Signal, Image and Video Processing

, Volume 13, Issue 7, pp 1311–1318 | Cite as

Underdetermined DOA estimation using coprime array via multiple measurement sparse Bayesian learning

  • Yanhua Qin
  • Yumin LiuEmail author
  • Zhongyuan Yu
Original Paper


Underdetermined direction of arrival (DOA) estimation with coprime array is discussed in the framework of multiple measurement sparse Bayesian learning (MSBL). Exploiting the extended difference coarray, a larger number of degrees of freedom can be obtained for locating more sources than sensors. A linear operation and a prewhitening procedure are incorporated into the sparse signal recovery model to eliminate the influence of noise. Then, MSBL employs an empirical Bayesian strategy to resolve \(l_{0}\) minimization problem. Simulation results show the superiority of the MSBL algorithm in underdetermined DOA detection performance, resolution ability and estimation accuracy when there are multiple measurement vectors for on-grid and off-grid sources, respectively.


Coprime array Direction of arrival estimation Degrees of freedom Multiple measurement sparse Bayesian learning 



The authors would like to thank the anonymous reviewers for their many insightful comments and suggestions, which help improve the quality and readability of this paper.


  1. 1.
    Ma, W.K., Hsieh, T.H., Chi, C.Y.: DOA estimation of quasi-stationary signals with less sensors than sources and unknown spatial noise covariance: a Khatri–Rao subspace approach. IEEE Trans. Signal Process. 58, 2168–2180 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Shen, Q., Liu, W., Cui, W., Wu, S.: Underdetermined DOA estimation under the compressive sensing framework: a review. IEEE Access 4, 8865–8878 (2016)CrossRefGoogle Scholar
  3. 3.
    Mohammadzadeh, S., Kukrer, O.: Robust adaptive beamforming based on covariance matrix and new steering vector estimation. SIViP 8, 1–8 (2019)Google Scholar
  4. 4.
    Pal, P., Vaidyanathan, P.P.: Nested arrays: a novel approach to array processing with enhanced degrees of freedom. IEEE Trans. Signal Process. 58, 4973–4973 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Vaidyanathan, P.P., Pal, P.: Sparse sensing with co-prime samplers and arrays. IEEE Trans. Signal Process. 59, 573–587 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Qin, Y., Liu, Y., Liu, J., Yu, Z.: Underdetermined wideband DOA estimation for off-grid sources with coprime array using sparse Bayesian learning. Sensors 18, 253–264 (2018)CrossRefGoogle Scholar
  7. 7.
    Nannuru, S., Koochakzadeh, A., Gemba, K.L., Pal, P., Gerstoft, P.: Sparse Bayesian learning for beamforming using sparse linear arrays. J. Acoust. Soc. Am. 144, 2719–2729 (2018)CrossRefGoogle Scholar
  8. 8.
    Ciuonzo, D.: On time-reversal imaging by statistical testing. IEEE Signal Process. Lett. 24, 1024–1028 (2017)CrossRefGoogle Scholar
  9. 9.
    Ciuonzo, D., Romano, G., Solimene, R.: Performance analysis of time-reversal MUSIC. IEEE Trans. Signal Process. 63, 2650–2662 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ciuonzo, D., Rossi, P.S.: Noncolocated time-reversal MUSIC: high-SNR distribution of null spectrum. IEEE Signal Process. Lett. 24, 397–401 (2017)CrossRefGoogle Scholar
  11. 11.
    Vaidyanathan, P.P., Pal, P.: Why does direct-MUSIC on sparse-arrays work? In: Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA (2013)Google Scholar
  12. 12.
    Tan, Z., Nehorai, A.: Sparse direction of arrival estimation using co-prime arrays with off-grid targets. IEEE Signal Process. Lett. 21, 26–29 (2014)CrossRefGoogle Scholar
  13. 13.
    Shen, Q., Liu, W., Cui, W., Wu, S., Zhang, Y.D., Amin, M.G.: Low-complexity direction-of-arrival estimation based on wideband co-prime arrays. IEEE Trans. Audio Speech Lang. Process. 23, 1445–1456 (2015)CrossRefGoogle Scholar
  14. 14.
    Li, J., Li, D., Jiang, D., Zhang, X.: Extended-aperture unitary root MUSIC-based DOA estimation for coprime array. IEEE Commun. Lett. 22, 752–755 (2018)CrossRefGoogle Scholar
  15. 15.
    Zhou, C., Gu, Y., He, S., Shi, Z.: A robust and efficient algorithm for coprime array adaptive beamforming. IEEE Trans. Veh. Technol. 67, 1099–1112 (2018)CrossRefGoogle Scholar
  16. 16.
    Zhou, C., Gu, Y., Fan, X., Shi, Z., Mao, G., Zhang, Y.D.: Direction-of-arrival estimation for coprime array via virtual array interpolation. IEEE Trans. Signal Process. 66, 5956–5971 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Shi, Z., Zhou, C., Gu, Y., Goodman, N.A., Qu, F.: Source estimation using coprime array: a sparse reconstruction perspective. IEEE Sens. J. 17, 755–765 (2017)CrossRefGoogle Scholar
  18. 18.
    Tan, Z., Eldar, Y.C., Nehorai, A.: Direction of arrival estimation using co-prime arrays: a super resolution viewpoint. IEEE Trans. Signal Process. 62, 5565–5576 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Balkan, O., Kreutz-Delgado, K., Makeig, S.: Localization of more sources than sensors via jointly-sparse Bayesian learning. IEEE Signal Process. Lett. 21, 131–134 (2014)CrossRefGoogle Scholar
  20. 20.
    Wipf, D.P., Rao, B.D.: Sparse Bayesian learning for basis selection. IEEE Trans. Signal Process. 52, 2153–2164 (2004)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zhang, Z., Rao, B.D.: Sparse signal recovery in the presence of correlated multiple measurement vectors. In: IEEE International Conference on Acoustics, Speech and Signal Processing, Dallas, TX, USA (2010)Google Scholar
  22. 22.
    Zhang, Z., Rao, B.D.: Sparse signal recovery with temporally correlated source vectors using sparse Bayesian learning. IEEE J. Sel. Top. Signal Proces. 5, 912–926 (2011)CrossRefGoogle Scholar
  23. 23.
    Zhang, Z., Rao, B.D.: Extension of SBL algorithms for the recovery of block sparse signals with intra-block correlation. IEEE Trans. Signal Process. 61, 2009–2015 (2013)CrossRefGoogle Scholar
  24. 24.
    Tipping, M.E.: Sparse Bayesian learning and the relevance vector machine. J. Mach. Learn. Res. 1, 211–244 (2001)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Tan, X., Roberts, W., Li, J., Stoica, P.: Sparse learning via iterative minimization with application to MIMO radar imaging. IEEE Trans. Signal Process. 59, 1088–1101 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Addabbo, P., Aubry, A., Maio, A.D., Pallotta, L., Ullo, S.L.: High range resolution profile estimation using sparse learning via iterative minimization. IET Radar Sonar Navig. 13, 512–521 (2019)CrossRefGoogle Scholar
  27. 27.
    Cotter, S.F., Rao, B.D., Engan, K., Kreutz-Delgado, K.: Sparse solutions to linear inverse problems with multiple measurement vectors. IEEE Trans. Signal Process. 53, 2477–2488 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wipf, D.P., Rao, B.D.: An empirical Bayesian strategy for solving the simultaneous sparse approximation problem. IEEE Trans. Signal Process. 55, 3704–3716 (2007)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Qin, S., Zhang, Y.D., Amin, M.G.: Generalized coprime array configurations for direction-of-arrival estimation. IEEE Trans. Signal Process. 63, 1377–1390 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Hu, N., Sun, B., Zhang, Y., Dai, J., Wang, J., Chang, C.: Underdetermined DOA estimation method for wideband signals using joint nonnegative sparse Bayesian learning. IEEE Signal Process. Lett. 24, 535–539 (2017)CrossRefGoogle Scholar
  31. 31.
    He, Z.Q., Shi, Z.P., Huang, L., So, H.C.: Underdetermined DOA estimation for wideband signals using robust sparse covariance fitting. IEEE Signal Process. Lett. 22, 435–439 (2015)CrossRefGoogle Scholar
  32. 32.
    Pal, P., Vaidyanathan, P.P.: Coprime sampling and the MUSIC algorithm. In: Digital Signal Processing and Signal Processing Education Meeting (DSP/SPE), Sedona, AZ, USA (2011)Google Scholar
  33. 33.
    Tibshirani, R.: Regression shrinkage and selection via the LASSO. J. R. Stat. Soc. Ser. B 58, 267–288 (1994)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Yang, Z., Xie, L., Zhang, C.: Off-grid direction of arrival estimation using sparse Bayesian inference. IEEE Trans. Signal Process. 61, 38–43 (2013)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Zhang, Z., Rao, B.D.: Iterative reweighted algorithms for sparse signal recovery with temporally correlated source vectors. In: IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Prague, Czech Republic (2011)Google Scholar
  36. 36.
    Stoica, P., Nehorai, A.: Performance study of conditional and unconditional direction-of-arrival estimation. IEEE Trans. Acoust. Speech Signal Process. 38, 1783–1795 (1990)CrossRefGoogle Scholar
  37. 37.
    Trees, H.L.Van: Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory. Wiley, New York (2002)CrossRefGoogle Scholar
  38. 38.
    Shaghaghi, M., Vorobyov, S.A.: Cramér–Rao bound for sparse signals fitting the low-rank model with small number of parameters. IEEE Signal Process. Lett. 22, 1497–1501 (2015)CrossRefGoogle Scholar
  39. 39.
    Chun-Lin, L., Vaidyanathan, P.P.: Cramér bounds for coprime and other sparse arrays, which find more sources than sensors. Digit. Signal Process. 61, 43–61 (2017)CrossRefGoogle Scholar
  40. 40.
    Wang, M., Nehorai, A.: Coarrays, MUSIC, and the Cramér bound. IEEE Trans. Signal Process. 65, 933–946 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Institute of Information Photonics and Optical CommunicationsBeijing University of Posts and TelecommunicationsBeijingChina

Personalised recommendations