Extraction of analytic eigenvectors from a parahermitian matrix

Weiss, Stephan and Proudler, Ian K. and Coutts, Fraser K. and Deeks, Julian (2020) Extraction of analytic eigenvectors from a parahermitian matrix. In: International Conference on Sensor Signal Processing for Defence, 2020-09-15 - 2020-09-16. (https://doi.org/10.1109/SSPD47486.2020.9272001)

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The space-time covariance matrix derived from broadband multichannel data admits — unless the data emerges from a multiplexing operation — a parahermitian matrix eigenvalue decomposition with analytic eigenvalues and analytic eigenvectors. The extraction of analytic eigenvalues has been solved previously in the discrete Fourier transform (DFT) domain; this paper addresses the approximation of analytic eigenvectors in the DFT domain. This is a two-stage process — in the first instance, we identify eigenspaces in which analytic eigenvectors can reside. This stage resolves ambiguities at frequencies where eigenvalues have algebraic mulitplicities greater than one. In a second stage, the phase ambiguity of eigenvectors is addressed by determining a maximally smooth phase response. Finally, a metric for the approximation error is derived, which allows us to increase the DFT length and iterate the two stages until a desired accuracy is reached.