[1] Fu Tuo, Gao Xiqi,. Joint eigenvalue estimationby balanced simultaneous Schur decomposition [J]. Journal of Southeast University (English Edition), 2006, 22 (4): 445-450. [doi:10.3969/j.issn.1003-7985.2006.04.001]
Copy
Joint eigenvalue estimationby balanced simultaneous Schur decomposition(
)
基于平衡同时Schur分解的联合特征值估计
)Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]
- Volumn:
- 22
- Issue:
- 2006 4
- Page:
- 445-450
- Research Field:
- Information and Communication Engineering
- Publishing date:
- 2006-12-30
Info
- Title:
- Joint eigenvalue estimationby balanced simultaneous Schur decomposition
- 基于平衡同时Schur分解的联合特征值估计
- Author(s):
- Fu Tuo, Gao Xiqi
- National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China
- Keywords:
- direction of arrival; multi-dimensional harmonic retrieval; joint eigenvalue; simultaneous Schur decomposition; balance algorithm
- PACS:
- TN911.7
- DOI:
- 10.3969/j.issn.1003-7985.2006.04.001
- Abstract:
- The problem of joint eigenvalue estimation for the non-defective commuting set of matrices A is addressed.A procedure revealing the joint eigenstructure by simultaneous diagonalization of A with simultaneous Schur decomposition(SSD)and balance procedure alternately is proposed for performance considerations and also for overcoming the convergence difficulties of previous methods based only on simultaneous Schur form and unitary transformations.It is shown that the SSD procedure can be well incorporated with the balancing algorithm in a pingpong manner, i.e., each optimizes a cost function and at the same time serves as an acceleration procedure for the other.Under mild assumptions, the convergence of the two cost functions alternately optimized, i.e., the norm of A and the norm of the left-lower part of A, is proved.Numerical experiments are conducted in a multi-dimensional harmonic retrieval application and suggest that the presented method converges considerably faster than the methods based on only unitary transformation for matrices which are not near to normality.
- 讨论可交换单纯矩阵族A的联合特征值估计问题.为了克服基于同时Schur分解和酉变换算法的收敛和性能分析缺陷, 提出了一种基于同时相似对角化的联合特征结构估计算法.该算法通过对A交替进行同时Schur分解和范数平衡来实现矩阵族的对角化.该算法的有效性在于:每个子过程在优化自身代价函数的同时, 还对另一子过程的收敛起到加速作用.在适当的假设条件下, 可以证明该算法交替优化的2个代价函数(矩阵族范数和矩阵族下三角元素范数)的收敛性.基于多维谐波提取的数值仿真显示该算法在矩阵族偏离正规阵时收敛速度显著快于基于同时Schur分解和酉变换算法, 并且联合特征值的估计性能可以进行简洁的闭式分析.
References:
[1] van der Veen A J, Ober P B, Deprettere E F.Azimuth and elevation computation in high resolution DOA estimation [J].IEEE Trans Signal Proc, 1992, 40(7):1828-1832.
[2] Lemma A N, van der Veen A J, Deprettere E F.Analysis of joint angle-frequency estimation using ESPRIT [J].IEEE Trans Signal Proc, 2003, 51(5):1264-1283.
[3] Haardt M, Nossek J A.Simultaneous Schur decomposition of several nonsymmetric matrices to achieve automatic pairing in multidimensional harmonic retrieval problems [J].IEEE Trans Signal Proc, 1998, 46(1):161-169.
[4] Zoltowski M D, Haardt M, Mathews C P.Closed-form 2-D angle estimation with rectangular arrays in element space or beam space via unitary ESPRIT [J].IEEE Trans Signal Proc, 1996, 44(2):316-328.
[5] Hua Y B, Abed-Meraim K.Techniques of eigenvalues estimation and association [J].Digital Signal Proc, 1997, 7(4):253-259.
[6] Abed-Meraim K, Hua Y B.A least-squares approach to joint Schur decomposition [A].In:Proc IEEE ICASSP’98 [C].Seattle, USA, 1998, 4:2541-2544.
[7] Froberg C E.On triangularization of complex matrices by two-dimensional unitary transformations [J].BIT Numer Math, 1965, 5:230-234.
[8] Huang C P.A Jacobi-type method for triangularizing an arbitrary matrix [J].SIAM J Numer Anal, 1975, 12:566-570.
[9] Strobach P.Bi-iteration multiple invariance subspace tracking and adaptive ESPRIT [J].IEEE Trans Signal Proc, 2000, 48(2):442-456.
[10] Golub G H, Loan C F V.Matrix computations [M].Johns Hopkins University Press, 1996.
[11] Yeredor A.Non-orthogonal joint diagonalization in the least-squares sense with application in blind source separation [J].IEEE Trans Signal Proc, 2002, 50(7):1545-1553.
[12] Rutishauser H.Une méthod pour le calcul des valeurs propres des matrices non symétriques [J].Comptes Rendus des Séances de l’Academie des Sciences, 1964, 259:2758.
[13] Fu T, Gao X Q.Simultaneous diagonalization with similarity transformation for non-defective matrices [A].In:Proc IEEE ICASSP’2006 [C].Toulouse, France, 2006, 4:1137-1140.
[14] Horn R A, Johnson C R.Matrix analysis [M].Cambridge University Press, 1985.52.
[15] Osborne E E.On pre-conditioning of matrices [J].Journal of the Association for Computing Machinery, 1960, 7(4):338-345.
[16] Henrici P.Bounds for iterates, inverses, spectral variation, and fields of values of non-normal matrices [J].Numer Math, 1962, 4:24-40.
[17] Bunse-Gerstner A, Byers R, Mehrmann V.Numerical methods for simultaneous diagonalization [J].SIAM J Matrix Anal Appl, 1993, 14(4):927-949.
[18] Mascarenhas W F.On the convergence of the Jacobi method for arbitrary orderings [J].SIAM J Matrix Anal Appl, 1995, 16(4):1197-1209.
[19] Mirsky L.On the minimization of matrix norms [J].Amer Math Monthly, 1958, 65:106-107.
[20] Mathews C P, Haardt M, Zoltowski M D.Performance analysis of closed-form, ESPRIT based 2-D angle estimator for rectangular arrays [J].IEEE Signal Proc Letters, 1996, 3(4):124-126.
Memo
- Memo:
- Biographies: Fu Tuo(1977—), male, doctor, associate research fellow, futuo@seu.edu.cn;Gao Xiqi(1967—), male, doctor, professor, xqgao@seu.edu.cn.