A DFT-based approximate eigenvalue and singular value decomposition of polynomial matrices

Abstract

In this article, we address the problem of singular value decomposition of polynomial matrices and eigenvalue decomposition of para-Hermitian matrices. Discrete Fourier transform enables us to propose a new algorithm based on uniform sampling of polynomial matrices in frequency domain. This formulation of polynomial matrix decomposition allows for controlling spectral properties of the decomposition. We set up a nonlinear quadratic minimization for phase alignment of decomposition at each frequency sample, which leads to a compact order approximation of decomposed matrices. Compact order approximation of decomposed matrices makes it suitable in filterbank and multiple-input multiple-output (MIMO) precoding applications or any application dealing with realization of polynomial matrices as transfer function of MIMO systems. Numerical examples demonstrate the versatility of the proposed algorithm provided by relaxation of paraunitary constraint, and its configurability to select different properties.

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).…”

“…Also, (Tohidian et al, 2013) have presented a frequency domain algorithm which can favour analytic over spectrally majorised solutions (Coutts et al, 2017b;Coutts et al, 2018). A further route of investigation is the impact which estimation errors in the space-time covariance matrix have on the accuracy of the factorisation (Delaosa et al, 2018).…”

“…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).…”

Identification of Broadband Source-Array Responses from Sensor Second Order Statistics

“…Instead of a single campaign, now multiple measurement campaigns have been performed and the decompositions (17) and (18) are available for i = 1 . .…”

“…In case that eigenvalues intersect on the unit circle, spectral majorisation will enforce the approximation of non-analytic eigenvalues and -vectors. To guarantee analytic eigenvalues even in case of eigenvalues intersecting on the unit circle, algorithms different from SBR2 or SMD are required, with a DFT-base approach in [17] a first step.…”