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Volume 9 Issue 11
Nov.  2022

IEEE/CAA Journal of Automatica Sinica

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Y. B. Gao, “Adaptive generalized eigenvector estimating algorithm for hermitian matrix pencil,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 11, pp. 1967–1979, Nov. 2022. doi: 10.1109/JAS.2021.1003955
Citation: Y. B. Gao, “Adaptive generalized eigenvector estimating algorithm for hermitian matrix pencil,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 11, pp. 1967–1979, Nov. 2022. doi: 10.1109/JAS.2021.1003955

Adaptive Generalized Eigenvector Estimating Algorithm for Hermitian Matrix Pencil

doi: 10.1109/JAS.2021.1003955
Funds:  This work was supported by the National Natural Science Foundation of China (62106242, 61903375), and in part by the Natural Science Foundation of Shaanxi Province, China (2020JM-356)
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  • Generalized eigenvector plays an essential role in the signal processing field. In this paper, we present a novel neural network learning algorithm for estimating the generalized eigenvector of a Hermitian matrix pencil. Differently from some traditional algorithms, which need to select the proper values of learning rates before using, the proposed algorithm does not need a learning rate and is very suitable for real applications. Through analyzing all of the equilibrium points, it is proven that if and only if the weight vector of the neural network is equal to the generalized eigenvector corresponding to the largest generalized eigenvalue of a Hermitian matrix pencil, the proposed algorithm reaches to convergence status. By using the deterministic discrete-time (DDT) method, some convergence conditions, which can be satisfied with probability 1, are also obtained to guarantee its convergence. Simulation results show that the proposed algorithm has a fast convergence speed and good numerical stability. The real application demonstrates its effectiveness in tracking the optimal vector of beamforming.


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    • To estimate the first generalized eigenvector, a novel algorithm is proposed when the matrix pencil is explicitly provided
    • To prove the convergence result of the proposed algorithm, the fixed stability of the proposed algorithm is analyzed by the Lyapunov function approach
    • The convergence analysis is accomplished by the DDT method and some convergence conditions are also obtained
    • An online adaptive algorithm is derived when the generalized eigenvector is needed to be calculated directly from the input signal or data sequences


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