A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
Volume 9 Issue 11
Nov.  2022

IEEE/CAA Journal of Automatica Sinica

  • JCR Impact Factor: 7.847, Top 10% (SCI Q1)
    CiteScore: 13.0, Top 5% (Q1)
    Google Scholar h5-index: 64, TOP 7
Turn off MathJax
Article Contents
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)
More Information
  • 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.

     

  • loading
  • [1]
    J. H. Qian, Z. S. He, W. Zhang, Y. L. Huang, N. Fu, and J. Chambers, “Robust adaptive beamforming for multiple-input multiple-output radar with spatial filtering techniques,” Signal Process., vol. 143, pp. 152–160, Feb. 2018. doi: 10.1016/j.sigpro.2017.09.004
    [2]
    C. A. Chen, Z. F. Yang, C. E. Chen, and Y. H. Huang, “A generalized eigenvalue decomposition processor for multi-user MIMO precoding,” in Proc. IEEE Asia Pacific Conf. Circuits and Systems, Jeju, Korea (South), 2016, pp. 281–284.
    [3]
    X. Y. Kong, B. Y. Du, X. W. Feng, and J. Y. Luo, “Unified and self-stabilized parallel algorithm for multiple generalized eigenpairs extraction,” IEEE Trans. Signal Process., vol. 68, pp. 3644–3659, May 2020. doi: 10.1109/TSP.2020.2997803
    [4]
    S. L. Sun, X. J. Xie, and C. Dong, “Multiview learning with generalized eigenvalue proximal support vector machines,” IEEE Trans. Cybern., vol. 49, no. 2, pp. 688–697, Feb. 2019. doi: 10.1109/TCYB.2017.2786719
    [5]
    B. Y. Du, X. Y. Kong, and X. W. Feng, “Generalized principal component analysis-based subspace decomposition of fault deviations and its application to fault reconstruction,” IEEE Access, vol. 8, pp. 34177–34186, Feb. 2020. doi: 10.1109/ACCESS.2020.2971507
    [6]
    V. D. Nguyen, K. Abed-Meraim, N. Linh-Trung, and R. Weber, “Generalized minimum noise subspace for array processing,” IEEE Trans. Signal Process., vol. 65, no. 14, pp. 3789–3802, Jul. 2017. doi: 10.1109/TSP.2017.2695457
    [7]
    S. Ouyang, T. Lee, and P. C. Ching, “A power-based adaptive method for eigenanalysis without square-root operations,” Digit. Signal Process., vol. 17, no. 1, pp. 209–224, Jan. 2007. doi: 10.1016/j.dsp.2006.02.003
    [8]
    R. Wang, M. L. Yao, D. M. Zhang, and H. X. Zou, “A novel orthonormalization matrix based fast and stable DPM algorithm for principal and minor subspace tracking,” IEEE Trans. Signal Process., vol. 60, no. 1, pp. 466–472, Jan. 2012. doi: 10.1109/TSP.2011.2169406
    [9]
    M. Arjomandi-Lari and M. Karimi, “Generalized YAST algorithm for signal subspace tracking,” Signal Process., vol. 117, pp. 82–95, 2015. doi: 10.1016/j.sigpro.2015.04.025
    [10]
    C. Pehlevan, T. Hu, and D. B. Chklovskii, “A Hebbian/anti-Hebbian neural network for linear subspace learning: A derivation from multidimensional scaling of streaming data,” Neural Comput., vol. 27, no. 7, pp. 1461–1495, Jul. 2015. doi: 10.1162/NECO_a_00745
    [11]
    U. K. Kalla, “Minor component analysis based anti-hebbian neural network scheme of decoupled voltage and frequency controller (DVFC) for nanohydro system,” in Proc. IEEE 7th Power India Int. Conf., Bikaner, India, 2016, pp. 1–6.
    [12]
    X. W. Feng, X. Y. Kong, H. G. Ma, and X. S. Si, “A novel unified and self-stabilizing algorithm for generalized eigenpairs extraction,” IEEE Trans. Neural Netw. Learn. Syst., vol. 28, no. 12, pp. 3032–3044, Dec. 2017. doi: 10.1109/TNNLS.2016.2614130
    [13]
    Y. B. Gao, X. Y. Kong, Z. X. Zhang, and L. A. Hou, “An adaptive self-stabilizing algorithm for minor generalized eigenvector extraction and its convergence analysis,” IEEE Trans. Neural Netw. Learn. Syst., vol. 29, no. 10, pp. 4869–4881, Oct. 2018. doi: 10.1109/TNNLS.2017.2783360
    [14]
    Y. B. Gao, X. Y. Kong, C. H. Hu, H. Z. Li, and L. A. Hou, “A generalized information criterion for generalized minor component extraction,” IEEE Trans. Signal Process., vol. 65, no. 4, pp. 947–959, Feb. 2017. doi: 10.1109/TSP.2016.2631444
    [15]
    R. Möller, Derivation of Coupled PCA and SVD Learning Rules From a Newton Zero-Finding Framewor. Computer Engineering, Faculty of Technology, Bielefeld University, Berlin, Germany, 2017.
    [16]
    T. Tanaka, “Fast generalized eigenvector tracking based on the power method,” IEEE Signal Process. Lett., vol. 16, no. 11, pp. 969–972, Nov. 2009. doi: 10.1109/LSP.2009.2027667
    [17]
    A. Valizadeh and M. Najibi, “A constrained optimization approach for an adaptive generalized subspace tracking algorithm,” Comput. Electr. Eng., vol. 36, no. 4, pp. 596–602, Jul. 2010. doi: 10.1016/j.compeleceng.2008.11.015
    [18]
    S. Attallah and K. Abed-Meraim, “A fast adaptive algorithm for the generalized symmetric eigenvalue problem,” IEEE Signal Process. Lett., vol. 15, pp. 797–800, Nov. 2008. doi: 10.1109/LSP.2008.2006346
    [19]
    L. J. Liu, H. M. Shao, and D. Nan, “Recurrent neural network model for computing largest and smallest generalized eigenvalue,” Neurocom-puting, vol. 71, pp. 16–18, Oct. 2008.
    [20]
    J. Yang, H. Hu, and H. S. Xi, “Weighted non-linear criterion-based adaptive generalised eigendecomposition,” IET Signal Process., vol. 7, no. 4, pp. 285–295, Jun. 2013. doi: 10.1049/iet-spr.2012.0212
    [21]
    W. T. Zhang, S. T. Lou, and D. Z. Feng, “Adaptive quasi-Newton algorithm for source extraction via CCA approach,” IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 4, pp. 677–689, Apr. 2014. doi: 10.1109/TNNLS.2013.2280285
    [22]
    T. D. Nguyen and I. Yamada, “Adaptive normalized quasi-Newton algorithms for extraction of generalized eigen-pairs and their convergence analysis,” IEEE Trans. Signal Process., vol. 61, no. 6, pp. 1404–1418, Mar. 2013. doi: 10.1109/TSP.2012.2234744
    [23]
    X. W. Feng, X. Y. Kong, H. G. Ma, and H. M. Liu, “Unified and coupled self-stabilizing algorithms for minor and principal eigen-pairs extraction,” Neural Process. Lett., vol. 45, no. 1, pp. 197–222, Feb. 2017. doi: 10.1007/s11063-016-9520-3
    [24]
    T. D. Nguyen, N. Takahashi, and I. Yamada, “An adaptive extraction of generalized eigensubspace by using exact nested orthogonal complement structure,” Multidim. Syst. Signal Process., vol. 24, no. 3, pp. 457–483, Sept. 2013. doi: 10.1007/s11045-012-0172-9
    [25]
    H. Z. Li, B. Y. Du, X. Y. Kong, Y. B. Gao, C. H. Hu, and X. H. Bian, “A generalized minor component extraction algorithm and its analysis,” IEEE Access, vol. 6, pp. 36771–36779, Jul. 2018. doi: 10.1109/ACCESS.2018.2852060
    [26]
    T. D. Nguyen and I. Yamada, “Necessary and sufficient conditions for convergence of the DDT systems of the normalized PAST algorithms,” Signal Process., vol. 94, pp. 288–299, Jan. 2014. doi: 10.1016/j.sigpro.2013.06.017
    [27]
    P. J. Zufiria, “On the discrete-time dynamics of the basic Hebbian neural network node,” IEEE Trans. Neural Netw., vol. 13, no. 6, pp. 1342–1352, Nov. 2002. doi: 10.1109/TNN.2002.805752
    [28]
    T. D. Nguyen and I. Yamada, “A unified convergence analysis of normalized PAST algorithms for estimating principal and minor components,” Signal Process., vol. 93, no. 1, pp. 176–184, Jan. 2013. doi: 10.1016/j.sigpro.2012.07.020
    [29]
    M. Ye, “Global convergence analysis of a self-stabilizing MCA learning algorithm,” Neurocomputing, vol. 67, pp. 321–327, Aug. 2005. doi: 10.1016/j.neucom.2005.01.002
    [30]
    M. Ye, Y. G. Liu, H. Wu, and Q. H. Liu, “A few online algorithms for extracting minor generalized eigenvectors,” in Proc. IEEE Int. Joint Conf. Neural Networks, Hong Kong, China, 2008, pp. 1714–1720.
    [31]
    X. Y. Kong, Q. S. An, H. G. Ma, C. Z. Han, and Q. Zhang, “Convergence analysis of deterministic discrete time system of a unified self-stabilizing algorithm for PCA and MCA,” Neural Netw., vol. 36, pp. 64–72, Dec. 2012. doi: 10.1016/j.neunet.2012.08.016
    [32]
    Y. B. Gao, X. Y. Kong, C. H. Hu, H. H. Zhang, and L. A. Hou, “Convergence analysis of Möller algorithm for estimating minor component,” Neural Process. Lett., vol. 42, no. 2, pp. 355–368, Oct. 2015. doi: 10.1007/s11063-014-9360-y
    [33]
    X. W. Feng, X. Y. Kong, Z. S. Duan, and H. G. Ma, “Adaptive generalized eigen-pairs extraction algorithms and their convergence analysis,” IEEE Trans. Signal Process., vol. 64, no. 11, pp. 2976–2989, Jun. 2016. doi: 10.1109/TSP.2016.2537260
    [34]
    G. O’Regan, “Matrix theory,” in Guide to Discrete Mathematics: An Accessible Introduction to the History, Theory, Logic and Applications. G. O’Regan, Ed. Cham, Germany: Springer Int. Publishing, 2016.
    [35]
    G. Cirrincione and M. Cirrincione, Neural-based Orthogonal Data Fitting: The EXIN Neural Networks. Hoboken, USA: John Wiley & Sons, 2010.
    [36]
    Y. Chen and G. Chen, “Stability analysis of systems with timevarying delay via a novel Lyapunov functional,” IEEE/CAA J. Autom. Sinica, vol. 6, no. 4, pp. 1068–1073, Jul. 2019. doi: 10.1109/JAS.2019.1911597
    [37]
    S. Ouyang and Y. B. Hua, “Bi-iterative least-square method for sub-space tracking,” IEEE Trans. Signal Process., vol. 53, no. 8, pp. 2984–2996, Aug. 2005. doi: 10.1109/TSP.2005.851102
    [38]
    L. H. Zhang, “On optimizing the sum of the Rayleigh quotient and the generalized Rayleigh quotient on the unit sphere,” Comput. Optim. Appl., vol. 54, no. 1, pp. 111–139, Jan. 2013. doi: 10.1007/s10589-012-9479-6
    [39]
    Z. Z. Xiang and M. X. Tao, “Robust beamforming for wireless information and power transmission,” IEEE Wirel. Commun. Lett., vol. 1, no. 4, pp. 372–375, Jun. 2012. doi: 10.1109/WCL.2012.053112.120212
    [40]
    D. K. Zhu, B. Y. Li, and P. Liang, “A novel hybrid beamforming algorithm with unified analog beamforming by subspace construction based on partial CSI for massive MIMO-OFDM systems,” IEEE Trans. Communications, vol. 65, no. 99, pp. 594–607, 2017.
    [41]
    K.-J. Kim, J. H. Kang, J.-H. Hwang, and K.-H. Ahn, “Hybrid beamforming architecture and wide bandwidth true-time delay for future high speed communications 5G and beyond 5G beamforming system,” in Proc. 3rd IEEE Int. Conf. Integrated Circuits and Microsystems, Shanghai, China, 2018, pp. 331–335.
    [42]
    G. D. M. Del Campo, Y. V. Shkvarko, and D. L. T. Román, “Multi-polarimetric SAR adapted virtual beamforming-based techniques for feature enhanced tomography of forested areas,” in Proc. IEEE MTT-S Latin America Microwave Conf., Puerto Vallarta, Mexico, 2016, pp. 1–3.
    [43]
    N. Nonaka, Y. Kakishima, and K. Higuchi, “Investigation on beamforming control methods in base station cooperative multiuser MIMO using block-diagonalized beamforming matrix,” in Proc. 80th IEEE Vehicular Technology Conf., Vancouver, BC, Canada, 2014, pp. 1–5.
    [44]
    G. P. Huang, J. Benesty, I. Cohen, and J. D. Chen, “A simple theory and new method of differential beamforming with uniform linear microphone arrays,” IEEE/ACM Trans. Audio,Speech,Language Processing, vol. 28, pp. 1079–1093, 2020.
    [45]
    H. Alwazani, A. Kammoun, A. Chaaban, M. Debbah, and M.-S. Alouini, “Intelligent reflecting surface-assisted multi-user MISO communication: Channel estimation and beamforming design,” IEEE Open J. Communications Society, vol. 1, pp. 661–680, 2020.
    [46]
    S. E. Lin, B. X. Zheng, G. C. Alexandropoulos, M. W. Wen, M. D. Renzo, and F. J. Chen, “Reconfigurable intelligent surfaces with reflection pattern modulation: Beamforming design and performance analysis,” IEEE Trans. Wireless Communications, vol. 20, no. 2, pp. 741–754, 2020.
    [47]
    E. Ali, M. Ismail, R. Nordin, and N. F. Abdulah, “Beamforming techniques for massive MIMO systems in 5G: Overview, classification, and trends for future research,” Frontiers of Information Technology Electronic Engineering, vol. 18, no. 6, pp. 753–772, 2017.
    [48]
    B. Y. Di, H. L. Zhang, L. Y. Song, Y. H. Li, Z. Han, and H. V. Poor, “Hybrid beamforming for reconfigurable intelligent surface based multi-user communications: Achievable rates with limited discrete phase shifts,” IEEE J. Selected Areas in Communications, vol. 38, no. 8, pp. 1809–1822, 2020.
    [49]
    B. Y. Di, H. L. Zhang, L. L. Li, L. Y. Song, Y. H. Li, and Z. Han, “Practical hybrid beamforming with finite-resolution phase shifters for reconfigurable intelligent surface based multi-user communications,” IEEE Trans. Vehicular Technology, vol. 69, no. 4, pp. 4565–4570, 2020.
    [50]
    W. J. Yan, X. J. Yuan, Z.-Q. He, and X. Y. Kuai, “Passive beamforming and information transfer design for reconfigurable intelligent surfaces aided multiuser MIMO systems,” IEEE J. Selected Areas in Communications, vol. 38, no. 8, pp. 1793–1808, 2020.
    [51]
    Z. Lin, M. Lin, J.-B. Wang, T. De Cola, and J. Z. Wang, “Joint beamforming and power allocation for satellite-terrestrial integrated networks with non-orthogonal multiple access,” IEEE J. Selected Topics in Signal Processing, vol. 13, no. 3, pp. 657–670, 2019.
    [52]
    P. S. Babu, P. V. Naganjaneyulu, and K. S. Prasad, “Adaptive beamforming of MIMO system using optimal steering vector with modified neural network for channel selection,” Int. J. Wavelets,Multiresolution Information Processing, vol. 19, no. 4, pp. 377–392, 2019.
    [53]
    N. K. Srivastava, R. Parihar, and S. K. Raghuwanshi, “Efficient photonic beamforming system incorporating a unique featured tunable chirped fiber bragg grating for application extended to the Ku-band,” IEEE Trans. Microwave Theory Techniques, vol. 3, no. 99, pp. 1851–1857, 2020.
    [54]
    P. S. R. Diniz, Adaptive Filtering: Algorithms and Practical Implementation. 4th ed. New York, USA: Springer, 2013.
    [55]
    B. Farhang-Boroujeny, “Sensor array processing,” Adaptive Filters: Theory and Applications, 2nd ed. B. Farhang-Boroujeny, Ed. Chichester, UK: John Wiley & Sons, 2013, pp. 659–694.

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Figures(8)  / Tables(1)

    Article Metrics

    Article views (725) PDF downloads(54) Cited by()

    Highlights

    • 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

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return