Fixed Beamformer Design Using Polynomial Eigenvalue Decomposition

Neo, Vincent W.; d'Olne, Emilie; Moore, Alastair H.; Naylor, Patrick A.

doi:10.1109/iwaenc53105.2022.9914799

Cited by 1 publication

(1 citation statement)

References 32 publications

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“…This CSD matrix can assist in formulating and solving broadband array problems, including e.g. beamforming [2]- [4], blind source separation [5], multichannel coding [6], [7], speech enhancement [8]- [10], or MIMO system design [11]- [13].…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue Decomposition of a Parahermitian Matrix: Extraction of Analytic Eigenvectors

Weiss

Proudler

Coutts

et al. 2023

IEEE Trans. Signal Process.

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An analytic parahermitian matrix admits in almost all cases an eigenvalue decomposition (EVD) with analytic eigenvalues and eigenvectors. We have previously defined a discrete Fourier transform (DFT) domain algorithm which has been proven to extract the analytic eigenvalues. The selection of the eigenvalues as analytic functions guarantees in turn the existence of unique one-dimensional eigenspaces in which analytic eigenvectors can exist. Determining such eigenvectors is not straightforward, and requires three challenges to be addressed. Firstly, one-dimensional subspaces for eigenvectors have to be woven smoothly across DFT bins where a non-trivial algebraic multiplicity causes ambiguity. Secondly, with the one-dimensional eigenspaces defined, a phase smoothing across DFT bins aims to extract analytic eigenvectors with minimum time domain support. Thirdly, we need to check whether the DFT length, and thus the approximation order, is sufficient. We propose an iterative algorithm for the extraction of analytic eigenvectors and prove that this algorithm converges to the best of a set of stationary points. We provide a number of numerical examples and simulation results, in which the algorithm is demonstrated to extract the ground truth analytic eigenvectors arbitrarily closely.

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