Frequency-selective Vandermonde decomposition of Toeplitz matrices with applications
The classical result of Vandermonde decomposition of positive semidefinite Toeplitz matrices, which dates back to the early twentieth century, forms the basis of modern subspace and recent atomic norm methods for frequency estimation. In this paper, we study the Vandermonde decomposition in which th...
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sg-ntu-dr.10356-1419672020-06-12T05:36:59Z Frequency-selective Vandermonde decomposition of Toeplitz matrices with applications Yang, Zai Xie, Lihua School of Electrical and Electronic Engineering Engineering::Electrical and electronic engineering Frequency-selective Vandermonde Decomposition Toeplitz Matrix The classical result of Vandermonde decomposition of positive semidefinite Toeplitz matrices, which dates back to the early twentieth century, forms the basis of modern subspace and recent atomic norm methods for frequency estimation. In this paper, we study the Vandermonde decomposition in which the frequencies are restricted to lie in a given interval, referred to as frequency-selective Vandermonde decomposition. The existence and uniqueness of the decomposition are studied under explicit conditions on the Toeplitz matrix. The new result is connected by duality to the positive real lemma for trigonometric polynomials nonnegative on the same frequency interval. Its applications in the theory of moments and line spectral estimation are illustrated. In particular, it provides a solution to the truncated trigonometric K-moment problem. It is used to derive a primal semidefinite program formulation of the frequency-selective atomic norm in which the frequencies are known a priori to lie in certain frequency bands. Numerical examples are also provided. MOE (Min. of Education, S’pore) 2020-06-12T05:36:58Z 2020-06-12T05:36:58Z 2018 Journal Article Yang, Z., & Xie, L. (2018). Frequency-selective Vandermonde decomposition of Toeplitz matrices with applications. Signal Processing, 142, 157-167. doi:10.1016/j.sigpro.2017.07.024 0165-1684 https://hdl.handle.net/10356/141967 10.1016/j.sigpro.2017.07.024 2-s2.0-85025645580 142 157 167 en Signal Processing © 2017 Elsevier B.V. All rights reserved. |
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Engineering::Electrical and electronic engineering Frequency-selective Vandermonde Decomposition Toeplitz Matrix Yang, Zai Xie, Lihua Frequency-selective Vandermonde decomposition of Toeplitz matrices with applications |
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The classical result of Vandermonde decomposition of positive semidefinite Toeplitz matrices, which dates back to the early twentieth century, forms the basis of modern subspace and recent atomic norm methods for frequency estimation. In this paper, we study the Vandermonde decomposition in which the frequencies are restricted to lie in a given interval, referred to as frequency-selective Vandermonde decomposition. The existence and uniqueness of the decomposition are studied under explicit conditions on the Toeplitz matrix. The new result is connected by duality to the positive real lemma for trigonometric polynomials nonnegative on the same frequency interval. Its applications in the theory of moments and line spectral estimation are illustrated. In particular, it provides a solution to the truncated trigonometric K-moment problem. It is used to derive a primal semidefinite program formulation of the frequency-selective atomic norm in which the frequencies are known a priori to lie in certain frequency bands. Numerical examples are also provided. |
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School of Electrical and Electronic Engineering |
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School of Electrical and Electronic Engineering Yang, Zai Xie, Lihua |
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Article |
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Yang, Zai Xie, Lihua |
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Yang, Zai |
title |
Frequency-selective Vandermonde decomposition of Toeplitz matrices with applications |
title_short |
Frequency-selective Vandermonde decomposition of Toeplitz matrices with applications |
title_full |
Frequency-selective Vandermonde decomposition of Toeplitz matrices with applications |
title_fullStr |
Frequency-selective Vandermonde decomposition of Toeplitz matrices with applications |
title_full_unstemmed |
Frequency-selective Vandermonde decomposition of Toeplitz matrices with applications |
title_sort |
frequency-selective vandermonde decomposition of toeplitz matrices with applications |
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2020 |
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https://hdl.handle.net/10356/141967 |
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