Convergences of adaptive block simultaneous iteration method for eigenstructure decomposition

Abstract Eigenstructure decomposition of data correlation matrices is of basic interest in many modern signal processing problems. In practical situations, often the data environment is changing or nonstationary, and thus it may be advantageous to use adaptive algorithms to obtain the desired eigenvectors and eigenvalues. Recently, we have shown the adaptive block simultaneous iteration method (SIM) has an efficient implmentation using a systolic processor array architecture for evaluating some or all of the eigenvectors of the data correlation matrix. In this paper, the convergence and optimality properties of this adaptive algorithm are considered. We first review the basic properties of the adaptive SIM algorithm and consider its application to the MUSIC direction-of-arrival problem. Then we discuss the convergence in the mean and the asymptotic convergence of the adaptive SIM eigendecomposition solutions to the true solutions, based upon two standard assumptions in the analysis of adaptive algorithm. Furthermore, we consider the optimality of the algorithm under the minimum variance criterion. We show the asymptotic equivalence of the Kalman algorithm approach and the adaptive SIM algorithm approach in estimating the smallest eigenvector using a transversal filter. Our analytical results on the adaptive SIM algorithm confirm and extend various optimality results reported by Karhunen based on simulation results.

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