Wideband Source Localization Using a Distributed Acoustic Vector-Sensor Array

Transcription

1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 6, JUNE Wideband Source Localization Using a Distributed Acoustic Vector-Sensor Array Malcolm Hawkes, Member, IEEE, and Arye Nehorai, Fellow, IEEE Abstract We derive fast wideband algorithms, based on measurements of the acoustic intensity, for determining the bearings of a target using an acoustic vector sensor (AVS) situated in free space or on a reflecting boundary. We also obtain a lower bound on the mean-square angular error (MSAE) of such estimates. We then develop general closed-form weighted least-squares (WLS) and reweighted least-squares algorithms that compute the three-dimensional (3-D) location of a target whose bearing to a number of dispersed locations has been measured. We devise a scheme for adaptively choosing the weights for the WLS routine when measures of accuracy for the bearing estimates, such as the lower bound on the MSAE, are available. In addition, a measure of the potential estimation accuracy of a distributed system is developed based on a two-stage application of the Cramér Rao bound. These 3-D results are quite independent of how bearing estimates are obtained. Naturally, the two parts of the paper are tied together by examining how well distributed arrays of AVSs located on the ground, seabed, and in free space can determine the 3-D position of a target The results are relevant to the localization of underwater and airborne sources using freely drifting, moored, or ground sensors. Numerical simulations illustrate the effectiveness of our estimators and the new potential performance measure. I. INTRODUCTION ACOUSTIC emissions from battlefield or underwater sources can provide an invaluable signature by which to detect, locate, and track hostile units. The passivity of an acoustic surveillance system allows it to monitor the battlefield or ocean without giving away its own presence. Passive acoustic surveillance has long been used underwater, but its battlefield application [1] is more recent. The feasibility of acoustic localization and tracking on the battlefield has been demonstrated in [2] [5]. We propose using a distributed array of acoustic vector sensors (AVSs) to perform the surveillance function in three separate scenarios: sensors free floating in the water column, sensors located on the seabed, and aero-acoustic vector sensors located on the ground for battlefield surveillance. These sensors measure the (scalar) acoustic pressure and all three components of the acoustic particle velocity vector at a given point Manuscript received April 6, 2001; revised December 3, This work was supported by the Air Force Office of Scientific Research under Grants F and F , the National Science Foundation under Grant CCR , and the Office of Naval Research under Grants N and N The associate editor coordinating the review of this paper and approving it for publication was Dr. Fulvio Gini. M. Hawkes was with the Department of Electrical Engineering and Computer Science, University of Illinois, Chicago, IL USA. He is now with Ronin Capital LLC, Chicago, IL USA. A. Nehorai is with the Department of Electrical and Computer Engineering, University of Illinois, Chicago, IL USA. Digital Object Identifier /TSP and possess a number of advantages over arrays of pressure sensors [6] [10]. The ability of a single AVS to rapidly determine the bearing of a wideband source makes them especially attractive for the present problem. Vector sensors for underwater applications have already been constructed [11], [12], and sea tested [13], [14]. The Swallow float sensor described in [11] is a freely drifting device that is perfectly suited to this application s free-space scenario. The above-referenced devices use highly sensitive moving-coil geophones to measure velocity, but other velocity sensors have been based on physical principles such as the change in inductance of a metallic glass strip [15], piezoceramics [16], and fiber-optic interferometry [17]. Recently, a new aero-acoustic velocity sensor called the Microflown [18], [19] has become commercially available, from Microflown Technologies, B.V. [20] in the Netherlands, that would be appropriate for the battlefield context. It measures the differential resistance between two micro-machined metallic strips in an air-current to determine velocity. Thus, a miniature, lightweight, portable AVS could be constructed. We have analyzed the use of vector sensors near a boundary in [9] and [10], and they have been tested on a mock vessel hull [21] and at the seabed [22]. In this paper (see also [23] and [24]), we develop a fast, wideband algorithm for finding the bearing of an acoustic source using a single aero-acoustic AVS located on the ground. Similar bearing estimation algorithms, which we will make use of here, that are applicable to the free-space and seabed scenarios were presented in [6] and [10], respectively. We also develop wideband, closed-form algorithms that combine bearing estimates from several arbitrary locations to determine a source s three-dimensional (3-D) position. The bearing estimate is based on the measured acoustic intensity vector. We derive an optimal bound on its mean-square angular error (MSAE) [25], [26] and use it to obtain a data-based measure of the bearing estimator s accuracy. Each AVS transmits its bearing estimate, the estimate of its variability, and its current location, to a central processor (CP), which uses them to determine 3-D position. We propose a weighted least-squares (WLS) method to estimate position using the variability measures supplied by the sensors as weights. We also develop a reweighted least-squares (RWLS) algorithm to account for the different ranges of the sensors from the target. The WLS estimator is closed form and the RWLS estimator requires just a single iteration of the WLS algorithm. Consequently, they are both very computationally efficient. Since the individual bearing estimates are also closed form, the system provides a very fast estimate of the location of a wideband source. Note that these 3-D position estimation algorithms are independent of how the bearing estimates are obtained; they could come from subarrays of pressure sensors or X/03$ IEEE

2 1480 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 6, JUNE 2003 other direction finders (passive or active). Finally, we derive a novel measure of potential performance for a distributed system such as this, based on a two-stage calculation of Cramér Rao bounds (CRBs). Like the CRB, this measure is estimator independent and can be used as a benchmark against which to compare distributed processing techniques and as a criterion for array design. As each AVS transmits only a bearing estimate rather than all measurements to the CP, this is a decentralized processing scheme [27], [28]. The resulting 3-D position estimator is suboptimal because it does not make use of correlations between different locations, but it has numerous advantages: Sensor placement is arbitrary and need not be fixed (although it must be known) so sensors can be dropped (from the air or sea surface) and may be used in a dynamic context, free floating like the sensors in [11], or carried by battlefield units for example; each sensor provides local target bearing information (especially valuable in the dynamic context) without the need to communicate with the CP. Even when communication is made, minimal data is sent, hence, minimizing the risk of detection and telemetry requirements; last, the algorithms are wideband and very computationally efficient as they require no numerical optimization. Of the previous work on distributed arrays, [27] supposes that the source is in the far field of all subarrays, i.e., all bearings from different locations are the same; therefore, no 3-D estimate of position can be made. The approach taken in [28] could be adapted to our current situation; however, it requires that transmission of the covariance matrix of each subarray, resulting in a somewhat greater communication burden. Furthermore, both require numerical search algorithms so greatly increasing computational complexity and use subarrays of standard omni-directional sensors. The fact that each subarray in our method is an AVS contained in a single sensor package gives it great flexibility in terms of deployment and usage. In Section II, we present the mathematical model for the sensor measurements, and Section III develops an algorithm to rapidly estimate bearing using a single vector sensor. In Section IV, we develop weighted and reweighted least-squares algorithms for determining 3-D source position given the bearing estimates from each sensor, construct an estimator to determine the weights, and give an expression for a lower bound on the bearing estimator from each sensor. We propose the new potential performance measure for a distributed system in Section IV-C and numerically illustrate the efficacy of the proposed algorithms in Section V. Section VI concludes the paper. II. MEASUREMENT MODEL In the following, bold-face characters represent vectors, and upper case characters represent matrices. We assume that there is a single bandlimited acoustic source, whose location is donated by, radiating bandlimited spherical waves into an isotropic homogeneous whole-space or half-space. The signal is received by vector sensors at arbitrary distinct locations. The scenario (for ground or seabed) is illustrated in Fig. 1. As long as the source to sensor distance is more Fig. 1. Schematic illustration. Source at emits spherical waves, sensors on boundary at r ;...; r estimate bearing vectors u ;...; u. than a few times the maximum wavelength and the sensor s dimensions are small compared with the minimum wavelength, the wavefront arriving at each sensor is essentially planar. The acoustic particle velocity and the acoustic pressure at any point that is more than a few (maximum) wavelengths from the source are related by Euler s equation [29] where is the velocity vector, is the pressure, is the density of the medium, is the speed of sound, and is the unit vector from the source to. In the free-space scenario, the output of an AVS located at is the four-element vector (see [6]) for, where and are the complex envelopes of the acoustic pressure and particle velocity, respectively (the latter normalized by ), is the complex envelope of the pressure signal at, and represents noise. The vector is the unit vector pointing from the sensor to the source s location at time, where is the propagation delay between the source and sensor at time. Since this propagation delay cannot be compensated for, we will simply refer to the target s location at time as its position. Note that we implicitly assume that the observation interval is short enough relative to the inverse of the source s speed such that is approximately constant for. For the ground and seabed scenarios, we assume that the ground or seabed, which defines the -plane, forms a flat planar boundary on which all sensors are located. If the source is not too close to the boundary, the total field at any point on or above the interface may be obtained as the superposition of the sound fields arising from the original source and an image source (see Fig. 2). The image source is obtained by reflecting the original source in the boundary and has amplitude and phase determined by the boundary characteristics and the point of interest. The fields from the two sources are summed as if the boundary were not present. This is known as the ray acoustics or geometrical optics approximation [30]. Note that if the source is located on or very close to the boundary, ground and surface waves may exist [31]. The resultant field may still be obtained using an image source for locally reacting surfaces (1) (2)

3 HAWKES AND NEHORAI: WIDEBAND SOURCE LOCALIZATION 1481 Fig. 2. Field at r is sum of real and image fields and satisfies boundary condition Z = 0p=v at r. but now the image source must be somewhat modified [32]. Although not explicitly shown here, the algorithm for determining the azimuth of the source relative to each sensor (see Section III) is valid under any boundary conditions that ensure that the ratio of the signals at the two horizontal velocity components is equal to the tangent of the azimuth. We therefore assume that the source is far enough away from the boundary that the resulting field can be regarded as arising from a point source radiating spherically symmetric waves and a simple point image with (complex) amplitude, relative to the true source. Consider an AVS located at a point on the boundary. The bearing of the image source relative to the sensor is, which is obtained by negating the -component of. The output of the AVS is thus (3) much less stringent requirement than the standard narrowband array processing assumption. Thus, we refer to our algorithms as wideband. We model the ground as a locally reacting surface, i.e., one for which is independent of the incidence angle. The reflection coefficient is then given by (5), with a fixed complex constant. Experiments in [31] showed that various ground surfaces behave as if they are locally reacting. Such surfaces also arise in architectural acoustics with porous sound-absorbing materials [29]. A locally reacting surface may be characterized as one in which the sound disturbance transmitted into the lower medium does not travel along its boundary (actually, this is only strictly true for plane waves since, as mentioned above, ground waves and surface waves may exist when the source is very close to the boundary), and therefore, the normal velocity at each point is completely determined by the pressure at this point [30]. The seabed is modeled as the interface between two liquid layers, one of which is absorptive. This model is a reasonable approximation for a water packed sandy bottom [30, pp. 11], although the reflective properties of many seafloor terrains will undoubtedly be more complicated. The reflection coefficient is given by where is the ratio of sand density to water density, and is the index of refraction. Absorption is accounted for by allowing the index of refraction to be complex, i.e.,, with (6). For the sandy ocean bottom typical values are,, and [30, pp. 11]. Note that as required, does not depend on frequency in these models (6). It follows from (2) and (3) that for each of the scenarios, the measurement of a single AVS may be written for, where is the complex envelope of the pressure signal that would arrive at if the boundary were not present. At the interface, the condition, where is the -component of the total (unnormalized) velocity field and is the specific acoustic impedance of the surface at, must be satisfied [29]. Therefore where for the free-space scenario, and (7) (8) where is the elevation of the source with respect to the sensor; therefore The quantity is known as the reflection coefficient. In general, and, hence, is a function of both incidence angle, i.e., the angle between and the -axis, and frequency. Our measurement model implicitly assumes that is frequency independent. Therefore, the signal bandwidth must be such that for any given incidence angle, is approximately constant over the frequency range. This is the only bandwidth restriction that we require. Since the distributed system of sensors may extend over many hundreds of times the smallest wavelength, this is a (4) (5) where is the source s azimuth relative to the sensor for the ground and seabed cases. In fact, (7) and (8) form the same measurement model as for a single vector sensor located on a boundary when illuminated by a planewave from bearing (see [10]). This occurs because of the point-like sensor assumption and the ray acoustic approximation. Of course, unlike the planewave model, the bearing and, hence, the reflection coefficient will differ from one location to another. These expressions for assume that the three velocity components are aligned with the three coordinate axes or that the orientation of the sensor is known and that the data have been rotated to achieve the same effect. The Swallow floats described in [11] contain a fluxgate compass to provide information on the horizontal components alignment and are carefully trimmed before deployment to minimize tilt. For the ground and seabed sensors, it should not be difficult to design a sensor package for which it is easy to align

4 1482 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 6, JUNE 2003 the vertical component, even when deployed by air or surface vessel. Again, a compass could provide horizontal alignment. A. Statistical Assumptions We assume that the signal and noise processes and are zero-mean uncorrelated processes with finite second-order moments and that (9) (10) for, where is the Kronecker delta function, is the identity matrix, the overbar represents conjugation, and superscript represents complex conjugation and transposition. The assumption of spatially white noise is consistent with internal sensor noise. In the free-space case, we have shown it to be consistent with isotropic and even certain anisotropic ambient noise fields [33]. The assumption of independent time samples is consistent with spectra that are symmetric about their center frequency and sampled at the first zero of their autocorrelation function. It is included for ease of exposition and is not necessary for the following algorithms to be implemented. If there is time correlation, however, more samples will be required to achieve a given level of accuracy. Letting be the pressure signal at the origin, the complete measurement model for the -sensor distributed array is (11) for,, where is the th sensor s steering vector, is the distance of the source from the origin, are the source to sensor distances, and are the differential time delays with respect to the origin. The term accounts for spherical spreading loss of the signal so that the signal power will vary between sensors. The account for the differential Doppler shifts between each sensor and the origin. In this model, it is implicitly assumed that each points toward the same location. Thus, the differential time delays between all sensors and the observation interval must be short enough, relative to the inverse s of the target s speed, such that and remain approximately constant. The algorithms in the following sections do not make use of intersensor correlations; therefore, there is no similar requirement on and. No further statistical assumptions are required regarding the noise at different sensors. Indeed, it may vary in power and even be correlated between sensors at different locations. Note that for the ground and seabed scenarios, the reflection coefficients, on which depend, will generally vary from one location to another as a result of the different bearings. However, variability in the can also be used to account for different reflection characteristics of the surface at different locations. III. BEARING ESTIMATION In this section, we derive fast wideband algorithms to estimate the bearing of the source from a single vector sensor. As well as providing the information on which the 3-D position estimator is based, this information may be of use in its own right. For example, in a mobile battlefield array, where each sensor is carried by a battlefield unit, such as a soldier, this estimate provides vital target information to the unit without the need for any communication, thereby minimizing detection risk. Acoustic intensity is a vector quantity defined as the product of pressure and velocity. For the free-space problem, it is parallel to ; therefore, it may be estimated to determine the bearing [6]. In the two boundary problems, since the and components of the intensity vector are the same for both real and image sources, the acoustic intensity vector is parallel to the projection of onto the -plane. However, the 3-D intensity vector is not parallel to ; therefore, the same method cannot be used to find the elevation. A. Free Space For the free-space scenario, the bearing estimate, based on the intensity vector, is (see [6]) Re (12) The asymptotic, normalized MSAE is defined as (13) MSAE (14) For a very large class of estimators, MSAE by (see [6] and [26]) is lower bounded MSAE CRB CRB (15) where CRB is the CRB. In addition, if is an unbiased unit length estimator of, MSAE is also a lower bound for the normalized finite-sample MSAE. In [6], it was shown that when the signals and noise are Gaussian, the estimator in (12) and (13) has MSAE, where is the signal-tonoise ratio (SNR) at the sensor. Is was also shown that MSAE (16) under the same distributional assumptions. B. Ground and Seabed The horizontal component of acoustic intensity is Thus, under the noise model of (10) (17) (18) Since this is purely real, we let Re, and by the strong law of large numbers,. Thus, we can estimate azimuth from (19)

5 HAWKES AND NEHORAI: WIDEBAND SOURCE LOCALIZATION 1483 Note that (19) is independent of, and therefore, we can use this estimator to determine the azimuth even without knowing the local reflective properties of the ground. Neither is a normalcomponent velocity sensor required, of course. Furthermore, even when the source is close to the boundary, so that model (3) does not hold, (19) will still produce a consistent estimator of the azimuth as long as the and components of the velocity signal are proportional to and, respectively, and that the constant of proportionality is the same for both. When this is the case, (19) may be used to find the bearing of targets located on the boundary. Since the magnitude of the horizontal component of acoustic intensity depends strongly on the elevation, so will the accuracy of. With appropriate modification, we can apply the analysis of [6, App. B] to this azimuthal estimator to show that its asymptotic MSAE is (see Appendix B) (20) where is the SNR. Obtaining an estimator of the elevation requires that the functional form of be known. The vertical component of acoustic intensity has expected value Using (18), we see that is a function of alone, which we estimate from the statistic (21) (22) seabed problem and the azimuthal estimator were developed in the context of planewave reflection in [10]. The same estimators arise here because the ray acoustics approximation results in a similar model for the AVS s steering vector. It can be seen from the above that the bearing is estimated via the three components of intensity. Therefore, we could in theory use three orthogonally oriented single-axis intensity probes. Microflown Technologies B.V. already packages its novel aeroacoustic velocity sensor with a pressure sensor to form such an intensity probe. Intensity probes are available from other manufacturers such as Brüel and Kjær, although those currently available use two closely spaced pressure sensors, instead of true velocity sensors, to determine the velocity. As discussed in [6], such sensors are not considered appropriate for vector-sensor processing. There is no known expression for the MSAE of the full bearing estimator in the ground and seabed scenarios. However, in Appendix A, we show that under the assumption of Gaussian signals and noise MSAE where (28) (29) (30) Re (23) Re (31) The elevation estimate is then the solution to (24) Substituting (5) into (22), we obtain. Hence, for the ground surface, the elevation can be estimated from Re (25) For the seabed, we substitute (6) into (22) to obtain and therefore, we estimate using (26) Re (27) Note that the estimates produced by (25) and (27) lie between and, hence, incorporating the a priori information that the source lies above the surface. The bearing estimator for the (32) and is the derivative of with respect to. Thus, the MSAE is a function of the SNR and the elevation but not the azimuth. Compare this with the free-space situation in which MSAE is solely a function of the SNR. 1) Multiple Sources: The intensity-based bearing estimators outlined above are only effective when there is one dominant source: They may not be used in the presence of a strong interfering course. A detailed analysis of the case of multiple sources is beyond the scope of this paper; however, we make the following observations and suggestions for future work. It has been shown that an AVS can identify the directions of up to two sources [34]. Therefore, the case of a source and single interfering source can be handled with the same array structure, i.e., a single AVS at each location. To deal with the two source situation, we suggest the following adaptations of conventional and minimum variance beamforming-based estimators: Form the spectra (33)

6 1484 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 6, JUNE 2003 (34) where is the sample covariance matrix, and and denote the conventional and minimum-variance spectra, respectively. The expressions for the spectra are similar to the usual conventional and minimum-variance beamforming spectra with the array steering vector replaced by, except that an additional normalizing term of is added to compensate for the fact that in the boundary scenarios, the magnitude of is not constant but depends on the elevation angle. The conventional (minimum-variance) bearing estimates are the values of corresponding to the two largest local maxima of. The vector is independent of frequency because all four AVS components are co-located; therefore, unlike traditional pressure-sensor array frequency-domain beamforming estimators, these estimators are wideband. Of course, a 2-D numerical search is required so that they are not as fast as the intensity-based algorithms. The beamforming estimates can be used when there is one or two sources, and in the former case, the intensity-based estimate could be used to initialize the search. An investigation of the properties of these estimators, for one or two sources, would be an interesting direction for future research. When there are more than two sources, a small subarray of two or more spatially separated AVSs would be required at each location in order to estimate the bearings to all sources. IV. POSITION ESTIMATION We now consider the problem of how to combine the decentralized estimates of the target bearings to obtain a 3-D estimate of its location. The algorithms of this section apply to all scenarios. Each sensor transmits its local estimate of the direction from its location to the source, say, as well as its own location. The location could be determined from a lightweight GPS receiver packaged with the sensor, for example. In practice, both the local bearing and position will contain errors; however, we assume that the bearing estimate is the dominant source of error and ignore possible inaccuracies in the location. Therefore, a total of five quantities (four if all sensors are at the same altitude) need to be transmitted three to describe the location and two for the bearing no matter how long an observation window is used. This is a huge advantage over a centralized processing scheme in terms of communications overhead, where every single data sample from every single sensor must be sent to a central processor. The results of this section are applicable to bearing estimates obtained from any type of subarray and not just individual AVSs. In particular, many sources may be handled by these techniques if each subarray consists of multiple traditional or vector sensors and can therefore estimate the bearing to more than two sources. A. Weighted Least Squares If all the were without error, the collection of lines passing through each with direction would intersect at the true source position. Therefore, we want to choose the estimate of the source s location to be a point that is in some sense closest to all these lines. We will choose to minimize a weighted sum of the minimum squared distances from to each line. By doing so, we will derive a closed-form solution for the location estimate, avoiding the need for a complex computational search. Any point along the line defined by the th sensor s location and bearing estimate is defined by the vector for some. For fixed, the point of closest approach occurs when, i.e., is the projection of the vector from to onto the direction. Thus, we propose a weighted least squares (WLS) estimate of, which is given by (35) where is a weight corresponding to the accuracy of each. Expanding (35) and rearranging, we obtain (36) where we have dropped terms independent of. Note that is the projection matrix onto the plane orthogonal to. Differentiating with respect to and setting the result equal to zero gives Hence, we arrive at the closed-form solution (37) (38) where, diag,, and (39) 1) Choice of Weights: In general, the accuracy of the bearing estimates will be different from sensor to sensor due to a number of factors. There may be local variations in background noise level or ground reflectivity, signal strength will differ between sensors that are different distances from the source because of spherical spreading loss or partial occlusion, and the accuracy of the estimation algorithm may depend on the true, which differs between sensors. Thus, it is important for each sensor to transmit a measure of accuracy along with its bearing estimate to the central processor. A very natural measure in this situation is the MSAE. Since no finite sample expression is known for the MSAE, we consider instead the bound MSAE (28). In the free-space problem, if the signal and noise are Gaussian, we could also use the previously given expression for MSAE ; however, in our simulations, we found no discernible difference in the resulting accuracy of. Let us suppose the signal and noise are Gaussian, so that MSAE is given by (16) or (28). It then depends on the unknown quantities and, for the boundary scenarios,, so that it must be estimated by plugging in estimates of the unknowns. The bearing estimator itself already provides us with an estimate of ; therefore, remains to be estimated. If we knew the maximum-likelihood (ML) estimate of, say,, then we could

7 HAWKES AND NEHORAI: WIDEBAND SOURCE LOCALIZATION 1485 use closed-form expressions (see, e.g., [35]) to find the ML estimates of and. Since we do not know, we will use our actual estimate of. Therefore, we have Re tr (40) (41) where, and is the sample covariance matrix. Our estimate of the SNR is. Finally, we plug and the elevation estimate into (16) or (28) to obtain MSAE. This calculation is made locally at each sensor, and the weight sent to the central processor and used in determining is then MSAE. Note that this method of choosing weights can be used with bearing estimators other than those developed in Section III. If the signal and noise are not Gaussian, (28) may not always be easy to compute. If it is particularly intractable, we expect that using the above procedure based on the Gaussian assumption will still lead to better estimates of than would uniform weighting for many distributions. B. Reweighted Least Squares Errors in the bearing estimates from sensors far from the source have a much greater effect upon than those from sensors nearby. The contribution of the th bearing estimate to the squared error criterion is approximately, where is the angular error of, and (as before) is the distance from each measurement location to the source. Although we do not know the, we do have an estimate of them after we have estimated using the above WLS procedure. Therefore, we propose a reweighted least-squares (RWLS) estimator constructed as follows: Find using the weights MSAE. Using this, estimate the distances from each sensor to the source as, and then, construct a reweighted estimate, again using WLS but now with weights MSAE. This RWLS estimator can be thought of as extension of Stansfield s estimator [36] to three dimensions and to the case of unknown angular error variance and lengths, as we now show. In two dimensions, the bearings can be represented by a single angle. Suppose each estimated bearing is an unbiased Gaussian distributed estimate of the true bearing with known variance, say, and that the bearing estimates from different locations are uncorrelated. The ML estimate of would be (42) where is the angular error of the th bearing estimate. Since this has no closed-form solution, Stansfield proposed replacing with its sine, i.e., let (43) which does have a closed-form solution, provided that the lengths are known as well as the angular error variances. Now, in two or three dimensions, is equal to the ratio of the length of the projection of onto the subspace orthogonal to to the length of, i.e., (44) However,. Substituting this into (44), then (43), and comparing with (35), we see that Stansfield s estimator is a WLS estimator with. C. Distributed Potential Performance Calculation of the CRB on for the entire array based on all measurements made by all sensors, i.e., (11), quickly becomes computationally infeasible for even a moderate number of samples because of the wideband nature of the source signals. Even for a narrowband source, however, it leads to a completely unrealistic assessment of the potential of the distributed system because it implicitly supposes that all possible cross-correlations between measurements at different locations can be estimated. Therefore, in this section, we develop an estimatorindependent indicator of the potential performance achievable with a distributed system. It is based on a two-stage calculation of CRBs, made under the assumption of Gaussian signals and noise, but is not itself a CRB. Nevertheless, in the examples of Section V, it does lower bound the variance of the estimate of and is attained at high SNRs. Therefore, we expect it to provide a good benchmark against which to assess the performance of a particular distributed estimation scheme and an effective criterion for system design. Each sensor sends a bearing estimate and an estimate of its accuracy based on the estimated values of signal power and noise power at its location. We could, of course, send the signal power and noise power estimates separately, so let us suppose that each sensor transmits the vector to the CP. At the first stage, we calculate the CRB on using only those measurements made by the th vector sensor, i.e., using model (7). This may be done using Bangs formula for zero-mean complex Gaussian data, which gives the entries of the Fisher information matrix (FIM) (see [37, pp. 525], for example). Thus, the th entry of FIM is FIM tr (45) where is the 4 4 covariance matrix of the measurement data at the th AVS, and is the th entry of. In the second stage, we consider the as measurements that are Gaussian distributed with mean and covariance CRB. The ML estimate of will asymptotically have these properties [38]. In addition, we suppose that and are mutually uncorrelated for. For a finite number of samples, this latter assumption is unlikely to be exactly true since the measurements are correlated between different sensor locations, although it may well be true asymptotically. Nevertheless, we expect that ignoring possible correlations between different will tend to lead to a lower bound on the variance of since setting the off-diagonal elements of a positive definite matrix to zero generally increases its determinant and, hence, reduces the diagonal

8 1486 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 6, JUNE 2003 elements of its inverse. Stacking the vectors for in a single vector, and similarly creating from for, we have the measurement model (46) where is a zero-mean (real) Gaussian distributed vector that has a block-diagonal covariance blkdiag CRB CRB (47) The vector of unknowns at the CP, say, consists of the location, the source power at the origin, and the noise powers at each of the sensors, i.e.,. Note that in defining the unknowns thusly, it is implicitly assumed that the noise at different AVSs is uncorrelated. We calculate CRB, using the equivalent of Bangs formula for real Gaussian data (see [37, pp. 47]); therefore, the th entry of the FIM [FIM ]is Fig. 3. Underwater scenario: Performance MSAE of fast bearing estimator (solid), and bound MSAE (dashed), for 3-dB source with 175 snapshots. Mean estimated bound MSAE (dash-dotted) plus and minus three empirical standard deviations (dotted) is also shown. Five hundred realizations were used. FIM tr (48) We then invert FIM and extract the appropriate 3 3 block corresponding to CRB. We denote this 3 3 matrix by DPP. Note that (48) requires determination of the derivative of each CRB with respect to each component of, which is, in most cases, unlikely to be analytically tractable. Therefore, it may have to be computed numerically, which is what was done in the examples in Section V. When is expressed in Cartesian coordinates, these may be converted to polar coordinates by DPP DPP (49) where contains the range, azimuth, and elevation, respectively, of the source relative to the origin. This is similar to the CRB parameter transformation formula (see e.g., [37, pp. 45]). The potential mean-square range error (MSRE, see [26]) of the distributed system, which is denoted MSRE, is just that entry of DPP corresponding to. The potential MSAE is, by analogy with the MSAE [see (15)] MSAE DPP DPP (50) D. Multiple Sources The issue of multiple sources is beyond the scope of the paper; however, we believe the WLS and RWLS position estimation methods could be extended for use with multiple sources. If the CP can correctly associate bearing estimates from each location with the bearing estimates from all other locations, that is, it correctly decides which bearing and source power estimates correspond to the first source, which to the second and so on, each source s position could be estimated independently of the others with the algorithms described above. Methods of association would form a very interesting topic for further work. Perhaps matching frequency spectra or signatures of the various signals would be fruitful. The problem Fig. 4. Ground scenario: Performance MSAE of fast bearing estimator (solid) and bound MSAE (dashed) for 20-dB source with 350 snapshots. Mean estimated bound MSAE (dash-dotted) plus and minus three empirical standard deviations (dotted) is also shown. Five hundred realizations were used. may also be related to the problem of eigenvalue association in ESPRIT. Alternatively, the WLS could be extended to jointly find the positions of all sources by searching all associations and choosing that which minimizes the sum of the individual WLS criteria over the number of sources. Extension of the formulae for the distributed potential performance measure to multiple sources should be relatively straightforward. It clearly remains a lower bound as its calculation would assume that the associations are correct at the central processor. V. NUMERICAL EXAMPLES A. Single-Sensor Bearing Estimation For the purposes of simulation, we use a Gaussian distributed signal and noise. For the seabed case, we use the parameters values given directly below (6), which is a 3-dB source, and 175 snapshots. For the ground problem, we take, which was measured by [31] for grass-covered flat ground at 215 Hz, and use a 20-dB source with 350 snapshots. The performance of the bearing estimate in free space is studied in [6]. Figs. 3 and 4 summarize the results, which are independent of azimuth, as a function of incidence angle, say (i.e., ). The seabed estimator is rather more accurate

9 HAWKES AND NEHORAI: WIDEBAND SOURCE LOCALIZATION 1487 TABLE I ANGLES AND RANGES FROM THE SENSORS TO THE SOURCE Fig. 5. Signal gain resulting from reflection and sensor directionality for pressure (solid), in-plane (dashed), and normal (dash-dotted) components of a vector sensor at a locally reacting boundary. Sensor located on ground with normalized input impedance is Z = i. than the ground estimator at all angles (note the higher SNR and greater number of snapshots used in the latter example). The reason for this is the substantially lower density of air relative to the ground than that of water relative to the seabed. The large differential causes the ground surface to appear almost rigid, i.e., there is very little motion normal to the boundary and, as a result, very low signals at the normal velocity sensor, which is responsible for much of the directional sensitivity of the AVS at the boundary. In both cases is close to the bound at normal incidence,, i.e., with the source directly above the sensor. As incidence increases, the performance, though not the bound, worsens steadily for the ground sensor. For the seabed case, performance stays approximately constant and close to the bound before worsening rapidly around, which is the same time that the bound actually decreases. In both cases, performance and bound tend to as the incidence reaches grazing because, and therefore, so does the SNR. Except at large incidence, the quantity MSAE is seen to estimate MSAE well. This is especially true for the battlefield problem and is probably due to the higher SNR in the simulation. It is possible that on any run,, with finite probability if the argument of the inverse cosine in (25) or (27) is greater than one. In this case, our technique fails to yield an estimate of MSAE because it is theoretically infinite if really is zero as. In our simulation, this never occurred below 66 incidence, and the chances of it occurring rose to about 50% within a few degrees of grazing incidence. However, we do not expect that such large incidences will need to be measured in this application, and therefore, this should not be a problem in practice. If it does occur at one sensor in the array, a solution would be to use the average of the MSAE obtained from the other sensors as its weight in the WLS location procedure. Examination of the standard deviation of the elevation and azimuthal estimates separately (not shown) reveals that the majority of the angular error, especially in the battlefield scenario, is due to errors in the elevation rather than the azimuth. Thus, the 3-D location system can determine that the -coordinates (ground track) of a target can be rather more accurate than its 3-D position or height. The reason for this may be seen in Fig. 5. It shows the signal gain resulting from reflection at the boundary and the incoming signal s direction for each sensor component, i.e., the squared magnitudes of the entries of, for the ground sensor. Actually, the illustrated in-plane gain is the gain of the sum of the in-plane components or, equivalently, the gain of one in-plane component when is in the same plane as its axis. It is seen that the normal component gain is much lower than both pressure and in-plane gain for almost all angles. When the normal gain is larger than the in-plane, very near normal incidence, large errors in the azimuthal estimate have little effect on the MSAE because of the inherent singularity in the spherical coordinate system. This poor normal gain is mainly due to the size of the input impedance, if the input impedance were small, i.e., the surface were acoustically more pliable, the normal gain would improve relative to the in-plane and pressure gains, resulting in better estimation of elevation but poorer azimuthal estimation. B. Position Estimation We now give examples of the performance of the WLS and RWLS position estimators for a wideband signal. We use a stationary target with a Gaussian signal and six vector sensors. In the free-space example, sensors are located at,,,,, and. In the seabed and ground examples, they are located on the surface with coordinates,,,,, and. In all examples, the source is at. The resulting angles and ranges are shown in Table I. We also assume the sampling frequency in each case is equal to the speed of sound (1500 m/s in water and 330 m/s in air) so that the differential delay between sensors is an exact multiple of the sampling period, thereby avoiding the need to implement fractional delays in the simulation. The

10 1488 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 6, JUNE 2003 p Fig. 6. Angular estimation performance MSAE for LS (dotted), WLS (dash-dotted), and RWLS (solid) position estimators. MSAE (dashed) is also shown. Free-space scenario. p Fig. 9. Range estimation performance MSRE for LS (dotted), WLS (dash-dotted), and RWLS (solid) position estimators. MSRE (dashed) is also shown. Seabed scenario. p Fig. 7. Range estimation performance MSRE for LS (dotted), WLS (dash-dotted), and RWLS (solid) position estimators. MSRE (dashed) is also shown. Free-space scenario. p Fig. 10. Angular estimation performance MSAE for LS (dotted), WLS (dash-dotted), and RWLS (solid) position estimators. MSAE (dashed) is also shown. Ground scenario. p Fig. 8. Angular estimation performance MSAE for LS (dotted), WLS (dash-dotted), and RWLS (solid) position estimators. MSAE (dashed) is also shown. Seabed scenario. source signal is uncorrelated from one snapshot to the next. This would occur if, for example, it has a flat spectral density and its bandwidth was equal to the sampling frequency. Noise is of equal power and uncorrelated between different sensors. The SNR is defined as the ratio of the signal power at the origin to the common noise power. In keeping with model (11), the signal power at any point is determined by spherical p Fig. 11. Range estimation performance MSRE for LS (dotted), WLS (dash-dotted), and RWLS (solid) position estimators. MSRE (dashed) is also shown. Ground scenario. spreading. For the boundary problems, the SNR is defined as the ratio of the signal power that would exist at the origin if the boundary were not present to the common noise power. The same reflection properties as in Section V-A are used for the ground and seabed. Figs show the MSAE and mean-square range error (MSRE), which are defined by, of the location estimated for the WLS estimator and the RWLS estimator and for an

11 HAWKES AND NEHORAI: WIDEBAND SOURCE LOCALIZATION 1489 un-weighted least-squares (LS) algorithm, i.e., the WLS procedure with equal weights. A total of 350 snapshots (at each AVS) were used and the results averaged over 1500 Monte Carlo trials. The improvement obtained by using WLS and RWLS over LS depends on the scenario; it is most noticeable in the ground problem, and least in the seabed. In addition, the source s direction is more accurately determined than its range. This is not too surprising. It seems obvious that small errors in the local bearings will cause a larger error in the estimated range than in the estimated direction to the source, especially if the source is very far from the sensors. Indeed, it is known that the CRB on the variance of range estimators for a passive sensor array increases as the fourth power of the range [39]. We also show MSAE and MSRE, which are the angular and range potential accuracy measures computed using the distributed potential performance (DPP) measure of Section IV-C. The DPP measures lower bound the actual MSAE and MSRE in all cases. In the free-space scenario, they are essentially achieved by the RWLS estimate at high SNRs. In all cases, the MSAE of all estimates comes closer to MSAE than does the MSRE to MSRE, indicating that the source s bearing is more efficiently estimated than its range. The actual performance is worst, relative to the DPP in the ground case. This is not surprising. In the free-space situation, the MSAE of each individual bearing estimate is quite close to the MSAE, regardless of the actual bearing (see expressions in Section III-A), and achieves it at high SNR. In the seabed problem (see Fig. 3), the MSAE is close to MSAE at most angles. However, in the ground case (see Fig. 4), the two quantities are only close near normal incidence. VI. CONCLUSION We developed a fast, wideband decentralized processing scheme for 3-D source localization of targets in three dimensions using a distributed array of acoustic vector sensors. This method requires minimal communication between the sensors of the array and is very adaptable to a changing array configuration. We examined the cases of sensors located in free space and on two different types of surface: the seabed and the ground. For the latter case, we proposed a new fast wideband algorithm to determine the bearing from a single AVS to the source and gave a lower bound on its performance. We also showed how to estimate this bound from the data. We constructed a weighted least squares and 3-D position estimator based on combining bearings and an estimated accuracy of each bearing from various locations. We also proposed a reweighted least-squares method to take into account the different distances from the source to the various locations and showed the relationship between it and Stansfield s 2-D means of combining bearings [36]. We also developed a distributed potential performance measure, based on a two-step calculation of Cramér Rao bounds, to use as a benchmark for assessing the efficacy of various distributed estimators and as a criterion for array design. Numerical examples illustrated our results. We note several particularly interesting extensions of this work: We believe the DPP measure to be an important tool for determining the optimal performance of distributed systems in general. Further study of its theoretical properties would be very useful. We also note that the vertical stratification of the speed of sound in the ocean causes refraction of acoustic waves. As a result, when there is considerable variability in the sound speed profile between the source and the sensor, the vector sensor s direction estimate will not point directly at the target. In such a case, the direction estimates can be used to provide boundary conditions to ray-tracing algorithms. Furthermore, the location estimate can be used as an initial estimate in the iterative numerical optimizations that would be required to locate a source from simultaneous ray tracings. Development of algorithms along these lines could prove very fruitful. Finally, we note that the algorithms herein do not require the measurement of the source over a period of time long enough such that is has moved a substantially difference. Methods involving measuring the changing acoustic (pressure) signature of a source from a number of locations as it moves through space in order to determine location and to track sources have been developed (see [40] for example). We believe that the spatial information provided by the distributed vector-sensor approach could be combined very effectively with temporal techniques to produce accurate and fast source tracking systems. APPENDIX A In this Appendix, we derive the MSAE for a single vector sensor located on a reflecting boundary. Under the assumption of Gaussian signals and noise, the single sensor measurement model (3) satisfies the requirements of [10, Th. 3.1]. Now, from the definition of the vector-sensor steering vector [see (8)], noting that the reflection coefficient is a function of but not, we have that (51) (52) where is the derivative of with respect to. From these equations, we see that. Therefore, [10, Th. 3.1] gives the CRB as CRB (53) The MSAE (28) then follows from (15), and (29) (32) follow from (8), (51), and (52).

12 1490 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 6, JUNE 2003 APPENDIX B In this Appendix, we determine the performance of the fast, wideband azimuthal estimator. Consider the model for measurements made by a single vector sensor in free space given by (2). In [6], the following bearing estimator was proposed for use with a single AVS Re Re (54) where and are defined in (2). In [6, App. B], an expression is presented for the MSAE of this estimator, which holds under the statistical assumptions of this paper, including the Gaussianity assumption (see [6, eq. (B.5)]). This expression may also be written as MSAE tr (55) where and are the SNRs at the pressure and velocity sensors, respectively, and is the unit-length vector being estimated. It is clear that the analysis of [6, App. B] holds no matter what the dimension of the unit vector [and ]. Furthermore, the MSAE only depends on the SNRs and not the absolute values of signal and noise powers. Consequently, the expression is applicable to the current azimuthal estimator. Making the following substitutions in (55) gives the result (20). (56) (57) (58) REFERENCES [1] D. Lake. (1998, Jan.) Battlefield acoustic signal processing and target identification. CSI/Stat. Colloq., George Mason Univ., Fairfax, VA. [Online]. Available: [2] D. Lake and D. Keenan, Maximum likelihood estimation of geodesic subspace trajectories using approximate methods and stochastic optimization, in Proc. 9th IEEE SP Workshop Statistical Signal Array Process., Portland, OR, Sept. 1998, pp [3] T. Pham and B. Sadler, Adaptive wideband aeroacoustic array processing, in Proc. 8th IEEE SP Workshop Statistical Signal Array Process., Corfu, Greece, June 1996, pp [4] B. G. Ferguson, Time-delay estimation techniques applied to the acoustic detection of jet aircraft transits, J. Acoust. Soc. Amer., vol. 106, no. 1, pp , July [5] B. G. Ferguson and B. G. 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13 HAWKES AND NEHORAI: WIDEBAND SOURCE LOCALIZATION 1491 [37] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, NJ: Prentice-Hall, [38] T. S. Ferguson, A Course in Large Sample Theory. New York: Chapman & Hall, [39] Y. Rockah, Array processing in the presence of uncertainty, Ph.D. dissertation, Yale Univ., Hew Haven, CT, [40] C. Y. Chong, K. C. Chang, and S. Mori, Tracking multiple air targets with distributed acoustic sensors, in Proc. Amer. Contr. Conf., Minneapolis, MN, June 1997, pp Malcolm Hawkes (S 95 M 00) was born in Stockton-on-Tees, U.K., in 1970 and grew up in Swansea, U.K. He received the B.A. degree in electrical and information science from the University of Cambridge, Cambridge, U.K., in 1992, the M.Sc. degree in applied statistics from the University of Oxford, Oxford, U.K., in 1993, and the Ph.D. degree in electrical engineering from Yale University, New Haven, CT, in In 1988, he won a scholarship from GEC-Marconi Research Centre, Chelmsford, U.K., and worked there from 1988 to 1989, and again in 1990 and 1991, on a variety of projects. He won scholarships at Emmanuel College, Cambridge, from 1990 to He was awarded a U.K. Medical Research Council grant to pursue a Master s program in 1992 and a Yale University Fellowship in From 1996 to 2000, he was a Visiting Scholar at the University of Illinois, Chicago, where he received the John and Grace Nuveen International Scholar Award in He is now a Research Associate with Ronin Capital, LLC, Chicago. His research interests include statistical signal processing and time series analysis with applications in finance, array processing, and biomedicine. Arye Nehorai (S 80 M 83 SM 90 F 94) received the B.Sc. and M.Sc. degrees in electrical engineering from the Technion Israel Institute of Technology, Haifa, Israel, in 1976 and 1979, respectively, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in After graduation, he worked as a Research Engineer for Systems Control Technology, Inc., Palo Alto, CA. From 1985 to 1995, he was with the Department of Electrical Engineering, Yale University, New Haven, CT, where he became an Associate Professor in In 1995, he joined the Department of Electrical Engineering and Computer Science, University of Illinois at Chicago (UIC), as a Full Professor. From 2000 to 2001, he was Chair of the department s Electrical and Computer Engineering (ECE) Division, which is now a full department. In 2001, he was named a University Scholar by the University of Illinois. He holds a joint professorship with the ECE and Bioengineering Departments at UIC. His research interests are in signal processing, communications, and biomedicine. Dr. Nehorai is Vice President Publications of the IEEE Signal Processing Society and was Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from January 2000 to December He is currently a member of the Editorial Boards of Signal Processing, the IEEE SIGNAL PROCESSING MAGAZINE, and The Journal of the Franklin Institute. He has previously been an Associate Editor of the IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, the IEEE SIGNAL PROCESSING LETTERS, the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, the IEEE JOURNAL OF OCEANIC ENGINEERING, and Circuits, Systems, and Signal Processing. He served as Chairman of the Connecticut IEEE Signal Processing Chapter from 1986 to 1995 and was a Founding Member, Vice-Chair, and later Chair of the IEEE Signal Processing Society s Technical Committee on Sensor Array and Multichannel (SAM) Processing from 1998 to He was the co-general Chair of the First and Second IEEE SAM Signal Processing Workshops, held in 2000 and He was co-recipient, with P. Stoica, of the 1989 IEEE Signal Processing Society s Senior Award for Best Paper. He received the Faculty Research Award from UIC College of Engineering in 1999 and was Advisor for the UIC Outstanding Ph.D. Thesis Award that went to Aleksander Dogandzic in This year, he was elected Distinguished Lecturer of the IEEE Signal Processing Society for the years 2004 and He has been a Fellow of the Royal Statistical Society since 1996.