. Bohme. Multidimensional Rank Re-duction Estimator for Parametric Mimo Channel Models. EURASIP Journal onApplied Signal Processing. 2004, 2004(9):1354–1363
22 B. Friedlander, A. J. Weiss. Direction Finding for Wideband Signals Using anInterpolated Array. Asilomar Conference on Circuits, Systems and Computers.Pacific Grove, CA, USA, 1991, 1:583–587
23 B. Friedlander, A. J. Weiss. Direction Finding Using Spatial Smoothing withInterpolated Arrays. IEEE Transactions on Aerospace and Electronic Systems.1992, 28(2):574–587
24 A. J. Weiss, B. Friedlander. Performance Analysis of Spatial Smoothing withInterpolated Arrays. IEEE Transactions on Signal Processing. 1993, 41(5):1881–1892
25 B. Friedlander, A. J. Weiss. Direction Finding for Wide-band Signals Using anInterpolated Array. IEEE Transactions on Signal Processing. 1993, 41(4):1618–1634
26 H. S. Hung, K. R. Whu. Direction Finding Using ESPRIT with Array ManifoldInterpolation. Proceedings of the SPIE - The International Society for OpticalEngineering. 1992, 1770:255–265
27 R. H. Roy. ESPRIT: Estimation of Signal Parameters via Rotational InvarianceTechniques. Ph.D. thesis, Stanford University, CA. 1987:1–283
28 J. Eriksson, M. Viberg. Data Reduction in Spatially Colored Noise Using a Vir-tual Uniform Linear Array. ICASSP, IEEE International Conference on Acous-tics, Speech and Signal Processing - Proceedings. Istanbul, Turkey, 5
29 Z. Ming, Z. Zhao-Da. Direction Finding of Coherent Sources with NonlinearArrays. Circuits, Systems, and Signal Processing. 1996, 15(1):137–144
30 L. I. Vaskelainen. Virtual Array Synthesis Method for Planar Array Antennas.IEEE Transactions on Antennas and Propagation. 1998, 46(3):391–396
31 M. Buhren, M. Pesavento, J. E. Bohme. A New Approach to Array Interpolationby Generation of Artificial Shift Invariances: Interpolated ESPRIT. Proceed-ings - ICASSP, IEEE International Conference on Acoustics, Speech and SignalProcessing. Hong Kong, China, 2003, 5:205–208
32 T. P. Bronez. Sector Interpolation of Non-uniform Arrays for Efficient HighResolution Bearing Estimation. Proceedings - ICASSP, IEEE InternationalConference on Acoustics, Speech and Signal Processing. New York, NY, US,1988:2885–2888
33 F. Belloni, A. Richter, V. Koivunen. Extension of Root-MUSIC to Non-ULAArray Configurations. ICASSP, IEEE International Conference on Acoustics,Speech and Signal Processing - Proceedings. Toulouse, France, 2006, 4:897–900
34 F. Belloni, A. Richter, V. Koivunen. Reducing Excess Variance in BeamspaceMethods for Uniform Circular Array. IEEE Workshop on Statistical Signal Pro-cessing Proceedings. Bordeaux, France, 2005, 2005:940–943
35 F. Belloni, A. Richter, V. Koivunen. DOA Estimation via Manifold Separationfor Arbitrary Array Structures. IEEE Transactions on Signal Processing. 2007,55(10):4800–4810
36 A. L. Swindlehurst, T. Kailath. A Performance Analysis of Subspace-basedMethods in the Presence of Model Errors–I: The MUSIC Algorithm. IEEETransactions on Signal Processing. 1992, 40(7):1758–1774
37 A. L. Swindlehurst, T. Kailath. Performance Analysis of Subspace-base Methodsin the Presence of Model Errors: Part II - Multidimensional Algorithms. IEEETransactions on Signal Processing. 1993, 41(9):2882–2890
38 B. D. Rao, K. Hari. Performance Analysis of Subspace Based Methods. 4thAnnual ASSP Workshop on Spectrum Estimation and Modeling. Minneapolis,MN, USA, 1988:92–97
39 B. R. Mahafza, A. Z. Elsherbeni. Matlab Simulations for Radar Systems Design.Boca Raton, FL: Chapman & Hall/CRC, 2004
40 R. A. Fisher. On the Mathematical Foundations of Theoretical Statistics. Phil.Trans. R. Soc., London, Ser. A. 1922, 222:309–368
41 R. A. Fisher. Theory of Statistical Estimation. Proc. Cambridge Phil. Soc. 1925,22:700
42 H. Crame′r. Mathematical Methods of Statistics. Princeton: Princeton U. P., 1946
43 C. R. Rao. Linear Statistical Inference and its Applications. 2nd edn. New York:Wiley-Interscience, 1973
44 J. Capon. High-resolution Frequency-wavenumber Spectrum Analysis. Proceed-ings of the IEEE. 1969, 57(8):1408–1418
45 J. P. Burg. Maximum Entropy Spectral Analysis. Ph.D. thesis, Stanford Univer-sity. 1975
46 R. O. Schmidt. Multiple Emitter Location and Signal Parameter Estimation.IEEE Transactions on Antennas and Propagation. 1986, AP-34(3):276–280
47 B. Friedlander. On the Crame′r-Rao Bound for Time Delay and Doppler Estima-tion. IEEE Transactions on Information Theory. 1984, IT-30(3):575–580
48 A. Richter, F. Belloni, V. Koivunen. DOA and Polarization Estimation UsingArbitrary Polarimetric Array Configurations. 2006 IEEE Sensor Array and Mul-tichannel Signal Processing Workshop Proceedings, SAM 2006. Waltham, MA,United States, 2006:55–59
49 A. L. Swindlehurst, B. Ottersten, R. Roy, et al. Multiple Invariance ESPRIT.IEEE Transactions on Signal Processing. 1992, 40(4):867–881
50 R. O. Schmidt. Multilinear Array Manifold Interpolation. IEEE Transactions onSignal Processing. 1992, 40(4):857–866
2 E. H. Attia, K. Abend. An Experimental Demonstration of a Distributed ArrayRadar. AP-S International Symposium (Digest), IEEE Antennas and PropagationSociety. London, Ont, Can, 1991, 3:1720–1723
3 R. J. Kozick, B. M. Sadler. Source Localization with Distributed Sensor Arraysand Partial Spatial Coherence. IEEE Transactions on Signal Processing. 2004,52(3):601–616
4 A. Farina. Target Tracking with Bearings-only Measurements. Signal Processing.1999, 78(1):61–78
5 J. Levine, R. Marino. Constant-speed Target Tracking via Bearings-only Mea-surements. IEEE Transactions on Aerospace and Electronic Systems. 1992,28(1):174–182
6 M. Wax, T. Kailath. Optimum Localization of Multiple Sources by Passive Ar-rays. IEEE Transactions on Acoustics, Speech, and Signal Processing. 1983,ASSP-31(5):1210–1218
7 B. D. Van Veen, K. M. Buckley. Beamforming: A Versatile Approach to SpatialFiltering. IEEE Acoustics, Speech, and Signal Processing Magazine. 1988,5(2):4–24
8 H. L. Van Trees. Optimum Array Processing, Pt. IV of Detection, Estimation andModulation Theory. Wiley-Interscience, 2002
9 P. Hyberg, M. Jansson, B. Ottersten. Array Interpolation and Bias Reduction.IEEE Transactions on Signal Processing. 2004, 52(10):2711–2720
10 P. Hyberg, M. Jansson, B. Ottersten. Array Interpolation and DOA MSE Reduc-tion. IEEE Transactions on Signal Processing. 2005, 53(12):4464–4471
11 E. Weinstein. Decentralization of the Gaussian Maximum Likelihood Estimatorand its Applications to Passive Array Processing. IEEE Transactions on Acous-tics, Speech, and Signal Processing. 1981, ASSP-29(5):945–951
12 M. Wax, T. Kailath. Decentralized Processing in Sensor Arrays. IEEE Trans-actions on Acoustics, Speech, and Signal Processing. 1985, ASSP-33(5):1123–1129
13 P. Stoica, A. Nehorai, T. Soderstrom. Decentralized Array-processing Using theMode Algorithm. Circuits Systems and Signal Processing. 1995, 14(1):17–38
14 R. J. Kozick, B. M. Sadler. Source Localization with Distributed Sensor Arraysand Partial Spatial Coherence. Proceedings of SPIE - The International Societyfor Optical Engineering. 2000, 4040:142–153
15 R. J. Kozick, B. M. Sadler. Distributed Source Localization with Multiple Sen-sor Arrays and Frequency-selective Spatial Coherence. IEEE Signal ProcessingWorkshop on Statistical Signal and Array Processing. Pennsylvania, PA, USA,2000:419–423
16 R. J. Kozick, B. M. Sadler. Tracking Moving Acoustic Sources with a Networkof Sensors. Proceedings of SPIE - The International Society for Optical Engi-neering. 2002, 4743:20–31
17 W. F. Walker, G. E. Trahey. A Fundamental Limit on Delay Estimation UsingPartially Correlated Speckle Signals. IEEE Transactions on Ultrasonics Ferro-electrics and Frequency Control. 1995, 42(2):301–308
18 H. Krim, M. Viberg. Two Decades of Array Signal Processing Research - theParametric Approach. IEEE Signal Processing Magazine. 1996, 13(4):67–94
19 B. Friedlander. Root-MUSIC Algorithm for Direction Finding with InterpolatedArrays. Signal Processing. 1993, 30(1):15–29
20 C. P. Mathews, M. D. Zoltowski. Eigenstructure Techniques for 2-D Angle Esti-mation with Uniform Circular Arrays. IEEE Transactions on Signal Processing.1994, 42(9):2395–2407
21 M. Pesavento, C. F. Mecklenbrauker, J. F. Bohme. Multidimensional Rank Re-duction Estimator for Parametric Mimo Channel Models. EURASIP Journal onApplied Signal Processing. 2004, 2004(9):1354–1363
22 B. Friedlander, A. J. Weiss. Direction Finding for Wideband Signals Using anInterpolated Array. Asilomar Conference on Circuits, Systems and Computers.Pacific Grove, CA, USA, 1991, 1:583–587
23 B. Friedlander, A. J. Weiss. Direction Finding Using Spatial Smoothing withInterpolated Arrays. IEEE Transactions on Aerospace and Electronic Systems.1992, 28(2):574–587
24 A. J. Weiss, B. Friedlander. Performance Analysis of Spatial Smoothing withInterpolated Arrays. IEEE Transactions on Signal Processing. 1993, 41(5):1881–1892
25 B. Friedlander, A. J. Weiss. Direction Finding for Wide-band Signals Using anInterpolated Array. IEEE Transactions on Signal Processing. 1993, 41(4):1618–1634
26 H. S. Hung, K. R. Whu. Direction Finding Using ESPRIT with Array ManifoldInterpolation. Proceedings of the SPIE - The International Society for OpticalEngineering. 1992, 1770:255–265
27 R. H. Roy. ESPRIT: Estimation of Signal Parameters via Rotational InvarianceTechniques. Ph.D. thesis, Stanford University, CA. 1987:1–283
28 J. Eriksson, M. Viberg. Data Reduction in Spatially Colored Noise Using a Vir-tual Uniform Linear Array. ICASSP, IEEE International Conference on Acous-tics, Speech and Signal Processing - Proceedings. Istanbul, Turkey, 5
29 Z. Ming, Z. Zhao-Da. Direction Finding of Coherent Sources with NonlinearArrays. Circuits, Systems, and Signal Processing. 1996, 15(1):137–144
30 L. I. Vaskelainen. Virtual Array Synthesis Method for Planar Array Antennas.IEEE Transactions on Antennas and Propagation. 1998, 46(3):391–396
31 M. Buhren, M. Pesavento, J. E. Bohme. A New Approach to Array Interpolationby Generation of Artificial Shift Invariances: Interpolated ESPRIT. Proceed-ings - ICASSP, IEEE International Conference on Acoustics, Speech and SignalProcessing. Hong Kong, China, 2003, 5:205–208
32 T. P. Bronez. Sector Interpolation of Non-uniform Arrays for Efficient HighResolution Bearing Estimation. Proceedings - ICASSP, IEEE InternationalConference on Acoustics, Speech and Signal Processing. New York, NY, US,1988:2885–2888
33 F. Belloni, A. Richter, V. Koivunen. Extension of Root-MUSIC to Non-ULAArray Configurations. ICASSP, IEEE International Conference on Acoustics,Speech and Signal Processing - Proceedings. Toulouse, France, 2006, 4:897–900
34 F. Belloni, A. Richter, V. Koivunen. Reducing Excess Variance in BeamspaceMethods for Uniform Circular Array. IEEE Workshop on Statistical Signal Pro-cessing Proceedings. Bordeaux, France, 2005, 2005:940–943
35 F. Belloni, A. Richter, V. Koivunen. DOA Estimation via Manifold Separationfor Arbitrary Array Structures. IEEE Transactions on Signal Processing. 2007,55(10):4800–4810
36 A. L. Swindlehurst, T. Kailath. A Performance Analysis of Subspace-basedMethods in the Presence of Model Errors–I: The MUSIC Algorithm. IEEETransactions on Signal Processing. 1992, 40(7):1758–1774
37 A. L. Swindlehurst, T. Kailath. Performance Analysis of Subspace-base Methodsin the Presence of Model Errors: Part II - Multidimensional Algorithms. IEEETransactions on Signal Processing. 1993, 41(9):2882–2890
38 B. D. Rao, K. Hari. Performance Analysis of Subspace Based Methods. 4thAnnual ASSP Workshop on Spectrum Estimation and Modeling. Minneapolis,MN, USA, 1988:92–97
39 B. R. Mahafza, A. Z. Elsherbeni. Matlab Simulations for Radar Systems Design.Boca Raton, FL: Chapman & Hall/CRC, 2004
40 R. A. Fisher. On the Mathematical Foundations of Theoretical Statistics. Phil.Trans. R. Soc., London, Ser. A. 1922, 222:309–368
41 R. A. Fisher. Theory of Statistical Estimation. Proc. Cambridge Phil. Soc. 1925,22:700
42 H. Crame′r. Mathematical Methods of Statistics. Princeton: Princeton U. P., 1946
43 C. R. Rao. Linear Statistical Inference and its Applications. 2nd edn. New York:Wiley-Interscience, 1973
44 J. Capon. High-resolution Frequency-wavenumber Spectrum Analysis. Proceed-ings of the IEEE. 1969, 57(8):1408–1418
45 J. P. Burg. Maximum Entropy Spectral Analysis. Ph.D. thesis, Stanford Univer-sity. 1975
46 R. O. Schmidt. Multiple Emitter Location and Signal Parameter Estimation.IEEE Transactions on Antennas and Propagation. 1986, AP-34(3):276–280
47 B. Friedlander. On the Crame′r-Rao Bound for Time Delay and Doppler Estima-tion. IEEE Transactions on Information Theory. 1984, IT-30(3):575–580
48 A. Richter, F. Belloni, V. Koivunen. DOA and Polarization Estimation UsingArbitrary Polarimetric Array Configurations. 2006 IEEE Sensor Array and Mul-tichannel Signal Processing Workshop Proceedings, SAM 2006. Waltham, MA,United States, 2006:55–59
49 A. L. Swindlehurst, B. Ottersten, R. Roy, et al. Multiple Invariance ESPRIT.IEEE Transactions on Signal Processing. 1992, 40(4):867–881
50 R. O. Schmidt. Multilinear Array Manifold Interpolation. IEEE Transactions onSignal Processing. 1992, 40(4):857–866