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1.E. J. Sullivan and J. V. Candy, “Space-time array processing: A model-based approach,” J. Acoust. Soc. Am. 102, No. 5, 28092820 (1997).
2.E. J. Sullivan, J. D. Holmes, W. M. Carey, and J. F. Lynch, “Broadband passive synthetic aperture: Experimental results,” J. Acoust. Soc. Am. 120-EL, 4952 (2006).
3.S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Addison-Wesley, Reading, MA, 1991), Chap. 3.
4.It should be pointed out that in practice, the joint estimate can still outperform the stationary array. This indicates that the CRLB is a highly conservative measure.
5.S. Stergiopoulos, “Optimum bearing resolution for a moving towed array and extension of its physical aperture,” J. Acoust. Soc. Am. 87(5), 21282140 (1990).
6.G. S. Edelson, “On the Estimation of Source Location Using a Passive Towed Array,” Ph.D. dissertation, University of Rhode Island, 1993.
7.Since we are assuming that the bearing is deterministic, it has no contribution to the prior Fisher matrix. Bell and Van Trees (Ref. 8) refer to this case as the “hybrid” case.
8.K. L. Bell and H. L. Van Trees, “Posterior Cramér-Rao Bound for Tracking Target Bearings,” in Proceedings of the Adaptive Sensor Array Processing Workshop, MIT Lincoln Labs, 7–8 June (2005).
9.The velocity signals do not explicitly contain the acoustic impedance, since we assume that the measured velocities have already been scaled by the impedance, thus converting them to their equivalent pressure units.
10.N. K. Naulai and G. C. Lauchle, “Acoustic intensity methods and their applications to vector sensor use and design,” Penn State Report. No. 2006-01 (2006).
11.The factors of 3 and are a consequence of the inherent directivity of the vector sensors.

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By properly including the forward motion of the array in the signal model, improved bearing estimation performance for a towed line array can be obtained. The improvement is a consequence of utilizing the bearing information contained in the Doppler. In this paper, it is shown by use of the Cramér-Rao lower bound that, as the array moves forward, the variance on the bearing estimate for an array of pressure sensors decreases, and that if an array of pressure-vector sensors is used, a significant improvement over that obtained for the array using pressure sensors only is obtained.


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