Space-time covariance matrix estimation : loss of algebraic multiplicities of Eigenvalues

Khattak, Faizan A. and Weiss, Stephan and Proudler, Ian K. and McWhirter, John G. (2022) Space-time covariance matrix estimation : loss of algebraic multiplicities of Eigenvalues. In: 56th Asilomar Conference on Signals, Systems, and Computers, 2021-11-30 - 2022-11-03, Pacific Grove, CA.

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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 space-time 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.