Vandermonde Decomposition of Multilevel Toeplitz Matrices With Application to Multidimensional Super-Resolution

The Vandermonde decomposition of Toeplitz matrices, discovered by Carathéodory and Fejér in the 1910s and rediscovered by Pisarenko in the 1970s, forms the basis of modern subspace methods for 1-D frequency estimation. Many related numerical tools have also been developed for multidimensional (MD)...

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Veröffentlicht in:IEEE transactions on information theory Jg. 62; H. 6; S. 3685 - 3701
Hauptverfasser: Yang, Zai, Xie, Lihua, Stoica, Petre
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.06.2016
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Algorithms
Atomic measurements
atomic norm
Blocking
Computer simulation
Covariance matrices
Decomposition
Electronic mail
Estimating techniques
Frequency distribution
Frequency estimation
Mathematical models
Matrix
Matrix decomposition
multidimensional frequency estimation
Multilevel
multilevel Toeplitz matrix
Norms
Numerical simulation
Signal resolution
Subspace methods
superresolution
The Vandermonde decomposition
multilevel Toeplitz matrix
The Vandermonde decomposition
super-resolution
multidimensional frequency estimation
atomic norm
ISSN:0018-9448, 1557-9654, 1557-9654
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Zusammenfassung:The Vandermonde decomposition of Toeplitz matrices, discovered by Carathéodory and Fejér in the 1910s and rediscovered by Pisarenko in the 1970s, forms the basis of modern subspace methods for 1-D frequency estimation. Many related numerical tools have also been developed for multidimensional (MD), especially 2-D, frequency estimation; however, a fundamental question has remained unresolved as to whether an analog of the Vandermonde decomposition holds for multilevel Toeplitz matrices in the MD case. In this paper, an affirmative answer to this question and a constructive method for finding the decomposition are provided when the matrix rank is lower than the dimension of each Toeplitz block. A numerical method for searching for a decomposition is also proposed when the matrix rank is higher. The new results are applied to study the MD frequency estimation within the recent super-resolution framework. A precise formulation of the atomic \ell _{0} norm is derived using the Vandermonde decomposition. Practical algorithms for frequency estimation are proposed based on the relaxation techniques. Extensive numerical simulations are provided to demonstrate the effectiveness of these algorithms compared with the existing atomic norm and subspace methods.
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ISSN:0018-9448
1557-9654
1557-9654
DOI:10.1109/TIT.2016.2553041