Bearing estimation with acoustic vectorsensor arrays

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Bearing Estimation with Acoustic Vector-Sensor Arrays Malcolm Hawkes and Arye Nehorai Department of Electrical Engineering, Yale University, New Haven, CT 06511

A b s t r a c t . We consider direction-of-arrival (DOA) estimation using arrays of acoustic vector sensors in free space, and derive expressions for the Cram@r-Rao bound on the DOA parameters when there is a single source. The vector-sensor array is seen to have improved performance over the traditional scalar-sensor pressure-sensor) array for two distinct reasons: its elements have an inherent rectional sensitivity and the array makes a greater number of measurements. The improvement is greatest for small array apertures and low signal-to-noise ratios. Examination of the conventional beamforming and Capon DOA estimators shows that vector-sensor arrays can completely resolve the bearing, even with a linear array, and can remove the ambiguities associated with spatial undersampling. We also propose and analyze a diversely-oriented array of velocity sensors that possesses some of the advantages of the vector-sensor array without the increase in hardware and computation. In addition, in certain scenarios it can avoid problems with spatially correlated noise that the vector-sensor array may suffer.

1 INTRODUCTION T h e p a s s i v e d i r e c t i o n - o f - a r r i v a l ( D O A ) e s t i m a t i o n p r o b l e m , in which t h e b e a r i n g s of a n u m b e r of far-field a c o u s t i c sources are d e t e r m i n e d , is of g r e a t i m p o r t a n c e in m a n y u n d e r w a t e r a p p l i c a t i o n s . T r a d i t i o n a l l y , t h e solution is to use a s p a t i a l l y d i s t r i b u t e d a r r a y of o m n i d i r e c t i o n a l pressure-sensors, a n d m a n y e s t i m a t i o n t e c h n i q u e s have evolved for this scenario. As e m p h a s i s has shifted to low i n - w a t e r costs, d e m a n d for b e t t e r p e r f o r m a n c e at lower signal-to-noise r a t i o s (SNRs) a n d w i t h s m a l l e r a r r a y s has increased. This has l e a d to t h e i d e a of using a c o u s t i c v e c t o r sensors. A v e c t o r sensor m e a s u r e s t h e acoustic p r e s s u r e a n d all t h r e e c o m p o n e n t s of t h e a c o u s t i c p a r t i c l e velocity at a p a r t i c u l a r p o i n t in space. A r r a y s of a c o u s t i c v e c t o r sensors are, therefore, in c o n t r a s t to t r a d i t i o n a l p r e s s u r e - s e n s o r a r r a y s since t h e y s a m p l e t h e v e l o c i t y field, as well as t h e p r e s s u r e field, at a n u m b e r of fixed locations. T h e i r n a m e derives from t h e fact t h a t t h e y m e a s u r e a c o m p l e t e v e c t o r p h y s i c a l q u a n t i t y . By m a k i n g use of m o r e of t h e available i n f o r m a t i o n in t h e a c o u s t i c field, vector-sensor a r r a y s are a b l e to i m p r o v e source l o c a l i z a t i o n accuracy. In p a r t i c u l a r , since each v e c t o r sensor m e a s u r e s t h e s t r u c t u r e of t h e

© 1996AmericanInstituteof Physics 345


acoustic field as well as its magnitude, the improvement is more than a simple increase in SNR associated with taking a greater number of measurements. A theoretical framework for DOA estimation using acoustic vector-sensor arrays has recently been developed in [1] and preliminary studies for free-space and hull-mounted arrays made in [2] and [3]. In addition, acoustic vector sensors have already been constructed [4] and linear arrays of them built and been subject to sea trials [5], [6]. We consider an array of vector sensors, illuminated by n narrowband Gaussian signals in spatially and temporally uncorrelated Gaussian noise. The Cram6r-Rao bound (CRB) for a single source scenario is derived and examined. We are able to distinguish and quantify two distinct improvements of the vector-sensor array over the pressure-sensor array: one associated with the greater number of measurements and one with the inherent directionality of the vector sensors themselves. Examination of these expressions shows that the use of vector sensors is most advantageous in small array, low SNR situations. We examine the performance of the conventional beamforming and Capon [7] DOA estimators, when used with the vector-sensor array. It is seen that even the simplest yector-sensor array can fully resolve both azimuth and elevation, and further advantages are revealed with regard to reduced aliasing and removal of ambiguities in undersampled arrays. A diversely-oriented array consisting of spatially distributed velocity sensors with various orientations is proposed and analyzed. Although only having the same hardware and computational requirements as a pressure-sensor array, it gains some of the directional capabilities of a vector-sensor array at the expense of SNR. In addition, it may be used in the presence of non-isotropic spatially correlated noise and still provide uncorrelated outputs if the correlation distance is shorter than the inter-sensor spacing or the sensors are placed at nulls of the spatial correlation function. This is in contrast to the vectorsensor array where the noise of co-located measurements will be correlated in the presence of a non-isotropic noise field, thus reducing performance.

2 THE MODEL Consider n narrowband planewaves, with common center frequency w, traveling in an isotropic, quiescent, homogeneous medium, and impinging on an array of rn acoustic sensors, located far from any reflecting boundary (see [1] for details). The problem is to determine the DOA parameter vector 0 = [O1,..., 8,~] T, where Ok = [q~k,~bk]T, and ~k and ~bk are the azimuth and elevation of the kth source. Let u be the unit vector pointing from the array towards a source. Then u = [cos ff cos ~, sin q~cos ~b, sin ~b]T, where q~ E [0, 2~r) and ~b E [-~r/2, r/2].

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The linear acoustic equation [8] (also known as Euler's equation) relates acoustic pressure and acoustic particle velocity; for planewaves it becomes

v(r,t) -

P(r't)u,

poc

(1)

where v ( r , t ) and p ( r , t ) denote the (vector) acoustic particle velocity and (scMar) acoustic pressure at position r and time t, po the ambient pressure, and c the speed of sound in the medium.

2.1 Vector-Sensor Array Each vector sensor consists of a co-located hydrophone (pressure-sensor) and triad of orthogonal geophones (velocity sensors) [1]-[5]. We assume that each geophone is aligned with a coordinate axis, its output is proportional to the cosine of the angle between u and the axis, and that all sensors measure the complex envelope (phasor representation) of the appropriate quantity - the complex envelope is obtained by preprocessing with the known center frequency w and, for narrowband sources, is equivalent to taking the Fourier Transform [9]. For a single source, the output of an array of m vector-sensors, located at *'1 • .. rm, is a 4m vector given by

y(t) = a(O)x(t) + e(t),

(2)

where x(t) is the complex envelope of the source's acoustic pressure and e(t) is additive (complex) noise. The array's steering vector a(O) is its response to a unit amplitude sinewave from direction 0. For the vector-sensor array, it is given by the Kronecker product of a positional steering vector and an orientational steering vector,

a(0) = av(0) ~ ap(0) ® h(O),

(3)

where the positional steering vector ap(O) is the steering vector of a regular pressure-sensor array, while the orientational steering vector h(O) is the steering vector of a single vector sensor located at the origin. Thus

ap(O) = [ei~(rTu)/c,..., ei~(rTu)/c]T,

(4)

h(O) = [1, u(0)] r,

(5)

where -(r~u)/c is the differential time delay between the origin and the kth sensor. So ap(O) depends on both the relative location of the sensors and the DOA while h(O) depends solely on the DOA. The latter accounts for the directional response of each component: omni-directional for the pressure sensor and cosine for the velocity sensors, and its form is clear from equation (1) if we

347


assume that the velocity measurements are normalized by the characteristic impedance poc. With n sources

y(t) = A(O)$(t) + e(t),

(6)

where the transfer matrix A(O) = [av(01),..., av(0n)], and ~(t) contains the complex envelope representations of the acoustic pressures of the n sources. We assume that both the signal $(t) and the noise e(t) are independent identically distributed multivariate Gaussian processes [10], with zero-mean. In addition, we assume that ~(t) and e(s) are independent for all s and t. These two processes are completely characterized by their covariance matrices,

E{ze(t)x'(s)} E{e(t)e~(s)}

=

PSt,,,

=Im ®

(7)

ap o

,St,~,

(8)

where the superscript ~ represents conjugate transposition, a~ and a~ are the pressure-sensor and velocity-sensor noise variances respectively, and Im is the mth order identity matrix. The assumed structure of the noise covariance is consistent with internal sensor noise and isotropic ambient noise, provided the correlation distance is less than the inter-sensor spacing or the sensors are placed at nulls of the correlation function. For example, the model is appropriate for a uniform linear array with half-wavelength spacing in spherically isotropic noise. We allow the noise variance to differ between the pressure and velocity sensors to account for their different construction. The unknowns are the 2n DOA parameters, 0 . . [0~', r r , the signal covariance, P and the noise variances, crp~ and . . ,0hi 2 Note that the standard pressure-sensor array model is obtained by setting h=l. O" v .

2.2 Diversely-Oriented

Array

The diversely-oriented array consists of a single velocity sensor at each location r l . . . , rm, with the axis of each velocity sensor oriented differently from one position to the next. While requiring the same hardware and computational power as a pressure-sensor array, it possesses some of the added directional capabilities of the vector-sensor array at the expense of a lower effective SNR (see Section 3.2). In addition, it may often be the case that measurements at different, co-located sensors will be correlated due to a non-isotropic noise field, but that the noise correlation distance is less than the inter-sensor spacing. In this scenario the diversely-oriented array still maintains uncorrelated measurements, while the vector-sensor array's performance is impaired by the correlation within each vector sensor.

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We assume that there are m = 3k sensors in the array, with k sensors aligned with each of the coordinate axes. Thus, there are three groups of k sensors that we shall call x-sensors, y-sensors and z-sensors according to their alignment. The output of the array, an m element vector, may be written as in equation (2), but with steering vector

a(O) = aD(O) ~ Q(O)ap(O),

(9)

where Q(O) is an m × m block diagonal matrix Q(O) zx Ik ® diag{u}, Ik is the kth order identity matrix and diag{u} is the 3 x 3 diagonal matrix with the entries of u down the diagonal. When there are n sources, equation (6) is the applicable model with A(O) = [aD(01),...,aD(0n)] r. The statistical assumptions on the signal, x(t), and noise e(t), are as in Section 2.1, except that the noise covariance becomes

E{e(t)e"(,)}

(10)

= c%IrnSt,~,

since all sensors are of the same type and there are only m measurements.

3 CRAM]~R-RAO B O U N D S The CRB is a lower bound on the variance of all unbiased estimators of some parameter or set of parameters. Under mild regularity conditions, the maximum-likelihood estimator, and often many other estimators, asymptotically attain the CRB. Indeed, when there is a single source, Capon and beamforming techniques are known to give asymptotically unbiased estimates of the DOA parameters, and to asymptotically attain the CRB.

3.1 Vector-Sensor Array Consider m element pressure-sensor and vector-sensor arrays, with arbitrary geometry, illuminated by a single source. Under the assumptions of Section 2, the CRBs on the source's azimuth and elevation are given by the diagonal entries of CRBp(0)

-

1 ( 1 )j~_l, 2Npp 1 + mpp

CRBv(O)

_

1 ( 1 ) 2Np~ 1 + (r 2 +-1)rap:

(11)

avl'

(12)

where (11) is the hydrophone and (12) the vector-sensor expression. The 2 2 matrices Jp and Jv are the respective Fisher information matrices, pp = a s/crp,

349


2 2 and r 2 = pp/pv, where a82 is the signal power. Taking the origin pv = as/av, of the coordinate system to be the array centroid,

COS¢

cos¢ Jv

]

= (r2+l)Jp+D,

(13) (14)

where the sums are over the number of sensors, m. The vectors V 1 and v2 are defined by vl = (Ou/O¢)/cos ¢ and v2 = Ou/O¢, and in fact (u, Vl, v2) is a right orthonormal triad. The matrix D is given by rn

cos2¢ 0

0]1 "

(15)

Comparison of (11) with (12) and (13) with (14) shows that these differ in two ways: there is an extra factor of (r 2 + 1) in the vector-sensor equations and an extra additive term, D, in (14). These differences are due to separate phenomena. The (r2+ 1) term is the result of an effective increase in SNR, due to the greater number of measurements made by the vector-sensor array, while D is due entirely to the inherent directional sensitivity of each vector-sensor, and is unaffected by sensor location. Consider the SNR increase in more detail. Firstly, note that r 2 is a measure of the efficiency of the velocity sensors relative to the pressure sensors - - it is the ratio of hydrophone SNR to geophone SNR. In practice, we expect this value to be around unity but it may indeed be lower, as is the case for spherically isotropic noise where it is 1/3 [11]. Rewrite (12) as r2 CRBv(0) - 2Npv(r 2 + 1)

) (jp +r--~--~ 1 D) -1 • r2 1 + mpp(r2 + 1)

(16)

Ignoring the directional term, D, comparison of (16) and (11) shows that the effect of the extra measurements is to increase the SNR from pp at each of the m pressure sensors to pv(r 2 + 1)/r 2 at each of the m vector sensors. This factor of (r 2 + 1)/r 2 is not what we might first expect. For example, if r 2 = 1 we would expect a fourfold increase in SNR since there are four times as many measurements, yet the increase is only twofold. The reason for this lies in the directional sensitivity of the geophones. The signal power at an individual velocity sensor is only the same as at the pressure sensor when the planewave is propagating in the direction of its axis, at any other angle it is less. The response of each vector sensor is proportional to h and Ihl 2 = 2. The output of the velocity-sensor triad is proportional to u, which is a unit vector. Thus, half of the vector sensor's output signal power comes from the pressure sensor and half from the set of velocity sensors. Once the relative efficiency of the

350


two types of sensors is taken into account, we see that each vector sensor has a SNR that is (r 2 + 1)/r 2 times that of each hydrophone. Let rlv denote the ratio of the entries of CRBv to the entries of CRBp, again ignoring the directional term, D. Assuming, r 2 = 1, and thus p~ = pp = p,

=

1(2mp+1 X -p Tf/-

(17)

If mp > 1, r/v ~- 1/2. So if rap, the so-called array SNR, is very poor (low input SNR and a small number of sensors), the improvement due to the extra measurements of the vector-sensor array is twice as great as in the case of a very good array SNR. In both cases, CRBv will be further reduced relative to CRBp once D is taken into account, and this effect will be significant whenever the entries of Jp are on the order of m/(r 2 + 1) or less. Now we turn to the effect of the vector sensors' inherent directionality. Examination of equation (13) shows that the diagonal entries of Jp depend on the projection of the array's squared aperture onto directions orthogonal to the DOA. Thus, Jv is made up of two very different components (see equations (14), (15)), (r 2 + 1)Je that depends on the array's aperture size and geometry, and D that depends solely on the number of sensors. The aperture component contains only the information resulting from the differential time delay between sensors, while D contains only the inherent directional information from each sensor and has no relationship to the array geometry. Clearly, from equation (14), the directionality will have a significant effect whenever l~ = (r 2 --~ 1)((M/c)2 E j ( I ' y V l ) 2 or l~ ---- (r 2 + 1)(to/c) 2x-" z-,jr' t rjy 21,2 , is on the order of the number of sensors or less. From these expressions we see that l, is a measure of the squared aperture along Vl, which lies in the x, y plane orthogonal to u, while l~ is a measure of the squared aperture along v2, which is orthogonal to u and Vl. The farther apart sensors are placed the lower the CRB, however, the maximum spacing between adjacent sensors is usually limited to half a wavelength in order to avoid problems with aliasing and grating lobes (ambiguities). With fixed inter-sensor spacing, it is obvious upon reflection that as the number of sensors, and hence the aperture, increases, the ratios l~/m and l~/m increase. This leads to the conclusion that the inherent directionality of vector sensors is most effective in arrays that do not have large numbers of elements. Alternatively, if the array is constrained to have a small aperture in at least one dimension, either 12¢/m or l~/m may be small for many values of u; for example, l~ = 0 for all u in the case of a vertical linear array. Thus the directionality will be effective for such arrays no matter how large. Finally, note that l¢ and l¢ decrease with frequency, thus, vector-sensors will be more advantageous at lower frequencies if the array geometry is held fixed. Clearly,

351


the point at which vector sensors cease to have much of a directional effect, is also dependent upon r 2. If r 2 is very large, then the effect is negligible in all but the smallest of arrays. If r 2 is unity or less, the directional effect will be considerable for much larger arrays. Combining these comments with those on the SNR, we see that the improvement afforded by the use of a vector-sensor array is greatest in small array, low SNR situations. However, the improvement may well be of practical value in many other situations situations with medium to large arrays, particularly if a one or two dimensional array is required. 3.2

Diversely-Oriented

Array

We now consider the optimum performance bounds for the diversely-oriented array introduced in Section 2.2. The advantage of the diversely-oriented array is that it requires the same amount of processing and hardware as the regular pressure-sensor array, yet its sensors have an inherent directional capability. We pay for this with a reduction in SNR relative to the pressure-sensor array, and in this section the tradeoff between directionality and SNR is examined in detail. We make the following definitions: Q is the Schur-Hadmard (elementwise) product, and the subscript x, y or z on a sum indicates that the sum is over those m/3 sensors parallel to the x, y or z-axis respectively. Let Uu

ix

rvl

A =

T

=

~%vl

E~ rjTvt

U@U, /X

, r~ 2

=

~yrTv2 E~ r~v2

Ex(rjVl) ZX =

rvlvl

T ~y(rjVl) T

~z(9"jVl)

v

,

Ex(rjv2) 2

,

/X =

rv2v2

V" [~.,T v "~2 / - . y k j 2) ~ f~,T v ~2

2

Z.~z I, j

,

2}

:

For an arbitrary geometry and a single source the CRB is CRBI)(0)_

1 2Np,

(

1 ) 1 + (m/--3)p~ j~-a,

(18)

where the Fisher information matrix is given by 2

J.

=

COS2 ~ ) { U u r v l v l

T COS ~/){"~urVlU2

1

T

-1 7T( u

2

}

- - T ' l l u T ' v ~ ' aTu r v 2 }

T a T )2 u~r,~,~ - T(u~r,~

J -[-

5 D'

(19)

352


where D was defined in equation (15). The diversely-oriented array consists of three equal-sized subarrays (corresponding to the x-sensors, y-sensors and z-sensors). Assuming that each subarray has the same centroid and that this is the origin of the coordinate system, 2 cos 2¢uurvlv I cos¢ r 1 = ~ u~rv,~ + (20) JD COS¢ Uurvlv2 T "~'U ~ V2V2

I

This is similar to the assumption in Section 3.1 that the array centroid lay at the origin; however, now the restriction is more stringent because we require it to be true for three separate subarrays. We are now in a position to again examine the SNR and directional effects separately. Note that D again accounts for the inherent directional sensitivity of each sensor. Without, D the entries of JD are on the order one third of the entries of Jr. To see this suppose r~lv~, rv2~2 and r ~ are each proportional to [1, 1, 1]T, i.e. all three subarrays have the same squared projections onto va and v2. Then the entries of JD are exactly one third the corresponding entries of Jp. For example, if the entries of r ~ are equal, the top left entry becomes JD** =

cos2 ~) E ry'dl + -~ cos2 ¢ 59

=

2

1 Tf,~ 1 + rn 1 m 2 c°s2 ¢ 3 = 3- c°s2 ¢ = ~Jv** + ~ cos ¢.

(21)

Assuming JD = (1/3)Jp + D, CRBD(0) -

3r2 ( 3 r 2 ] (Jp + D) -1 . 2Npp 1 + mpp ]

(22)

Comparison of (22) with (11) shows that apart from the directional term D, the diversely-oriented array has the same optimum performance as an m-sensor pressure-sensor array with a SNR of (pp/3r 2) at each sensor. So we have forsaken some SNR relative to the pressure-sensor array for the benefit of inherent directional sensitivity. With r 2 = 1, thus pv = pp = p, and ignoring D, the ratio of the entries of CRBD to the entries of CRBp is ,o

= 3 (m.

\mp]

(23)

When m p > 1, r/D - 3. Thus, in terms of SNR, we wish to use the diversely oriented array when the array SNR ratio is large. However, it is clear from (22) that, as with the vector-sensor array, the directional terms will be most effective when the entries of Jp are on the order rn or smaller. As before, this implies a small array size or one that has no

353


significant aperture in at least one dimension. So, the ideal conditions for use of a diversely-oriented array are slightly different to those of a vector-sensor array - - the array size should be small or dimensionally constrained, but the SNR at each sensor should be high.

3.3 Uniform Linear Array The uniform linear array (ULA) is one of the most commonly used in practice. For convenience, we shall choose our coordinate system such that the array lies along the z-axis (i.e. the array is vertical). The position vector of the jth sensor is just rj = [O,O, vjz] T and the matrices JF, dv and JD become

&

[0

0

]

Jv = (~ + 1)g~ + D, Jo =

(

)2cos2

2 Ez ri2 ,2 rjz,

(25) u.

+ ~D.

(26)

Note that Jp is singular, corresponding to the fact that a linear pressure-sensor array can only resolve up to a conic angle. In contrast, Jv and J , are always non-singular (except at I¢] = r / 2 , when azimuth is undefined). Thus, both the vector-sensor and diversely-oriented arrays can fully determine the source bearing with a linear arrangement. This is a great advantage when geometry is constrained to be linear or nearly linear - - when using a towed or hull mounted array for exaxnple - - yet full bearing information is desired. The ability to estimate azimuth depends only on the number of sensors and cos 2 ¢, the latter due to the nature of the spherical coordinate system, and is completely independent of the spacing between sensors. The time delays provide no information on the azimuth, hence aperture is irrelevant and the only information comes from the directional sensitivity of the vector sensors and geophones. For the pressure-sensor and vector-sensor arrays, the ability to estimate elevation only ;:lepends on the squared aperture ~ j ~ l rj~ and cos: ¢. For the diversely-oriented array, it is also a function of azimuth through u~,. Some manipulations show that if we can arrange for ~ , rj2z ---- )-'~u r/2z, it becomes a function of elevation alone. With a half-wavelength spacing this requires a nine element array. If we can make all three sums in (26) equal, which requires a 27 element array using half-wavelength spacing, the CRB on elevation has the same dependence on ¢ as for the vector-sensor array. As the source nears endfire, i.e. I¢1 ~ ~/2, the entries of Jv and JD remain non-zero. Thus the CRB on elevation remains finite for these two arrays while it tends to infinity for the hydrophone array. This suggests, at least

354


for (asymptotically) unbiased estimators, that estimates of beams close to endfire will be much more accurate using the vector-sensor or diversely-oriented arrays.

4 BEAMFORMING

AND

CAPON

ESTIMATION

Conventional beamforming and Capon's method [7] of DOA estimation, involve estimating a spatial spectrum and maximizing over all possible directions [12]. The beamforming and Capon spectra are

fB(O)

=

fc(0)

=

aH(O)Ra(O),

(27)

[aH(O)R-la(O)] - l ,

(28)

respectively, where R is the data covariance matrix and a(0) is the steering vector. In practice, the maximum likelihood estimate of R (i.e. the sample covariance) is used. When there is a single source from direction 00, the beamforming spectra for pressure-sensor and vector-sensor arrays, normMized such that the maximum is unity, are

L,p(O) = PPla~(O)a'(O°)]2 + m ppm 2 + m

A,v(O) =

pp(1 + cos~,)2la~(O)a~(Oo)[ ~ + m(r 2 + 1) 4rn2pp + m(r 2 + 1) ,

(29) (30)

respectively, where 7 is the angle between the direction to the source, u0, and the direction of look, u. Clearly, fB,v(0) __ fB,p(0) for all 0 C O, with equality holding if and only if 0 = 0o. So, for vector-sensor arrays, the beamforming spectrum has a sharper peak and uniformly lower sidelobes, leading to better resolution and smaller estimation errors. This is partly due to the SNR improvement that produces the factor (r 2 + 1) in (30), but also due to the directionality which results in the term (1 + cosT) 2. This term further reduces the vector-sensor spectrum relative to the pressure-sensor spectrum for bearings away from the true bearing. The reduction is greatest when cos 7 is small or negative, thus the directionality suppresses the spectrum most far from the source. This suggests that the presence of a second, well separated source affects performance much less for the vector-sensor array than the pressure-sensor array but that the ability to resolve closely spaced sources is not profoundly altered. However, the term ensures that fn,v(0) < fB,v(00) unless 0 = 00, not matter what the array geometry, and thus allows us to always completely resolve the bearing. Another benefit of the directionality is reduction of the effects of spatial aliasing. In particular, if the wavefield is undersampled, grating lobes, which

355


are secondary peaks that do not correspond to actual sources, appear in the spectra. For the pressure-sensor array they are the same height as the mainlobe and cause unresolveable ambiguities. However, it is clear that for the vector-sensor array, their height will be reduced relative to the mainlobe as a result of the directional term (1 + cosT) 2, thereby allowing the DOA to be unambiguously determined. The most well known example of this is the ULA with the usual half-wavelength spacing, which (as well as being unable to determine azimuth) is unable to distinguish between the two endfire directions. This ambiguity may be overcome by shifting one of the sensors so that it is less than half a wavelength from its neighbor, however, the reduction in the height of the grating lobe is only slight and performance close to endfire remains poor. If a vector-sensor array is used instead, not only can we completely determine the bearing, but the grating lobe does not appear at all. As well as allowing us to determine from which end an endfire source is coming, this also greatly improves the ability to accurately resolve sources close to endfire, since there will be no danger of making an estimate coming from the wrong end. This property allows us to use larger apertures by undersampling the wavefield, so further reducing the CRB. As we increase the spacing beyond half a wavelength, however, the grating lobe moves closer to the mainlobe and the (1 + cos 7) 2 term is no longer able to completely suppress it. The result is that the threshold SNR, the point at which large errors can occur, increases, so setting a limit on how much we may lengthen the aperture. The SNR change in the beamforming spectrum for the diversely-oriented array, IB,D(O) = Pvta~(O)Q(O)Q(O°)ap(O°)I2 + k pvk 2 + k

'

(31)

is easy to see, however, the effect of the directionality is somewhat hidden in the matrix product Q(O)Q(Oo) = Ik ® diag{u ® u0}. It is true that, in general, there is a lowering of the spectrum relative to fB,r(O) away from the true source due to the inherent directional sensitivity, and that this effect is stronger farther away from the true source. Indeed, the spectrum is a function of azimuth as well as elevation and azimuth may usually be determined, however, the diversely-oriented array can give one or two surprising ambiguities. For example, although the nine element ULA with half-wavelength spacing can resolve azimuth, it cannot distinguish endfire beams and, if the source is broadside, it cannot tell whether the azimuth is ¢0 or ¢0 + r. Thus, the use of larger apertures which deliberately undersample the wavefield, which was suggested above for the vector-sensor array, is not feasible with the diversely-oriented array if unambiguous results are desired. For brevity, we do not include the very similar expressions for the Capon spectra, however, the above discussion applies verbatim to that case.

356


5 CONCLUSION By examining the CRB and the beamforming and Capon direction estimators we have shown the vector-sensor array to have improved performance due to its sensors' inherent directionality, as well as the greater number of measurements. The improvement is most pronounced for small arrays and low SNR, or when the array has no significant spatial extent in at least one dimension. As well as reducing the CRB, so permitting smaller arrays, the effect is to allow greater observability with simple structures, for example the ULA can be used to find azimuth as well as elevation, and to reduce the effects of spatial aliasing. In particular, the height of grating lobes relative to the mainlobe is reduced, thus allowing the use of spatially undersampled arrays with a lower CRB. A diversely-oriented array has been proposed which offers some of the directional advantages of vector-sensor arrays, but requires the same processing power and hardware as the standard hydrophone array. Again, there is an improvement due to the sensors directional sensitivity but we lose SNR relative to the pressure-sensor array. This array works best in a small array, high SNR setting, especially when the array has no significant spatial extent in at least one dimension. Although the greater directional capabilities again allow, for example, the ULA to resolve both azimuth and elevation, certain ambiguities may exists depending on the DOA. Finally, it can provide uncorrelated measurements in certain non-isotropic noise fields in which the performance of the vector-sensor array is reduced due to intra-sensor correlation. We are continuing to investigate the use of vector sensor arrays for source localization particularly in the important hull-mounted scenario [3].

ACKNOWLEDGEMENTS This work was supported by the Office of Naval Research under Grant numbers N00014-95-1-1101 and N00014-91-J-1298.

REFERENCES 1. A. Nehorai and E. Paldi, "Acoustic vector-sensor array processing," IEEE Trans. on Sig. Proc., vol. 42, no. 9, pp. 2481-2491, Sept. 1994. A short version appeared in Proc. 26th Asilomar Conf. on Signals, Sysl. Comput., Pacific Grove, CA, Oct. 1992, pp. 192-198. 2. M. Hawkes and A. Nehorai, "Acoustic vector-sensor beamforming and Capon direction estimation," Proc. Intl Conf. on Acous., Speech and Sig. Proc. (ICASSP95), Detroit, MI, May 1995, pp. 1673-1676. 3. M. Hawkes and A Nehorai, "Hull-mounted acoustic vector-sensor processing," to be presented at 29th Asilomar Conf. on Signals, Syst. Comput., Pacific Grove, CA, Oct. 1995.

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4. G.L. D'Spain and W. S. Hodgkiss, "The simultaneous measurement of infrasonic acoustic particle velocity and acoustic pressure in the ocean by freely drifting swallow floats," IEEE Jour. of Oceanic Eng., vol. 16, no. 2, pp. 195-207, Apr. 1991. 5. J.C. Nickles et al, "A vertical array of directional acoustic sensors," Proc. Mast. Oceans thru Tech. (Oceans 92), Newport,RI, Oct. 1992, pp. 340-345. 6. G.L. D'Spain et al, "Initial analysis of the data from the vertical DIFAR array," Proc. Mast. Oceans thru Tech. (Oceans 92), Newport,RI, Oct. 1992, pp. 346-351. 7. J. Capon, "High resolution frequency-wavenumber spectrum analysis," Proc. IEEE, vol. 57, no. 8, pp. 1408-1418, Aug. 1969. 8. A.D. Pierce, Acoustics--An Introduction to its Physical Principles and Applications. New York: McGraw-Hill, 1981. 9. D. Starer, "Algorithms for polynomial-based signal processing," Ph.D. Dissertation, Yale University, New Haven, CT, May 1990. 10. N.R. Goodman, "Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)," Ann. Math. Star., vol. 34, pp. 152-177, 1963. 11. H. Cox and R.M. Zeskind, "Adaptive cardiod processing," Proc. 26th Asilomar Conf. on Signals, Syst. Cornput., Pacific Grove, CA, Oct. 1992, pp. 1058-1061. 12. D.H. Johnson, "The application of spectral estimation methods to bearing estimation problems," Proc. IEEE, vol. 70, no. 9, Sept. 1982.

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Bearing estimation with acoustic vectorsensor arrays

Bearing estimation with acoustic vectorsensor arrays

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