Support Estimation of Analytic Eigenvectors of Parahermitian Matrices

Abstract

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.

Generalized Polynomial Power Method

“…The underlying idea is to extend the polynomial power method, proposed in [19] for para-Hermitian matrices, to general polynomial matrices for the extraction of the dominant singular vectors and singular value. Thus this section provides a brief summary of the polynomial power method.…”

“…This normalisation is carried out in the DFT domain. If c 1 (e jΩ ) possesses any spectral nulls for some Ω, the resulting division by zero in the normalisation process can be avoided by regularization [19]. For a sufficiently large k, the normalized vectors become…”

“…To limit the order growth of x (k) (z), truncation can be applied. This can be achieved by either limiting the order to the estimated support of the eigenvector obtained from [24] by shifted-truncation [25], or removing any trailing coefficients below a small threshold. The iterations are stopped once a suitable defined difference between consecutive polynomial vectors falls below some threshold.…”

“…In order to overcome the deficiencies of the above PSVD algorithms, in this document, we extend the polynomial power method [19] from a para-Hermitian matrix to a general polynomial matrix for the computation of the dominant leftand right-singular vector and the singular value. The polynomial power method is an extension of the ordinary power method [20] to para-Hermitian matrices where a polynomial vector is repeatedly multiplied by a para-Hermitian matrix, and the resulting vector converges to the dominant eigenvector provided that the matrix is spectrally majorised.…”

Space-Time Covariance Matrix Estimation: Loss of Algebraic Multiplicities of Eigenvalues

“…It is interesting to note that the loss of algebraic multiplicities or the strict spectral majorisation of eigenvalues cannot be alleviated by enhancing estimates. This includes, for example, limiting the perturbation of eigenvalues through optimum support estimation [6], [10]. Bypassing some estimation errors through performing a system identification of the source model [11] generally still retains some finite error, for example due to observation noise.…”