高分辨测向阵列几何结构研究
摘要
在通信、雷达、声纳、地震勘测及生物医学工程等领域里，对辐射源的方向估计是一项重要的课题。传感器阵列可利用辐射源信号抵达不同阵元的波程差进行测向，因而成为当今测向的主要工具。
来自不同方向的信号呈现出不同的空间频率，测向可以通过对这些信号空间频率的估计来实现。阵列各阵元是安置在空间的采样点，可以通过这些采样点上获得的信号样本来估计信号的空间频率，进而获知信号的波达方向，可见这些采样点的布局方式，也就是阵列的(几何)结构，对系统的测向性能有着重要影响。因此对阵列结构的研究非常必要。
阵列结构对系统的影响还与系统所采用的测向算法有关。目前最流行的一类算法是基于子空间分解的算法，该方法具有分辨力高、计算复杂度小等优点。但是由于它的高度非线性，采用该算法的系统性能与阵列结构的关系较为复杂。作为阵列结构的设计优化准则，需要对这种关系进行研究，进而得到适用的设计方法。
困扰子空间类算法的一个重要问题是模糊问题。流形模糊发生的原因是阵列流形上出现了线性相关的导向矢量。因为阵列流形是阵列结构的函数，所以阵列结构对模糊有着直接的影响。对模糊问题的研究可分两方面：
1、分析阵列结构与模糊的关系。研究某种结构的阵列中是否存在模糊、模糊都发生在哪些角度上。
2、当模糊发生后，研究这些模糊是否可解，如果可解，找到解决模糊问题的方法。
针对上述阵列结构问题，本文给出如下一些新的研究结果：
1、提出一种基于阵列流形长度的指标来比较不同结构线阵的测向精度。
传统上评价一个阵列的测向性能一般是考察其孔径的大小，但是除了阵列的孔径之外，阵元的数量、各阵元的分布位置等都是影响阵列测向性能的重要因素。作者所提出的指标可将这些因素通盘考虑以评价阵列的测向精度。
2、推出了立体阵对波达方向估计的CRLB。
平面阵可用来测入射波的方位角和仰角，但是在低仰角区域，它对仰角的估计变差。立体阵可克服这一缺点。我们推出了立体阵测向的CRLB与其阵列结构的直接关系，证实了立体阵在这方面的优点。
3、本文给出了线阵、面阵的阵列结构误差与其所造成的子空间算法的测向误差之间的关系。讨论了若干阵列对结构误差的敏感性。
由于实际阵列都不可避免地存在结构误差，这种误差会降低子空间算法的测向性能。本文通过对阵列流形的研究，给出了结构误差与测向误差之间的解析关系。
4、提出一种基于二维角度测向性能的面阵设计技术。
现有的面阵设计方法只考虑到仰角的性能，而面阵可完成二维角度测向，所以，在面阵设计时最好能考虑到二维角度上的测向性能。作者在把波达方向的方位角、仰角坐标系转化为锥角坐标后，提出了这种能兼顾二维方向性能的面阵设计技术。
5、给出了任意线阵、面阵在仰角方向上无秩二模糊的条件。由于阵列流形的高度非线性，至今对阵列结构与流形模糊的研究仍不完善。鉴于低阶模糊对测向系统的影响更大，本文就任意线阵、面阵在仰角方向上的秩二模糊进行了研究，给出了无秩二模糊的条件。
6、提出了一种采用最陡下降优化技术的解模糊方法。
采用稀疏阵列可以提高系统测向性能、节约成本，但是必然会产生流形模糊，因此需要解模糊。由于现有的解模糊方法对系数误差较为敏感，故提出采用了最陡下降优化方法的解模糊算法，实验证明这种方法更为稳健。
In such fields as communication,radar,sonar,seismology and biomedicalengineering,the estimation of Direction of Arrival(DOA) is a very important subject.Sensors array is often employed as the equipment of Direction Finding(DF) because thephase difference of impinging signals between the sensors of array can be used toestimate DOA.
The impinging signals coming from different DOAs exhibit different spatialfrequencies,the estimation of DOAs can be implemented by estimating spatialfrequencies.The sensors of array are sampling points arranged spatially,samples fromthe sensors can be used to estimate the spatial frequencies,i.e.,the DOAs,so,thearrangement of the spatial sampling points,namely,the array geometry,exerts greatinfluence on the performance of DF system.The study of array geometry is an essentialtopic in array signal processing.
The influence of array geometry on the performance of DF system depends onDOA estimation algorithms.Subspace algorithms are popular for their high resolutionand low complexity.For the nonlinearity of those algorithms,the relationship betweenarray geometry and the performance of the DF system adopting the algorithms iscomplex.Even though,as the criteria of design and optimization of array geometry,therelationship needs to be studied.
One problem which subspace algorithms suffer is ambiguity.Manifoldambiguities occur when there are linear dependent steer vectors on array manifold.Manifold is a function of array geometry,so the ambiguities are determined by arraygeometry.The study of ambiguity consists of two parts:
1.Studying the relationship between the array geometry and ambiguity;Findingthe existence condition of ambiguity on certain array geometry and locating theambiguities.
2.Determining if the ambiguities can be resolved;Designing method to resolvethe ambiguities.
About the geometry of array,the following novel results are presented in this thesis.
1.Based on minimal manifold length,an index is suggested,the index can beused to evaluate the accuracy performance of given linear array.
Conventionally,aperture is often used as an index to compare the performance ofdifferent linear arrays,but it does not take into account the number of sensor and thearray geometry.We propose an index to evaluate accuracy of linear array,the indexconsiders the whole information of array geometry.
2.The CRLB of DOA estimation on volume array is deduced.
Planar array can be used to estimate azimuth and elevation,but in the zone of lowelevation,the estimation of elevation becomes worse.To overcome the shortcoming,the application of volume array is suggested.In this thesis,we deduce the relationshipbetween the CRLB of DOA estimation on volume array and the geometry of the volumearray.The CRLB verify the experience that volume array outperform planar array inlow elevation zone
3.The relationship between the geometry errors of array and errors of DOAsestimated by subspace algorithm is deduced.The sensitivity of several planar arrayswith respect to geometry errors is studied.
In practice,the arrays suffer geometry errors,this leads to DF errors,so therelationship between them is very important in DF system implementation.Based on thestudy of array manifold,we derive the relationship.
4.Based on specifications on two independent cone angles,an approach of planararray design is suggested.
The planar array can estimate two dimensional independent angles,but knowndesign technique of planar array only considers specifications of elevation,so the designapproach that considers the specifications of two dimensional angles is needed.Afterparameters transform from azimuth-elevation to cone angles,one such design approachis proposed.
5.We identify the linear array free of rank-2 ambiguity and planar array free ofrank 2 elevation ambiguity on given azimuth.
Because the array manifold is nonlinear function of array geometry,the resultsabout relationship between array geometry and ambiguity are not sufficient.Low rankambiguity is more harmful to DF system,so we identify the linear array free of rank 2 ambiguity,and then,we identify the planar array free of rank 2 elevation ambiguity ongiven azimuth.
6.Based on steepest descent method,a robust approach to resolve ambiguity isproposed.
Sparse arrays have advantages of high performance and low cost,but they arecertain to suffer ambiguities,so the methods resolving manifold ambiguities arenecessary.Because the known methods are based on linear programming technique,they are sensitive to coefficient error,a robust method based on steepest descent methodis proposed,simulation results verify the robustness of the proposed method.
来自不同方向的信号呈现出不同的空间频率，测向可以通过对这些信号空间频率的估计来实现。阵列各阵元是安置在空间的采样点，可以通过这些采样点上获得的信号样本来估计信号的空间频率，进而获知信号的波达方向，可见这些采样点的布局方式，也就是阵列的(几何)结构，对系统的测向性能有着重要影响。因此对阵列结构的研究非常必要。
阵列结构对系统的影响还与系统所采用的测向算法有关。目前最流行的一类算法是基于子空间分解的算法，该方法具有分辨力高、计算复杂度小等优点。但是由于它的高度非线性，采用该算法的系统性能与阵列结构的关系较为复杂。作为阵列结构的设计优化准则，需要对这种关系进行研究，进而得到适用的设计方法。
困扰子空间类算法的一个重要问题是模糊问题。流形模糊发生的原因是阵列流形上出现了线性相关的导向矢量。因为阵列流形是阵列结构的函数，所以阵列结构对模糊有着直接的影响。对模糊问题的研究可分两方面：
1、分析阵列结构与模糊的关系。研究某种结构的阵列中是否存在模糊、模糊都发生在哪些角度上。
2、当模糊发生后，研究这些模糊是否可解，如果可解，找到解决模糊问题的方法。
针对上述阵列结构问题，本文给出如下一些新的研究结果：
1、提出一种基于阵列流形长度的指标来比较不同结构线阵的测向精度。
传统上评价一个阵列的测向性能一般是考察其孔径的大小，但是除了阵列的孔径之外，阵元的数量、各阵元的分布位置等都是影响阵列测向性能的重要因素。作者所提出的指标可将这些因素通盘考虑以评价阵列的测向精度。
2、推出了立体阵对波达方向估计的CRLB。
平面阵可用来测入射波的方位角和仰角，但是在低仰角区域，它对仰角的估计变差。立体阵可克服这一缺点。我们推出了立体阵测向的CRLB与其阵列结构的直接关系，证实了立体阵在这方面的优点。
3、本文给出了线阵、面阵的阵列结构误差与其所造成的子空间算法的测向误差之间的关系。讨论了若干阵列对结构误差的敏感性。
由于实际阵列都不可避免地存在结构误差，这种误差会降低子空间算法的测向性能。本文通过对阵列流形的研究，给出了结构误差与测向误差之间的解析关系。
4、提出一种基于二维角度测向性能的面阵设计技术。
现有的面阵设计方法只考虑到仰角的性能，而面阵可完成二维角度测向，所以，在面阵设计时最好能考虑到二维角度上的测向性能。作者在把波达方向的方位角、仰角坐标系转化为锥角坐标后，提出了这种能兼顾二维方向性能的面阵设计技术。
5、给出了任意线阵、面阵在仰角方向上无秩二模糊的条件。由于阵列流形的高度非线性，至今对阵列结构与流形模糊的研究仍不完善。鉴于低阶模糊对测向系统的影响更大，本文就任意线阵、面阵在仰角方向上的秩二模糊进行了研究，给出了无秩二模糊的条件。
6、提出了一种采用最陡下降优化技术的解模糊方法。
采用稀疏阵列可以提高系统测向性能、节约成本，但是必然会产生流形模糊，因此需要解模糊。由于现有的解模糊方法对系数误差较为敏感，故提出采用了最陡下降优化方法的解模糊算法，实验证明这种方法更为稳健。
In such fields as communication,radar,sonar,seismology and biomedicalengineering,the estimation of Direction of Arrival(DOA) is a very important subject.Sensors array is often employed as the equipment of Direction Finding(DF) because thephase difference of impinging signals between the sensors of array can be used toestimate DOA.
The impinging signals coming from different DOAs exhibit different spatialfrequencies,the estimation of DOAs can be implemented by estimating spatialfrequencies.The sensors of array are sampling points arranged spatially,samples fromthe sensors can be used to estimate the spatial frequencies,i.e.,the DOAs,so,thearrangement of the spatial sampling points,namely,the array geometry,exerts greatinfluence on the performance of DF system.The study of array geometry is an essentialtopic in array signal processing.
The influence of array geometry on the performance of DF system depends onDOA estimation algorithms.Subspace algorithms are popular for their high resolutionand low complexity.For the nonlinearity of those algorithms,the relationship betweenarray geometry and the performance of the DF system adopting the algorithms iscomplex.Even though,as the criteria of design and optimization of array geometry,therelationship needs to be studied.
One problem which subspace algorithms suffer is ambiguity.Manifoldambiguities occur when there are linear dependent steer vectors on array manifold.Manifold is a function of array geometry,so the ambiguities are determined by arraygeometry.The study of ambiguity consists of two parts:
1.Studying the relationship between the array geometry and ambiguity;Findingthe existence condition of ambiguity on certain array geometry and locating theambiguities.
2.Determining if the ambiguities can be resolved;Designing method to resolvethe ambiguities.
About the geometry of array,the following novel results are presented in this thesis.
1.Based on minimal manifold length,an index is suggested,the index can beused to evaluate the accuracy performance of given linear array.
Conventionally,aperture is often used as an index to compare the performance ofdifferent linear arrays,but it does not take into account the number of sensor and thearray geometry.We propose an index to evaluate accuracy of linear array,the indexconsiders the whole information of array geometry.
2.The CRLB of DOA estimation on volume array is deduced.
Planar array can be used to estimate azimuth and elevation,but in the zone of lowelevation,the estimation of elevation becomes worse.To overcome the shortcoming,the application of volume array is suggested.In this thesis,we deduce the relationshipbetween the CRLB of DOA estimation on volume array and the geometry of the volumearray.The CRLB verify the experience that volume array outperform planar array inlow elevation zone
3.The relationship between the geometry errors of array and errors of DOAsestimated by subspace algorithm is deduced.The sensitivity of several planar arrayswith respect to geometry errors is studied.
In practice,the arrays suffer geometry errors,this leads to DF errors,so therelationship between them is very important in DF system implementation.Based on thestudy of array manifold,we derive the relationship.
4.Based on specifications on two independent cone angles,an approach of planararray design is suggested.
The planar array can estimate two dimensional independent angles,but knowndesign technique of planar array only considers specifications of elevation,so the designapproach that considers the specifications of two dimensional angles is needed.Afterparameters transform from azimuth-elevation to cone angles,one such design approachis proposed.
5.We identify the linear array free of rank-2 ambiguity and planar array free ofrank 2 elevation ambiguity on given azimuth.
Because the array manifold is nonlinear function of array geometry,the resultsabout relationship between array geometry and ambiguity are not sufficient.Low rankambiguity is more harmful to DF system,so we identify the linear array free of rank 2 ambiguity,and then,we identify the planar array free of rank 2 elevation ambiguity ongiven azimuth.
6.Based on steepest descent method,a robust approach to resolve ambiguity isproposed.
Sparse arrays have advantages of high performance and low cost,but they arecertain to suffer ambiguities,so the methods resolving manifold ambiguities arenecessary.Because the known methods are based on linear programming technique,they are sensitive to coefficient error,a robust method based on steepest descent methodis proposed,simulation results verify the robustness of the proposed method.
引文
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