An approximate polynomial matrix eigenvalue decomposition algorithm for para-hermitian matrices

Soydan Redif, Stephan Weiss, John G. McWhirter

Research output: Chapter in Book/Report/Conference proceedingConference contribution book

10 Citations (Scopus)

Abstract

In this paper, we propose an algorithm for computing an approximate polynomial matrix eigenvalue decomposition (PEVD). The PEVD of a para-Hermitian matrix yields a factorisation into a polynomial matrix product consisting of a spectrally majorised diagonal matrix that is pre- and post-multiplied by paraunitary (PU) matrices. All current PEVD algorithms, such as the second order sequential best rotation (SBR2) algorithm, perform a factorisation whereby diagonalisation and spectral majorisation are only achieved in approximation. The purpose of this paper is to present a new iterative approach which constitutes a “Householder-like” version of SBR2 and is akin to Tkacenko's approximate EVD (AEVD); however, unlike the AEVD, the proposed method carries out the diagonalisation successively by applying arbitrary-degree, finite impulse response PU matrices. We show an application of our algorithm to the design of signal-adapted PU filter banks for subband coding. Simulation results for the proposed approach show very close agreement with the behaviour of the infinite order principal component filter banks and demonstrate its superiority compared to state-of-the-art algorithms in terms of strong decorrelation and spectral majorisation.

LanguageEnglish
Title of host publicationProceedings of IEEE International Symposium on Signal Processing and Information Technology (ISSPIT), 2011
EditorsS. Weiss, J.G. McWhirter
Place of PublicationNew York
PublisherIEEE
Pages421-425
Number of pages5
ISBN (Print)9781467307529
DOIs
Publication statusPublished - 2011
Event11th IEEE International Symposium on Signal Processing and Information Technology - Bilbao, Spain
Duration: 14 Dec 201117 Dec 2011

Conference

Conference11th IEEE International Symposium on Signal Processing and Information Technology
CountrySpain
CityBilbao
Period14/12/1117/12/11

Fingerprint

Polynomials
Decomposition
Filter banks
Factorization
Impulse response

Keywords

  • para-hermitian matrices
  • polynomial matrix
  • eigenvalue decomposition algorithm
  • Algorithm design and analysis
  • Polynomials
  • Matrix decomposition
  • Approximation algorithms
  • encoding
  • eigenvalues and eigenfunctions

Cite this

Redif, S., Weiss, S., & McWhirter, J. G. (2011). An approximate polynomial matrix eigenvalue decomposition algorithm for para-hermitian matrices. In S. Weiss, & J. G. McWhirter (Eds.), Proceedings of IEEE International Symposium on Signal Processing and Information Technology (ISSPIT), 2011 (pp. 421-425). New York: IEEE. https://doi.org/10.1109/ISSPIT.2011.6151599
Redif, Soydan ; Weiss, Stephan ; McWhirter, John G. . / An approximate polynomial matrix eigenvalue decomposition algorithm for para-hermitian matrices. Proceedings of IEEE International Symposium on Signal Processing and Information Technology (ISSPIT), 2011 . editor / S. Weiss ; J.G. McWhirter. New York : IEEE, 2011. pp. 421-425
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abstract = "In this paper, we propose an algorithm for computing an approximate polynomial matrix eigenvalue decomposition (PEVD). The PEVD of a para-Hermitian matrix yields a factorisation into a polynomial matrix product consisting of a spectrally majorised diagonal matrix that is pre- and post-multiplied by paraunitary (PU) matrices. All current PEVD algorithms, such as the second order sequential best rotation (SBR2) algorithm, perform a factorisation whereby diagonalisation and spectral majorisation are only achieved in approximation. The purpose of this paper is to present a new iterative approach which constitutes a “Householder-like” version of SBR2 and is akin to Tkacenko's approximate EVD (AEVD); however, unlike the AEVD, the proposed method carries out the diagonalisation successively by applying arbitrary-degree, finite impulse response PU matrices. We show an application of our algorithm to the design of signal-adapted PU filter banks for subband coding. Simulation results for the proposed approach show very close agreement with the behaviour of the infinite order principal component filter banks and demonstrate its superiority compared to state-of-the-art algorithms in terms of strong decorrelation and spectral majorisation.",
keywords = "para-hermitian matrices , polynomial matrix , eigenvalue decomposition algorithm, Algorithm design and analysis, Polynomials , Matrix decomposition , Approximation algorithms, encoding, eigenvalues and eigenfunctions",
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Redif, S, Weiss, S & McWhirter, JG 2011, An approximate polynomial matrix eigenvalue decomposition algorithm for para-hermitian matrices. in S Weiss & JG McWhirter (eds), Proceedings of IEEE International Symposium on Signal Processing and Information Technology (ISSPIT), 2011 . IEEE, New York, pp. 421-425, 11th IEEE International Symposium on Signal Processing and Information Technology, Bilbao, Spain, 14/12/11. https://doi.org/10.1109/ISSPIT.2011.6151599

An approximate polynomial matrix eigenvalue decomposition algorithm for para-hermitian matrices. / Redif, Soydan ; Weiss, Stephan; McWhirter, John G. .

Proceedings of IEEE International Symposium on Signal Processing and Information Technology (ISSPIT), 2011 . ed. / S. Weiss; J.G. McWhirter. New York : IEEE, 2011. p. 421-425.

Research output: Chapter in Book/Report/Conference proceedingConference contribution book

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AB - In this paper, we propose an algorithm for computing an approximate polynomial matrix eigenvalue decomposition (PEVD). The PEVD of a para-Hermitian matrix yields a factorisation into a polynomial matrix product consisting of a spectrally majorised diagonal matrix that is pre- and post-multiplied by paraunitary (PU) matrices. All current PEVD algorithms, such as the second order sequential best rotation (SBR2) algorithm, perform a factorisation whereby diagonalisation and spectral majorisation are only achieved in approximation. The purpose of this paper is to present a new iterative approach which constitutes a “Householder-like” version of SBR2 and is akin to Tkacenko's approximate EVD (AEVD); however, unlike the AEVD, the proposed method carries out the diagonalisation successively by applying arbitrary-degree, finite impulse response PU matrices. We show an application of our algorithm to the design of signal-adapted PU filter banks for subband coding. Simulation results for the proposed approach show very close agreement with the behaviour of the infinite order principal component filter banks and demonstrate its superiority compared to state-of-the-art algorithms in terms of strong decorrelation and spectral majorisation.

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Redif S, Weiss S, McWhirter JG. An approximate polynomial matrix eigenvalue decomposition algorithm for para-hermitian matrices. In Weiss S, McWhirter JG, editors, Proceedings of IEEE International Symposium on Signal Processing and Information Technology (ISSPIT), 2011 . New York: IEEE. 2011. p. 421-425 https://doi.org/10.1109/ISSPIT.2011.6151599