Sensor array calibration with joint-block-sparsity in the presence of multiple separable observations

  • Ahmet M. ElbirEmail author
Original Paper


In sparsity-based optimization problems, one of the major issue is computational complexity, especially when the unknown signal is represented in multi-dimensions such as in the problem of 2-D (azimuth and elevation) direction-of-arrival (DOA) estimation. In this paper, a low-complexity sparsity-based method is proposed for DOA estimation in the presence of array imperfections such as mutual coupling. In order to reduce the complexity of the optimization problem, this paper introduces a new sparsity structure that can be used to model the optimization problem in case of multiple data snapshots and multiple separable observations where the dictionary can be decomposed into two parts: azimuth and elevation dictionaries. The proposed sparsity structure is called joint-block-sparsity which exploits the sparsity in both multiple dimensions, namely azimuth and elevation, and data snapshots. In order to model the joint-block-sparsity in the optimization problem, triple mixed norms are used. In the simulations, the proposed method is compared with both sparsity-based techniques and subspace-based methods as well as the Cramer–Rao lower bound. It is shown that the proposed method effectively calibrates the sensor array with significantly low complexity and sufficient accuracy.


Direction of arrival estimation Mutual coupling Joint-block-sparsity Separable observations Triple mixed norms 



  1. 1.
    Krim, H., Viberg, M.: Two decades of array signal processing research: the parametric approach. IEEE Signal Process. Mag. 13, 67–94 (1996)CrossRefGoogle Scholar
  2. 2.
    Tuncer, T.E., Friedlander, B.: Classical and Modern Direction-of-Arrival Estimation. Academic Press, Cambridge (2009)Google Scholar
  3. 3.
    Schmidt, R.: Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas Propag. 34, 276–280 (1986)CrossRefGoogle Scholar
  4. 4.
    Ziskind, I., Wax, M.: Maximum likelihood localization of multiple sources by alternating projection. IEEE Trans. Acoust. Speech Signal Process. 36, 1553–1560 (1988)CrossRefzbMATHGoogle Scholar
  5. 5.
    Balanis, C.A.: Antenna Theory: Analysis and Design. Wiley, Hoboken (2005)Google Scholar
  6. 6.
    Elbir, A.M., Tuncer, T.E.: 2D-DOA and mutual coupling coefficient estimation for arbitrary array structures with single and multiple snapshots. Digit. Signal Process. 54, 75–86 (2016)CrossRefGoogle Scholar
  7. 7.
    A, Elbir: Direction finding in the presence of direction-dependent mutual coupling. IEEE Antennas Wirel. Propag. Lett. PP(99), 1–1 (2017)Google Scholar
  8. 8.
    Friedlander, B., Weiss, A.: Direction finding in the presence of mutual coupling. IEEE Trans. Antennas Propag. 39, 273–284 (1991)CrossRefGoogle Scholar
  9. 9.
    Liao, B., Zhang, Z.-G., Chan, S.-C.: DOA estimation and tracking of ULAs with mutual coupling. IEEE Trans. Aerosp. Electron. Syst. 48, 891–905 (2012)CrossRefGoogle Scholar
  10. 10.
    Ye, Z., Liu, C.: 2-D DOA estimation in the presence of mutual coupling. IEEE Trans. Antennas Propag. 56, 3150–3158 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Wang, M., Ma, X., Yan, S., Hao, C.: An autocalibration algorithm for uniform circular array with unknown mutual coupling. IEEE Antennas Wirel. Propag. Lett. 15, 12–15 (2016)CrossRefGoogle Scholar
  12. 12.
    Elbir, A.M.: A novel data transformation approach for doa estimation with 3-d antenna arrays in the presence of mutual coupling. IEEE Antennas Wirel. Propag. Lett. PP(99), 1–1 (2017)Google Scholar
  13. 13.
    Candès, E., Wakin, M.: An introduction to compressive sampling. IEEE Signal Process. Mag. 25, 21–30 (2008)Google Scholar
  14. 14.
    Candès, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20, 33–61 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    He, Z.-Q., Shi, Z.-P., Huang, L.: Covariance sparsity-aware DOA estimation for nonuniform noise. Digital Signal Process. 28, 75–81 (2014)CrossRefGoogle Scholar
  17. 17.
    Porter, R., Tadic, V., Achim, A.: Sparse bayesian learning for non-Gaussian sources. Digital Signal Proce. 45, 2–12 (2015)CrossRefGoogle Scholar
  18. 18.
    Dehkordi, M.B., Abutalebi, H.R., Taban, M.R.: Sound source localization using compressive sensing-based feature extraction and spatial sparsity. Digital Signal Process. 23(4), 1239–1246 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Malioutov, D., Cetin, M., Willsky, A.: A sparse signal reconstruction perspective for source localization with sensor arrays. IEEE Trans. Signal Process. 53, 3010–3022 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Liu, Z.-M., Zhou, Y.-Y.: A unified framework and sparse Bayesian perspective for direction-of-arrival estimation in the presence of array imperfections. IEEE Trans. Signal Process. 61, 3786–3798 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cands, E.J., Fernandez-Granda, C.: Towards a mathematical theory of super-resolution. Commun. Pure Appl. Math. 67(6), 906–956 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Elbir, A. M., Tuncer, T. E.: Single snapshot DOA estimation in the presence of mutual coupling for arbitrary array structures. In: 2016 IEEE sensor array and multichannel signal processing workshop (SAM), pp. 1–5 (2016)Google Scholar
  23. 23.
    Hyder, M., Mahata, K.: Direction-of-arrival estimation using a mixed \(l_{2,0}\)-norm norm approximation. IEEE Trans. Signal Process. 58, 4646–4655 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wu, X., Zhu, W.P., Yan, J.: Direction of arrival estimation for off-grid signals based on sparse Bayesian learning. IEEE Sens. J. 16, 2004–2016 (2016)CrossRefGoogle Scholar
  25. 25.
    Yang, Z., Li, J., Stoica, P., Xie, L.: Chapter 11—Sparse Methods for Direction-of-Arrival Estimation vol 7 of Academic Press Library in Signal Processing. Elsevier, Hoboken (2017)Google Scholar
  26. 26.
    Zhao, G., Shi, G., Shen, F., Luo, X., Niu, Y.: A sparse representation-based doa estimation algorithm with separable observation model. IEEE Antennas Wirel. Propag. Lett. 14, 1586–1589 (2015)CrossRefGoogle Scholar
  27. 27.
    Stoica, P., Moses, R.: Spectral Analysis of Signals. Prentice Hall, Upper Saddle River (2005)Google Scholar
  28. 28.
    Svantesson, T.: Modeling and estimation of mutual coupling in a uniform linear array of dipoles. In: 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 5, pp. 2961–2964 (1999)Google Scholar
  29. 29.
    Ye, Z., Liu, C.: On the resiliency of MUSIC direction finding against antenna sensor coupling. IEEE Trans. Antennas Propag. 56, 371–380 (2008)CrossRefGoogle Scholar
  30. 30.
    Gondzio, J.: Interior point methods 25 years later. Eur. J. Oper. Res. 218, 587 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Cotter, S., Rao, B., Engan, K., Kreutz-Delgado, K.: Sparse solutions to linear inverse problems with multiple measurement vectors. IEEE Trans. Signal Process. 53, 2477–2488 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Rao, B.D., Kreutz-Delgado, K.: An affine scaling methodology for best basis selection. IEEE Trans. Signal Process. 47, 187–200 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Aktas, M., Tuncer, T. E.: HOS based online calibration. In: 2011 19th European Signal Processing Conference, pp. 604–608 (2011)Google Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Electrical and Electronics EngineeringDuzce UniversityDuzceTurkey

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