Extraction of Analytic Eigenvectors From a Parahermitian Matrix

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.