Eigenvalue decomposition of a parahermitian matrix: extraction of analytic eigenvalues

Stephan Weiss; Ian K. Proudler; Fraser K. Coutts

doi:10.1109/TSP.2021.3049962

Eigenvalue decomposition of a parahermitian matrix: extraction of analytic eigenvalues

Stephan Weiss, Ian K. Proudler, Fraser K. Coutts

Electronic And Electrical Engineering

Research output: Contribution to journal › Article › peer-review

36 Citations (Scopus)

123 Downloads (Pure)

Abstract

An analytic parahermitian matrix admits an eigenvalue decomposition (EVD) with analytic eigenvalues and eigenvectors except in the case of multiplexed data. In this paper, we propose an iterative algorithm for the estimation of the analytic eigenvalues. Since these are generally transcendental, we find a polynomial approximation with a defined error. Our approach operates in the discrete Fourier transform (DFT) domain and for every DFT length generates a maximally smooth association through EVDs evaluated in DFT bins; an outer loop iteratively grows the DFT order and is shown, in general, to converge to the analytic eigenvalues. In simulations, we compare our results to existing approaches.

Original language	English
Pages (from-to)	722-737
Number of pages	16
Journal	IEEE Transactions on Signal Processing
Volume	69
Early online date	8 Jan 2021
DOIs	https://doi.org/10.1109/TSP.2021.3049962
Publication status	Published - 28 Feb 2021

Keywords

eigenvalue
eigenfunctions
matrix decomposition
covariance matrices
approximation algorithms

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10.1109/TSP.2021.3049962

Weiss-etal-IEEE-TOPS-2021-Eigenvalue-decomposition-of-a-parahermitian-matrixAccepted author manuscript, 1.22 MB

1 Finished

Signal Processing in the Information Age (UDRC III)
Weiss, S. (Principal Investigator) & Stankovic, V. (Co-investigator)
EPSRC (Engineering and Physical Sciences Research Council)
1/07/18 → 31/03/24
Project: Research

36 Citations
2 Article
1 Conference contribution book
1 Paper

Scalable extraction of analytic eigenvalues from a parahermitian matrix
Khattak, F. A., Proudler, I. K. & Weiss, S., 30 Aug 2024, 32nd European Signal Processing Conference: EUSIPCO 2024. Piscataway, NJ: IEEE, p. 1317-1321 5 p. 2031
Research output: Chapter in Book/Report/Conference proceeding › Conference contribution book

Open Access
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2 Citations (Scopus)

7 Downloads (Pure)
Polynomial eigenvalue decomposition for multichannel broadband signal processing: a mathematical technique offering new insights and solutions
Neo, V. W., Redif, S., McWhirter, J. G., Pestana, J., Proudler, I. K., Weiss, S. & Naylor, P. A., 8 Nov 2023, In: IEEE Signal Processing Magazine. 40, 7, p. 18-37 20 p.
Research output: Contribution to journal › Article › peer-review

Open Access
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14 Citations (Scopus)

152 Downloads (Pure)
Eigenvalue decomposition of a parahermitian matrix: extraction of analytic Eigenvectors
Weiss, S., Proudler, I., Coutts, F. K. & Khattak, F. A., 24 Apr 2023, (E-pub ahead of print) In: IEEE Transactions on Signal Processing. 71, p. 1642-1656 15 p.
Research output: Contribution to journal › Article › peer-review

Open Access
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18 Citations (Scopus)

82 Downloads (Pure)

Cite this

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title = "Eigenvalue decomposition of a parahermitian matrix: extraction of analytic eigenvalues",

abstract = "An analytic parahermitian matrix admits an eigenvalue decomposition (EVD) with analytic eigenvalues and eigenvectors except in the case of multiplexed data. In this paper, we propose an iterative algorithm for the estimation of the analytic eigenvalues. Since these are generally transcendental, we find a polynomial approximation with a defined error. Our approach operates in the discrete Fourier transform (DFT) domain and for every DFT length generates a maximally smooth association through EVDs evaluated in DFT bins; an outer loop iteratively grows the DFT order and is shown, in general, to converge to the analytic eigenvalues. In simulations, we compare our results to existing approaches. ",

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PY - 2021/2/28

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N2 - An analytic parahermitian matrix admits an eigenvalue decomposition (EVD) with analytic eigenvalues and eigenvectors except in the case of multiplexed data. In this paper, we propose an iterative algorithm for the estimation of the analytic eigenvalues. Since these are generally transcendental, we find a polynomial approximation with a defined error. Our approach operates in the discrete Fourier transform (DFT) domain and for every DFT length generates a maximally smooth association through EVDs evaluated in DFT bins; an outer loop iteratively grows the DFT order and is shown, in general, to converge to the analytic eigenvalues. In simulations, we compare our results to existing approaches.

AB - An analytic parahermitian matrix admits an eigenvalue decomposition (EVD) with analytic eigenvalues and eigenvectors except in the case of multiplexed data. In this paper, we propose an iterative algorithm for the estimation of the analytic eigenvalues. Since these are generally transcendental, we find a polynomial approximation with a defined error. Our approach operates in the discrete Fourier transform (DFT) domain and for every DFT length generates a maximally smooth association through EVDs evaluated in DFT bins; an outer loop iteratively grows the DFT order and is shown, in general, to converge to the analytic eigenvalues. In simulations, we compare our results to existing approaches.

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KW - eigenfunctions

KW - matrix decomposition

KW - covariance matrices

KW - approximation algorithms

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DO - 10.1109/TSP.2021.3049962

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JO - IEEE Transactions on Signal Processing

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ER -

Eigenvalue decomposition of a parahermitian matrix: extraction of analytic eigenvalues

Abstract

Keywords

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Projects

Signal Processing in the Information Age (UDRC III)

Research output

Scalable extraction of analytic eigenvalues from a parahermitian matrix

Polynomial eigenvalue decomposition for multichannel broadband signal processing: a mathematical technique offering new insights and solutions

Eigenvalue decomposition of a parahermitian matrix: extraction of analytic Eigenvectors

Cite this