3946

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 7, JULY 2010

The Acoustic Vector-Sensor’s Near-Field Array-Manifold Yue Ivan Wu, Kainam Thomas Wong, and Siu-Kit Lau

Abstract—The acoustic vector-sensor is a practical and versatile soundmeasurement system, for applications in-room, open-air, or underwater. Its far-field measurement model has been introduced into signal processing over a decade ago; and many direction-finding algorithms have since been developed for acoustic vector-sensors, but only for far-field sources. Missing in the literature is a near-field measurement model for the acoustic vectorsensor. This correspondence fills this literature gap. Index Terms—Acoustic arrays, acoustic position measurement, acoustic signal processing, array signal processing, direction of arrival estimation, underwater acoustic arrays.

I. INTRODUCTION An acoustic vector-sensor (a.k.a. vector-hydrophone) consists of three identical, but orthogonally oriented, acoustic velocity-sensors, plus an acoustic pressure-sensor—all spatially co-located in a point-like geometry. Each acoustic velocity-sensor measures one Cartesian component of the incident acoustic particle-field vector. The entire acoustic vector-sensor thus distinctly measures all three Cartesian components of the particle-velocity vector plus the pressure scalar. This contrasts with a customary microphone or hydrophone, measuring only the acoustic pressure. More precisely: for a point-source incident with unit-power from the far field, an acoustic vector-sensor (located at the Cartesian coordinates’ origin) has this array manifold, [3], [5],

afar

=

def

u( s ; s ) v( s ; s ) w( s )

1

sin s cos s = sincoss sins s 1

def

(1)

where 0  s   symbolizes the elevation-angle measured from the vertical z -axis, 0  s < 2 denotes the azimuth-angle measured from the positive x-axis, u( s ; s ) refers to the direction-cosine along the x-axis, v ( s ; s ) refers to the direction-cosine along the y -axis, and w(s ) refers to the direction-cosine along the z -axis. Specifically, the first, second, and third components in (1) correspond to the acoustic velocity-sensors aligned along the x-axis, the y -axis, and the z -axis, respectively. These three Cartesian components of particle-velocity field-vector has a Euclidean norm (i.e., [u( s ; s )]2 + Manuscript received July 28, 2009; accepted March 14, 2010. Date of publication April 5, 2010; date of current version June 16, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Shahram Shahbazpanahi. This work was supported by the Internal Competitive Research Grant G-YG67 from the Hong Kong Polytechnic University. Y. I. Wu is with the School of Communication and Information Engineering, University of Electronic Science & Technology of China, Chengdu, Sichuan, China (e-mail: [email protected]). K. T. Wong is with the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong (e-mail: [email protected]). S.-K. Lau is with the Charles W. Durham School of Architectural Engineering and Construction, University of Nebraska-Lincoln, Omaha, NE 68182 USA (e-mail: [email protected]). Color versions of one or more of the figures in this correspondence are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2010.2047393

[v(

s .

s ; s

)] + [w( s )] ) equal to the unity in pressure, for all 2

2

s

and

The acoustic vector-sensor concept is versatile for direction-finding, due to these properties: i) A single acoustic vector-sensor intrinsically possesses a two-dimensional azimuth-elevation directivity, because all three Cartesian components of the acoustic velocity-vector-field are simultaneously measured. ii) Multiple incident sources’ azimuth-angles and the elevation-angles may be estimated and automatically matched with only one acoustic vector-sensor. Please refer to [8] for an extended literature survey on acoustic vector-sensor based direction-finding algorithms, target-tracking algorithms, beam-pattern analysis, hardware implementations, sea trials, atmospheric trials, in-room trials, and suggested applications. Though the above far-field measurement model in (1) was first introduced to the signal-processing literature by [3] and [5] over a decade ago, the corresponding near-field measurement-model has not been investigated. This overlooked issue is herein investigated. As Sections II–IV will show, the far-field measurement model’s independence of the signal frequency, the source/sensor distance, and the propagation-medium in (1) is invalid for the near-field case. II. MATHEMATICAL DERIVATION OF THE MAIN RESULT A. Review of the Pressure-Field Wave-Equation & the Particle-Velocity Wave-Equation This Section II-A reviews the basic mathematics inter-relating the acoustic pressure-field wave-equation to the acoustic particle-velocity field wave-equation.1 This Section II-A will adopt the spherical coordinates customary in the acoustics literature, locating the emitter at the origin r0s = (0; 0 ; 0 ) of the coordinates (R0 ; 0 ; 0 ). These new coordinates differ from the (R; ; ) coordinates of Sections I and II-B, customary to the direction-finding literature, with the sensor at the origin. The aforementioned source emits into a quiescent isotropic homogeneous fluid, such as air or water. The resulting acoustic particle-velocity vector-field v(r0 ; t)2 is related by Euler’s equation3 to the corresponding pressure scalar field p(r0 ; t) as follows:

@ v(r0 ; t) @t 1 @p(r0 ; t) c2 0 @t 0

= 0 rp(r ; t)

(2)

= 0 r 1 v(r ; t)

(3)

0

0

where 0 refers to the ambient fluid density, c symbolizes to the soundwave propagation-speed, r represents the gradient operator, and r1 signifies the divergence operator. To obtain the pressure-field wave-equation (namely[1, d’Alembert’s eq. (1.30)]): Take the partial derivative of (3) with respect to t, and combine with (2), giving

1 @ p(r ; t) = r p(r ; t): 2

c2

0

@t2

2

0

(4)

1The authors would like to thank an anonymous reviewer for suggesting some of the materials in this Section II-A. 2I.e., a vector function at any spatial location r ;  ; and any time t. R 

[cos sin

3Please

sin sin

cos ]

=

see [1, equations (1.58) to (1.59)], and [2, equation (1–3.7)].

1053-587X/$26.00 © 2010 IEEE Authorized licensed use limited to: University of Electronic Science and Tech of China. Downloaded on July 03,2010 at 09:08:08 UTC from IEEE Xplore. Restrictions apply.

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 7, JULY 2010

3947

To obtain the particle-velocity-field wave-equation ([1, eq. (1.33)]): Take the gradient of (2), and combine with (3), giving

1 @ 2 v(r0 ; t)

c2

@t2

=

r r 1 v(r ; t) 0

:

(5)

To avoid distraction from the present focus on the array-manifold, a simple signal-model will be used: Let the emitted signal be a pure tone of angular frequency ! and complex-amplitude A. That is, p(r0 ; t) = p(r0 )ej!t and v(r0 ; t) = v(r0 )ej!t . Then, the wave-equations become the Helmholz equations (i.e., [1, eq. (1.101), (1.102)]) ! 2 0 p(r ; t) = c ! 2 v(r0 ; t) = c

0 0

r2 p(r ; t) r r 1 v(r ; t) 0

0

(6) :

(7)

One solution to the above Helmholz equations is the spherical sinusoidal wave: p(r ; t) = 0

A exp R0

j

!t

0 2 R

0

(8)

B. To Derive the Acoustic Vector-Sensor Near-Field Array-Manifold Fig. 1 shows the sensor-centric spherical coordinates (R; ; ) customarily used in the direction-finding literature, with the acoustic vector-sensor at the origin r0 = [0; ; ]T . Let Rs denote the distance between the acoustic vector-sensor and an emitting source located at rs = (Rs ; s ; s ).4 Hence, r0s = (0; 0 ; 0 ) (in the emitter-centric coordinates of (R0 ; 0 ; 0 ) from the acoustics literature in Section II-A) would correspond to rs in the sensor-centric coordinates of (R; ; ) in Section II-B here from the direction-finding literature. The gradient of the pressure-field at the acoustic vector-sensor can be expressed as

=

@p(r; t) @R r=r

rs +

@p(r; t) s  @ r=r 1 @p(r; t) s + R @ r=r

1

R sin

j 2R + 1 exp @p(r; t)  = A exp(j!t) @R Rs2 r=r @p(r; t) =0 @ r=r

0j 2

R 

(10)

4Or

r

dinates

s

= R [cos  sin ; sin  sin ; cos ]

00 c

=

00 c

1

j 2R +1 

A exp Rs

j 2R 

1

 0 j 2R

s

j

!t

0 2R 

s

0j 2

R 

rs : (13)

in the Cartesian coor-

r

p (r0 ; t)  rs :

(14)

rs , (14) beUsing the definition of the source’s direction-vector  comes 1+

s

2



2R

00 c

s

 2 exp 0j arctan 2R

s

:

(15)

From (15), the acoustic vector-sensor near-field array-manifold equals5,6:

anear =

cos s sin sin s sin cos s

0

s s

:

c

 exp j arctan 2R

1+

Substituting these three identities into (9), the gradient of the pressurefield at r0 is obtained as

R2

1

=

(9)

(12)

j 2R + 1 exp 

r; t)jr=r 0 rp(j! 0

(16)

(11)

@p(r; t) = 0: @ r=r

= A exp(j!t)

v(r0 ; t) =

j1j

cos s sin v(r0 ; t) = p(r0 ; t) sin s sin cos s

 s and  s , respectively, denote the unit-vectors along the where  azimuth-angle coordinate and the elevation-angle coordinate. Especially, in the spherical coordinate system r0 = [0; s ; s ]T . Using the MATLAB Symbolic Math Toolbox, the following identities are found:

rp(r; t)jr=r

Likewise, the particle-velocity vector v(r0 ; t) may be represented as j!t at the spatial location r0 and time t, where refers s to the Euclidean norm of the vector inside the pair of vertical lines. Hence, @ v(r0 ; t)=@t = j! v(r0 ; t). Combining (2) and (13)

jv(r0 ; t)jr e

where  signifies the signal wavelength.

rp(r; t)jr=r

Fig. 1. The spatial geometry relating the emitter and the acoustic vector-sensor. Shown here is the sensor-centric spherical coordinates, customarily used in the direction-finding literature.

A complex-phase difference thus exists between the velocity-sensor triad measurements and the pressure-sensor measurement in the nearfield measurement-model in (16). This phase-difference depends on the wavelength-normalized source/sensor distance Rs = and the propagation-medium’s 0 c, but does not depend on the azimuth-elevation arrival-angles. 5Concerning the “—” sign before the above  c, it arises from how the direction is defined above for the unit-vector r. This “—” sign may not appear in the acoustics literature. 6In the far-field case, where R , it holds that =j R . Thus, the pressure scalar field would relate to the particle-velocity vector-field as in (2), which presumes a planar wavefront upon the acoustic vector-sensor.





2

!0

Authorized licensed use limited to: University of Electronic Science and Tech of China. Downloaded on July 03,2010 at 09:08:08 UTC from IEEE Xplore. Restrictions apply.

3948

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 7, JULY 2010

As the wavelength-normalized distance Rs = array-manifold converges to

 

cos s sin s sin s sin s cos s

0 c

! 1, the near-field

 = s ; s ; Rs ; !; ; v2; p2

. The resulting 7 tion Matrix, J, would have the (i; j )th entry7

Ji;j = 2<

:

(17)

0

0

The above is consistent with the far-field array-manifold [3, eq. (2.5)] which normalizes the pressure-sensor gain from the above 0 c to become unity.


2

(18)

where Ts refers to the time-sampling period, and n(t) denotes a 4 1 vector of additive zero-mean spatiotemporally uncorrelated Gaussian noise, with an unknown deterministic covariance-matrix of 0 0 = diag v2 ; v2 ; v2 ; p2 . That is, v2 represents the noise-variance at each velocity-sensor, and p2 symbolizes the noise-variance at the pressure-sensor. The velocity-sensor and the pressure-sensor likely have different noise-variances, because of their different hardware implementations and the distinct physical wave-pressure measured [4]–[6]. With M number of time-samples, the collected 4 M data-set equals

2

2

T

Ts ))T ; . . . ; (z(MTs))T T T = s anear + (n(Ts )) ; . . . ; (n(MTs )) = =

N

2

01 +Tr 0

@0 01 @0 @ [ ]i 0 @ [ ]j

1

f1g

J

= 1;1 =

2

(21) (22) (23) 2

(24) (25) (26) (27)

(28) (29)

(30)

s

T



(19)

where s = ej ejT ! ; ej 2T ! ; . . . ; ejM T ! , symbolizes the Kronecker product,  represents a 4M 1 noise vector with a spatio-temporal covariance matrix of 0 = IM 0 0 , and IM denotes an M M identity matrix. Therefore,  (; 0). The near-field source-localization problem is to estimate the azimuth-elevation arrival-angles s and s plus the radial distance Rs , based on the 4M 1 collected data  . T

@ @ [ ]j

01 0

M 2 v2 sin s ; J ; = J2;2 = 2M2 ; v !0 c JR ;R = J3;3 = M p2 Rs k1 J!;! = J4;4 = 2M2 k12 + !Rc k1 0 k2 s p 2M (M + 1) 0 c + p2 fs 1 + c 2 k2 !R M (M + 1)(2M + 1) k3 + 3fs J; = J5;5 = 2Mk3 J ; = J6;6 = 3M4 ; v M J ; = J7;7 = 4 ; p JR ;! = J3;4 = J4;3 = M p2 2 2 0k1 Rcs 0 ! (M2fs+ 1) 2 JR ; = J3;5 = J5;3 = 0 M p2 ! 0 k1 ; J!; = J4;5 = J5;4 = M p2 !Rs 0 k1 M (M + 1) k3 + f

To further characterize the acoustic vector-sensor’s array-manifold, this section will derive the Cramér-Rao bound for near-field (three-dimensional) source-localization by an acoustic vector-sensor. To avoid unnecessary distraction from focusing on the near-field array-manifold, a very simple signal statistical model will be used here: The emitted signal s(t) = ej (!t+) is a pure tone at angular frequency ! as before, now with an initial phase of . Both ! and  are deterministic unknown constants. At the mth time-instant t = mTs , a 4 1 data-vector z(mTs ) is collected by the four-component acoustic vector-sensor

(z(

H

(20)

J ;

A. Defining the Statistical Data Model

=

@ @ [ ]i

2 7 Fisher Informa-

where signifies the real-value part of the entity inside the curly denotes the trace operation, and [ ]i symbolizes the ith brackets, Tr element of the vector inside the square brackets. Straightforward calculus manipulations can express the Fisher information matrix entries in terms of the measurement-model parameters and statistical data-model parameters, as follows:

III. CRAMÉR-RAO BOUND ANALYSIS OF THE NEAR-FIELD MEASUREMENT MODEL

z(mTs ) = anear s(mTs ) + n(mTs )

T

k1 = (20 Rs (c=!Rs )3 = 1 + (c=!Rs )2 2 ), k2 2 2 = (1=v2 (0 Rs (c=!Rs ) =1 + (c=!Rs ) ), and k3 2 2 2 2 0 c = 1 + (c=!Rs ) p ). where

All other entries are zero in the Fisher information matrix. As a consequence

2

2

J1;1 0

J=

B. Deriving the Cramér-Rao Bound for Near-Field Source-Localization by an Acoustic Vector-Sensor

0 0 0 0 0

2

In the statistical data model in Section III-A, there exist seven deterministic unknown entities, which are here collected into a 7 1 vector,

= +

0

J2;2 0 0 0 0

0

0 0

0 0

0 0

J3;3 J3;4 J3;5 J3;4 J4;4 J4;5 J3;5 J4;5 J5;5 0

0

0

0

0

0

0 0

0 0

0 0 0

0 0 0 0

J6;6 0

J7;7

is block-diagonal. 7Please

see [7, eq. (8.34)].

Authorized licensed use limited to: University of Electronic Science and Tech of China. Downloaded on July 03,2010 at 09:08:08 UTC from IEEE Xplore. Restrictions apply.

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 7, JULY 2010

3949

TABLE I DEGENERATE FORMS OF CRB (R ) FOR SPECIAL CASES IN THE DATA MODEL:

3 Equations (31) and (32) here for these near-field CRB (s ) and CRB ( s ) are identical to their far-field counterparts in [6, Eqs. (63) and (64)]. For CRB (Rs ) of (35), Fig. 2(a) plots (2=)2 2M=p2 [CRB (Rs )] versus v2 =p2 , for ranges of Rs = and for a value of 0 c that are typical for air-acoustics applications. Fig. 2(b) does the same for underwater acoustics applications. Table I shows how CRB (Rs ) may be approximated, under various degenerate cases in the data model. The following qualitative trends may be observed. f4g CRB (Rs ) is independent of the azimuth-elevation arrival direction. 2 2 f5g In both Fig. 2(a) and (b), (2=) 2M=p [CRB (Rs )] decreases almost linearly with decreasing Rs =, when they are both in logarithmic scale, until the near-field condition of Rs = < 1=2 applies. This graphic observation concurs with the approximation in Table I’s second row, which shows a rough proportionality to (Rs =) raised to the fourth to sixth power (depending on the magnitude of v =p 0 c). Then in the near field, (2=)2 CRB (Rs ) becomes largely constant with respect to Rs =. This graphic observation concurs with the approximation in Table I’s second row, which is independent of Rs =. f6g In the air-acoustic Fig. 2(a) where Rs = < 1=2 (for near-field situations), (2=)2 2M=p2 [CRB (Rs )] is hardly dependent on the ratio v =p between the two noise-variances. This graphic observation concurs with the approximation in Table I’s second row, which is independent of v =p . However, (2=)2 2M=p2 [CRB (Rs )] would decrease with decreasing on v =p , for Rs = > 1=2 . Moreover, this dependence becomes more pronounced as Rs = increases, when source lies more into the far field. This graphic observation also concurs with the approximation in Table I’s first row. There, (2=)2 2M=p2 [CRB (Rs )] would be roughly proportional to v =p for large Rs =. 2 2 f7g In the underwater acoustic Fig. 2(b), (2=) 2M=p [CRB (Rs )] is not notably dependent on v =p , over the entirety of the

Hence, the Cramér-Rao bounds of the emitter’s azimuth-elevation arrival-angles are

f g

2

 CRB(s ) =J1011 = 2M sin 2 v

;

(31) s

2

CRB( ) =J2021 = 2M : s

v

;

(32)

As for the radial distance Cramér-Rao bound, see (33) at the bottom of the page, where [1]i;j represents the (i; j )th entry of the matrix inside the square brackets. Substitution of (21) to (30) in (33) gives 2 2 2 3 2 CRB(R ) = 2M ! !R2 c4+2c 0 !2 R2 20 c2 2 + !2 R2 + c2 2 + O(M 01 ) : !2 R2 20 c4 2 + (!2 R2 + c2 )2 2 + O(M 01 ) s

p

s

2

s

s

v

s

v

p

s

(34)

p

By overlooking the O(M 01 ) terms, which contain multiples of M 01 ; M 02 ; . . . ; the following approximation may be obtained8: see (35) at the bottom of the page. IV. QUALITATIVE OBSERVATIONS The following qualitative trends may be observed of CRB (s ) from (31) and CRB ( s ) from (32). f1g Both CRB (s ) and CRB ( s ) are independent of the signal frequency ! , the source-sensor distance Rs , the propagation-medium’s 0 c, the source’s azimuth-angle s , and the pressure-sensor’s noise-variance p (as the pressure-sensor offers no arrival-angle information). f2g CRB ( s ) is unaffected also by the source’s elevation-angle s. 8For an M as little as 100, simulations show that the approximate CRB (R ) in (35) can be graphically indistinguishable from the exact expression in (34).

CRB(R ) = J01 s

2 

2

CRB(R ) s



2

J4 4 J5 5 J4 5 3 3 = J3 3 J4 4 J55 + 2J3 4 J4 5 J3 5 J3 3 J42 5 J5 5 J32 4 J4 4 J32 5

p2 2M

;

0

;

0

;

2R

2



;

;

;

1

 c

2

;

;

2R

2



+1

2R 

2

;

3



 

+



1

 c

2

;

0

;

+ 2R 

;

;

2

1

 c

2

0

2R 

+1

2

;

(33)

;

2

+1 (35)

Authorized licensed use limited to: University of Electronic Science and Tech of China. Downloaded on July 03,2010 at 09:08:08 UTC from IEEE Xplore. Restrictions apply.

3950

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 7, JULY 2010

Fig. 2. (a) (2=) 2M= CRB (R ) is plotted at  c = 413:3 kg=m s, a value typical of air-acoustics applications. (b) (2=) 2M= CRB (R ) is plotted at  c = 1:48 10 kg=m s, a value typical of underwater acoustics applications.

2

relevant range of Rs =. This graphic observation also concurs with the approximation in Table I’s first row, which applies for Rs = 1. There, the far right fraction would approximate to unity at the underwater acoustics’ very large 0 c, thereby allowing an independence from v =p . For Rs = < 1=2 , applicable is Table I’s second row, which is also independent of v =p . Fig. 3(a) and (b) reveals the dependence of CRB (Rs ) on the acoustic medium’s 0 c. These two figures plot (2=)2 CRB (Rs ) under the degenerate condition of v2 = p2 =  2 at 20 dB SNR at each component-sensor, over ranges of Rs = and 0 c relevant, respectively, to air-acoustics and underwater acoustics applications.9 The following qualitative trends may be observed. 8 For the air-acoustic ranges of Rs = and 0 c, Fig. 3(a) shows that (2=)2 CRB (Rs ) does not vary much with 0 c, until the



fg

2

2

9For air-acoustic applications, !=2 [20; 2 10 ] Hz, c = 343:3 m=s, the air density  = 1:204 kg=m at 20 C, and hence  c 413:3 kg=m s. In contrast, underwater acoustic applications have these typical values: !=2 [10 ; 10 ] Hz, c = 1481 m=s, the water density  = 998:2 kg=m at 20 C, and, hence,  c 1:48 10 kg=m s.



2



2

Fig. 3. (a) (2=) CRB (R ) is plotted at 20 dB SNR ( =  = 0:1), over those ranges of  c and R = that are most relevant to air-acoustic applications. (b) (2=) CRB (R ) is plotted at 20 dB SNR ( =  = 0:1), over those ranges of  c and R = that are most relevant to underwater acoustic applications.

near-field condition of Rs = < 1=2 applies. There in the near field, (2=)2 CRB (Rs ) decreases with increasing 0 c. 9 For the underwater acoustic Fig. 3(b), like the air-acoustic Fig. 3(a), shows that (2=)2 CRB (Rs ) here decreases very slightly with increasing 0 c whether inside of outside the near field.

fg

V. CONCLUSION This correspondence is first to derive the near-field array-manifold for an acoustic vector-sensor. Unlike the far-field array-manifold, a complex-phase is found to exist between the pressure measurement and the particle-velocity vector measurement. This phase-difference depends on the source/sensor wavelength-normalized distance Rs = and the propagation-medium’s 0 c, but not on the azimuth-elevation arriving angles. For three-dimensional source-localization, the azimuthelevation arrival-angle estimation accuracy could remain the same for the near-field case as for the far-field case. However, the distance-estimation could have a wavelength-normalized accuracy that decreases

Authorized licensed use limited to: University of Electronic Science and Tech of China. Downloaded on July 03,2010 at 09:08:08 UTC from IEEE Xplore. Restrictions apply.

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 7, JULY 2010

almost linearly with decreasing Rs = outside the near field, but becomes largely flat inside the near field. Furthermore, this distance-estimation could also be independent of the source’s azimuth-elevation arrival direction.

REFERENCES [1] M. Rossi, Acoustics and Electroacoustics. Norwood, MA: Artech House, 1988. [2] A. D. Pierce, Acoustics—An Introduction to its Physical Principles and Applications. New York: McGraw-Hill, 1989. [3] A. Nehorai and E. Paldi, “Acoustic vector-sensor array processing,” IEEE Trans. Signal Process., vol. 42, pp. 2481–2491, Sep. 1994.

3951

[4] M. Hawkes and A. Nehorai, “Acoustic vector-sensor beamforming and capon direction estimation,” IEEE Trans. Signal Process., vol. 46, pp. 2291–2304, Sep. 1998. [5] M. Hawkes and A. Nehorai, “Effects of sensor placement on acoustic vector-sensor array performance,” IEEE J. Ocean. Eng., vol. 24, pp. 33–40, Jan. 1999. [6] P. Tichavský, K. T. Wong, and M. D. Zoltowski, “Near-field/far-field azimuth and elevation angle estimation using a single vector hydrophone,” IEEE Trans. Signal Process., vol. 49, pp. 2498–2510, Nov. 2001. [7] H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part IV: Optimum Array Processing. New York: Wiley, 2002. [8] P. K. Tam and K. T. Wong, “Cramér-Rao bounds for direction finding by an acoustic vector-sensor under non-ideal gain-phase responses, non-collocation, or non-orthogonal orientation,” IEEE Sens. J., vol. 9, pp. 969–982, Aug. 2009.

Authorized licensed use limited to: University of Electronic Science and Tech of China. Downloaded on July 03,2010 at 09:08:08 UTC from IEEE Xplore. Restrictions apply.