Enforcing Eigenvector Smoothness for a Compact DFT-Based Polynomial Eigenvalue Decomposition

Abstract

A variety of algorithms have been developed to compute an approximate polynomial matrix eigenvalue decomposition (PEVD). As an extension of the ordinary EVD to polynomial matrices, the PEVD will generate paraunitary matrices that diagonalise a parahermitian matrix. While iterative PEVD algorithms that compute a decomposition in the time domain have received a great deal of focus and algorithmic improvements in recent years, there has been less research in the field of frequency-based PEVD algorithms. Such algorithms have shown promise for the decomposition of problems of finite order, but the state-of-the-art requires a priori knowledge of the length of the polynomial matrices required in the decomposition. This paper presents a novel frequency-based PEVD algorithm which can compute an accurate decomposition without requiring this information. Also presented is a new metric for measuring a function's smoothness on the unit circle, which is utilised within the algorithm to maximise eigenvector smoothness for a compact decomposition, such that the polynomial eigenvectors have low order. We demonstrate through the use of simulations that the algorithm can achieve superior levels of decomposition accuracy to a state-of-the-art frequency-based method.

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“…Therefore, the proposed metric offers some good properties for the extraction of analytic factors for, for example, the EVD of an analytic, parahermitian matrix [12], [13]. Analyticity in turn offers the opportunity of Laurent polynomial approximations that can be siginificantly lower in order than for factors that are obtained by current time domain algorithms favouring spectral majorisation [4], [5], [7], [9].…”

“…Thus, for constraints C1 and C2, the overall constraint in (11) will be drawn from (13) and (12), such that G ∈ R 2(L N +K)×2N and b ∈ R 2(L N +K) . For K > N − L N , the constraint equation will be an overdetermined system, and it will either be possible to condense the constraint equation Gf = b via an SVD similar to robust MVDR beamforming [23], or in case it is approximately full rank, entirely via f = G † b, with {•} † denoting the pseudo-inverse.…”

“…DFT-domain algorithms do not naturally possess the frequency domain coherence that has motivated time domain approaches [4], [5], [7], and therefore require association across frequency bins. In [10]- [12] this association is based on the continuity of eigenvectors, which in principle is easier to detect than a non-differentiability of eigenvalues. The association decisions are most crucial near Q-fold algebraic multiplicities of eigenvalues, where eigenvectors can be arbitrarily selected as an orthogonal basis within a Q-dimensional subspace [2], thus creating challenges for an eigenvector-driven association [13].…”

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“…This association is driven by the bin-wise eigenvectors. However, in case if a Q-fold multiplicity of eigenvalues, the associated eigenvectors can form an arbitrary basis in a Q-fold subspace; similarly, for closely-spaced eigenvalues, the calculation of individual eigenvectors may therefore be ill-conditioned [57,46,48]. The approach in [47] therefore establishes associations across frequency bins solely based on the eigenvalues, driven by a smoothness criterion of the interpolated functions [49,50].…”

“…Additionally, the computational complexity of these algorithms has been addressed by various means, including linear algebraic approximations of the EVD [31,32], the truncation of large polynomials [33,34,35,36], reduction of the optimisation parameter space [37,38,39,40] as well as the exploitation of the symmetry of R(z) [41], and the parallelisation of algorithms in [41,42,43,44]. For the PhEVD in (3), and initial approximation has been provided in [45,46], with further developments that exploit the analyticity of the extracted solution for eigenvalues [47] and eigenvectors [48], based in smoothness criteria in [49,50].…”