IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 9, SEPTEMBER 1998 2291 Acoustic Vector-Sensor Beamforming and Capon Direction Estimation Malcolm Hawkes, Student Member, IEEE, and Arye Nehorai, Fellow, IEEE Abstract— We examine the improvement attained by using acoustic vector sensors for direction-of-arrival (DOA) estimation, instead of traditional pressure sensors, via optimal performance bounds and particular estimators. By examining the Cram´ er–Rao bound in the case of a single source, we show that a vector- sensor array’s smaller estimation error is a result of two distinct phenomena: 1) an effective increase in signal-to-noise ratio due to a greater number of measurements of phase delays between sensors and 2) direct measurement of the DOA information contained in the structure of the velocity field due to the vector sensors’ directional sensitivity. Separate analysis of these two phenomena allows us to determine the array size, array shape, and SNR conditions under which the use of a vector-sensor array is most advantageous and to quantify that advantage. By extending the beamforming and Capon direction estimators to vector sensors, we find that the vector sensors’ directional sensitivity removes all bearing ambiguities. In particular, even simple structures such as linear arrays can determine both azimuth and elevation, and spatially undersampled regularly spaced arrays may be employed to increase aperture and, hence, performance. Large sample approximations to the mean-square error matrices of the estimators are derived and their validity is assessed by Monte Carlo simulation. I. INTRODUCTION T HE PASSIVE direction-of-arrival (DOA) estimation problem, in which the bearings of a number of far- field acoustic sources are determined, is of great importance in many underwater applications. The traditional solution is to use a spatially distributed array of pressure sensors, and many estimation techniques have evolved for this scenario. As demand for smaller arrays that perform better at a lower signal- to-noise ratio (SNR) has increased, the idea of measuring particle velocity, as well as pressure, has arisen [1], [2]. This has coincided with a surge of interest in particle velocity sensors and improvements in fabrication techniques [3] to make vector-sensor arrays a practical reality [4]. An acoustic vector sensor measures the acoustic pressure and all three components of the acoustic particle velocity at a single point in space. Thus, whereas standard pressure Manuscript received December 2, 1996; revised December 19, 1997. This work was supported by the Air Force Office of Scientific Research under Grant F49620-97-1-0481, the National Science Foundation under Grant MIP- 9615590, and the Office of Naval Research under Grant N00014-96-1-1078. The associate editor coordinating the review of this paper and approving it for publication was Prof. Victor A. N. Barroso. M. Hawkes is with the Department of Electrical Engineering and Computer Science, University of Illinois, Chicago, IL 60607 USA, on leave from the Department of Electrical Engineering, Yale University, New Haven, CT 06520 USA. A. Nehorai is with the Department of Electrical Engineering and Computer Science, University of Illinois, Chicago, IL 60607 USA. Publisher Item Identifier S 1053-587X(98)05953-4. sensors can only utilize the directional information present in the propagation delays between sensors, each vector sensor can extract further directional information directly from the structure of the velocity field. This directional information permits, for example, a single vector-sensor to identify up to two sources [5]. By making use of this extra information, arrays of vector sensors are able to improve source localization accuracy without increasing array aperture. Both theoretical and practical work has been published on the topic of acoustic vector sensors. A model for acoustic vector sensors, a general expression for the Cram´ er–Rao bound (CRB), and a simple DOA estimation algorithm for a single sensor were introduced in [1] and [2]; preliminary theoretical performance analyses appear in [1], [2], [6], and [7]. In addition, ESPRIT and Root-MUSIC algorithms have recently been applied to arrays of velocity-sensor triads [8]–[10]. Meanwhile, acoustic vector sensors have been constructed [11] and linear arrays of them built and subjected to sea trials, [4], [12]–[14]. In this paper (see also [6] and [7]), we investigate and quantify the factors that lead to the improved DOA estimation performance of a vector-sensor array over a pressure-sensor array. By examining optimum performance (via the CRB) in detail for a single source, we are able to discern two distinct factors: 1) an effective increase in SNR due to extra measurements of the phase delays between noncoincident sensors and 2) a further reduction in the bound due to direct measurement of the structure of the velocity field by each vector sensor. Separate consideration of the two phenomena shows that the increase in estimation accuracy obtained by using vector sensors is greatest for linear or planar arrays [as opposed to three-dimensional (3-D) geometries], small numbers of sensors, and low SNR’s. We then consider the conventional beamforming and Capon [15] (which is also known as minimum-variance distortionless response beamforming) direction estimators. We show that any vector-sensor array and, hence, the popular linear array can be used to unambiguously determine both the azimuth and eleva- tion. This has significant practical value: For example, when using a linear towed array of vector sensors to determine the location of surface ships, the left/right ambiguity problem does not arise, avoiding the need to rotate the vessel through 90 between readings. Furthermore, we show that a vector-sensor array suffers less from spatial aliasing in an undersampled wavefield. In particular, grating lobes do not appear in the beampattern. This allows the use of undersampled regular arrays in order to extend the spatial aperture and, hence, 1053–587X/98$10.00  1998 IEEE