Circuits, Systems, and Signal Processing

, Volume 37, Issue 9, pp 3862–3887 | Cite as

Pseudo-beam-Forming for Direction-of-Arrival Estimation with Difference Co-array of Co-prime Array

  • Yong JiaEmail author
  • Chuan Chen
  • Xiaoling Zhong
  • Yong Guo


This paper focuses on direction finding using a co-prime array from the view of a difference co-array. According to the corresponding relationship between the correlation lag and virtual element position of the difference co-array, from the correlation matrix of the co-prime array, the desired correlation units are extracted as single-snapshot data of the virtual co-array elements of the co-prime array and are then coherently accumulated into a pseudo-beam pattern. Because the difference co-array of a co-prime array consists of a group of contiguous virtual elements and multiple non-uniform virtual elements, this paper considers pseudo-beam-forming using only contiguous virtual elements, as in existing studies, and all of the virtual elements. Compared with the existing sub-beam multiplication method, pseudo-beam-forming reduces the negative effect from grating lobes and resolves more uncorrelated sources than the number of physical elements. Moreover, application of non-uniform virtual elements improves the resolvable source number, angle resolution and noise immunity, which are analyzed quantitatively based on the proposed distribution characteristic of virtual elements. Finally, to suppress side-lobe interference caused by the non-uniform virtual elements, we introduce and evaluate three coherence weighting factors, namely coherence factor (CF), phase coherence factor (PCF) and sign coherence factor (SCF), where CF is proved to be ineffective and SCF is optimal in suppression and computation performance.


Co-prime array Pseudo-beam-forming Difference co-array Side-lobe suppression Coherence weighting factor 



The authors would like to thank the editor and anonymous reviewers for their valuable comments. This work is supported financially by the National Natural Science Foundation of China under Grants 61501062, 41574136 and 41304117 and the Program of Sichuan Education Department under Grant 15ZB0082.


  1. 1.
    K. Adhikari, J.R. Buck, K.E. Wage, Beamforming with extended co-prime sensor arrays, in 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (Canada, 2013), pp 4183–4186Google Scholar
  2. 2.
    K. Adhikari, J.R. Buck, K.E. Wage, Extending coprime sensor arrays to achieve the peak side lobe height of a full uniform linear array. EURASIP J. Adv. Signal Process. 148, 1–17 (2014)Google Scholar
  3. 3.
    G.S. Bloom, S.W. Golomb, Application of numbered undirected graphs. Proc. IEEE 65, 562–570 (1977)CrossRefGoogle Scholar
  4. 4.
    E. BouDaher, F. Ahmad, M.G. Amin, Sparse reconstruction for direction-of-arrival estimation using multi-frequency co-prime arrays. EURASIP J. Adv. Signal Process. 168, 1–11 (2014)Google Scholar
  5. 5.
    E. BouDaher, Y. Jia, F. Ahmad, M.G. Amin, Direction-of-arrival estimation using multi-frequency co-prime arrays, in 22nd European Signal Processing Conference (Portugal, 2014), pp 1034–1038Google Scholar
  6. 6.
    E. BouDaher, Y. Jia, F. Ahmad, M.G. Amin, Multi-frequency co-prime arrays for high-resolution direction-of-arrival estimation. IEEE Trans. Signal Process. 63, 3797–3808 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D. Bush, N. Xiang, J.E. Summers, Experimental investigations on coprime microphone arrays for direction-of-arrival estimation. J. Acoust. Soc. Am. 136, 2214 (2014)CrossRefGoogle Scholar
  8. 8.
    D. Bush, N. Xiang, Broadband implementation of coprime linear microphone arrays for direction of arrival estimation. J. Acoust. Soc. Am. 138, 447–456 (2015)CrossRefGoogle Scholar
  9. 9.
    R.J. Burkholder, K.E. Browne, Coherence factor enhancement of through-wall radar images. IEEE Antennas Wirel. Propag. Lett. 9, 842–845 (2010)CrossRefGoogle Scholar
  10. 10.
    J. Camacho, M. Parrilla, C. Fritsch, Grating-lobes reduction by application of phase coherence factors, in 2009 IEEE International Ultrasonics Symposium (Italy, 2009), pp 341–344Google Scholar
  11. 11.
    J. Camacho, M. Parrilla, C. Fritsch, Phase coherence imaging. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 56, 958–974 (2009)CrossRefGoogle Scholar
  12. 12.
    C.L. Liu, P.P. Vaidyanathan, Remarks on the spatial smoothing step in coarray MUSIC. IEEE Signal Process. Lett. 22, 1438–1442 (2015)CrossRefGoogle Scholar
  13. 13.
    Y. Ma, B. Chen, M. Yang, Y. Wang, A novel ESPRIT-based algorithm for DOA estimation with distributed subarray antenna. Circuits Syst. Signal Process. 34, 2951–2972 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    A. Moffet, Minimum-redundancy linear arrays. IEEE Trans. Antennas Propag. 16, 172–175 (1968)CrossRefGoogle Scholar
  15. 15.
    P. Pal, P.P. Vaidyannathan, Nested arrays: a novel approach to array processing with enhanced degrees of freedom. IEEE Trans. Signal Process. 58, 4167–4181 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    P. Pal, P.P. Vaidyannathan, Coprime sampling and the MUSIC algorithm, in 2011 Digital Signal Processing and Signal Processing Education Meeting (Sedona, AZ, 2011), pp 289–294Google Scholar
  17. 17.
    S. Qin, Y.D. Zhang, M.G. Amin, Generalized coprime array configurations for direction-of-arrival estimation. IEEE Trans. Signal Process. 63, 1377–1390 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Q. Shen, W. Liu, W. Cui, S. Wu, Y.D. Zhang, M.G. Amin, Low-complexity direction-of-arrival estimation based on wideband co-prime arrays. IEEE/ACM Trans. Audio Speech Lang. Process. 23, 1445–1456 (2015)CrossRefGoogle Scholar
  19. 19.
    P. Stoica, A. Nehorai, MUSIC, maximum likelihood, and Cramer–Rao bound. IEEE Trans. Acoust. Speech Signal Process. 37, 720–741 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    P. Stoica, A. Nehorai, MUSIC, maximum likelihood, and Cramer–Rao bound: further results and comparisons. IEEE Trans. Acoust. Speech Signal Process. 38, 2140–2150 (1990)CrossRefGoogle Scholar
  21. 21.
    Z. Tan, Y.C. Eldar, A. Nehorai, Direction of arrival estimation using co-prime arrays: a super resolution viewpoint. IEEE Trans. Signal Process 62, 5565–5576 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    P.P. Vaidyannathan, P. Pal, Sparse sensing with co-prime samplers and arrays. IEEE Trans. Signal Process 59, 573–586 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    N. Xiang, D. Bush, J.E. Summers, Experimental validation of a coprime linear microphone array for high-resolution direction-of-arrival measurements. J. Acoust. Soc. Am. 137, 261–266 (2015)CrossRefGoogle Scholar
  24. 24.
    K. Yang, Z. Zhao, J. Liu, Q. Liu, Robust adaptive beamforming using an iterative FFT algorithm. Signal Process. 96, 253–260 (2014)CrossRefGoogle Scholar
  25. 25.
    J. Yang, G. Liao, J. Li, Robust adaptive beamforming in nested array. Signal Process. 114, 143–149 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Yong Jia
    • 1
    Email author
  • Chuan Chen
    • 1
  • Xiaoling Zhong
    • 1
  • Yong Guo
    • 1
  1. 1.College of Information Science and TechnologyChengdu University of TechnologyChengduChina

Personalised recommendations