Simultaneous Schur decomposition of several matrices to achieve automatic pairing in multidimensional harmonic retrieval problems

This paper presents a new Jacobi-type method to calculate a simultaneous Schur decomposition (SSD) of several real-valued, non-symmetric matrices by minimizing an appropriate cost function. Thereby, the SSD reveals the “average eigenstructure” of these non-symmetric matrices. This enables an R-dimensional extension of Unitary ESPRIT to estimate several undamped R-dimensional modes or frequencies along with their correct pairing in multidimensional harmonic retrieval problems. Unitary ESPRIT is an ESPRIT-type high-resolution frequency estimation technique that is formulated in terms of real-valued computations throughout. For each of the R dimensions, the corresponding frequency estimates are obtained from the real eigenvalues of a real-valued matrix. The SSD jointly estimates the eigenvalues of all R matrices and, thereby, achieves automatic pairing of the estimated R-dimensional modes via a closed-form procedure, that neither requires any search nor any other heuristic pairing strategy. Finally, we show how R-dimensional harmonic retrieval problems (with R ≥ 3) occur in array signal processing and model-based object recognition applications.