矢量水听器超复数模型及其DOA估计算法
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摘要
在水声信号处理领域,传统的声压传感器由于不能直接地利用声场的空间信息,因此,需要用多个声压传感器在空间布成不同几何形式的阵列,进行空间采样来确定声波到达角(波达角,DOA,Direction of Arrival)。这给应用带来了极大的不便。近年来,一种新的传感器——矢量水听器获得了人们的广泛关注。矢量水听器可以测量声场中某点的声压和三个正交方向的速度分量,因此有四个输出分量。矢量水听器较传统的声压传感器具有诸多的优点。可是,传统的声矢量信号处理方法通常是将矢量传感器中的每个输出分量拼接起来,然后用标量信号的处理方法进行处理。这种处理方式破坏了矢量传感器输出分量之间的内在关系,因而具有一定的局限性。
矢量传感器的四个输出分量形式和Hamilton于1843年提出的超复数形式非常相似。因此将超复数代数用于矢量信号处理能够很好的保持矢量信号各分量之间内在的联系。本文在研究对超复数代数结构及矢量水听器输出结构的基础上,提出了矢量水听器的超复数测量模型,并将该超复数测量模型进一步推广至矢量水听器阵列,从而推导出了矢量水听器阵列的超复数测量模型。
基于提出的矢量水听器超复数测量模型,本文提出一种基于超复数的DOA估计新算法。给出了在超复数代数结构下矢量水听器信号能量的定义,并根据该定义给出了在超复数空间中估计水声信号波达角的算法。该算法具有明确的物理意义,即水声信号沿着其传播方向能量分量最大。同时,第一次给出了矢量水声信号在超复数代数结构下信号能量与超复数不同基之间的关系,并证明了估计值的无偏性和超复数分解轴无偏性的关系。
在上述矢量水听器DOA估计研究的基础上,本文进行了利用高分辨率的超复数MUSIC(Multiple Signal Classification)矢量水听器阵列的DOA估计的研究。分析表明,采用超复数整体描述的矢量水听器阵列的模型,借助于超复数信号处理算法,在同样的估计精度下,不但可以节约75%的内存量,而且可以还减少了近75%的除法操作。上述结果表明:矢量水听器的超复数测量模型为矢量水听器的信号处理开辟了一个新的天地。
In the field of acoustic signal processing, classic hydrophone can't utilize the spatial information in a single sensor mode. So it is required to deploy several sensors in some geometrical form to determine the DOA (DOA Direction of Arrival). Recently, a novel acoustic sensor has attracted widely interests, namely the acoustic vector sensor. Acoustic vector sensor has four output signals which correspond to the pressure and the three velocity components along three orthogonal axes. Acoustic vector sensor has many advantages compared with scalar acoustic sensor. However traditional technology of processing the vector sensor signals is to concatenate the four components, then process the signals with methods using in the processing of hydrophone signals. Because these traditional signal processing methods violate the relationship of the four components themselves, these methods have some disadvantages.
The form of acoustic vector sensor signal is similar to the hypercomplex algebra which was proposed by Hamilton in 1843. Hypercomplex deals the four components holistic in the process of operation, so the relationship of every component can be maintained. Due to this fact, representing the outputs of acoustic vector sensor signals will bring many advantages. Based on the structure of Hypercomplex algebra and output signals of vector acoustic sensor, this paper proposes the hypercomplex measurement equation of the vector acoustic sensor and the hypercomplex measurement is extended to the acoustic vector sensor array.
Based on the hypercomplex measurement equation, a novel DOA estimation algorithm is proposed. The concepts of energy and energy distribution under hypercomplex algebra are defined in this paper. The DOA algorithm is proposed through studying the energy distribution over the hypercomplex space. The proposed DOA algorithm has explicit physical meaning that the energy reaches its maximum value along its direction of propagation. Furthermore the relationship of acoustic vector signal energy under hypercomplex algebra and the base of simplicity decomposition is given. Finally the relationship between estimated value and the hypercomplex decomposition axis is also given.
Based on our hypercomplex model, the hypercomplex MUSIC-like algorithm can be used to estimate the directions of arrival of the sound sources. The analytical results show that, using some signal processing algorithm with the proposed hypercomplex model of the acoustic vector sensor, the memory requirements for representing the data covariance model are reduced 75% while the division requirements are reduced by 75%, for equivalent performance when compared with the long vector method. The results show that the hypercomplex model of acoustic vector sensor breaks a new path for the signal processing field.
矢量传感器的四个输出分量形式和Hamilton于1843年提出的超复数形式非常相似。因此将超复数代数用于矢量信号处理能够很好的保持矢量信号各分量之间内在的联系。本文在研究对超复数代数结构及矢量水听器输出结构的基础上,提出了矢量水听器的超复数测量模型,并将该超复数测量模型进一步推广至矢量水听器阵列,从而推导出了矢量水听器阵列的超复数测量模型。
基于提出的矢量水听器超复数测量模型,本文提出一种基于超复数的DOA估计新算法。给出了在超复数代数结构下矢量水听器信号能量的定义,并根据该定义给出了在超复数空间中估计水声信号波达角的算法。该算法具有明确的物理意义,即水声信号沿着其传播方向能量分量最大。同时,第一次给出了矢量水声信号在超复数代数结构下信号能量与超复数不同基之间的关系,并证明了估计值的无偏性和超复数分解轴无偏性的关系。
在上述矢量水听器DOA估计研究的基础上,本文进行了利用高分辨率的超复数MUSIC(Multiple Signal Classification)矢量水听器阵列的DOA估计的研究。分析表明,采用超复数整体描述的矢量水听器阵列的模型,借助于超复数信号处理算法,在同样的估计精度下,不但可以节约75%的内存量,而且可以还减少了近75%的除法操作。上述结果表明:矢量水听器的超复数测量模型为矢量水听器的信号处理开辟了一个新的天地。
In the field of acoustic signal processing, classic hydrophone can't utilize the spatial information in a single sensor mode. So it is required to deploy several sensors in some geometrical form to determine the DOA (DOA Direction of Arrival). Recently, a novel acoustic sensor has attracted widely interests, namely the acoustic vector sensor. Acoustic vector sensor has four output signals which correspond to the pressure and the three velocity components along three orthogonal axes. Acoustic vector sensor has many advantages compared with scalar acoustic sensor. However traditional technology of processing the vector sensor signals is to concatenate the four components, then process the signals with methods using in the processing of hydrophone signals. Because these traditional signal processing methods violate the relationship of the four components themselves, these methods have some disadvantages.
The form of acoustic vector sensor signal is similar to the hypercomplex algebra which was proposed by Hamilton in 1843. Hypercomplex deals the four components holistic in the process of operation, so the relationship of every component can be maintained. Due to this fact, representing the outputs of acoustic vector sensor signals will bring many advantages. Based on the structure of Hypercomplex algebra and output signals of vector acoustic sensor, this paper proposes the hypercomplex measurement equation of the vector acoustic sensor and the hypercomplex measurement is extended to the acoustic vector sensor array.
Based on the hypercomplex measurement equation, a novel DOA estimation algorithm is proposed. The concepts of energy and energy distribution under hypercomplex algebra are defined in this paper. The DOA algorithm is proposed through studying the energy distribution over the hypercomplex space. The proposed DOA algorithm has explicit physical meaning that the energy reaches its maximum value along its direction of propagation. Furthermore the relationship of acoustic vector signal energy under hypercomplex algebra and the base of simplicity decomposition is given. Finally the relationship between estimated value and the hypercomplex decomposition axis is also given.
Based on our hypercomplex model, the hypercomplex MUSIC-like algorithm can be used to estimate the directions of arrival of the sound sources. The analytical results show that, using some signal processing algorithm with the proposed hypercomplex model of the acoustic vector sensor, the memory requirements for representing the data covariance model are reduced 75% while the division requirements are reduced by 75%, for equivalent performance when compared with the long vector method. The results show that the hypercomplex model of acoustic vector sensor breaks a new path for the signal processing field.
引文
[1]A.Nehorai,E.Paldi,"Acoustic Vector-Sensor Array Processing",IEEE transactions on signal processing,vol.42,no.9,pp.2481-2489,1994
[2]A.Nehorai,E.Paldi,"Vector-Sensor Array Processing for Electromagnetic Source Location",IEEE transactions on signal processing,vol.42,no.2,pp.376-398,1994
[3]陈洪娟 矢量传感器,哈尔滨工程大学出版社,哈尔滨
[4]姚直象 硕士学位论文:单矢量水听器信号处理研究 哈尔滨工程大学
[5]RayleighJ,"Device for measuring the intensity of air borne oscillations" Pb 1.M ag.,1882,14:186-188
[6]Olsen HF,"System responsive to energy flow in sound waves",U.S.Patent,1932;No.1 89 2644.
[7]Hawkes M et al,"Acoustic vectorsensor beamforming and Capon direction estimation",IEEE Trans.on Signal Processing,1998,46(9):2291-230
[8]Gordienko Vet al,"Vector phase method in acoustics",Moscow,Nauka,1989
[9]Shchurov V A,"Vector acoustics of the ocean",Vladivostok,Dalhauka,2003
[10]Shchurov V A et al,"The interaction of energy flows of under water ambient noise and local source",J.Acoust.Soc.Am.1991,90(2):1002-1004
[11]D,Spain G.L.et al,"Energeties of the deep ocean's infrasonic sound field",J.Acoust.Soc.Am.,1991,89(3):1134-1158
[12]Wong K T et al,"Uni-vector-sensor ESPRIT for mu 1tisource azimuth,elevation,and polarization estimation,"IEEE Trans.on Antennas and Propagation,1997,45(10):1467-1470
[13]Wong K T et al,"Closed form underwater acoustic direction-finding with arbitrarily spaced vector hydrophones at unknown locations",IEEE J.of oceanic Engineering,1997,22(4):649-658
[14]Zoltowski M D et al,"Closed form eigenstructure-based direction finding using arbitrary but identical subarrys on a sparse uniform Cartesian array grid," IEEE Trans.on signal processing,2000,48(8):2205-2210
[15]陈华伟,赵俊渭,基于矢量传感器复声强测量的低空日标二维波达方向估计,声学学报,2004,29(3):277-282.
[16]田坦,齐娜,孙大军,矢量水听器阵波束域MVDR方法研究,哈尔滨工程大学学报,2004,25(3):295-298
[17] Jian Li, Compton R.T, Jr. "Angle and polarization estimation using ESPRIT with a polarization sensitive array," IEEE Trans. AP,1991,39(9): 1376-1383
[18] Jian Li, Compton R.T, Jr. 'Two-dimensional angle and estimation using the ESPRIT algorithm," IEEE Trans.AP,1992,40(5):550-555
[19] Hochwald B, Nehorai A. "Identifiability in array processing models with vector-sensor application," IEEE Trans, signal processing, 1996 44(1):83-95
[20] Kah-Chye Tan, KwokChiang Ho, Arye Nehorai. "Uniqueness study of measurements obtainable with arrays of electromagnetic vector sensors," IEEE Trans. signal processing, 1996,44(4): 1036-1039
[21] P.M.Morse and K.U.Ingard, Theoretical Acoustics.New York:McGraw-Hill,1968
[22] A.D.Pierce. Acoustic-An Introduction to Its Physical Principles and Applications. New York:McGraw-Hill,1981
[23] O.H.Bjor , H.J.Krystad, "A velocity microphones for sound intensity measurement," proc. Autumn Conf. Inst.Acoust.1982(Edinburgh. Scotland), 1982, pp.B7.1-B7.5.
[24] S.A.Nordby and O.H.Bjor," Measurement of sound intensity by use of a dual channel real-time analyzer and a special sound intensity microphone," in Proc. Inter-Noise 84(New York: Noise Control Foundation), Dec. 1984, pp.1107-1110
[25] G.Rasmussen, "Intensity-Its measurement and uses," Sound Bibrat., vol.23, pp12-21,Mar.1989
[26] G.Rasmussen,"Source location using vector intensity measurements," Sound Vibrat, vol.23,pp.28-33,Mar.1989
[27] Schmidt R O. "Multiple emitter location and signal parameter estimation, "IEEE Trans, on AP,1986, 34(3):276-280
[28] Stoica P, Nehorai A. "MUSIC, maximum likelihood ,and Cramer-Rao bound, " IEEE Trans, on ASSP,1989,37(5):720-741
[29] Porat B, Friedlander B. "Analysis of the asymptotic relative efficiency of the MUSIC algorithm," IEEE Trans, on ASSP,1988,36(4):532-543
[30] 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
[31] Stoica P,Nehorai A. "MUSIC,maximum likelihood and Cramer-Rao bound : further results and comparisons," IEEE Trans. on ASSP, 1990, 38(12): 2140-2150
[32]Stoica P,Nehorai A."Performance comparison of subspace rotation and MUSIC methods for direction estimation," IEEE Trans.on signal process,1991,39(2):446-453
[33]Stoica P,SOderstrom T."Statistical analysis of MUSIC and subspace rotation estimates of sinusoidal frequencies,"IEEE Trans.on signal process,1991,39(8):1836-1847
[34]Stoica P,Nehorai "A.MUSIC,maximum likelihood,and Cramer-Rao bound:further results and comparisons." In Proc.ICASSP,1989:2605-2608
[35]Xu X L,Buckley K M."Bias analysis of the MUSIC location estimator," IEEE Trans.on signal process,1992,40(10):2259-2569
[36]Zhou C,Haber F,Jaggard D L."A resolution measure for the MUSIC algorithm and sits application to plane wave arrivals contaminated by coherent interference,"IEEE Trans.on signal process,1991,39(2):454-463
[37]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
[38]Paulraj A,Roy R,Kailath T."Estimation of signal parameters via rotational invariance techniques-ESPRIT," In proc.19st Asilomar Conf.on Signals,Systems,and computers,Pacific Grove,CA,1985,83-89
[39]Roy R,Paulraj A,Kailath T."Comparative performance of ESPRIT and MUSIC for direction-of-arrival estimation," In proc.20st Asilomar Conf.Circuits,Syst.,Computation,Asilomar.CA,1987,12:2344-2347
[40]Roy R,Kailath T."ESPRIT-estimation of signal parameters via rotational invariance techniques," IEEE Trans.on ASSP,1989,37(7):984-995
[41]王永良 陈辉 彭应宁 等 空间谱估计理论与算法 清华大学出版社 北京,2004
[42]Sebastian Miron,Nicolas Le Behan,Jerome I Mars," Quatemion-MUSIC for Vector-Sensor Array Processing",IEEE transactions on signal processing,vol.54,no.4,pp.1218-1229,2006
[43]Sebastian Miron,Nicolas Le Behan,Jerome I Mars,"High Resolution Vector-Sensor Array Processing Based on Biquaternions",ICASSP,2006
[44]Sebastian Miron,Nicolas Le Behan,Jerome I Mars," Singular value decomposition of quaternion matrices:a new tool for vector-sensor signal processing",Signal Processing 84,pp.1177-1199,2004
[45]Todd A.Ell,Stephen J.Sangwine,"Hypercomplex Fourier Transforms of Color Images",IEEE transactions on image processing,vol.16,no.2,pp.1-14,2006
[46]李闻亮,四元数矩阵,国防科技大学出版社,北京
[47]Petre Stoica,Peter Handel,Torsten Soderstrom,"study of Capon method for array signal processing" circuits system signal process VOL.14,No.6,1995,pp.749-770
[48]Yanwei Wang,Jian Li,IEEE,Petre Stoica,"Rank-Deficient Robust Capon Filter Bank Approach to Complex Spectral Estimation,"IEEE transactions on signal processing,vol.53,no.8,Aug.2005,pp.2173-2726
[49]Petre Stoica,Zhisong Wang,Jian Li," Robust Capon Beamforming," IEEE signal processing letters,vol.10,no.6,June 2003,pp.172-175
[50]Jian Li,Petre Stoica," An adpative Filtering Approach to Spectral Estimation and SAR Imaging," IEEE transactions on signal processing,vol.44,no.6,June 1996,pp.1469-1484
[51]Petre Stoica,Hongbin Li,and Jian Li," A New Derivation of the APES Filter,"IEEE signal processing letters,vol.6,no.8,Aug.1999,pp.205-206
[52]Petre Stoica,Hongbin Li,and Jian Li," Amplitude Estimation of Sinusoidal Signals:Survey,New Results,and an Application," IEEE Trans.on signal processing,vol.48,no.2,Feb.2000,pp.338-352
[53]Andreas Jakobsson,Petre Stoica," Combing Capon and APES for estimation of spectral lines,"circuits systems signal process vol.19,no.2,2000,pp.159-169
[54]Zheng-She Liu,Hong Bin Li,Jian Li,"Efficient Implementationi of Capon and APES for Spectral Estimation," IEEE Trans.on Aerospace and electronic systems vol.34,no.4 Oct.1998,pp.1314-1319
[2]A.Nehorai,E.Paldi,"Vector-Sensor Array Processing for Electromagnetic Source Location",IEEE transactions on signal processing,vol.42,no.2,pp.376-398,1994
[3]陈洪娟 矢量传感器,哈尔滨工程大学出版社,哈尔滨
[4]姚直象 硕士学位论文:单矢量水听器信号处理研究 哈尔滨工程大学
[5]RayleighJ,"Device for measuring the intensity of air borne oscillations" Pb 1.M ag.,1882,14:186-188
[6]Olsen HF,"System responsive to energy flow in sound waves",U.S.Patent,1932;No.1 89 2644.
[7]Hawkes M et al,"Acoustic vectorsensor beamforming and Capon direction estimation",IEEE Trans.on Signal Processing,1998,46(9):2291-230
[8]Gordienko Vet al,"Vector phase method in acoustics",Moscow,Nauka,1989
[9]Shchurov V A,"Vector acoustics of the ocean",Vladivostok,Dalhauka,2003
[10]Shchurov V A et al,"The interaction of energy flows of under water ambient noise and local source",J.Acoust.Soc.Am.1991,90(2):1002-1004
[11]D,Spain G.L.et al,"Energeties of the deep ocean's infrasonic sound field",J.Acoust.Soc.Am.,1991,89(3):1134-1158
[12]Wong K T et al,"Uni-vector-sensor ESPRIT for mu 1tisource azimuth,elevation,and polarization estimation,"IEEE Trans.on Antennas and Propagation,1997,45(10):1467-1470
[13]Wong K T et al,"Closed form underwater acoustic direction-finding with arbitrarily spaced vector hydrophones at unknown locations",IEEE J.of oceanic Engineering,1997,22(4):649-658
[14]Zoltowski M D et al,"Closed form eigenstructure-based direction finding using arbitrary but identical subarrys on a sparse uniform Cartesian array grid," IEEE Trans.on signal processing,2000,48(8):2205-2210
[15]陈华伟,赵俊渭,基于矢量传感器复声强测量的低空日标二维波达方向估计,声学学报,2004,29(3):277-282.
[16]田坦,齐娜,孙大军,矢量水听器阵波束域MVDR方法研究,哈尔滨工程大学学报,2004,25(3):295-298
[17] Jian Li, Compton R.T, Jr. "Angle and polarization estimation using ESPRIT with a polarization sensitive array," IEEE Trans. AP,1991,39(9): 1376-1383
[18] Jian Li, Compton R.T, Jr. 'Two-dimensional angle and estimation using the ESPRIT algorithm," IEEE Trans.AP,1992,40(5):550-555
[19] Hochwald B, Nehorai A. "Identifiability in array processing models with vector-sensor application," IEEE Trans, signal processing, 1996 44(1):83-95
[20] Kah-Chye Tan, KwokChiang Ho, Arye Nehorai. "Uniqueness study of measurements obtainable with arrays of electromagnetic vector sensors," IEEE Trans. signal processing, 1996,44(4): 1036-1039
[21] P.M.Morse and K.U.Ingard, Theoretical Acoustics.New York:McGraw-Hill,1968
[22] A.D.Pierce. Acoustic-An Introduction to Its Physical Principles and Applications. New York:McGraw-Hill,1981
[23] O.H.Bjor , H.J.Krystad, "A velocity microphones for sound intensity measurement," proc. Autumn Conf. Inst.Acoust.1982(Edinburgh. Scotland), 1982, pp.B7.1-B7.5.
[24] S.A.Nordby and O.H.Bjor," Measurement of sound intensity by use of a dual channel real-time analyzer and a special sound intensity microphone," in Proc. Inter-Noise 84(New York: Noise Control Foundation), Dec. 1984, pp.1107-1110
[25] G.Rasmussen, "Intensity-Its measurement and uses," Sound Bibrat., vol.23, pp12-21,Mar.1989
[26] G.Rasmussen,"Source location using vector intensity measurements," Sound Vibrat, vol.23,pp.28-33,Mar.1989
[27] Schmidt R O. "Multiple emitter location and signal parameter estimation, "IEEE Trans, on AP,1986, 34(3):276-280
[28] Stoica P, Nehorai A. "MUSIC, maximum likelihood ,and Cramer-Rao bound, " IEEE Trans, on ASSP,1989,37(5):720-741
[29] Porat B, Friedlander B. "Analysis of the asymptotic relative efficiency of the MUSIC algorithm," IEEE Trans, on ASSP,1988,36(4):532-543
[30] 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
[31] Stoica P,Nehorai A. "MUSIC,maximum likelihood and Cramer-Rao bound : further results and comparisons," IEEE Trans. on ASSP, 1990, 38(12): 2140-2150
[32]Stoica P,Nehorai A."Performance comparison of subspace rotation and MUSIC methods for direction estimation," IEEE Trans.on signal process,1991,39(2):446-453
[33]Stoica P,SOderstrom T."Statistical analysis of MUSIC and subspace rotation estimates of sinusoidal frequencies,"IEEE Trans.on signal process,1991,39(8):1836-1847
[34]Stoica P,Nehorai "A.MUSIC,maximum likelihood,and Cramer-Rao bound:further results and comparisons." In Proc.ICASSP,1989:2605-2608
[35]Xu X L,Buckley K M."Bias analysis of the MUSIC location estimator," IEEE Trans.on signal process,1992,40(10):2259-2569
[36]Zhou C,Haber F,Jaggard D L."A resolution measure for the MUSIC algorithm and sits application to plane wave arrivals contaminated by coherent interference,"IEEE Trans.on signal process,1991,39(2):454-463
[37]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
[38]Paulraj A,Roy R,Kailath T."Estimation of signal parameters via rotational invariance techniques-ESPRIT," In proc.19st Asilomar Conf.on Signals,Systems,and computers,Pacific Grove,CA,1985,83-89
[39]Roy R,Paulraj A,Kailath T."Comparative performance of ESPRIT and MUSIC for direction-of-arrival estimation," In proc.20st Asilomar Conf.Circuits,Syst.,Computation,Asilomar.CA,1987,12:2344-2347
[40]Roy R,Kailath T."ESPRIT-estimation of signal parameters via rotational invariance techniques," IEEE Trans.on ASSP,1989,37(7):984-995
[41]王永良 陈辉 彭应宁 等 空间谱估计理论与算法 清华大学出版社 北京,2004
[42]Sebastian Miron,Nicolas Le Behan,Jerome I Mars," Quatemion-MUSIC for Vector-Sensor Array Processing",IEEE transactions on signal processing,vol.54,no.4,pp.1218-1229,2006
[43]Sebastian Miron,Nicolas Le Behan,Jerome I Mars,"High Resolution Vector-Sensor Array Processing Based on Biquaternions",ICASSP,2006
[44]Sebastian Miron,Nicolas Le Behan,Jerome I Mars," Singular value decomposition of quaternion matrices:a new tool for vector-sensor signal processing",Signal Processing 84,pp.1177-1199,2004
[45]Todd A.Ell,Stephen J.Sangwine,"Hypercomplex Fourier Transforms of Color Images",IEEE transactions on image processing,vol.16,no.2,pp.1-14,2006
[46]李闻亮,四元数矩阵,国防科技大学出版社,北京
[47]Petre Stoica,Peter Handel,Torsten Soderstrom,"study of Capon method for array signal processing" circuits system signal process VOL.14,No.6,1995,pp.749-770
[48]Yanwei Wang,Jian Li,IEEE,Petre Stoica,"Rank-Deficient Robust Capon Filter Bank Approach to Complex Spectral Estimation,"IEEE transactions on signal processing,vol.53,no.8,Aug.2005,pp.2173-2726
[49]Petre Stoica,Zhisong Wang,Jian Li," Robust Capon Beamforming," IEEE signal processing letters,vol.10,no.6,June 2003,pp.172-175
[50]Jian Li,Petre Stoica," An adpative Filtering Approach to Spectral Estimation and SAR Imaging," IEEE transactions on signal processing,vol.44,no.6,June 1996,pp.1469-1484
[51]Petre Stoica,Hongbin Li,and Jian Li," A New Derivation of the APES Filter,"IEEE signal processing letters,vol.6,no.8,Aug.1999,pp.205-206
[52]Petre Stoica,Hongbin Li,and Jian Li," Amplitude Estimation of Sinusoidal Signals:Survey,New Results,and an Application," IEEE Trans.on signal processing,vol.48,no.2,Feb.2000,pp.338-352
[53]Andreas Jakobsson,Petre Stoica," Combing Capon and APES for estimation of spectral lines,"circuits systems signal process vol.19,no.2,2000,pp.159-169
[54]Zheng-She Liu,Hong Bin Li,Jian Li,"Efficient Implementationi of Capon and APES for Spectral Estimation," IEEE Trans.on Aerospace and electronic systems vol.34,no.4 Oct.1998,pp.1314-1319