Impact of Space-Time Covariance Estimation Errors on a Parahermitian Matrix EVD

Abstract

This paper studies the impact of estimation errors in the sample space-time covariance matrix on its parahermitian matrix eigenvalue decomposition. We provide theoretical bounds for the perturbation of the ground-truth eigenvalues and of the subspaces of their corresponding eigenvectors. We show that for the eigenvalues, the perturbation depends on the norm of the estimation error in the space-time covariance matrix, while the perturbation of eigenvector subspaces can additionally be influenced by the distance between the eigenvalues. We confirm these theoretical results by simulations. Index Terms-broadband array processing; space-time covariance estimation; parahermitian matrix; eigenvalue decomposition.

Sample Space-time Covariance Matrix Estimation

“…The mean square modelling error is a metric that has previously been established in [25] to perturb the eigenvalues and eigenspaces of the space-time covariance matrix; therefore, the results, particularly (9), now directly link this perturbation to the sample size and the ground truth matrix. Fig.…”

“…Similarly, the PhEVDR(z) = U(z)Λ(z)Û P (z) will deviate from that in (2). In [25], the norm of the modelling error…”

“…The perturbation of the subspace angle between a particular eigenspace of U (z) and ofÛ (z) can in turn be linked to the norm of E(z) and the distance to the nearest eigenvalue, i.e. near eigenvalues with algebraic multiplicity greater than one, subspaces can undergo a larger perturbation [25].…”

“…, which forms the basis for the perturbation analysis of the eigenvalues and eigenvector in [25]. Since the sample size N and the specific ground truth R[τ ] are given for a particular problem, the only way to minimise this perturbation is the judicious selection of the range of lags, |τ | ≤ T , over whichR[τ ] is evaluated.…”

“…In applications, the space-time covariance or the CSD matrix generally have to be estimated from data. While the accuracy of the decomposition itself has been investigated in [19,20], and limiting factors due to algorithm-internal order reductions [8,[21][22][23] and the conditioning of the underlying source model [24] are known, it is only recently that the effect of estimating the space time covariance matrix from a finite data set has been addressed [25]. While [25] linked the estimation error to the perturbation of the eigenvectors and eigenspaces of the CSD matrix, the formulation of this estimation error had still been missing.…”