Impact of space-time covariance matrix estimation on bin-wise eigenvalue and eigenspace perturbations

In the context of broadband multichannel signal processing, problems can often be formulated using a space-time covariance matrix, and solved using a diagonalisation of this quantity via a polynomial or analytic eigenvalue decomposition (EVD). In this paper, we address the impact that an estimation of the space-time covariance has on the factors of such a decomposition. In order to address this, we consider a linear unbiased estimator based on Gaussian distributed data, and characterise the variance of this estimate, as well as the variance of the error between the estimate and the ground truth. These quantities in turn enable to find expressions for the bin-wise perturbation of the eigenvalues, which depends on the error variance of the estimate, and for the bin-wise perturbation of the eigenspaces, which depends on both the error variance but also on the eigenvalue distance. We adapt a number of known bounds for ordinary matrices and demonstrate the fit of these bounds in simulations. In order to minimise the error variance of the estimate, and hence the perturbation of the EVD factors, we discuss a way to optimise the lag support of the space-time covariance estimate without access to the ground truth on which the estimate is based.