应用压缩感知求解宽角度激励下三维电磁散射问题
|
摘要
对三维目标在宽角度激励下的电磁散射问题进行快速计算一直以来都是计算电磁学(CEM)的一个难点。矩量法(M0M)作为数值方法中积分方程方法的代表,尽管凭借其高计算精确度而成为分析电磁散射问题最主要的方法之一,且己发展出了多类快速算法,然而在面对宽角度问题时,至今却也仍无法有效实现快速求解。传统矩量法需针对每个入射角度分别反复迭代运算,因此耗时长,效率低。本文利用压缩感知(CS)这一信号处理领域近年来所提出的新技术,通过将其引入传统矩量法中,实现了对宽角度激励下三维目标电磁散射问题的快速求解,并在此基础上进一步的就优化该快速算法和将其应用到工程实践中进行了探索。主要工作及创新之处有：
(1)在对传统矩量法求解三维电磁散射问题和压缩感知技术进行前期调研的基础上,针对宽角度散射问题进行建模。
(2)设计一种新的入射源,该入射源包含宽角度下的丰富入射信息。通过该入射源的引入,使得在传统矩量法所原有矩阵方程的基础上,实现了对待求宽角度电流信号进行压缩感知计算所需的数次观测。由这数次观测得到的若干个含丰富入射角度信息的电流观测值,再结合CS中的稀疏转换基和恢复算法,便能高精度的还原出所有入射角度下的电流向量。该方法仅通过少量的观测即可完成对各入射角度下电流的最终求解,从而有效的实现了对宽角度散射问题的快速计算。
(3)对优化该快速算法的研究探索。将勒让德多项式、切比雪夫多项式、第二类切比雪夫多项式、拉盖尔多项式、埃尔米特多项式等正交多项式分别进行逐阶离散化,构造相应的正交基,并与傅里叶基、离散余弦变换基等常用稀疏转换基一起作为该算法中所使用的稀疏转换基,应用于各类不同形状的具体三维散射目标进行测试比较。实验结果表明,不同稀疏转换基下该快速算法所需观测次数不同。因此,通过构造更为良好的稀疏转换基可有效减少观测次数,从而优化该算法。此外,测试比较在使用各类随机观测矩阵和确定性观测矩阵时算法所需的观测次数,从选取观测矩阵的角度进一步对算法实施优化。
(4)对该快速算法所需先验知识的建立。提出一种基于物理光学法(PO)的可对宽角度下电流信号在进行稀疏表示时的投影稀疏度以及该快速算法所需的观测次数进行预估的先验方法,使得在运行该快速算法前便能获取各不同稀疏转换基下其分别所需观测次数的先验知识,从而为算法的实际运行特别是其中稀疏转换基的筛选提供了指导依据。先验技术的实现令该快速算法具备了真王意义上的实际应用能力,为最终的工程实践奠定了良好基础。
Fast algorithm for three dimensional (3D) electromagnetic scattering problem over a wide angle is always a difficult problem in computational electromagnetism (CEM). As a representative of integral equation methods, method of moments (MoM) is applied extensively since its advantage of high calculation accuracy to solve electromagnetic scattering problem, and many fast algorithms based on MoM have already been proposed and improved, however, there is no efficient technique for fast solving wide angle scattering problems as so far. Traditional MoM needs to calculate iteration each time as incident angle changes, so the operation time is long and the efficiency is low. Compressed sensing (CS) is a new technology proposed in the area of signal processing in recent years. By introducing it into traditional MoM,3D electromagnetic scattering problems over a wide angle can be solved rapidly, and based on this, some further improvement work of the new method and the topic of how to apply it into engineering practice are studied. The main work can be expanded as:
Firstly, on the basis of investigation of traditional MoM and CS theory, modeling for scattering problem over a wide angle is constructed for3D electromagnetic scattering problems.
Secondly, a new kind of excitation is constructed, which includes plentiful incident information over a wide angle. By these new excitations, and on the foundation of original matrix equation of traditional MoM, several measurements of unknown currents over the wide angle which is needed by CS computing are achieved. By these measured values which include plentiful information about incident angles and the other two element factors of CS-sparse transform matrix and recovery algorithm, currents over each angle of incidence can be reconstructed accurately. While using this method, currents over any incident angles can be calculated after only several measurements, thus fast calculation of scattering problem over a wide angle is finished.
Thirdly, some improvement of this new fast algorithm are researched and discovered. Five new sparse transform matrices are set up by discretization of five types of classical orthogonal polynomials-Legendre, Chebyshev, the second kind of Chebyshev, Laguerre and Hermite polynomials. By applying these matrcis and some common basis such as Fourier basis, discrete cosine transform basis into the new method as the sparse transform matrix to calculate several3D objects which have different shapes and comparing their performances, numerical results show that the number of times of measurement is different as sparse transform matrix changes. So a conclusion that the fast algorithm can be improved by constructing better sparse transform matrices is obtained. On the other hand, by comparing numbers of times of measurement while using different kinds of random or certain measurement matrix, the method is further improved from the point of view of measurement matrix.
Fourthly, apriori knowledges for the new method are established. A priori technique based on physical optics (PO) is proposed. By this technique, sparsity of the projection of current coefficients over the wide angle and the number of times of measurement can be pre-estimated, thus total number of measurements and choice of sparse transform can be acquired before operating the algorithm. This priori technique makes the fast algorithm possess abilities of practical application, and with the help of apriori knowledges, the method can be applied into engineering practice in a real sense.
(1)在对传统矩量法求解三维电磁散射问题和压缩感知技术进行前期调研的基础上,针对宽角度散射问题进行建模。
(2)设计一种新的入射源,该入射源包含宽角度下的丰富入射信息。通过该入射源的引入,使得在传统矩量法所原有矩阵方程的基础上,实现了对待求宽角度电流信号进行压缩感知计算所需的数次观测。由这数次观测得到的若干个含丰富入射角度信息的电流观测值,再结合CS中的稀疏转换基和恢复算法,便能高精度的还原出所有入射角度下的电流向量。该方法仅通过少量的观测即可完成对各入射角度下电流的最终求解,从而有效的实现了对宽角度散射问题的快速计算。
(3)对优化该快速算法的研究探索。将勒让德多项式、切比雪夫多项式、第二类切比雪夫多项式、拉盖尔多项式、埃尔米特多项式等正交多项式分别进行逐阶离散化,构造相应的正交基,并与傅里叶基、离散余弦变换基等常用稀疏转换基一起作为该算法中所使用的稀疏转换基,应用于各类不同形状的具体三维散射目标进行测试比较。实验结果表明,不同稀疏转换基下该快速算法所需观测次数不同。因此,通过构造更为良好的稀疏转换基可有效减少观测次数,从而优化该算法。此外,测试比较在使用各类随机观测矩阵和确定性观测矩阵时算法所需的观测次数,从选取观测矩阵的角度进一步对算法实施优化。
(4)对该快速算法所需先验知识的建立。提出一种基于物理光学法(PO)的可对宽角度下电流信号在进行稀疏表示时的投影稀疏度以及该快速算法所需的观测次数进行预估的先验方法,使得在运行该快速算法前便能获取各不同稀疏转换基下其分别所需观测次数的先验知识,从而为算法的实际运行特别是其中稀疏转换基的筛选提供了指导依据。先验技术的实现令该快速算法具备了真王意义上的实际应用能力,为最终的工程实践奠定了良好基础。
Fast algorithm for three dimensional (3D) electromagnetic scattering problem over a wide angle is always a difficult problem in computational electromagnetism (CEM). As a representative of integral equation methods, method of moments (MoM) is applied extensively since its advantage of high calculation accuracy to solve electromagnetic scattering problem, and many fast algorithms based on MoM have already been proposed and improved, however, there is no efficient technique for fast solving wide angle scattering problems as so far. Traditional MoM needs to calculate iteration each time as incident angle changes, so the operation time is long and the efficiency is low. Compressed sensing (CS) is a new technology proposed in the area of signal processing in recent years. By introducing it into traditional MoM,3D electromagnetic scattering problems over a wide angle can be solved rapidly, and based on this, some further improvement work of the new method and the topic of how to apply it into engineering practice are studied. The main work can be expanded as:
Firstly, on the basis of investigation of traditional MoM and CS theory, modeling for scattering problem over a wide angle is constructed for3D electromagnetic scattering problems.
Secondly, a new kind of excitation is constructed, which includes plentiful incident information over a wide angle. By these new excitations, and on the foundation of original matrix equation of traditional MoM, several measurements of unknown currents over the wide angle which is needed by CS computing are achieved. By these measured values which include plentiful information about incident angles and the other two element factors of CS-sparse transform matrix and recovery algorithm, currents over each angle of incidence can be reconstructed accurately. While using this method, currents over any incident angles can be calculated after only several measurements, thus fast calculation of scattering problem over a wide angle is finished.
Thirdly, some improvement of this new fast algorithm are researched and discovered. Five new sparse transform matrices are set up by discretization of five types of classical orthogonal polynomials-Legendre, Chebyshev, the second kind of Chebyshev, Laguerre and Hermite polynomials. By applying these matrcis and some common basis such as Fourier basis, discrete cosine transform basis into the new method as the sparse transform matrix to calculate several3D objects which have different shapes and comparing their performances, numerical results show that the number of times of measurement is different as sparse transform matrix changes. So a conclusion that the fast algorithm can be improved by constructing better sparse transform matrices is obtained. On the other hand, by comparing numbers of times of measurement while using different kinds of random or certain measurement matrix, the method is further improved from the point of view of measurement matrix.
Fourthly, apriori knowledges for the new method are established. A priori technique based on physical optics (PO) is proposed. By this technique, sparsity of the projection of current coefficients over the wide angle and the number of times of measurement can be pre-estimated, thus total number of measurements and choice of sparse transform can be acquired before operating the algorithm. This priori technique makes the fast algorithm possess abilities of practical application, and with the help of apriori knowledges, the method can be applied into engineering practice in a real sense.
引文
[1]W. Geraint, V. Rosser. Interpretation of Classical Electromagnetism [M]. Netherlands:Springer, 1997:1-426.
[2]盛新庆.计算电磁学要论[M].北京：科学出版社,2004：1-153.
[3]A. Bossavit. Computational Electromagnetism [M]. USA:Academic Press,1998:1-352.
[4]S. Rao. The Finite Element Method in Engineering [M]. USA:Butterworth-Heinemann,2004: 1-688.
[5]哈林顿.计算电磁场的矩量法[M].北京：国防工业出版社,1981：1-262.
[6]葛德彪,闫玉波.电磁波时域有限差分方法[M].西安：西安电子科技大学出版社,2005：1-376.
[7]S. Sauter, C. Schwab. Boundary Element Methods [M]. Germany:Springer Berlin Heidelberg, 2011:183-287.
[8]A. Mickelson. Physical Optics [M]. USA:Springer US,1992:1-345.
[9]赵为粮.目标散射的复射线分析及其误差控制技术[D].成都：电子科技大学,2002.
[10]吕英华.计算电磁学的数值方法[M].北京：清华大学出版社,2006：1-468.
[11]P. B. Zhou. Numerical Analysis of Electromagnetic Fields [M]. Germany:Springer Berlin Heidelberg,1993:1-406.
[12]陈毅乔.电大尺寸目标的电磁特性分析方法研究[D].成都：电子科技大学,2008.
[13]徐翠.高频算法在电磁散射计算中的应用[D].南京：南京理工大学,2009.
[14]R. Baraniuk. Compressive sensing [J]. IEEE Signal Processing Magazine,2007,24(4):118-121.
[15]E. Candes. Compressive Sampling [C]. European Mathematical Society. Proceedings of the International Congress of Mathematicians. Madrid: American Mathematical Society,2006: 1433-1452.
[16]J. Tropp, J. Laska, M. Duarte, J. Romberg and R. Baraniuk. Beyond Nyquist Efficient Sampling of Sparse Bandlimited Signals [J]. IEEE Transactions on Information Theory,2010,56(1):520-544.
[17]M. S. Aly, T. T. Y. Wong. Combined Mie Series and Asymptotic Formulation of Transient Scattered Field and Perfectly Conducting Sphere [C]. IEEE. Antennas and Propagation Society International Symposium. Syracuse, NY, USA:IEEE Press,1988:664-667.
[18]B. Fowler. Expansion of Mie-theory Phase Functions in Series of Legendre Polynomials [J]. Journal of the Optical Society of America,1983,73(1):19-22.
[19]M. Catedra, F. Ruiz, E. Gago. Analysis of Arbitrary Metallic Surfaces Conformed to a Circular Cylinder Using the Conjugate Gradient-fast Fourier Transform (CG-FFT) Method [J]. IEEE Transactions on Antennas and Propagation,1990,38(2):286-289.
[20]W. B. Ewe, L. W. Li, C. S. Chang, et al. AIM Analysis of Scattering and Radiation by Arbitrary Surface-wire Configurations [J]. IEEE Transactions Antennas Propagation,2007,55(1):162-166.
[21]K. Yoshida, N. Nishimura, S. Kobayashi. Application of New Fast Multipole Boundary Integral Equation Method to Crack Problems in 3D [J]. Engineering Analysis with Boundary Elements,2001, 25(4-5):239-247.
[22]J. M. Song, C. C. Lu, W. C. Chew. Multilevel Fast Multipole Algorithm for Electromagnetic Scattering by Large Complex Objects [J]. IEEE Transactions on Antennas and Propagation,1997, 45(10):1488-1493.
[23]彭朕,盛新庆.一种多层不完全LU分解预处理方法在合元极技术中的应用[J].电子学报,2008,36(2)：230-234.
[24]J. Zhu, R. S. Chen. SSOR Preconditioned CG Method for Formulation of Hierarchical Vector Finite Element [C]. IEEE. Antennas and Propagation Society International Symposium. Albuquerque, NM:IEEE Press,2006:1778-1781.
[25]P. L. Rui, R. S. Chen. An Efficient Sparse Approximate Inverse Preconditioning for FMM Implementation [J]. Microwave and Optical Technology Letters,2007,49(7):1746-1750.
[26]李月卉.电磁场数值求解中迭代方法与预条件技术研究[D].成都：电子科技大学,2011.
[27]柯涛,丁建军,丁大志,樊振宏,陈如山.特征谱双步预条件结合多分辨预条件技术快速分析电磁散射问题[C].中国电子学会.2007年全国微波毫米波会议论文集(上册).宁波：电子工业出版社,2007：170-173.
[28]X. C. Wei, E. P. Li. Wide-band EMC Analysis of On-platform Antennas Using Impedance-matrix Interpolation with the Moment Method-physical Optics Method [J]. IEEE Transactions on Electromagnetic Compatibility,2003,45(3):552-556.
[29]M. Burchett. Development of a Comfort Measurement Matrix [EB/OL]. http://www.lib.ncsu.edU/resolver/1840.16/197,2008-04-26.
[30]A. Atiya, M. Aly, A. Parlos. Sparse Basis Selection:New Results and Application to Adaptive Prediction of Video Source Traffic [J]. IEEE Transactions on Neural Networks,2005, 16(5):1136-1146.
[31]L. David, Y. Tasig. Extensions of Compressed sensing [J]. Signal Processing,2006, 86(3):533-548.
[32]E. Candes, J. Romberg, T. Tao. Robust Uncertainty Principles:Exact Signal Reconstruction from Highly Incomplete Frequency Information [J]. IEEE Transactions on Information Theory,2006, 52(2):489-509.
[33]L. David, D. Donoho. Compressed Sensing [J]. IEEE Transactions on Information Theory,2006, 52(4):1289-1306.
[34]E. Candes, T. Tao. Near Optimal Signal Recovery from Random Projections:Universal Encoding Strategies? [J]. IEEE Transactions on Information Theory,2006,52(12):5406-5425.
[35]R. Craigen. Regular Conference Matrices and Complex Hadamard Matrices [J]. Utilitas Mathematic,1994,45(1994):65-69.
[36]向锦.基于二分图邻接矩阵的压缩传感图像重建算法研究[D].长沙：湖南师范大学,2011.
[37]G. Konidaris, S. Osentoski. Value Function Approximation in Reinforcement Learning Using the Fourier Basis [R]. USA:Autonomous Learning Laboratory, Computer Science Department, University of Massachusetts Amherst,2008.
[38]N. Ahmed, T. Natarajan, K. Rao. Discrete Cosine Transform [J]. IEEE Transactions on Computers,1974,C-23(1):90-93.
[39]M. Unser, D. V. D. Ville. The Pairing of a Wavelet Basis With a Mildly Redundant Analysis via Subband Regression [J]. IEEE Transactions on Image Processing,2008,17(11):2040-2052.
[40]S. Mann, S. Haykin. The Chirplet Transform:A Generalization of Gabor's Logon Transform [J]. Vision Interface'91,1991,205-212.
[41]J. W. Ma, M. Y. Hussaini. Three-dimensional Curvelets for Coherent Vortex Analysis of Turbulence [J]. Applied Physics Letters,2007,91(18):184101-184101-3.
[42]R. Gribonval, R. Nielsen. Sparse Decompositions in Unions of Bases [J]. IEEE Trans Inform Theory,2003,49(12):3320-3325.
[43]S. J. Kim, K. Koh, M. Lustig, et al. An Interior-point Method for Large-scale L1 Regularized Least Squares [J]. IEEE Journal of Selected Topics in Signal Processing,2007, 1(4):606-617.
[44]A. Mario, R. Nowak, S. Wright Gradient Projection for Sparse Reconstruction:Application to Compressed Sensing and Other Inverse Problems [J]. IEEE Journal of Selected Topics in Signal Processing,2007, 1(4):586-597.
[45]J. Tropp, A. Gilbert. Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit [J]. IEEE Transactions on Theory,2007,53(12):4655-4666.
[46]张鹏.压缩传感中的阈值迭代重构算法的研究[D].天津：天津理工大学,2011.
[47]S. Rao, D. Wilton, and A. Glisson. Electromagnetic Scattering by Surfaces of Arbitrary Shape [J]. IEEE Transactions on Antennas and Propagation,1982,30(3):409-418.
[48]P. F. Guan, D. H. Phong. Partial Legendre Transforms of Non-linear Equations [J]. Proceedings of the American Mathematical Society,2012,140(11):3831-3842.
[49]W. Cody, C. Henry. Chebyshev Approximations for the Exponential Integral Ei(x) [J]. Mathematics of Computation,1969,23(106):289-303.
[50]D. Nicholson, P. Rabinowitz, N. Richter, D. Zeilberger. On the Error in the Numerical Integration of Chebyshev Polynomials [J]. Mathematics of Computation,1971,25(113):79-86.
[51]R. Lord. Integrals of Products of Laguerre Polynomials [J]. Mathematics of Computation,1960, 14(72):375-376.
[52]S. Thangavelu. Summability of Hermite Expansions II [J]. Transactions of the American Mathematical Society,1989,314(1):143-170.
[1]W. Geraint, V. Rosser. Interpretation of Classical Electromagnetism [M]. Netherlands:Springer, 1997:1-426.
[2]盛新庆.计算电磁学要论[M].北京：科学出版社,2004：1-153.
[3]A. Bossavit. Computational Electromagnetism [M]. USA:Academic Press,1998:1-352.
[4]吕英华.计算电磁学的数值方法[M].北京：清华大学出版社,2006：1-468.
[5]P. B. Zhou. Numerical Analysis of Electromagnetic Fields [M]. Germany:Springer Berlin Heidelberg,1993:1-406.
[6]文舸一.计算电磁学的进展与展望[M].电子学报,1995,23(10)：62-69.
[7]A. Auckenthaler, C. Bennett. Computer Solution of Transient and Time Domain Thin-Wire Antenna Problems [J]. IEEE Trans Microwave Theory Tech,1971,19(11):892-893.
[8]哈林顿.计算电磁场的矩量法[M].北京：国防工业出版社,1981：1-262.
[9]王洁,陈超波.基于MATLAB的静态场边值问题有限差分法的研究[J].微计算机应用,2010,31(3)：1-5.
[10]S. Rao. The Finite Element Method in Engineering[M]. USA: Butterworth-Heinemann,2004: 1-688.
[11]K. Yoshida, N. Nishimura, S. Kobayashi. Application of New Fast Multipole Boundary Integral Equation Method to Crack Problems in 3D [J]. Engineering Analysis with Boundary Elements,2001, 25(4-5):239-247.
[12]W. B. Ewe, L. W. Li, C. S. Chang, et al. AIM Analysis of Scattering and Radiation by Arbitrary Surface-wire Configurations [J]. IEEE Transactions Antennas Propagation,2007,55(1):162-166.
[13]M. Catedra, F. Ruiz, E. Gago. Analysis of Arbitrary Metallic Surfaces Conformed to a Circular Cylinder Using the Conjugate Gradient-fast Fourier Transform (CG-FFT) Method [J]. IEEE Transactions on Antennas and Propagation,1990,38(2):286-289.
[14]C. H. Chan, R. A. Kipp. An Improved Implementation of Triangular-domain Basis Functions for the Analysis of Microstrip Interconnects [J]. Journal of Electromagnetic Waves and Applications, 1994,8(6):779-795.
[15]蒋长锦.线性代数计算方法[M].合肥：中国科学技术大学出版社,2003：27-73.
[16]袁浩波.基于NURBS建模的高阶矩量法[D].西安：西安电子科技大学,2009.
[17]U. Jakobus, F. Landstorfer. Improved PO-MM Hybrid Formulation for Scattering from Three-dimensional Perfectly Conducting Bodies of Arbitrary Shape [J]. IEEE Transactions on Antennas and Propagation,1995,43(2):162-169.
[18]R. Baraniuk. Compressive sensing [J]. IEEE Signal Processing Magazine,2007,24(4):118-121.
[19]E. Candes. Compressive Sampling [C]. European Mathematical Society. Proceedings of the International Congress of Mathematicians. Madrid:American Mathematical Society,2006: 1433-1452.
[20]P. Yla-oijala, M. Taskinen. Calculation of CFIE Impedance Matrix Elements with RWG and n× RWG Functions [J]. IEEE Transactions on Antennas and Propagation,2003,51(8):1837-1846.
[21]R. Bates, J. James, I. Callett, R. Miller. An Overview of Point Matching [J]. Radio and Electronic Engineer,1973,43(3):193-200.
[22]马跃.金属多面体上多根线天线的特性分析[D].成都：电子科技大学,2002.
[23]C. Fletcher. Computational Galerkin methods [M]. USA:Springer-Verlag,1984:1-309.
[24]F. Andriulli, A. Tabacco, G. Vecchi. Solving the EFIE at Low Frequencies With a Conditioning That Grows Only Logarithmically With the Number of Unknowns [J]. IEEE Transactions on Antennas and Propagation,2010,58(5):1614-1624.
[25]J. Kataja, A. Polimeridis, J. Mosig, P. Yla-Oijala. Analytical Shape Derivatives of the MFIE System Matrix Discretized With RWG Functions [J]. IEEE Transactions on Antennas and Propagation,2013,61(2):985-988.
[26]M. Smith, A. Peterson. Numerical Solution of the CFIE Using Vector Bases and Dual Interlocking Meshes [J]. IEEE Transactions on Antennas and Propagation,2005,53(10):3334-3339.
[27]傅克祥.高斯积分离散法及其在电磁问题中的应用[J].扬州师院学报(自然科学版),1992,12(2)：41-49.
[28]冯守平.关于积分中值定理的证明方法[J].北京印刷学院学报,2011,19(2)：68-73.
[29]M. Murphy, M. Alley, J. Demmel, et al. Fast L1-Spirit Compressed Sensing Parallel Imaging MRI:Scalable Parallel Implementation and Clinically Feasible Runtime [J]. IEEE Transactions on Medical Imaging,2012,31(6):1250-1262.
[30]R. Justin. Imaging via Compressive Sampling [J]. IEEE Signal Processing Magazine,2008, 25(2):14-20.
[31]N. Ahmed, T. Natarajan, K. Rao. Discrete Cosine Transform [J]. IEEE Transactions on Computers,1974, C-23(1):90-93.
[32]M. Unser, D. V. D. Ville. The Pairing of a Wavelet Basis With a Mildly Redundant Analysis via Subband Regression [J]. IEEE Transactions on Image Processing,2008,17(11):2040-2052.
[33]G. Konidaris, S. Osentoski. Value Function Approximation in Reinforcement Learning Using the Fourier Basis [R]. USA:Autonomous Learning Laboratory, Computer Science Department, University of Massachusetts Amherst,2008.
[34]E. Candes, J. Romberg, T. Tao. Robust Uncertainty Principles:Exact Signal Reconstruction from Highly Incomplete Frequency Information [J]. IEEE Transactions on Information Theory,2006, 52(2):489-509.
[35]J. Tropp, A. Gilbert. Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit [J]. IEEE Transactions on Theory,2007,53(12):4655-4666.
[36]袁承武.3ds max7基础教程[M].北京：机械工业出版社,2006：1-77.
[1]W. Geraint, V. Rosser. Interpretation of Classical Electromagnetism [M]. Netherlands:Springer, 1997:1-426.
[2]盛新庆.计算电磁学要论[M].北京：科学出版社,2004：1-153.
[3]A. Bossavit. Computational Electromagnetism [M]. USA:Academic Press,1998:1-352.
[4]F. Vipiana, P. Pirinoli, G. Vecchi. Spectral Properties of the EFIE-MoM Matrix for Dense Meshes With Different Types of Bases [J]. IEEE Transactions on Antennas and Propagation,2007, 55(11):3229-3238.
[5]哈林顿.计算电磁场的矩量法[M].北京：国防工业出版社,1981：1-262.
[6]K. Yoshida, N. Nishimura, S. Kobayashi. Application of New Fast Multipole Boundary Integral Equation Method to Crack Problems in 3D [J]. Engineering Analysis with Boundary Elements,2001, 25(4-5):239-247.
[7]J. M. Song, C. C. Lu, W. C. Chew. Multilevel Fast Multipole Algorithm for Electromagnetic Scattering by Large Complex Objects [J]. IEEE Transactions on Antennas and Propagation,1997, 45(10):1488-1493.
[8]W. B. Ewe, L. W. Li, C. S. Chang, et al. AIM Analysis of Scattering and Radiation by Arbitrary Surface-wire Configurations [J]. IEEE Transactions Antennas Propagation,2007,55(1):162-166.
[9]M. Catedra, F. Ruiz, E. Gago. Analysis of Arbitrary Metallic Surfaces Conformed to a Circular Cylinder Using the Conjugate Gradient-fast Fourier Transform (CG-FFT) Method [J]. IEEE Transactions on Antennas and Propagation,1990,38(2):286-289.
[10]C. H. Chan, R. A. Kipp. An Improved Implementation of Triangular-domain Basis Functions for the Analysis of Microstrip Interconnects [J]. Journal of Electromagnetic Waves and Applications, 1994,8(6):779-795.
[11]R. Baraniuk. Compressive sensing [J]. IEEE Signal Processing Magazine,2007,24(4):118-121.
[12]E. Candes. Compressive Sampling [C]. European Mathematical Society. Proceedings of the International Congress of Mathematicians. Madrid:American Mathematical Society,2006: 1433-1452.
[13]E. Candes, J. Romberg, T. Tao. Robust Uncertainty Principles:Exact Signal Reconstruction from Highly Incomplete Frequency Information [J]. IEEE Transactions on Information Theory,2006, 52(2):489-509.
[14]G. Konidaris, S. Osentoski. Value Function Approximation in Reinforcement Learning Using the Fourier Basis [R]. USA: Autonomous Learning Laboratory, Computer Science Department, University of Massachusetts Amherst,2008.
[15]J. Tropp, A. Gilbert. Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit [J]. IEEE Transactions on Theory,2007,53(12):4655-4666.
[16]袁承武.3ds max7基础教程[M].北京：机械工业出版社,2006：1-77.
[17]L. Sevgi. Complex Electromagnetic Problems and Numerical Simulation Approaches [M]. USA: Wiley-IEEE Press,2003:179-219.
[18]F. de Adana, I. Gonzalez, O. Utierrez, M. Catedra. Asymptotic Method for Analysis of RCS of Arbitrary Targets Composed by Dielectric and/or Magnetic Materials [J]. IEE Radar, Sonar and Navigation,2003,150(5):375-378.
[1]P. F. Guan, D. H. Phong. Partial Legendre Transforms of Non-linear Equations [J]. Proceedings of the American Mathematical Society,2012,140(11):3831-3842.
[2]W. Cody, C. Henry. Chebyshev Approximations for the Exponential Integral Ei(x) [J]. Mathematics of Computation,1969,23(106):289-303.
[3]D. Nicholson, P. Rabinowitz, N. Richter, D. Zeilberger. On the Error in the Numerical Integration of Chebyshev Polynomials [J]. Mathematics of Computation,1971,25(113):79-86.
[4]R. Lord. Integrals of Products of Laguerre Polynomials [J]. Mathematics of Computation,1960, 14(72):375-376.
[5]S. Thangavelu. Summability of Hermite Expansions II [J]. Transactions of the American Mathematical Society,1989,314(l):143-170.
[6]王彪.压缩传感中的观测矩阵研究[D].天津：天津理工大学,2011.
[7]N. Ahmed, T. Natarajan, K. Rao. Discrete Cosine Transform [J]. IEEE Transactions on Computers,1974, C-23(1):90-93.
[8]R. Gribonval, R. Nielsen. Sparse Decompositions in Unions of Bases [J]. IEEE Trans Inform Theory,2003,49(12):3320-3325.
[9]L. David, D. Donoho. Compressed Sensing [J]. IEEE Transactions on Information Theory,2006, 52(4):1289-1306.
[10]R. Craigen. Regular Conference Matrices and Complex Hadamard Matrices [J]. Utilitas Mathematic,1994,45(1994):65-69.
[11]E. Candes, J. Romberg. Quantitative Robust Uncertainty Principles and Optimally Spare Decomposions [J]. Foundations of Computational Mathematics,2006,6(2):227-254.
[12]L. Gan, T. Do, T. Tran. Fast Compressive Imaging Using Scrambled Block Hadamard Ensemble [C/OL].http://dsp.rice.edu/sites/dsp.rice.edu/files/cs/srambled_blk_WHT.pdf,2009-08-16.
[13]R. Sun, H. L. Zhao, H. Xu. The Application of Improved Hadamard Measurement Matrix in Compressed Sensing [C]. Yantai University.2012 International Conference on Systems and Informatics (ICSAI). Germany:Springer-Verlag,2012:1994-1997.
[14]朱雪莹.CS测量矩阵及其成像应用研究[D].合肥：安徽大学电子信息工程学院,2012.
[15]W. Bajwa, J. Haupt, G. Raz, S. Wright, R. Nowak. Toeplitz-sturctured Compressed Sensing Matrices [C]. IEEE. Proceedings of the IEEE Workshop on Statistical Signal Processing. Washington D.C.:IEEE Press,2007:294-298.
[16]张成.物理可实现的压缩成像理论与方法研究[D].合肥：安徽大学,2012.
[17]C. Zhang, C. Shen, H. Cheng, H. R. Yang, S. Wei. Phase Mask of Sparse Trinary Toeplitz Matrix with Random Pitch in Compressive Imaging [J]. Acta Optica Sinica,2012,32(2):0211002-1-0211002-9.
[18]W. Yin, S. Morgan, J. Yang, Y. Zhang. Practical Compressive Sensing with Toeplitz and Circulant Matrices [R]. USA:Rice University,2010.
[19]李浩.用于压缩感知的确定性测量矩阵研究[D].北京：北京交通大学,2011.
[20]T. Do, T. Trany, L. Gan. Fast Compressive Sampling with Structurally Random Matrices [C]. IEEE. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing. Washington D.C.:IEEE Press,2008:3369-3372.
[21]叶蕾,孙林慧,杨震.基于压缩感知观测序列倒谱距离的语音端点检测算法[J].信号处理,2011,27(1)：67-72.
[22]方红,章权兵,韦穗.基于非常稀疏随机投影的图像重建方法[J].计算机工程与应用,2007,43(22)：25-27.
[23]S. B. Chen, D. Donoho, M. Saunders. Atomic Decomposition by Basis Pursuit [J]. SIAM Journal on Scientific Computing,1998,20(1):33-61.
[1]A. Mickelson. Physical Optics [M]. USA:Springer US,1992:1-345.
[2]G. Crabtree. A Numerical Quadrature Technique For Physical Optics Scattering Analysis [J]. IEEE Transactions on Magnetics,1991,27(5):4291-4294.
[3]陈毅乔.电大尺寸目标的电磁特性分析方法研究[D].成都：电子科技大学,2008.
[4]徐翠.高频算法在电磁散射计算中的应用[D].南京：南京理工大学,2009.
[5]Southall, J. Cocke. The Principles and Methods of Geometrical Optics [M]. USA:Nabu Press, 2010:1-654.
[6]V. Borovikov, V. Borovikov, B. Kinber, B. Kinber. Geometrical theory of diffraction [M]. Britain: The Institution of Engineering and Technology (IET),1994:1-390.
[7]R. Kouyoumjian, P. Pathak. A Uniform Geometrical Theory of Diffraction for an Edge in a Perfectly Conducting Surface [J]. Proceedings of the IEEE,1974,62(11):1448-1461.
[8]P. Ufrmtsev. Fundamentals of the Physical Theory of Diffraction [M]. USA:Wiley,2007:1-336.
[9]S. Lee, G. Deschamps. A Uniform Asymptotic Theory of Electromagnetic Diffraction by a Curved Wedge [J]. IEEE Transactions on Antennas and Propagation,1976,24(1):25-34.
[10]W. J. Zhao, Y. B. Gan, L. W. Li, C. F. Wang. Effects of an Electrically Large Airborne Radome on Radiation Patterns and Input Impedance of a Dipole Array [J]. IEEE Transactions on Antennas and Propagation,2007,55(8):2399-2402.
[11]B. Fischer, I. Lahaie, M. Punnett. Determining the Effect of an Electromagnetic Window or Radome Wall Upon a Circularly Polarized Transmission [J]. IEEE Antennas and Propagation Magazine,2007,49(1):122-128.
[12]A. Fathy. Design of a Near Field Protective Dielectric Radome "Window" for a Curved Phased Array Antenna-Circumferential Polarization Case [J]. IEEE Transactions on Antennas and Propagation,2006,54(11):3356-3366.
[13]H. Y. Chen, Z. X. Shen, E. Li. Analysis of a Three-dimensional Antenna Radiating Through a Two-dimensional Radome Using a Fast High-order Method [J]. IEEE Transactions on Magnetics, 2006,42(4):699-702.
[14]W. Hsu, R. Barakat. Stratton-Chu Vectorial Diffraction of Electromagnetic Fields by Apertures with Application to Small-Fresnel-number Systems [J]. Journal of the Optical Society of America A, 1994, 11(2):623-629.
[15]S. Y. He, G. Q. Zhu. A Hybrid MM-PO Method Combining UV Technique for Scattering from Two-dimensional Target above a Rough Surface [J]. Microwave and Optical Technology Letters, 2007,49(12):2957-2960.
[16]U. Jakobus, F. Landstorfer. Improved PO-MM Hybrid Formulation for Scattering from Three-Dimensional Perfectly Conducting Bodies of Arbitrary Shape [J]. IEEE Transactions on Antennas and Propagation,1995,43(2):162-169.
[17]向锋,王宝发.电流步进法实现低频区RCS可视化计算[J].系统工程与电子技术,2004,26(9)：1179-1183.
[18]A. Zaporozhets, M. Levy. Electromagnetic Scattering Calculations with Current Marching Technique [C]. The Institution of Engineering and Technology (IET). IEE National Conference on Antennas and Propagation,1999. UK:IEEE Press,1999:217-220.
[2]盛新庆.计算电磁学要论[M].北京：科学出版社,2004：1-153.
[3]A. Bossavit. Computational Electromagnetism [M]. USA:Academic Press,1998:1-352.
[4]S. Rao. The Finite Element Method in Engineering [M]. USA:Butterworth-Heinemann,2004: 1-688.
[5]哈林顿.计算电磁场的矩量法[M].北京：国防工业出版社,1981：1-262.
[6]葛德彪,闫玉波.电磁波时域有限差分方法[M].西安：西安电子科技大学出版社,2005：1-376.
[7]S. Sauter, C. Schwab. Boundary Element Methods [M]. Germany:Springer Berlin Heidelberg, 2011:183-287.
[8]A. Mickelson. Physical Optics [M]. USA:Springer US,1992:1-345.
[9]赵为粮.目标散射的复射线分析及其误差控制技术[D].成都：电子科技大学,2002.
[10]吕英华.计算电磁学的数值方法[M].北京：清华大学出版社,2006：1-468.
[11]P. B. Zhou. Numerical Analysis of Electromagnetic Fields [M]. Germany:Springer Berlin Heidelberg,1993:1-406.
[12]陈毅乔.电大尺寸目标的电磁特性分析方法研究[D].成都：电子科技大学,2008.
[13]徐翠.高频算法在电磁散射计算中的应用[D].南京：南京理工大学,2009.
[14]R. Baraniuk. Compressive sensing [J]. IEEE Signal Processing Magazine,2007,24(4):118-121.
[15]E. Candes. Compressive Sampling [C]. European Mathematical Society. Proceedings of the International Congress of Mathematicians. Madrid: American Mathematical Society,2006: 1433-1452.
[16]J. Tropp, J. Laska, M. Duarte, J. Romberg and R. Baraniuk. Beyond Nyquist Efficient Sampling of Sparse Bandlimited Signals [J]. IEEE Transactions on Information Theory,2010,56(1):520-544.
[17]M. S. Aly, T. T. Y. Wong. Combined Mie Series and Asymptotic Formulation of Transient Scattered Field and Perfectly Conducting Sphere [C]. IEEE. Antennas and Propagation Society International Symposium. Syracuse, NY, USA:IEEE Press,1988:664-667.
[18]B. Fowler. Expansion of Mie-theory Phase Functions in Series of Legendre Polynomials [J]. Journal of the Optical Society of America,1983,73(1):19-22.
[19]M. Catedra, F. Ruiz, E. Gago. Analysis of Arbitrary Metallic Surfaces Conformed to a Circular Cylinder Using the Conjugate Gradient-fast Fourier Transform (CG-FFT) Method [J]. IEEE Transactions on Antennas and Propagation,1990,38(2):286-289.
[20]W. B. Ewe, L. W. Li, C. S. Chang, et al. AIM Analysis of Scattering and Radiation by Arbitrary Surface-wire Configurations [J]. IEEE Transactions Antennas Propagation,2007,55(1):162-166.
[21]K. Yoshida, N. Nishimura, S. Kobayashi. Application of New Fast Multipole Boundary Integral Equation Method to Crack Problems in 3D [J]. Engineering Analysis with Boundary Elements,2001, 25(4-5):239-247.
[22]J. M. Song, C. C. Lu, W. C. Chew. Multilevel Fast Multipole Algorithm for Electromagnetic Scattering by Large Complex Objects [J]. IEEE Transactions on Antennas and Propagation,1997, 45(10):1488-1493.
[23]彭朕,盛新庆.一种多层不完全LU分解预处理方法在合元极技术中的应用[J].电子学报,2008,36(2)：230-234.
[24]J. Zhu, R. S. Chen. SSOR Preconditioned CG Method for Formulation of Hierarchical Vector Finite Element [C]. IEEE. Antennas and Propagation Society International Symposium. Albuquerque, NM:IEEE Press,2006:1778-1781.
[25]P. L. Rui, R. S. Chen. An Efficient Sparse Approximate Inverse Preconditioning for FMM Implementation [J]. Microwave and Optical Technology Letters,2007,49(7):1746-1750.
[26]李月卉.电磁场数值求解中迭代方法与预条件技术研究[D].成都：电子科技大学,2011.
[27]柯涛,丁建军,丁大志,樊振宏,陈如山.特征谱双步预条件结合多分辨预条件技术快速分析电磁散射问题[C].中国电子学会.2007年全国微波毫米波会议论文集(上册).宁波：电子工业出版社,2007：170-173.
[28]X. C. Wei, E. P. Li. Wide-band EMC Analysis of On-platform Antennas Using Impedance-matrix Interpolation with the Moment Method-physical Optics Method [J]. IEEE Transactions on Electromagnetic Compatibility,2003,45(3):552-556.
[29]M. Burchett. Development of a Comfort Measurement Matrix [EB/OL]. http://www.lib.ncsu.edU/resolver/1840.16/197,2008-04-26.
[30]A. Atiya, M. Aly, A. Parlos. Sparse Basis Selection:New Results and Application to Adaptive Prediction of Video Source Traffic [J]. IEEE Transactions on Neural Networks,2005, 16(5):1136-1146.
[31]L. David, Y. Tasig. Extensions of Compressed sensing [J]. Signal Processing,2006, 86(3):533-548.
[32]E. Candes, J. Romberg, T. Tao. Robust Uncertainty Principles:Exact Signal Reconstruction from Highly Incomplete Frequency Information [J]. IEEE Transactions on Information Theory,2006, 52(2):489-509.
[33]L. David, D. Donoho. Compressed Sensing [J]. IEEE Transactions on Information Theory,2006, 52(4):1289-1306.
[34]E. Candes, T. Tao. Near Optimal Signal Recovery from Random Projections:Universal Encoding Strategies? [J]. IEEE Transactions on Information Theory,2006,52(12):5406-5425.
[35]R. Craigen. Regular Conference Matrices and Complex Hadamard Matrices [J]. Utilitas Mathematic,1994,45(1994):65-69.
[36]向锦.基于二分图邻接矩阵的压缩传感图像重建算法研究[D].长沙：湖南师范大学,2011.
[37]G. Konidaris, S. Osentoski. Value Function Approximation in Reinforcement Learning Using the Fourier Basis [R]. USA:Autonomous Learning Laboratory, Computer Science Department, University of Massachusetts Amherst,2008.
[38]N. Ahmed, T. Natarajan, K. Rao. Discrete Cosine Transform [J]. IEEE Transactions on Computers,1974,C-23(1):90-93.
[39]M. Unser, D. V. D. Ville. The Pairing of a Wavelet Basis With a Mildly Redundant Analysis via Subband Regression [J]. IEEE Transactions on Image Processing,2008,17(11):2040-2052.
[40]S. Mann, S. Haykin. The Chirplet Transform:A Generalization of Gabor's Logon Transform [J]. Vision Interface'91,1991,205-212.
[41]J. W. Ma, M. Y. Hussaini. Three-dimensional Curvelets for Coherent Vortex Analysis of Turbulence [J]. Applied Physics Letters,2007,91(18):184101-184101-3.
[42]R. Gribonval, R. Nielsen. Sparse Decompositions in Unions of Bases [J]. IEEE Trans Inform Theory,2003,49(12):3320-3325.
[43]S. J. Kim, K. Koh, M. Lustig, et al. An Interior-point Method for Large-scale L1 Regularized Least Squares [J]. IEEE Journal of Selected Topics in Signal Processing,2007, 1(4):606-617.
[44]A. Mario, R. Nowak, S. Wright Gradient Projection for Sparse Reconstruction:Application to Compressed Sensing and Other Inverse Problems [J]. IEEE Journal of Selected Topics in Signal Processing,2007, 1(4):586-597.
[45]J. Tropp, A. Gilbert. Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit [J]. IEEE Transactions on Theory,2007,53(12):4655-4666.
[46]张鹏.压缩传感中的阈值迭代重构算法的研究[D].天津：天津理工大学,2011.
[47]S. Rao, D. Wilton, and A. Glisson. Electromagnetic Scattering by Surfaces of Arbitrary Shape [J]. IEEE Transactions on Antennas and Propagation,1982,30(3):409-418.
[48]P. F. Guan, D. H. Phong. Partial Legendre Transforms of Non-linear Equations [J]. Proceedings of the American Mathematical Society,2012,140(11):3831-3842.
[49]W. Cody, C. Henry. Chebyshev Approximations for the Exponential Integral Ei(x) [J]. Mathematics of Computation,1969,23(106):289-303.
[50]D. Nicholson, P. Rabinowitz, N. Richter, D. Zeilberger. On the Error in the Numerical Integration of Chebyshev Polynomials [J]. Mathematics of Computation,1971,25(113):79-86.
[51]R. Lord. Integrals of Products of Laguerre Polynomials [J]. Mathematics of Computation,1960, 14(72):375-376.
[52]S. Thangavelu. Summability of Hermite Expansions II [J]. Transactions of the American Mathematical Society,1989,314(1):143-170.
[1]W. Geraint, V. Rosser. Interpretation of Classical Electromagnetism [M]. Netherlands:Springer, 1997:1-426.
[2]盛新庆.计算电磁学要论[M].北京：科学出版社,2004：1-153.
[3]A. Bossavit. Computational Electromagnetism [M]. USA:Academic Press,1998:1-352.
[4]吕英华.计算电磁学的数值方法[M].北京：清华大学出版社,2006：1-468.
[5]P. B. Zhou. Numerical Analysis of Electromagnetic Fields [M]. Germany:Springer Berlin Heidelberg,1993:1-406.
[6]文舸一.计算电磁学的进展与展望[M].电子学报,1995,23(10)：62-69.
[7]A. Auckenthaler, C. Bennett. Computer Solution of Transient and Time Domain Thin-Wire Antenna Problems [J]. IEEE Trans Microwave Theory Tech,1971,19(11):892-893.
[8]哈林顿.计算电磁场的矩量法[M].北京：国防工业出版社,1981：1-262.
[9]王洁,陈超波.基于MATLAB的静态场边值问题有限差分法的研究[J].微计算机应用,2010,31(3)：1-5.
[10]S. Rao. The Finite Element Method in Engineering[M]. USA: Butterworth-Heinemann,2004: 1-688.
[11]K. Yoshida, N. Nishimura, S. Kobayashi. Application of New Fast Multipole Boundary Integral Equation Method to Crack Problems in 3D [J]. Engineering Analysis with Boundary Elements,2001, 25(4-5):239-247.
[12]W. B. Ewe, L. W. Li, C. S. Chang, et al. AIM Analysis of Scattering and Radiation by Arbitrary Surface-wire Configurations [J]. IEEE Transactions Antennas Propagation,2007,55(1):162-166.
[13]M. Catedra, F. Ruiz, E. Gago. Analysis of Arbitrary Metallic Surfaces Conformed to a Circular Cylinder Using the Conjugate Gradient-fast Fourier Transform (CG-FFT) Method [J]. IEEE Transactions on Antennas and Propagation,1990,38(2):286-289.
[14]C. H. Chan, R. A. Kipp. An Improved Implementation of Triangular-domain Basis Functions for the Analysis of Microstrip Interconnects [J]. Journal of Electromagnetic Waves and Applications, 1994,8(6):779-795.
[15]蒋长锦.线性代数计算方法[M].合肥：中国科学技术大学出版社,2003：27-73.
[16]袁浩波.基于NURBS建模的高阶矩量法[D].西安：西安电子科技大学,2009.
[17]U. Jakobus, F. Landstorfer. Improved PO-MM Hybrid Formulation for Scattering from Three-dimensional Perfectly Conducting Bodies of Arbitrary Shape [J]. IEEE Transactions on Antennas and Propagation,1995,43(2):162-169.
[18]R. Baraniuk. Compressive sensing [J]. IEEE Signal Processing Magazine,2007,24(4):118-121.
[19]E. Candes. Compressive Sampling [C]. European Mathematical Society. Proceedings of the International Congress of Mathematicians. Madrid:American Mathematical Society,2006: 1433-1452.
[20]P. Yla-oijala, M. Taskinen. Calculation of CFIE Impedance Matrix Elements with RWG and n× RWG Functions [J]. IEEE Transactions on Antennas and Propagation,2003,51(8):1837-1846.
[21]R. Bates, J. James, I. Callett, R. Miller. An Overview of Point Matching [J]. Radio and Electronic Engineer,1973,43(3):193-200.
[22]马跃.金属多面体上多根线天线的特性分析[D].成都：电子科技大学,2002.
[23]C. Fletcher. Computational Galerkin methods [M]. USA:Springer-Verlag,1984:1-309.
[24]F. Andriulli, A. Tabacco, G. Vecchi. Solving the EFIE at Low Frequencies With a Conditioning That Grows Only Logarithmically With the Number of Unknowns [J]. IEEE Transactions on Antennas and Propagation,2010,58(5):1614-1624.
[25]J. Kataja, A. Polimeridis, J. Mosig, P. Yla-Oijala. Analytical Shape Derivatives of the MFIE System Matrix Discretized With RWG Functions [J]. IEEE Transactions on Antennas and Propagation,2013,61(2):985-988.
[26]M. Smith, A. Peterson. Numerical Solution of the CFIE Using Vector Bases and Dual Interlocking Meshes [J]. IEEE Transactions on Antennas and Propagation,2005,53(10):3334-3339.
[27]傅克祥.高斯积分离散法及其在电磁问题中的应用[J].扬州师院学报(自然科学版),1992,12(2)：41-49.
[28]冯守平.关于积分中值定理的证明方法[J].北京印刷学院学报,2011,19(2)：68-73.
[29]M. Murphy, M. Alley, J. Demmel, et al. Fast L1-Spirit Compressed Sensing Parallel Imaging MRI:Scalable Parallel Implementation and Clinically Feasible Runtime [J]. IEEE Transactions on Medical Imaging,2012,31(6):1250-1262.
[30]R. Justin. Imaging via Compressive Sampling [J]. IEEE Signal Processing Magazine,2008, 25(2):14-20.
[31]N. Ahmed, T. Natarajan, K. Rao. Discrete Cosine Transform [J]. IEEE Transactions on Computers,1974, C-23(1):90-93.
[32]M. Unser, D. V. D. Ville. The Pairing of a Wavelet Basis With a Mildly Redundant Analysis via Subband Regression [J]. IEEE Transactions on Image Processing,2008,17(11):2040-2052.
[33]G. Konidaris, S. Osentoski. Value Function Approximation in Reinforcement Learning Using the Fourier Basis [R]. USA:Autonomous Learning Laboratory, Computer Science Department, University of Massachusetts Amherst,2008.
[34]E. Candes, J. Romberg, T. Tao. Robust Uncertainty Principles:Exact Signal Reconstruction from Highly Incomplete Frequency Information [J]. IEEE Transactions on Information Theory,2006, 52(2):489-509.
[35]J. Tropp, A. Gilbert. Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit [J]. IEEE Transactions on Theory,2007,53(12):4655-4666.
[36]袁承武.3ds max7基础教程[M].北京：机械工业出版社,2006：1-77.
[1]W. Geraint, V. Rosser. Interpretation of Classical Electromagnetism [M]. Netherlands:Springer, 1997:1-426.
[2]盛新庆.计算电磁学要论[M].北京：科学出版社,2004：1-153.
[3]A. Bossavit. Computational Electromagnetism [M]. USA:Academic Press,1998:1-352.
[4]F. Vipiana, P. Pirinoli, G. Vecchi. Spectral Properties of the EFIE-MoM Matrix for Dense Meshes With Different Types of Bases [J]. IEEE Transactions on Antennas and Propagation,2007, 55(11):3229-3238.
[5]哈林顿.计算电磁场的矩量法[M].北京：国防工业出版社,1981：1-262.
[6]K. Yoshida, N. Nishimura, S. Kobayashi. Application of New Fast Multipole Boundary Integral Equation Method to Crack Problems in 3D [J]. Engineering Analysis with Boundary Elements,2001, 25(4-5):239-247.
[7]J. M. Song, C. C. Lu, W. C. Chew. Multilevel Fast Multipole Algorithm for Electromagnetic Scattering by Large Complex Objects [J]. IEEE Transactions on Antennas and Propagation,1997, 45(10):1488-1493.
[8]W. B. Ewe, L. W. Li, C. S. Chang, et al. AIM Analysis of Scattering and Radiation by Arbitrary Surface-wire Configurations [J]. IEEE Transactions Antennas Propagation,2007,55(1):162-166.
[9]M. Catedra, F. Ruiz, E. Gago. Analysis of Arbitrary Metallic Surfaces Conformed to a Circular Cylinder Using the Conjugate Gradient-fast Fourier Transform (CG-FFT) Method [J]. IEEE Transactions on Antennas and Propagation,1990,38(2):286-289.
[10]C. H. Chan, R. A. Kipp. An Improved Implementation of Triangular-domain Basis Functions for the Analysis of Microstrip Interconnects [J]. Journal of Electromagnetic Waves and Applications, 1994,8(6):779-795.
[11]R. Baraniuk. Compressive sensing [J]. IEEE Signal Processing Magazine,2007,24(4):118-121.
[12]E. Candes. Compressive Sampling [C]. European Mathematical Society. Proceedings of the International Congress of Mathematicians. Madrid:American Mathematical Society,2006: 1433-1452.
[13]E. Candes, J. Romberg, T. Tao. Robust Uncertainty Principles:Exact Signal Reconstruction from Highly Incomplete Frequency Information [J]. IEEE Transactions on Information Theory,2006, 52(2):489-509.
[14]G. Konidaris, S. Osentoski. Value Function Approximation in Reinforcement Learning Using the Fourier Basis [R]. USA: Autonomous Learning Laboratory, Computer Science Department, University of Massachusetts Amherst,2008.
[15]J. Tropp, A. Gilbert. Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit [J]. IEEE Transactions on Theory,2007,53(12):4655-4666.
[16]袁承武.3ds max7基础教程[M].北京：机械工业出版社,2006：1-77.
[17]L. Sevgi. Complex Electromagnetic Problems and Numerical Simulation Approaches [M]. USA: Wiley-IEEE Press,2003:179-219.
[18]F. de Adana, I. Gonzalez, O. Utierrez, M. Catedra. Asymptotic Method for Analysis of RCS of Arbitrary Targets Composed by Dielectric and/or Magnetic Materials [J]. IEE Radar, Sonar and Navigation,2003,150(5):375-378.
[1]P. F. Guan, D. H. Phong. Partial Legendre Transforms of Non-linear Equations [J]. Proceedings of the American Mathematical Society,2012,140(11):3831-3842.
[2]W. Cody, C. Henry. Chebyshev Approximations for the Exponential Integral Ei(x) [J]. Mathematics of Computation,1969,23(106):289-303.
[3]D. Nicholson, P. Rabinowitz, N. Richter, D. Zeilberger. On the Error in the Numerical Integration of Chebyshev Polynomials [J]. Mathematics of Computation,1971,25(113):79-86.
[4]R. Lord. Integrals of Products of Laguerre Polynomials [J]. Mathematics of Computation,1960, 14(72):375-376.
[5]S. Thangavelu. Summability of Hermite Expansions II [J]. Transactions of the American Mathematical Society,1989,314(l):143-170.
[6]王彪.压缩传感中的观测矩阵研究[D].天津：天津理工大学,2011.
[7]N. Ahmed, T. Natarajan, K. Rao. Discrete Cosine Transform [J]. IEEE Transactions on Computers,1974, C-23(1):90-93.
[8]R. Gribonval, R. Nielsen. Sparse Decompositions in Unions of Bases [J]. IEEE Trans Inform Theory,2003,49(12):3320-3325.
[9]L. David, D. Donoho. Compressed Sensing [J]. IEEE Transactions on Information Theory,2006, 52(4):1289-1306.
[10]R. Craigen. Regular Conference Matrices and Complex Hadamard Matrices [J]. Utilitas Mathematic,1994,45(1994):65-69.
[11]E. Candes, J. Romberg. Quantitative Robust Uncertainty Principles and Optimally Spare Decomposions [J]. Foundations of Computational Mathematics,2006,6(2):227-254.
[12]L. Gan, T. Do, T. Tran. Fast Compressive Imaging Using Scrambled Block Hadamard Ensemble [C/OL].http://dsp.rice.edu/sites/dsp.rice.edu/files/cs/srambled_blk_WHT.pdf,2009-08-16.
[13]R. Sun, H. L. Zhao, H. Xu. The Application of Improved Hadamard Measurement Matrix in Compressed Sensing [C]. Yantai University.2012 International Conference on Systems and Informatics (ICSAI). Germany:Springer-Verlag,2012:1994-1997.
[14]朱雪莹.CS测量矩阵及其成像应用研究[D].合肥：安徽大学电子信息工程学院,2012.
[15]W. Bajwa, J. Haupt, G. Raz, S. Wright, R. Nowak. Toeplitz-sturctured Compressed Sensing Matrices [C]. IEEE. Proceedings of the IEEE Workshop on Statistical Signal Processing. Washington D.C.:IEEE Press,2007:294-298.
[16]张成.物理可实现的压缩成像理论与方法研究[D].合肥：安徽大学,2012.
[17]C. Zhang, C. Shen, H. Cheng, H. R. Yang, S. Wei. Phase Mask of Sparse Trinary Toeplitz Matrix with Random Pitch in Compressive Imaging [J]. Acta Optica Sinica,2012,32(2):0211002-1-0211002-9.
[18]W. Yin, S. Morgan, J. Yang, Y. Zhang. Practical Compressive Sensing with Toeplitz and Circulant Matrices [R]. USA:Rice University,2010.
[19]李浩.用于压缩感知的确定性测量矩阵研究[D].北京：北京交通大学,2011.
[20]T. Do, T. Trany, L. Gan. Fast Compressive Sampling with Structurally Random Matrices [C]. IEEE. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing. Washington D.C.:IEEE Press,2008:3369-3372.
[21]叶蕾,孙林慧,杨震.基于压缩感知观测序列倒谱距离的语音端点检测算法[J].信号处理,2011,27(1)：67-72.
[22]方红,章权兵,韦穗.基于非常稀疏随机投影的图像重建方法[J].计算机工程与应用,2007,43(22)：25-27.
[23]S. B. Chen, D. Donoho, M. Saunders. Atomic Decomposition by Basis Pursuit [J]. SIAM Journal on Scientific Computing,1998,20(1):33-61.
[1]A. Mickelson. Physical Optics [M]. USA:Springer US,1992:1-345.
[2]G. Crabtree. A Numerical Quadrature Technique For Physical Optics Scattering Analysis [J]. IEEE Transactions on Magnetics,1991,27(5):4291-4294.
[3]陈毅乔.电大尺寸目标的电磁特性分析方法研究[D].成都：电子科技大学,2008.
[4]徐翠.高频算法在电磁散射计算中的应用[D].南京：南京理工大学,2009.
[5]Southall, J. Cocke. The Principles and Methods of Geometrical Optics [M]. USA:Nabu Press, 2010:1-654.
[6]V. Borovikov, V. Borovikov, B. Kinber, B. Kinber. Geometrical theory of diffraction [M]. Britain: The Institution of Engineering and Technology (IET),1994:1-390.
[7]R. Kouyoumjian, P. Pathak. A Uniform Geometrical Theory of Diffraction for an Edge in a Perfectly Conducting Surface [J]. Proceedings of the IEEE,1974,62(11):1448-1461.
[8]P. Ufrmtsev. Fundamentals of the Physical Theory of Diffraction [M]. USA:Wiley,2007:1-336.
[9]S. Lee, G. Deschamps. A Uniform Asymptotic Theory of Electromagnetic Diffraction by a Curved Wedge [J]. IEEE Transactions on Antennas and Propagation,1976,24(1):25-34.
[10]W. J. Zhao, Y. B. Gan, L. W. Li, C. F. Wang. Effects of an Electrically Large Airborne Radome on Radiation Patterns and Input Impedance of a Dipole Array [J]. IEEE Transactions on Antennas and Propagation,2007,55(8):2399-2402.
[11]B. Fischer, I. Lahaie, M. Punnett. Determining the Effect of an Electromagnetic Window or Radome Wall Upon a Circularly Polarized Transmission [J]. IEEE Antennas and Propagation Magazine,2007,49(1):122-128.
[12]A. Fathy. Design of a Near Field Protective Dielectric Radome "Window" for a Curved Phased Array Antenna-Circumferential Polarization Case [J]. IEEE Transactions on Antennas and Propagation,2006,54(11):3356-3366.
[13]H. Y. Chen, Z. X. Shen, E. Li. Analysis of a Three-dimensional Antenna Radiating Through a Two-dimensional Radome Using a Fast High-order Method [J]. IEEE Transactions on Magnetics, 2006,42(4):699-702.
[14]W. Hsu, R. Barakat. Stratton-Chu Vectorial Diffraction of Electromagnetic Fields by Apertures with Application to Small-Fresnel-number Systems [J]. Journal of the Optical Society of America A, 1994, 11(2):623-629.
[15]S. Y. He, G. Q. Zhu. A Hybrid MM-PO Method Combining UV Technique for Scattering from Two-dimensional Target above a Rough Surface [J]. Microwave and Optical Technology Letters, 2007,49(12):2957-2960.
[16]U. Jakobus, F. Landstorfer. Improved PO-MM Hybrid Formulation for Scattering from Three-Dimensional Perfectly Conducting Bodies of Arbitrary Shape [J]. IEEE Transactions on Antennas and Propagation,1995,43(2):162-169.
[17]向锋,王宝发.电流步进法实现低频区RCS可视化计算[J].系统工程与电子技术,2004,26(9)：1179-1183.
[18]A. Zaporozhets, M. Levy. Electromagnetic Scattering Calculations with Current Marching Technique [C]. The Institution of Engineering and Technology (IET). IEE National Conference on Antennas and Propagation,1999. UK:IEEE Press,1999:217-220.