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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Main Authors: Yang, Zai, Xie, Lihua
Other Authors: School of Electrical and Electronic Engineering
Format: Article
Language:English
Published: 2020
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Online Access:https://hdl.handle.net/10356/141967
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Institution: Nanyang Technological University
Language: English
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spelling 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.
institution Nanyang Technological University
building NTU Library
country Singapore
collection DR-NTU
language English
topic Engineering::Electrical and electronic engineering
Frequency-selective Vandermonde Decomposition
Toeplitz Matrix
spellingShingle Engineering::Electrical and electronic engineering
Frequency-selective Vandermonde Decomposition
Toeplitz Matrix
Yang, Zai
Xie, Lihua
Frequency-selective Vandermonde decomposition of Toeplitz matrices with applications
description 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.
author2 School of Electrical and Electronic Engineering
author_facet School of Electrical and Electronic Engineering
Yang, Zai
Xie, Lihua
format Article
author Yang, Zai
Xie, Lihua
author_sort 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
publishDate 2020
url https://hdl.handle.net/10356/141967
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