空间谱估计技术研究
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摘要
空间谱估计是阵列信号处理的一个重要方向,在雷达、通信、军事上等众多领域有着极为广阔的前景。本文围绕空间谱测向技术这一主题,系统的总结了空间谱发展历程出现的经典算法,并对涉及到的大多数算法作了详细的仿真与对比性能分析。
本文的主要工作体现在以下三个方面:
1、系统的总结了空间谱算法中的经典方法。Music和ESPRIT方法是空间谱估计技术中最重要的技术方法。本文首先从成功概率、估计误差、估计方差三个方面比较了这两个算法的性能。高阶谱分析法是近几年出现的算法。本文给出了基于四阶谱的Music和ESPRIT算法的谱搜索公式,详细的分析了在非高斯噪声下两种算法的性能。同时,本文也考察了基于二阶谱的传统Music和ESPRIT在非高斯噪声下的性能,从而证明了高阶谱分析方法的优越性。
2、详细分析长基线角度估计的相位模糊问题。本文首先通过计算机仿真的方法给出了相位模糊的实例。在这个基础上,本文从空间谱分析的角度解释了产生相位模糊的原因,并且对其做了定量的数学推倒,给出了相位模糊数的公式。
3、在传统内插法的基础上,本文提出了基于长短基线的内插法。长短基线法借用内插法基本思想,将长短基线阵列映射到一个均匀直线阵列(ULA)上去,然后利用性能优良的现代谱估计方法得到波达角度。该方法的创新之处在于,将长短基线阵列与内插法相结合,既回避了长短基线中复杂烦冗的解相位模糊算法,又克服了内插法中扇区分区大小于精确度、覆盖范围的矛盾。同时又因为所引入了长短基线的结构优势,以最小的阵元个数代价获得了尽可能大的天线孔径,与同阵元的智能天线相比可以大大提高算法的分辨力。仿真结果表明,长短基线结合法在阵元数相等的情况下,能比传统内插算法的分辨力高出1倍以上。
Spatial spectrum estimation is an important area in array signal processing, which is widely used in radar, communication and military. In this paper we introduce this topic in five chapters.
In chapter one, we introduce the system of the radio DOA estimations and the research and practice of smart-antenna. The main content and structure was introduced in the end of this chapter.
In chapter two, first we introduce the history of the spatial spectrum technology. Second, we describe the conception and the nomenclature of spatial spectrum technology. We analysis the model of the radio propagation channel for the last chapters.
In chapter three, we introduce five classes of spatial spectrum algorithm. The CBF algorithm was introduced in the first section. For strictly, this algorithm is out of the spatial spectrum system, but it work on the smart antenna, just as the spatial spectrum algorithm. It can be looked as the origin of the modern spectrum algorithm, so it was arranged in this chapter. The MUSIC algorithm and the ESPRIT algorithm are the classic algorithm in the DOA estimation system. They were introduced in the second and the third section respectively. In the last section we introduce the smooth algorithm, it can not exist independency, it can be looked as the engineering edition of the foregoing algorithm.
The interpolation algorithm was introduced in the chapter four. This algorithm was proposed by Friedlander in 1993. But there are some difficulties in the algorithm are sill open problems. In section four, we introduce a new algorithm called long-short base algorithm. This algorithm is the innovation of the author. It overcomes the drawback of the traditional algorithm, and has great performance progressing. The performance analysis is in the last section.
Chapter five is the summarization of this paper. It reviews this paper and point out the drawback of the research at present, and proposed the research directions.
本文的主要工作体现在以下三个方面:
1、系统的总结了空间谱算法中的经典方法。Music和ESPRIT方法是空间谱估计技术中最重要的技术方法。本文首先从成功概率、估计误差、估计方差三个方面比较了这两个算法的性能。高阶谱分析法是近几年出现的算法。本文给出了基于四阶谱的Music和ESPRIT算法的谱搜索公式,详细的分析了在非高斯噪声下两种算法的性能。同时,本文也考察了基于二阶谱的传统Music和ESPRIT在非高斯噪声下的性能,从而证明了高阶谱分析方法的优越性。
2、详细分析长基线角度估计的相位模糊问题。本文首先通过计算机仿真的方法给出了相位模糊的实例。在这个基础上,本文从空间谱分析的角度解释了产生相位模糊的原因,并且对其做了定量的数学推倒,给出了相位模糊数的公式。
3、在传统内插法的基础上,本文提出了基于长短基线的内插法。长短基线法借用内插法基本思想,将长短基线阵列映射到一个均匀直线阵列(ULA)上去,然后利用性能优良的现代谱估计方法得到波达角度。该方法的创新之处在于,将长短基线阵列与内插法相结合,既回避了长短基线中复杂烦冗的解相位模糊算法,又克服了内插法中扇区分区大小于精确度、覆盖范围的矛盾。同时又因为所引入了长短基线的结构优势,以最小的阵元个数代价获得了尽可能大的天线孔径,与同阵元的智能天线相比可以大大提高算法的分辨力。仿真结果表明,长短基线结合法在阵元数相等的情况下,能比传统内插算法的分辨力高出1倍以上。
Spatial spectrum estimation is an important area in array signal processing, which is widely used in radar, communication and military. In this paper we introduce this topic in five chapters.
In chapter one, we introduce the system of the radio DOA estimations and the research and practice of smart-antenna. The main content and structure was introduced in the end of this chapter.
In chapter two, first we introduce the history of the spatial spectrum technology. Second, we describe the conception and the nomenclature of spatial spectrum technology. We analysis the model of the radio propagation channel for the last chapters.
In chapter three, we introduce five classes of spatial spectrum algorithm. The CBF algorithm was introduced in the first section. For strictly, this algorithm is out of the spatial spectrum system, but it work on the smart antenna, just as the spatial spectrum algorithm. It can be looked as the origin of the modern spectrum algorithm, so it was arranged in this chapter. The MUSIC algorithm and the ESPRIT algorithm are the classic algorithm in the DOA estimation system. They were introduced in the second and the third section respectively. In the last section we introduce the smooth algorithm, it can not exist independency, it can be looked as the engineering edition of the foregoing algorithm.
The interpolation algorithm was introduced in the chapter four. This algorithm was proposed by Friedlander in 1993. But there are some difficulties in the algorithm are sill open problems. In section four, we introduce a new algorithm called long-short base algorithm. This algorithm is the innovation of the author. It overcomes the drawback of the traditional algorithm, and has great performance progressing. The performance analysis is in the last section.
Chapter five is the summarization of this paper. It reviews this paper and point out the drawback of the research at present, and proposed the research directions.
引文
[1] Chryssomallis M. Smart Antennas [J].IEEE Antennas and Propagation Mag,2000,42(3): 129-136.
[2] Boukalov A O, H ggman S G System Aspects of Smart-Antenna Technology in Cellular Wireless Communications-An Overview. IEEE Tran. Microwave Theory and Tech. [J], 2000,48(6):919-929.
[3] Haykin S. Adaptive Filter Theory (Third Edition)[M]. 北京: 电子工业出版社, 1998.
[4] 吴伟龙 美专家设计出手机智能天线[EB] http://tech.sina.com.cn/h/n45521.thml, 2001.
[5] Krim H, Viberg M. Two decades of array signal processing research. IEEE Signal Processing Magzine, 1996,13(4): 67-94
[6] Kay S M , Marple S L. Spectrum analysis-a modern perspective. Proc. of IEEE,1981,11
[7] Burg J P. Maximum entropy spectral analysis. Proc. of the 37th meeting of the Annual Int. SEG Meeting, Oklahoma City, OK, 1967
[8] Capon J. High-resolution frequency-wavenumber spectrum analysis. Proc. of IEEE, 1969, 57(8): 1408-1418
[9] Kumaresan R, Tufts D W. Estimating the angles of arrival of multiple plane waves. IEEE Trans, on AES ,1983,19(1) 134-139
[10] Special issue on adaptive antennas. IEEE Trans. Antennas Propagat, vol. AP-12, Mar. 1964.
[11] Special issue on adaptive antennas. IEEE Trans. Antennas Propagat. vol. AP-24,Sept.1976.
[12] Special issue on adaptive systems and applications. IEEE Trans. Circuits Syst. vol. CAS-34, July 1987
[13] Schmidt R O. Multiple emitter location and signal parameters estimation. IEEE Trans. 1986,AP-34(3):276~280
[14] 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
[15] Rao B D, Hari K S. Performance analysis of Root-MUSIC. IEEE Trans, on ASSP, 1989,27(12):1939-1949
[16] Kung S Y, Arun 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
[17] Roy R ,Kailath T. ESPRIT-a subspace rotation approach to estimation of parameters of cissoids. IEEE Trans. on ASSP, 1986,34(10): 1340-1342
[18] Roy R, Kailath T. ESPRIT-estimation of signal parameters via rotational invariance techniques. IEEE Trans, on ASSP,1989,37(7):984-995
[19] Stoica P,Nehorai A. MUSIC, Maximum likelihood, and Cramer-Rao bound. In Proc. ICASSP, 1988, 2296-2299
[20] 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, Springer-Verlag, Berlin, 1993,99-151
[21] 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
[22] Krim H, Viberg M. Two decades of array signal processing research. IEEE signal processing magazine, 1996,13(4): 67-94
[23] Clergeot H, Tressens S, Ouamri A. Performance of high resolution frequencies estimation methods compared to the Cramer-Rao bound. IEEE Trans. on ASSP, 1989,37(11):1703～1720
[24] Ziskind I, Wax M, Maximum likelihood localization of multiple sources by alternating projection.IEEE Trans. on ASSP, 1988,36(10):1553～1559
[25] Ching-Sung Shieh, Chin-Teng Lin. Direction of arrival estimation based on phase differences usin neural fuzzy network. IEEE Transactions on Antennas and Propagation, 2000,48(7):1115～1124
[26] Maruyama T, Kuwahara Y. Estimation of propagation structure by means of Hopfield neural network. IEEE Antennas and Propagation Society International Symposium. Boston, MA,USA,2001,l:476～479
[27] Chen Hui, Wang Yongliang. A Modified Method of Frequency and 2-D Angle Estimation. IEEE antennas and propagation society international symposium, Columbus, OH, USA, 22-27, Jun, 2003.
[28] Chen Hui, Wang Yongliang, Wu Zhiwen. Frequency and 2-D Angle Estimation Based on Uniform Circular Array. IEEE international symposium on phased array systems and technology, Boston, Massachusetts, USA, 22-27, Oct. 2003.
[29] Ning Ma, Joo Thiam Goh. Ambiguity-function-based techniques to estimate DOA of broadband chirp signals.Signal Processing, IEEE Transactions on [see also Acoustics, Speech, and Signal Processing, IEEE Trans .Volume 54, Issue 5, May 2006 Page(s):1826 - 1839
[30] Tuan Do-Hong, Russer, P. Wideband direction-of-arrival estimation using spatially interpolated wideband beamformers. Antennas and Propagation Society International Symposium, 2004. IEEE, Volume 3, 20-25 June 2004 Page(s):2647 -2650 Vol.3.
[31] Sellone F. Robust wideband DOA estimation. Statistical Signal Processing, 2005 IEEE/SP 13th Workshop on 17-20 July 2005 Page(s):277 - 282.
[32] Hanchoon Park; Hyungmyung Kim. A new direction finding scheme for DS/CDMA systems with array antenna. MILCOM 2000. 21st Century Military Communications Conference Proceedings Volume 2, 22-25 Oct. 2000 Page(s):836 - 840.
[33] Tai Ching , Chiang Chang. DOA estimation in the asynchronous DS-CDMA system. Antennas and Propagation, IEEE Transactions on Volume 51, Issue 1, Jan. 2003 Page(s):40 - 47.
[34] Li Ping'an, An efficient DOA estimation technique for multi-user multi-path signals in TD-SCDMA systems. Microwave, Antenna, Propagation and EMC Technologies for Wireless Communications, 2005. MAPE 2005. IEEE International Symposium on Volume 2, 8-12 Aug. 2005 Page(s):1382 - 1385.
[35]Lau,B.K,Viberg,M,Leung,Y.H.Data-adaptive array interpolation for DOA estimation in correlated signal environments.Acoustics,Speech,and Signal Processing,2005.Proceedings.(ICASSP '05).IEEE International Conference on Volume 4,18-23 March 2005 Page(s):ⅳ/945 - ⅳ/948 Vol.4.
[36]Hyberg P,Jansson M,Ottersten B.Array interpolation and bias reduction Signal Processing,IEEE Transactions on[see also Acoustics,Speech,and Signal Processing,IEEE Transactions on Volume 52,Issue 10,Part 1,Oct.2004Page(s):2711-2720.
[37]Friedlander B.The Root-MUSIC algorithm for direction finding with interpolated arrays.IEEE Trans.on Signal Processing,1993,41(1):15-25
[38]Gold smith.Wireless Communications.First Edition.Cambridge University Press,2005.
[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]Stoica P,Nehorai A.MUSIC,maximum likelihood,and Cramer-Rao bound.IEEE Trans.on ASSP,1989,37(50):720-741.
[41]Zhou C,Haber F,Jaggard D L.A resolution measure for the MUSIC algorithm and its application to plane wave arrivals contaminated by coherent interference.IEEETrans.on SP,1991,39(2):454-463.
[42]Rao B D,Hari K V S.Weighted subspace methods and spatial smoothing:analysis and comparison.IEEE Trans.on SP,1993,41(2):788-803.
[43]Barabell A J.Improving the resolution performance of eigenstructure-based direction-finding algorithm.ICASSP,1983,336-339.
[44]Ren Q S,Willis A J.Fast Root-MUSIC algorithm,IEE Electronics Letters,1997,33(6):450-452.
[45]熊良芳,司锡才,刁鸣.“一种舰载任意形状天线阵列超分辨测向技术的研究”应用科技vol.28,Feb.,2001.
[46]刁鸣,孙鉴.“二维超分辨测向天线阵性能仿真与分析”,弹箭与制导学报,2005-08-29.
[47]Nemirovsky,S,Doron,M.A."Sensitivity of MUSIC and Root-MUSIC to gain calibration errors of 2D arbitrary array configuration." Sensor Array and Multichannel Signal Processing Workshop Proceedings,2004,18-21 July 2004Page(s):594-598
[48]Lau B K,Cook G J,Leung Y.H.An improved array interpolation approach to DOA estimation in correlated signal environments Acoustics,Speech,and Signal Processing,2004.Proceedings.(ICASSP '04).IEEE International Conference on Volume 2,17-21 May 2004 Page(s):237-240 vol.2.
[49]Seunghyeon Hwang,Sarkar T K.Direction of arrival(DOA)estimation using a transformation matrix through singular value decomposition.Wireless Communications and Applied Computational Electromagnetics,2005.IEEE/ACES International Conference on 3-7 April 2005 Page(s):353-356.
[50]赵勇慧.提高相位干涉仪测向精度与改善测角范围的探讨.火控雷达技术,第31卷,34-37.2002.6.
[51]沈福民,贾永康.相位测距中的解模糊技术.西安电子科技大学学报.vol.24, 52-57.1997.3.
[52]池庆玺,司锡才,卓志敏.相位测向系统中解模糊方法研究.弹箭与制导学报.第25卷第4期,267-270.2005.
[2] Boukalov A O, H ggman S G System Aspects of Smart-Antenna Technology in Cellular Wireless Communications-An Overview. IEEE Tran. Microwave Theory and Tech. [J], 2000,48(6):919-929.
[3] Haykin S. Adaptive Filter Theory (Third Edition)[M]. 北京: 电子工业出版社, 1998.
[4] 吴伟龙 美专家设计出手机智能天线[EB] http://tech.sina.com.cn/h/n45521.thml, 2001.
[5] Krim H, Viberg M. Two decades of array signal processing research. IEEE Signal Processing Magzine, 1996,13(4): 67-94
[6] Kay S M , Marple S L. Spectrum analysis-a modern perspective. Proc. of IEEE,1981,11
[7] Burg J P. Maximum entropy spectral analysis. Proc. of the 37th meeting of the Annual Int. SEG Meeting, Oklahoma City, OK, 1967
[8] Capon J. High-resolution frequency-wavenumber spectrum analysis. Proc. of IEEE, 1969, 57(8): 1408-1418
[9] Kumaresan R, Tufts D W. Estimating the angles of arrival of multiple plane waves. IEEE Trans, on AES ,1983,19(1) 134-139
[10] Special issue on adaptive antennas. IEEE Trans. Antennas Propagat, vol. AP-12, Mar. 1964.
[11] Special issue on adaptive antennas. IEEE Trans. Antennas Propagat. vol. AP-24,Sept.1976.
[12] Special issue on adaptive systems and applications. IEEE Trans. Circuits Syst. vol. CAS-34, July 1987
[13] Schmidt R O. Multiple emitter location and signal parameters estimation. IEEE Trans. 1986,AP-34(3):276~280
[14] 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
[15] Rao B D, Hari K S. Performance analysis of Root-MUSIC. IEEE Trans, on ASSP, 1989,27(12):1939-1949
[16] Kung S Y, Arun 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
[17] Roy R ,Kailath T. ESPRIT-a subspace rotation approach to estimation of parameters of cissoids. IEEE Trans. on ASSP, 1986,34(10): 1340-1342
[18] Roy R, Kailath T. ESPRIT-estimation of signal parameters via rotational invariance techniques. IEEE Trans, on ASSP,1989,37(7):984-995
[19] Stoica P,Nehorai A. MUSIC, Maximum likelihood, and Cramer-Rao bound. In Proc. ICASSP, 1988, 2296-2299
[20] 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, Springer-Verlag, Berlin, 1993,99-151
[21] 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
[22] Krim H, Viberg M. Two decades of array signal processing research. IEEE signal processing magazine, 1996,13(4): 67-94
[23] Clergeot H, Tressens S, Ouamri A. Performance of high resolution frequencies estimation methods compared to the Cramer-Rao bound. IEEE Trans. on ASSP, 1989,37(11):1703～1720
[24] Ziskind I, Wax M, Maximum likelihood localization of multiple sources by alternating projection.IEEE Trans. on ASSP, 1988,36(10):1553～1559
[25] Ching-Sung Shieh, Chin-Teng Lin. Direction of arrival estimation based on phase differences usin neural fuzzy network. IEEE Transactions on Antennas and Propagation, 2000,48(7):1115～1124
[26] Maruyama T, Kuwahara Y. Estimation of propagation structure by means of Hopfield neural network. IEEE Antennas and Propagation Society International Symposium. Boston, MA,USA,2001,l:476～479
[27] Chen Hui, Wang Yongliang. A Modified Method of Frequency and 2-D Angle Estimation. IEEE antennas and propagation society international symposium, Columbus, OH, USA, 22-27, Jun, 2003.
[28] Chen Hui, Wang Yongliang, Wu Zhiwen. Frequency and 2-D Angle Estimation Based on Uniform Circular Array. IEEE international symposium on phased array systems and technology, Boston, Massachusetts, USA, 22-27, Oct. 2003.
[29] Ning Ma, Joo Thiam Goh. Ambiguity-function-based techniques to estimate DOA of broadband chirp signals.Signal Processing, IEEE Transactions on [see also Acoustics, Speech, and Signal Processing, IEEE Trans .Volume 54, Issue 5, May 2006 Page(s):1826 - 1839
[30] Tuan Do-Hong, Russer, P. Wideband direction-of-arrival estimation using spatially interpolated wideband beamformers. Antennas and Propagation Society International Symposium, 2004. IEEE, Volume 3, 20-25 June 2004 Page(s):2647 -2650 Vol.3.
[31] Sellone F. Robust wideband DOA estimation. Statistical Signal Processing, 2005 IEEE/SP 13th Workshop on 17-20 July 2005 Page(s):277 - 282.
[32] Hanchoon Park; Hyungmyung Kim. A new direction finding scheme for DS/CDMA systems with array antenna. MILCOM 2000. 21st Century Military Communications Conference Proceedings Volume 2, 22-25 Oct. 2000 Page(s):836 - 840.
[33] Tai Ching , Chiang Chang. DOA estimation in the asynchronous DS-CDMA system. Antennas and Propagation, IEEE Transactions on Volume 51, Issue 1, Jan. 2003 Page(s):40 - 47.
[34] Li Ping'an, An efficient DOA estimation technique for multi-user multi-path signals in TD-SCDMA systems. Microwave, Antenna, Propagation and EMC Technologies for Wireless Communications, 2005. MAPE 2005. IEEE International Symposium on Volume 2, 8-12 Aug. 2005 Page(s):1382 - 1385.
[35]Lau,B.K,Viberg,M,Leung,Y.H.Data-adaptive array interpolation for DOA estimation in correlated signal environments.Acoustics,Speech,and Signal Processing,2005.Proceedings.(ICASSP '05).IEEE International Conference on Volume 4,18-23 March 2005 Page(s):ⅳ/945 - ⅳ/948 Vol.4.
[36]Hyberg P,Jansson M,Ottersten B.Array interpolation and bias reduction Signal Processing,IEEE Transactions on[see also Acoustics,Speech,and Signal Processing,IEEE Transactions on Volume 52,Issue 10,Part 1,Oct.2004Page(s):2711-2720.
[37]Friedlander B.The Root-MUSIC algorithm for direction finding with interpolated arrays.IEEE Trans.on Signal Processing,1993,41(1):15-25
[38]Gold smith.Wireless Communications.First Edition.Cambridge University Press,2005.
[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]Stoica P,Nehorai A.MUSIC,maximum likelihood,and Cramer-Rao bound.IEEE Trans.on ASSP,1989,37(50):720-741.
[41]Zhou C,Haber F,Jaggard D L.A resolution measure for the MUSIC algorithm and its application to plane wave arrivals contaminated by coherent interference.IEEETrans.on SP,1991,39(2):454-463.
[42]Rao B D,Hari K V S.Weighted subspace methods and spatial smoothing:analysis and comparison.IEEE Trans.on SP,1993,41(2):788-803.
[43]Barabell A J.Improving the resolution performance of eigenstructure-based direction-finding algorithm.ICASSP,1983,336-339.
[44]Ren Q S,Willis A J.Fast Root-MUSIC algorithm,IEE Electronics Letters,1997,33(6):450-452.
[45]熊良芳,司锡才,刁鸣.“一种舰载任意形状天线阵列超分辨测向技术的研究”应用科技vol.28,Feb.,2001.
[46]刁鸣,孙鉴.“二维超分辨测向天线阵性能仿真与分析”,弹箭与制导学报,2005-08-29.
[47]Nemirovsky,S,Doron,M.A."Sensitivity of MUSIC and Root-MUSIC to gain calibration errors of 2D arbitrary array configuration." Sensor Array and Multichannel Signal Processing Workshop Proceedings,2004,18-21 July 2004Page(s):594-598
[48]Lau B K,Cook G J,Leung Y.H.An improved array interpolation approach to DOA estimation in correlated signal environments Acoustics,Speech,and Signal Processing,2004.Proceedings.(ICASSP '04).IEEE International Conference on Volume 2,17-21 May 2004 Page(s):237-240 vol.2.
[49]Seunghyeon Hwang,Sarkar T K.Direction of arrival(DOA)estimation using a transformation matrix through singular value decomposition.Wireless Communications and Applied Computational Electromagnetics,2005.IEEE/ACES International Conference on 3-7 April 2005 Page(s):353-356.
[50]赵勇慧.提高相位干涉仪测向精度与改善测角范围的探讨.火控雷达技术,第31卷,34-37.2002.6.
[51]沈福民,贾永康.相位测距中的解模糊技术.西安电子科技大学学报.vol.24, 52-57.1997.3.
[52]池庆玺,司锡才,卓志敏.相位测向系统中解模糊方法研究.弹箭与制导学报.第25卷第4期,267-270.2005.