Space-time covariance matrix factorisation and estimation for broadband multichannel problems, part 1: background

Within the SSP'25 tutorial, this Part I specifically addresses the background of space-time covariance matrices, polynomial cross-spectral density matrices, and their properties and operations. Overall, the tutorial addresses recent developments in formulating and solving broadband multichannel problems through matrices of functions and their factorisations, such as the analytic eigenvalue decomposition of the space-time covariance, or an analytic singular value decomposition applied to a data model. This can generalise well known formulations of narrowband problems using covariance matrices, and of narrowband solutions via their diagonalisation, to the broadband case. We present theoretical background on the factorisation of matrices of functions, and show how the estimation of the statistical parameters impacts on the perturbation of the ground truth factors of a decomposition. We review a number of algorithms, and discuss some sample applications such as direction of arrival estimation, beamforming, weak transient signal and subspace detection, MIMO communications, speech enhancement, or source separation.