Article (Scientific journals)
On Properties and Structure of the Analytic Singular Value Decomposition
Weiss, Stephan; Proudler, Ian K.; Barbarino, Giovanni et al.
2024In IEEE Transactions on Signal Processing, 72, p. 2260 - 2275
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Keywords :
analytic functions; diagonalisation; Singular value decomposition; transfer function matrices; Analytic solution; Index; matrix; Matrix decomposition; Over sampling; Property; Rectangular matrix; Signal processing algorithms; Singular values; Unit circles; Signal Processing; Electrical and Electronic Engineering
Abstract :
[en] We investigate the singular value decomposition (SVD) of a rectangular matrix A(z) of functions that are analytic on an annulus that includes at least the unit circle. Such matrices occur, e.g., as matrices of transfer functions representing broadband multiple-input multiple-output systems. Our analysis is based on findings for the analytic SVD applicable to continuous time systems, and on the analytic eigenvalue decomposition. Using these, we establish two potentially overlapping cases where analyticity of the SVD factors is denied. Firstly, from a structural point of view, multiplexed systems require oversampling by the multiplexing factor in order to admit an analytic solution. Secondly, from an algebraic perspective, we state under which condition spectral zeros of any singular value require additional oversampling by a factor of two if an analytic solution is to be found. In all other cases, an analytic matrix admits an analytic SVD, whereby the singular values are unique up to a permutation, and the left- and right-singular vectors are coupled through a joint ambiguity w.r.t. an arbitrary allpass function. We demonstrate how some state-of-the-art polynomial matrix decomposition algorithms approximate this solution, motivating the need for dedicated algorithms.
Disciplines :
Mathematics
Author, co-author :
Weiss, Stephan ;  University of Strathclyde, Department of Electronic & Electrical Engineering, Glasgow, United Kingdom
Proudler, Ian K. ;  University of Strathclyde, Department of Electronic & Electrical Engineering, Glasgow, United Kingdom
Barbarino, Giovanni  ;  Université de Mons - UMONS > Recherche > Service ERC Unit - Matrix Theory and Optimization
Pestana, Jennifer ;  University of Strathclyde, Department of Mathematics and Statistics, Glasgow, United Kingdom
McWhirter, John G. ;  University of Strathclyde, Department of Electronic & Electrical Engineering, Glasgow, United Kingdom
Language :
English
Title :
On Properties and Structure of the Analytic Singular Value Decomposition
Publication date :
2024
Journal title :
IEEE Transactions on Signal Processing
ISSN :
1053-587X
Publisher :
Institute of Electrical and Electronics Engineers Inc.
Volume :
72
Pages :
2260 - 2275
Peer reviewed :
Peer Reviewed verified by ORBi
Research unit :
F151 - Mathématique et Recherche opérationnelle
Research institute :
R300 - Institut de Recherche en Technologies de l'Information et Sciences de l'Informatique
R450 - Institut NUMEDIART pour les Technologies des Arts Numériques
European Projects :
HE - 101085607 - eLinoR - Beyond Low-Rank Factorizations
Name of the research project :
5597 - eLinoR - Beyond Low-Rank Factorizations - Sources publiques européennes
Funders :
UE - Union Européenne
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since 11 December 2024

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