Sensor array calibration method in presence of gain/phase uncertainties and position perturbations using the spatial- and time-domain information of the auxiliary sources

  • Ding Wang
  • Ying Wu


This paper deals with the problem of active calibration under the existence of sensor gain/phase uncertainties and position perturbations. Unlike many existing eigenstructure-based (also called subspace-based) calibration methods which using the spatial-domain (i.e., angle) information of the auxiliary sources only, our proposed approach enables exploitation of both the spatial- and time-domain knowledge of the sources, and therefore yields better performance than the eigenstructure-based calibration technology. For the purpose of incorporating the time-domain knowledge of the sources into the error calibration, the maximum likelihood criterion is selected as the optimization principle, and a concentrated alternating iteration procedure (called algorithm II) is developed, which has rapid convergence rate and robustness. As a byproduct of this paper, we also provide an eigenstructure-based calibration approach (termed algorithm I), which alternatively minimizes the weighted signal subspace fitting cost function and weighted noise subspace fitting criterion to update the estimates for sensor position perturbations and gain/phase errors in each iteration, respectively. Similar to some previous subspace-based calibration algorithms in the literature, algorithm I is also asymptotically efficient but is more computationally convenient, and can be introduced as benchmark to be compared to algorithm II. Additionally, the Cramér–Rao bound (CRB) expressions for the sensor gain/phase errors and position perturbations estimates are presented for two situations: (a) the time-domain waveform information of the sources is unavailable, and (b) the time-domain waveform information of the sources is taken as prior knowledge into account. The CRBs for the two cases are also quantitatively compared, and the resulting conclusion demonstrates that by combining the time-domain waveform information of the sources into the calibration algorithm, a significant performance improvement can be achieved. The simulation experiments are conducted to corroborate the advantages of the proposed algorithms as well as the theoretical analysis in this paper.


Array error calibration Active calibration Maximum likelihood (ML) estimator Gain/phase errors Position perturbations Spatial-domain information Time-domain waveform information Cramér–Rao bound (CRB) 


  1. Aktas, M., & Tuncer, T. E. (2010). Iterative HOS-SOS (IHOSS) algorithm for direction-of-arrival estimation and sensor localization. IEEE Transactions on Signal Processing, 58(12), 6181–6194.MathSciNetCrossRefGoogle Scholar
  2. Bao, Q., Ko, C. C., & Zhi, W. (2005). DOA estimation under unknown mutual coupling and multipath. IEEE Transactions on Aerospace and Electronic Systems, 41(2), 565–573.CrossRefGoogle Scholar
  3. Cheng, Q., Hua, Y. B., & Stoica, P. (2000). Asymptotic performance of optimal gain-and-phase estimators of sensor arrays. IEEE Transactions on Signal Processing, 48(12), 3587–3590.MATHCrossRefGoogle Scholar
  4. Ferréol, A., Larzabal, P., & Viberg, M. (2006). On the asymptotic performance analysis of subspace DOA estimation in the presence of modeling errors: Case of MUSIC. IEEE Transactions on Signal Processing, 54(3), 907–920.CrossRefGoogle Scholar
  5. Ferréol, A., Larzabal, P., & Viberg, M. (2008). Performance prediction of maximum-likelihood direction-of-arrival estimation in the presence of modeling errors. IEEE Transactions on Signal Processing, 56(10), 4785–4793.MathSciNetCrossRefGoogle Scholar
  6. Ferréol, A., Larzabal, P., & Viberg, M. (2010). Statistical analysis of the MUSIC algorithm in the presence of modeling errors, taking into account the resolution probability. IEEE Transactions on Signal Processing, 58(8), 4156–4166.MathSciNetCrossRefGoogle Scholar
  7. Flanagan, B. P., & Bell, K. L. (2001). Array self-calibration with large sensor position errors. Signal Processing, 81(10), 2201–2214.CrossRefGoogle Scholar
  8. Jansson, M., Götansson, B., & Ottersten, B. (1999). A subspace method for direction of arrival estimation of uncorrelated emitter signals. IEEE Transactions on Signal Processing, 47(4), 945–956.CrossRefGoogle Scholar
  9. Jia, Y. K., Bao, Z., & Wu, H. (1996). A new calibration technique with signal sources for position, gain and phase uncertainty of sensor array. Acta Electronica Sinica, 24(3), 47–52.Google Scholar
  10. Jiang, J. J., Duan, F. J., Chen, J., Chao, Z., Chang, Z. J., & Hua, X. N. (2013). Two new estimation algorithms for sensor gain and phase errors based on different data models. IEEE Sensors Journal, 13(5), 1921–1930.CrossRefGoogle Scholar
  11. Leshem, A., & Veen, A. J. (1999). Direction-of-arrival estimation for constant modulus signals. IEEE Transactions on Signal Processing, 47(11), 3125–3129.MATHCrossRefGoogle Scholar
  12. Li, J., Halder, B., Stoica, P., & Viberg, M. (1995). Computationally efficient angle estimation for signals with known waveforms. IEEE Transactions on Signal Processing, 43(9), 2154–2163.CrossRefGoogle Scholar
  13. Li, Y. M., & Er, M. H. (2006). Theoretical analyses of gain and phase error calibration with optimal implementation for linear equispaced array. IEEE Transactions on Signal Processing, 54(2), 712–723.CrossRefGoogle Scholar
  14. Liu, A. F., Liao, G. S., Zeng, C., Yang, Z. W., & Xu, Q. (2011). An eigenstructure method for estimating DOA and sensor gain-phase errors. IEEE Transactions on Signal Processing, 59(12), 5944–5956.MathSciNetCrossRefGoogle Scholar
  15. Ng, B.C., Ser, W. (1992). Array shape calibration using sources in known locations. In: Proceedings of the ICCS/ISITA communications on the Moveapos. Singapore: IEEE Press, 1992, 2: 836–840.Google Scholar
  16. Ng, B. C., & Nehorai, A. (1995). Active array sensor localization. Signal Processing, 44(3), 309–327.MATHCrossRefGoogle Scholar
  17. Ottersten, B., Viberg, M., Stoica, P., Nehorai, A. (1993). Exact and large sample ML techniques for parameter estimation and detection in array processing. In: Haykin, Litva and Shepherd (Eds.) Radar array processing (pp. 99–151). Berlin: Springer.Google Scholar
  18. Park, H. Y., Lee, C. Y., Kang, H. G., & Youn, D. H. (2004). Generalization of subspace-based array shape estimations. IEEE Journal of Oceanic Engineering, 29(3), 847–856.CrossRefGoogle Scholar
  19. See, C. M. S., & Poth, B. K. (1999). Parametric sensor array calibration using measured steering vectors of uncertain locations. IEEE Transactions on Signal Processing, 47(4), 1133–1137.CrossRefGoogle Scholar
  20. See, C. M. S., & Gershman, A. B. (2004). Direction-of-arrival estimation in partly calibrated subarray-based sensor arrays. IEEE Transactions on Signal Processing, 52(2), 329–338.MathSciNetCrossRefGoogle Scholar
  21. Soon, V. C., Tong, L., Huang, Y. F., & Liu, R. (1994). A subspace method for estimating sensor gains and phases. IEEE Transactions on Signal Processing, 42(4), 973–976.CrossRefGoogle Scholar
  22. Stoica, P., & Larsson, E. G. (2001). Comments on “Linearization method for finding Cramér–Rao bounds in signal processing”. IEEE Transactions on Signal Processing, 49(12), 3168–3169.CrossRefGoogle Scholar
  23. Viberg, M., & Swindlehurst, A. L. (1994). A Bayesian approach to auto-calibration for parametric array signal processing. IEEE Transactions on Signal Processing, 42(12), 3495–3507.CrossRefGoogle Scholar
  24. Vu, D. T., Renaux, A., Boyer, R., & Marcos, S. (2013). A cramér rao bounds based analysis of 3D antenna array geometries made from ULA branches. Multidimensional Systems and Signal Processing, 24, 121–155.MATHMathSciNetCrossRefGoogle Scholar
  25. Wan, S., Chung, P. J., & Mulgrew, B. (2012). Maximum likelihood array calibration using particle swarm optimization. IET Signal Processing, 6(5), 456–465.MathSciNetCrossRefGoogle Scholar
  26. Wang, C. C., Cadzow, J. A. (1991). Direction-finding with sensor gain, phase and location uncertainty. In: Proceedings of the international conference on acoustics, speech and signal processing. Toronto, Ontario: IEEE Press, 1991, vol. 2, pp. 1429–1432.Google Scholar
  27. Wang, D., & Wu, Y. (2008). Self-calibration algorithm for DOA estimation in presence of sensor amplitude, phase uncertainties and sensor position errors. Chinese Journal of Data Acquisition & Processing, 23(2), 176–181.Google Scholar
  28. Wang, D., & Wu, Y. (2010). Array errors active calibration algorithm and its improvement. Science China Information Sciences, 53(5), 1016–1033.MathSciNetCrossRefGoogle Scholar
  29. Wang, D. (2013). Sensor array calibration in presence of mutual coupling and gain/phase errors by combining the spatial-domain and time-domain waveform information of the calibration sources. Circuits, Systems, and Signal Processing, 32(3), 1257–1292.MathSciNetCrossRefGoogle Scholar
  30. Weiss, A. J., & Friedlander, B. (1990). Eigenstructure methods for direction finding with sensor gain and phase uncertainties. Circuits Systems, Signal Processing, 9(2), 272–300.MathSciNetGoogle Scholar
  31. Weiss, A. J., & Friedlander, B. (1991). Array shape calibration using eigenstructure methods. Signal Processing, 22(3), 251–258.CrossRefGoogle Scholar
  32. Weiss, A. J., & Friedlander, B. (1996). “Almost Blind” steering vector estimation using second-order moments. IEEE Transactions on Signal Processing, 44(4), 1024–1027.CrossRefGoogle Scholar
  33. Wijnholds, S. J., Boonstra, A. J. (2006). A multisource calibration method for phased array radio telescopes. In: Proceedings of 4th IEEE workshop on sensor array and multi-channel processing. Waltham, MA: IEEE Press, 2006, 200–204.Google Scholar
  34. Wijnholds, S. J., & Veen, A. J. (2009). Multisource self-calibration for sensor arrays. IEEE Transactions on Signal Processing, 57(9), 3512–3522.MathSciNetCrossRefGoogle Scholar
  35. Zhang, X., Chen, C., Li, J., & Xu, D. (2014). Blind DOA and polarization estimation for polarization-sensitive array using dimension reduction MUSIC. Multidimensional Systems and Signal Processing, 25, 67–82.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Zhengzhou Information Science and Technology InstituteZhengzhouPeople’s Republic of China

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