Extracting analytic eigenvectors from parahermitian matrices relies on phase smoothing in the discrete Fourier transform (DFT) domain as its most expensive algorithmic component. Some algorithms require an a priori estimate of the eigenvector support and therefore the DFT length, while others iteratively increase the DFT. Thus in this document, we aim to complement the former and to reduce the computational load of the latter by estimating the time-domain support of eigenvectors. The proposed approach is validated via an ensemble of eigenvectors of known support, which the estimated support accurately matches.

Parahermitian matrices in almost all cases admit an eigenvalue decomposition (EVD) with analytic eigenvalues. This decomposition is key in order to extend the utility of the EVD from narrowband multichannel signal processing problems to the broadband case, where the EVD factors are frequency dependent. In the frequency domain, the ground truth analytic eigenvalues may intersect, in this paper we discuss why with estimated spacetime covariance matrices such algebraic multiplicities are lost, resulting with probability one in analytic, spectrally majorised eigenvalues that no longer intersect. We characterise this phenomenon and some of its profound consequences for broadband multichannel array signal processing.

The second order sequential best rotation (SBR2) algorithm is a popular algorithm to decompose a parahermitian matrix into approximated polynomial eigenvalues and eigenvectors. The work horse behind SBR2 is a Givens rotation interspersed by delay operations. In this paper, we investigate and analyse the application of a fast Givens rotation in order to reduce the computation complexity of SBR2. The proposed algorithm inherits the SBR2's proven convergence to a diagonalised and spectrally majorised solution for the polynomial eigenvalues. We provide some analysis and examples for the execution speed of this fast Givens-based SBR2 compared to a standard SBR2 implementation.

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.

The error inflicted on a space-time covariance estimate due to the availability of only finite data is known to perturb the eigenvalues and eigenspaces of its z-domain equivalent, i.e., the cross-spectral density matrix. In this paper, we show that a significantly more accurate estimate can be obtained if the source signals driving the signal model are also accessible, such that a system identication approach for the source model becomes viable. We demonstrate this improved accuracy in simulations, and discuss its dependencies on the sample size and the signal to noise ratio of the data.