Multi-step Knowledge-Aided Iterative Conjugate Gradient Algorithms for DOA Estimation

  • Silvio F. B. PintoEmail author
  • Rodrigo C. de Lamare


In this work, we present direction-of-arrival (DoA) estimation algorithms based on the Krylov subspace that effectively exploit prior knowledge of the signals that impinge on a sensor array. The proposed multi-step knowledge-aided iterative conjugate gradient (CG) (MS-KAI-CG) algorithms perform subtraction of the unwanted terms found in the estimated covariance matrix of the sensor data. Furthermore, we develop a version of MS-KAI-CG equipped with forward–backward averaging, called MS-KAI-CG-FB, which is appropriate for scenarios with correlated signals. Unlike current knowledge-aided methods, which take advantage of known DoAs to enhance the estimation of the covariance matrix of the input data, the MS-KAI-CG algorithms take advantage of the knowledge of the structure of the forward–backward smoothed covariance matrix and its disturbance terms. Simulations with both uncorrelated and correlated signals show that the MS-KAI-CG algorithms outperform existing techniques.


Large sensor arrays Knowledge-aided techniques Direction finding High-resolution parameter estimation 



  1. 1.
    A. Barabell, Performance of orthogonal matching pursuit for multiple measurement vectors with noise, in IEEE International Conference on Acoustics, Speech and Signal Processing (April 1983), pp. 336–339Google Scholar
  2. 2.
    G. Golub, C.V. Loan, Matrix Computations (Johns Hopkins University Press, Baltimore, 1996)zbMATHGoogle Scholar
  3. 3.
    R. Grover, D. Pados, M. Medley, Subspace direction finding with an auxiliary-vector basis. IEEE Trans. Signal Process. 55(2), 758–763 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J.L. Jr, T. Rappaport, Smart antennas for Wireless Communications: IS-95 and Third Generation CDMA Applications (Prentice Hall, Upper Saddle River, 1999)Google Scholar
  5. 5.
    P. Pal, P. Vaidyanathan, A novel approach to array processing with enhanced degrees of freedom. IEEE Trans. Signal Process. 58(8), 4167–4181 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    P. Pal, P. Vaidyanathan, Sparse sensing with co-prime samplers and arrays. IEEE Trans. Signal Process. 59(2), 573–586 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    S. Pillai, B. Kwon, Forward/backward spatial smoothing techniques for coherent signal identification. IEEE Trans. Acoust. Speech Signal Process. 37(1), 8–15 (1989)CrossRefzbMATHGoogle Scholar
  8. 8.
    S. Pinto, R. de Lamare, Two-step knowledge-aided iterative ESPRIT algorithm, in IEEE Twenty First ITG Workshop on Smart Antennas (Berlin, March 2017), pp. 1–5Google Scholar
  9. 9.
    S. Pinto, R. de Lamare, Multi-step knowledge-aided iterative ESPRIT for direction finding, in IEEE 22nd International Conference on Digital Signal Processing (London, August 2017), pp. 1–5Google Scholar
  10. 10.
    S. Pinto, R. de Lamare, Knowledge-aided parameter estimation based on Conjugate Gradient algorithms, in 35th Brazilian Communications and Signal Processing Symposium (Sao Pedro, SP, Brazil, August 2017), pp. 1–5Google Scholar
  11. 11.
    S. Pinto, R. de Lamare, Multi-step knowledge-aided iterative Conjugate Gradient for direction finding, in IEEE 22nd ITG Workshop on Smart Antennas (Bochum, March 2018), pp. 1–5Google Scholar
  12. 12.
    S. Pinto, R. de Lamare, Multi-step knowledge-aided iterative ESPRIT: design and analysis. IEEE Trans. Aerosp. Electron. Syst. 54(5), 2189–2201 (2018b)CrossRefGoogle Scholar
  13. 13.
    L. Qiu, R. de Lamare, M. Zhao, Reduced-rank DOA estimation algorithms based on alternating low-rank decomposition. IEEE Signal Process. Lett. 23(5), 565–569 (2016)CrossRefGoogle Scholar
  14. 14.
    J. Rissanen, Modeling by the shortest data description. Automatica 14(6), 465–471 (1978)CrossRefzbMATHGoogle Scholar
  15. 15.
    R. Roy, T. Kailath, Estimation of signal parameter via rotational invariance technique. IEEE Trans. Acoust. Speech Signal Process. 37(7), 984–995 (1989)CrossRefGoogle Scholar
  16. 16.
    S. Schell, W. Gardner, High Resolution Direction Finding—Handbook of Statistics (Elsevier, Amsterdam, 1993)Google Scholar
  17. 17.
    R. Schmidt, Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas Propag. 34, 276–280 (1986)CrossRefGoogle Scholar
  18. 18.
    H. Semira, H. Belkacemi, S. Marcos, High-resolution source localization algorithm based on the conjugate gradient. EURASIP J. Adv. Signal Process. 2007(2), 12 (2007)MathSciNetzbMATHGoogle Scholar
  19. 19.
    M. Shaghaghi, S. Vorobyov, Iterative root-MUSIC algorithm for DOA estimation, in IEEE 5th International Workshop on Computational Advances in Multisensor Adaptive Processing (St. Martin, France, December 2013), pp. 1–4Google Scholar
  20. 20.
    M. Shaghaghi, S. Vorobyov, Subspace leakage analysis and improved DOA estimation with small sample size. IEEE Trans. Signal Process. 63(12), 3251–3265 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Z. Shi, C. Zhou, Y. Gu, N.A. Goodman, F. Qu, Source estimation using coprime array: a sparse reconstruction perspective. IEEE Sens. J. 17(3), 755–765 (2017)CrossRefGoogle Scholar
  22. 22.
    J. Steinwandt, R. de Lamare, M. Haardt, Knowledge-aided direction finding based on Unitary ESPRIT, in IEEE 45th Asilomar Conference on Signals, Systems and Computers (Pacific Grove, CA, USA, November 2011), pp. 613–617Google Scholar
  23. 23.
    J. Steinwandt, R. de Lamare, M. Haardt, Beamspace direction finding based on the conjugate gradient algorithm, in 2011 International ITG Workshop on Smart Antennas (Aachen, Germany, February 2011), pp. 1–5Google Scholar
  24. 24.
    J. Steinwandt, R. de Lamare, M. Haardt, Beamspace direction finding based on the conjugate gradient and the auxiliary vector filtering algorithms. Signal Process. 93(4), 641–651 (2013)CrossRefGoogle Scholar
  25. 25.
    P. Stoica, A. Gershman, Maximum-likelihood DOA estimation by data-supported grid search. IEEE Signal Process. Lett. 6(10), 273–275 (1999)CrossRefGoogle Scholar
  26. 26.
    P. Stoica, A. Nehorai, MUSIC, maximum likelihood, and Cramér–Rao bound. IEEE Trans. Acoust. Speech Signal Process. 37(5), 720–741 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    P. Stoica, A. Nehorai, Performance study of conditional and unconditional direction-of-arrival estimation. IEEE Trans. Acoust. Speech Signal Process. 38(10), 1783–1795 (1990)CrossRefzbMATHGoogle Scholar
  28. 28.
    P. Stoica, X. Zhu, J. Guerci, On using a priori knowledge in space-time adaptive processing. IEEE Trans. Signal Process. 56(6), 2598–2602 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    A. Thakre, M. Haardt, K. Giridhar, Single snapshot spatial smoothing with improved effective array aperture. IEEE Signal Process. Lett. 16(6), 505–508 (2009)CrossRefGoogle Scholar
  30. 30.
    H.V. Trees, Optimum Array Processing. Part IV of Detection, Estimation and Modulation Theory (Wiley, New York, 2002)Google Scholar
  31. 31.
    L. Wang, R. de Lamare, M. Haardt, Direction finding algorithms based on joint iterative subspace optimization. IEEE Trans. Aerosp. Electron. Syst. 50(4), 2541–2553 (2014)CrossRefGoogle Scholar
  32. 32.
    C. Zhou, Y. Gu, Y. Zhang, Z. Shi, T. Jin, X. Wu, Compressive sensing-based coprime array direction-of-arrival estimation. IET Commun. 11(11), 1719–1724 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Pontifical Catholic University of Rio de Janeiro - CETUCRio de JaneiroBrazil

Personalised recommendations