Scalable Analytic Eigenvalue Extraction Algorithm

Abstract

Broadband sensor array problems can be formulated using parahermitian polynomial matrices, and the optimal solution to these problems can be based on the eigenvalue decomposition (EVD) of these matrices. An algorithm has been proposed in the past to extract analytic eigenvalues of parahermitian matrices, but it does not scale well with the temporal and spatial dimensions of the parahermitian matrix. This paper introduces a scalable analytical eigenvalue extraction algorithm for parahermitian polynomial matrices. The proposed algorithm operates in the discrete Fourier transform (DFT) domain, where an EVD is computed in each bin. Associations across bins are established based on properties of the analytic eigenvectors. The need to avoid problems with non-trivial algebraic multiplicities and control time-domain aliasing leads to an iterative algorithm that increases the DFT size until a suitable error criterion is satisfied. The algorithm can be shown to converge. Benchmarked against the existing algorithm, it performs accurately and with lower cost, and can successfully decompose matrices with dimensions much larger than previously had been feasible.

Keywords

• Analytic functions
• algebraic multiplicities
• space-time covariance
• discrete Fourier transform
• eigenvalue decomposition
• parahermitian matrix