Multiple shift second order sequential best rotation algorithm for polynomial matrix EVD

Abstract

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Extending Narrowband Descriptions and Optimal Solutions to Broadband Sensor Arrays

“…Even though eigenvalues and particularly eigenvectors are not guaranteed to exist as analytic functions in case of spectral majorisation, a number of algorithms targetting the McWhirter decomposition (5) have been created over the past decade (McWhirter and Baxter, 2004;McWhirter et al, 2007;Tkacenko and Vaidyanathan, 2006;Tkacenko, 2010;Redif et al, 2011;Tohidian et al, 2013;Corr et al, 2014c;Redif et al, 2015;Wang et al, 2015a). These all share the restriction of considering the EVD of a parahermitian matrix R(z) whose elements are Laurent polynomials, which may be enforced by estimating or approximating R[τ] over a finite lag windwo (Redif et al, 2011).…”

“…Even though many algorithms can be proven to converge, in the sense that they reduce off-diagonal energy of Γ (z) at each iteration, see e.g. Redif et al, 2011;Corr et al, 2014c;Redif et al, 2015;Wang et al, 2015a), there is no practical experience yet where these algorithms could not find a practicable factorisation.…”

“…Over the past decade, a number of algorithms have emerged that implement a polynomial EVD Redif et al, 2011;Tohidian et al, 2013;Corr et al, 2014c;Redif et al, 2015;Wang et al, 2015a), and also triggered a range of applications in the area of filter banks (Redif et al, 2011;Weiss et al, 2006), beamforming Koh et al, 2009;Alrmah et al, 2011;Weiss et al, 2013;Vouras and Tran, 2014;Weiss et al, 2015;Alzin et al, 2016), communications (Weiss et al, 2006;Davies et al, 2007;Ta and Weiss, 2007a;Sandmann et al, 2015;Ahrens et al, 2017), or generic theoretical problems such as blind source separation (Redif et al, 2017) or spectral factorisation (Wang et al, 2015b).…”

Multichannel spectral factorization algorithm using polynomial matrix eigenvalue decomposition

“…The PEVD can be approximated by an iterative process which transforms off-diagonal elements in R(z) onto the diagonal. To date, a number of iterative algorithms have been developed to compute the PEVD, including the SBR2 algorithm [12], the sequential matrix diagonalization (SMD) algorithm [16], multiple shift maximum element SMD (MSME-SMD) algorithm [17], and multiple shift SBR2 (MS-SBR2) algorithm [18] etc.…”

Complexity and search space reduction in cyclic-by-row PEVD algorithms

“…Algorithms to compute the PEVD include the original second order sequential best rotation (SBR2) algorithm [9], sequential matrix diagonalisation (SMD) [12] and various evolutions of the algorithm families [13]- [16]. All of these algorithms employ an iterative approach to approximately diagonalise the parahermitian matrix, stopping when some suitable threshold is reached.…”