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A Cramér Rao bounds based analysis of 3D antenna array geometries made from ULA branches

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Abstract

In the context of passive sources localization using antenna array, the estimation accuracy of elevation, and azimuth are related not only to the kind of estimator which is used, but also to the geometry of the considered antenna array. Although there are several available results on the linear array, and also for planar arrays, other geometries existing in the literature, such as 3D arrays, have been less studied. In this paper, we study the impact of the geometry of a family of 3D models of antenna array on the estimation performance of elevation, and azimuth. The Cramér-Rao Bound (CRB), which is widely spread in signal processing to characterize the estimation performance will be used here as a useful tool to find the optimal configuration. In particular, we give closed-form expressions of CRB for a 3D antenna array under both conditional, and unconditional observation models. Thanks to these explicit expressions, the impact of the third dimension to the estimation performance is analyzed. Particularly, we give criterions to design an isotropic 3D array depending on the considered observation model. Several 3D particular geometry antennas made from uniform linear array (ULA) are analyzed, and compared with 2D antenna arrays. The isotropy condition of such arrays is analyzed. The presented framework can be used for further studies of other types of arrays.

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References

  • Baysal U., Moses R. L. (2003) On the geometry of isotropic arrays. IEEE Transactions on Signal Processing 51(6): 1469–1477

    Article  MathSciNet  Google Scholar 

  • Bos A. V. D. (1994) A Cramer Rao lower bound for complex parameters. IEEE Transactions on Acoustics Speech, Signal Processing 42: 2859

    Google Scholar 

  • Cedervall M., Moses R. L. (1997) Efficient maximum likelihood DOA estimation for signals with known waveforms in presence of multipath. IEEE Transactions on Signal Processing 45: 808–811

    Article  Google Scholar 

  • Choi Y. H. (2004) Unified approach to Cramer-Rao bounds in direction estimation with known signal structures. Signal Processing 84(10): 1875–1882

    Article  MATH  Google Scholar 

  • Ferréol, A., & Chevalier, P. (2009). High resolution direction finding: From performance toward antenna array optimization -the mono-source case. In Proceedings of European signal processing conference (pp. 1973–1977). Glasgow, Scotland.

  • Filik, T., & Tuncer, T. E. (2008). Uniform and nonuniform V-shaped isotropic planar arrays. In Proceedings of sensor array and multichannel signal processing workshop (pp. 99–103). Germany: Darmstadt.

  • Gazzah H., Abed-Meraim K. (2009) Optimum ambiguity free directional and omni directional planar antenna arrays for DOA estimation. IEEE Transactions on Signal Processing 57(10): 3942–3953

    Article  MathSciNet  Google Scholar 

  • Gazzah H., Marcos S. (2006) Cramér-Rao bounds for antenna array design. IEEE Transactions on Signal Processing 54(1): 336–345

    Article  Google Scholar 

  • Gavish M., Weiss A. J. (1991) Array geometry for ambiguity resolution in direction finding. IEEE Transactions on Antennas Propagation 44(6): 143–146

    Google Scholar 

  • Godara L. C., Cantoni A. (1981) Uniqueness and linear independence of sterring vectors in array space. Journal of Acoustic Society of America 70(2): 467–475

    Article  MathSciNet  Google Scholar 

  • Hua Y., Sarkar T. K. (1991) A note on the Cramér-Rao bound for 2-D direction finding based on 2-D array. IEEE Transactions on Signal Processing 39(5): 1215–1218

    Article  Google Scholar 

  • Hua Y., Sarkar T. K., Weiner D. D. (1991) An L-shaped array for estimating 2D directions of wave arrival. IEEE Transactions on Antennas Propagation 39: 143–146

    Article  Google Scholar 

  • Huang, X., Reilly, J. P., & Wong M. (1991). Optimal design of linear array of sensors. In: Proceedings of IEEE international conference acoustics speech, signal processing, Vol. 2 (pp. 1405–1408). Canada: Toronto, Ont.

  • Kay S. M. (1993) Fundamentals of statistical signal processing, Vol. 1. Prentice Hall, NJ

    Google Scholar 

  • Leshem A., van der Veen A. -J. (1999) Direction-of-arrival estimation for constant modulus signals. IEEE Transactions on Signal Processing 47(11): 3125–3129

    Article  MATH  Google Scholar 

  • Li J., Compton R. T. (1993) Maximum likelihood angle estimation for signals with known waveforms. IEEE Transactions on Signal Processing 41: 2850–2862

    Article  MATH  Google Scholar 

  • Li J., Halder B., Stoica P., Viberg M. (1995) Computationally efficient angle estimation for signals with known waveforms. IEEE Transactions on Signal Processing 43: 2154–2163

    Article  Google Scholar 

  • Lo J. T. H., Marple S. L. Jr (1992) Observability conditions for multiple signal direction finding and array sensor localization. IEEE Transactions on Signal Processing 40(11): 2641–2650

    Article  MATH  Google Scholar 

  • Lui K. W. K., So H. C. (2009) A study of two-dmensional sensor placement using time-difference-of-arrival measurements. Digital Signal Processing 19: 650–659

    Article  Google Scholar 

  • Manikas A. (2004) Differential geometry in array processing. Imperial College Press, London

    Book  MATH  Google Scholar 

  • Mirkin A., Sibul L. H. (1991) Cramér-Rao bounds on angle estimation with a two-dimensional array. IEEE Transactions on Signal Processing 39: 515–517

    Article  Google Scholar 

  • Moffet A. T. (1968) Minimum redundancy linear arrays. IEEE Transactions on Antennas Propagation 16: 172–175

    Article  Google Scholar 

  • Nielsen R. O. (1994) Azimuth and elevation angle estimation with a three dimensional array. IEEE Journal of Oceanic Engineering 19(1): 84–86

    Article  Google Scholar 

  • Oktel U., Moses R. L. (2005) A Bayesian approach to array geometry design. IEEE Transactions on Signal Processing 53(5): 1919–1923

    Article  MathSciNet  Google Scholar 

  • Ottersten B., Viberg M., Stoica P., Nehorai A. (1993) Exact and large sample maximum likelihood techniques for parameter estimation and detection in array processing. In: Haykin S., Litva J., Shepherd T. J. (eds) Radar array processing, Chap. 4. Springer, Berlin, pp 99–151

    Chapter  Google Scholar 

  • Porat B., Friedlander B. (1988) Analysis of the asymptotic relative efficiency of the MUSIC algorithm. IEEE Transactions on Acoustics Speech, Signal Processing 36(4): 532–544

    Article  MathSciNet  MATH  Google Scholar 

  • Renaux A., Foster P., Chaumette E., Larzabal P. (2006) On the high-SNR conditional maximum-likelihood estimator full statistical characterization. IEEE Transactions on Signal Processing 54(12): 4840–4843

    Article  Google Scholar 

  • Renaux A., Foster P., Boyer E., Larzabal P. (2007) Unconditional maximum likelihood performance at finite number of samples and high signal-to-noise ratio. IEEE Transactions on Signal Processing 55(5): 2358–2364

    Article  MathSciNet  Google Scholar 

  • Sengupta D., Smith T., Larson R. (1968) Radiation characteristics of a spherical array of circularly polarized elements. IEEE Transactions on Antennas Propagation 16(1): 2–7

    Article  Google Scholar 

  • Stoica P., Nehorai A. (1989) MUSIC, maximum likelihood and the Cramér-Rao bound. IEEE Transactions Acoustics Speech, Signal Processing 37: 720–741

    Article  MathSciNet  MATH  Google Scholar 

  • Stoica P., Nehorai A. (1990) Performances study of conditional and unconditional direction of arrival estimation. IEEE Transactions on Acoustics Speech, Signal Processing 38: 1783–1795

    Article  MATH  Google Scholar 

  • Stoica P., Moses R. (2005) Spectral analysis of signals. Prentice Hall, NJ

    MATH  Google Scholar 

  • Tan K. C., Goh S. S., Tan E. C. (1996) A study of the rank-ambiguilty issues in direction-of-arrival estimation. IEEE Transactions on Signal Processing 44(4): 880–887

    Article  Google Scholar 

  • VanTrees H. L. (2002) Detection, estimation and modulation theory: Optimum array processing, Vol. 4. Wiley, New York

    Google Scholar 

  • Yang, B., & Scheuing, J. (2005). Cramér-Rao bound and optimum sensor array for source localization from the differences of arrival. In Proceedings of IEEE international conference acoustics speech, signal processing Vol. 4 (pp. 961–964). Philadenphia, USA.

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Authors and Affiliations

  1. Laboratory of Signals and Systems Supélec, University Paris-Sud 11, 3 rue Joliot-Curie, 91192, Gif-sur-Yvette Cedex, France

    Dinh Thang Vu, Alexandre Renaux, Rémy Boyer & Sylvie Marcos

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  1. Dinh Thang Vu
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  2. Alexandre Renaux
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  3. Rémy Boyer
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  4. Sylvie Marcos
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Correspondence to Dinh Thang Vu.

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Vu, D.T., Renaux, A., Boyer, R. et al. A Cramér Rao bounds based analysis of 3D antenna array geometries made from ULA branches. Multidim Syst Sign Process 24, 121–155 (2013). https://doi.org/10.1007/s11045-011-0160-5

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  • DOI: https://doi.org/10.1007/s11045-011-0160-5

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