Estimation and Compensation of RotationPerturbation in Linear 2D Acoustic Vector Sensor

ArrayHock Siong LIM

Thales Solutions Asia Pte Ltd28 Changi North RiseSingapore (498755)

Email: hocksiong.lim@asia.thalesgroup.com

Ketan RAMESHCHANDRAThales Solutions Asia Pte Ltd

28 Changi North RiseSingapore (498755)

Wei Siong TOHDSO National Laboratories

20 Science Park DriveSingapore (118230)

Abstract—Most beamforming algorithms for vector sensorarray are derived based on the assumption that the directivitypattern of all sensors are perfectly aligned. However this is nottrue in practice and calibration for rotational perturbations arerequired to avoid drastic degradation of beamforming perfor-mance. This paper presents two algorithms that are capable ofestimating and compensating rotational perturbation angles. Thefirst algorithm is based on steering vector error and the secondalgorithm is based on subspace criteria. Computer simulationsshow that performance of both algorithms approach theoreticalbound, are consistent and able to restore distorted spatial spectralin the presence of single and multiple sources.

I. INTRODUCTION

A. Motivation

Underwater acoustic vector sensor (AVS) and array (VSA)have attracted a lot of interest since its introduction to theresearch community in the 1990s [2][3]. Its inherent directivitypattern has many advantages over omni-directional sensors,such as higher array directivity[8] and resolving left/rightambiguity in linear array[12]. Subsequently, many adaptivebeamforming algorithms on both single vector sensor andvector sensor array [4][5][6][10][16][15] were reported. Thesebeamforming algorithms are derived based on the assumptionthat the directiviy pattern of all vector sensors in the arrayare perfectly aligned among themselves and to the physicalcoordinate system of the array. However, this assumptionis generally not true in practice, especially in underwaterarray and the performance of the beamforming algorithmswill degrade tremendously [9][11][12][13]. The most straightforward solution to this problem is to place rotational sensorsbut is not cost effective especially for large arrays. A data-driven method would be attractive but till date there have beenvery few reports on this topic and it is the motivation of thispaper to contribute in this area.

B. Problem Definition

In this paper, a uniform linear array of 2D vector sensors isconsidered. A 2D vector sensor contains a hydrophone whichsenses sound pressure and 2 particle velocity sensors placed incross-axis orientation. In this configuration, each vector sensor

can be considered as a subarray. Rotational perturbation occurswhen directivity patterns of vector sensors are not perfectlyaligned and the objective of this work is to derive algorithm toestimate and compensate for the perturbation angles, thereforeit is assumed that gain-phase and positional calibrations havebeen performed. The noise assumed in this work is whitegaussian noise and spatially uncorrelated.

C. Organization

This paper is organised as follows, section II will presentthe nominal narrowband signal model of a 2D vector sensorarray and is followed by the perturbation model in sectionIII. Section IV will present the formulation and derivation of2 proposed signal processing algorithms for estimation andcompensation of rotational perturbation. The performance ofthe algorithms is then presented in section V. The work willthen be concluded in section VI.

II. NOMINAL MODEL

The reference co-ordinate system and array geometry isshown in Fig 1. The polar angle is denoted by θ ∈ [0 : π)and azimuth angle is denoted by φ ∈ [0 : 2π).

Under monochromatic plane wave assumption, measure-ment model of particle velocity sensors, v subject to acousticpressure p is

v =

cos(θ) sin(φ)sin(θ) sin(φ)

cos(φ)

pρc. (1)

where ρ denotes density of medium and c denotes the prop-agation speed of acoustic wave. Denoting acoustic impedanceas

Z = ρc (2)

and considering a monochromatic plane wave impinging on avector sensor at direction (θ,φ) , the cosine response of a 2Dvector sensor to this field from (1) is given by

978-1-61284-4577-0088-0/11/$26.00 ©2011 IEEE

Fig. 1. Reference Geometry

u2D =

(sin θ cosφsin θ sinφ

)(3)

The overall response of a 2D vector sensor, including thepressure sensor, is then given by

v2D =

(1

u2D

Z

)(4)

Since Z is a constant, the assumption of normalization can bemade to remove the Z from the rest of equations that follows.Assuming a monochromatic plane wave with propagatingvector k0 impinging on a uniform linear array with N vectorsensors located at r0, r1, · · · , rN−1 in space, the signal sourceposition vector h and steering vector a2D towards the sourceare given by, respectively,

h =

1

e−jkT

0 .r1

e−jkT

0 .r2

...e−jk

T

0 .rN−1

(5)

a2D = h⊗(

1u2D

)(6)

and the symbol ⊗ denotes the kronecker product. The narrowband signal received on the array is then given by x(t)=a2Dp(t).

III. PERTURBATION MODEL

A. Rotational Perturbation

The rotational perturbation can be modeled by Euler’srotation theorem. Consider a perturbation angle denoted by γ,the orientation of this perturbation is an anti-clockwise rotationof the nominal axis (x, y) to the true position at (x’, y’) as

illustrated in Fig 2. A perturbation on a single nth elementaround the array axis z on x and y by an angle γn can bewritten as

v2D,n =

1 0 00 cos(γn) sin(γn)0 − sin(γn) cos(γn)

· v2D,n

= Q2D,n · v2D,n (7)

where the transformation matrix of each nth sensor is denotedas Q2D,n. Similarly, denoting the array rotation matrix asΦ2D the perturbed array steering vector is

a2D =

Q2D,0 0 0 0

0 Q2D,1 0 0

0 0. . . 0

0 0 0 Q2D,N−1

a2D

= Φ2D · a2D (8)

When there is no perturbation, Φ2D reduces to a 3N ×3N identity matrix. The nice property of Φ2D matrix isthat ΦT

2DΦ2D = I3Nx3N, hence compensation for rotationalperturbation can be achieved by multiplying the nominalsteering vector by Φ2D. Hence the remaining challenge isto find an estimate Φ2D to calibrate for the mismatches.

Fig. 2. Orientation of Perturbation

B. Gain-Phase-Roll Perturbation

Generally, in addition to rotation perturbations, vector sen-sor array also suffers from gain-phase perturbations due tonon-uniform characteristics of electronics components andsensors. The gain-phase perturbation model of a vector sensorarray needs to consider two folds of impact, on single vectorsensor and on the array. The single sensor model shall beconsidered first. Without loss of generality, using the pressurechannel as reference, the gain-phase-roll perturbation modelof a single nth vector sensor can be described by

vgpr,n =

1 0 00 gx,n 00 0 gy,n

· v2D,n (9)

where gx,n and gy,n denotes gain-phase perturbations of x

and y channels of nth sensor respectively. The second fold ofconsideration is on the array. It is well known that the gain-phase model of a conventional omni-directional array is givenby

h = diag([

1 gh,1 gh,2 · · · gh,N−1])· h (10)

where gh,n denotes the gain-phase perturbation of nth pressurechannel from the first pressure channel. Denoting

Gxy = diag([

1 gx,0 gx,0 · · · 1 gx,N−1 gy,N−1])

(11)

and

Gh = diag([

1 gh,1 gh,2 · · · gh,N−1]T ⊗ 13x1

)(12)

where 13×1 denotes a vector of 3 number of 1s as there are 3channels in a vector sensor. The gain-phase-roll perturbationmodel from (8), (11) and (12) can then be written as

a2D = Φ2D ·Gh ·Gxy · a2D (13)

(13) shows that there exist a gain-phase ambiguity betweenthe various gain, phase and roll perturbation, to estimatethe rotation pertubation matrix would require the gain-phasecalibrated sensors signal.

IV. ALGORITHMS

A. Formulation

It can be observed from (7) that the entries in Q2D,n arenon-linear cosine functions of perturbation angles and hencedirect estimation of γn in this form is not possible. To linearise,one can re-write expressions for cosine and sine functionsrespective as

cos(γn) =∞∑n=0

(−1)n

(2n)!γ2nn

= 1 + fi (γn) (14)

and

sin(γn) =∞∑n=0

(−1)n

(2n+ 1)!γ2n+1n

= fq (γn) (15)

where fi (γn) denotes the higher order terms γn in cosinefunction and fq (γn) denotes the sine function of γn. Hencean alternative expression can be written for Q2D,n as

Q2D,n =

1 0 00 1 + fi (γn) fq (γn)0 −fq (γn) 1 + fi (γn)

= I3x3 + ∆Q2D,n (16)

where I3x3 denotes the 3×3 identity matrix and ∆Q2D,n

can be considered as an error term. Using (16) in (7) and (8),one can obtain

v2D,n = v2D,n + ∆Q2D,n · v2D,n (17)

Φ2D = I3Nx3N + ∆Φ2D (18)

and

a2D = a2D + ∆Φ2D · a2D (19)

where

∆Φ2D =

∆Q2D,0 0 0 0

0 ∆Q2D,1 0 0

0 0. . . 0

0 0 0 ∆Q2D,N−1

(20)

It is obvious that estimation of ∆Q2D,n will lead to anestimation of Φ2D.From (17), the steering vector error of asingle nth vector sensor is

ε2D,n =

0 0 00 fi (γn) fq (γn)0 −fq (γn) fi (γn)

0unxuny

(21)

where unx and uny denote the x and y components of vectoru2D,n in (3). A simple transformation can be performedto obtain an alternative expression respectively for (21) asfollows

ε2D,n = U2D,n · f (γn)

=

0 0 00 unx uny0 uny −unx

0fi (γn)fq (γn)

(22)

It can be seen from (22) that the pressure information is notuseful in estimating the roll perturbation angles. Denoting

Ur2D,n =

(unx unyuny −unx

)(23)

and

fr2D,n =

(fi (γn)fq (γn)

)(24)

where (23) and (24) are the reduced matrix and vector of(22) and using results from (23) and (24) in (19) and (20)and ignoring the pressure information where the superscript rdenotes reduced matrix and vector, the array steering vectorerrors can be re-written as follows:

ε2D = a2D − a2D

= ∆Φ2D · a2D (25)

εr2D = ∆Φr2D · ar

2D

= Ur2D · fr2D (26)

where

Ur2D =

Ur2D,0 0 0

0. . . 0

0 0 Ur2D,N−1

· diag[h⊗

(11

)](27)

fr2D =

fr2D,0fr2D,1

...fr2D,N−1

(28)

ar2D = h⊗ ur

2D (29)

The expressions in (27), (28) and (22) enable the estimationof parameters vectors fr2D and f2D to form the compensationmatrix Φ2D.

B. Direct Least Square

The approach is to obtain an estimate of the roll perturbationparameters ˆfr2D from the nominal and measured array steeringvector. Using the superscript H to denote hermitian transpose,the least square estimation to (26) is given by

ˆfr2D =[(Ur

2D)H

(Ur2D)]−1

(Ur2D)

Hεr2D (30)

Notice that for N sensors, there are 2N unknowns and there are2N equations available. Hence the estimation can be performedwith 1 measurement of 1 known source. However it should benoted that a solution is only available if (Ur

2D) is full rank.Hence the location of the position of source must be placedsuch that θ 6= 0o, θ 6= π, φ 6= π

2 and φ 6= 3π2 .

C. Subspace Least Square

Another approach to estimate roll perturbation angles isvia the subspace method [1]. This approach exploits theproperty that the true steering vector is orthorgonal to the noisesubspace.

1) Single Vector Sensor: Denoting Ew,n as the noise sub-space of a nth vector sensor, one can express the relationshipbetween the noise subspace and single vector sensor responseas

(Ew,n)H · v2D,n = 0

(Ew,n)H · v2D,n + (Ew,n)

H ·∆Q2D,n · v2D,n = 0

(31)

The derivation of this algorithm takes a slightly differentapproach from section IV-B where the pressure channel wasomitted. This approach includes the pressure channel informa-tion to ensure that the noise subspace is computed from theentire array instead of just from the particle velocity channels.Therefore,

∆Q2D,n · v2D =

0 0 00 fi (γn) fq (γn)0 −fq (γn) fi (γn)

1unxuny

=

0fi (γn)unx + fq (γn)uny−fq (γn)unx + fi (γn)uny

=

X 0 00 unx uny0 uny −unx

0fi (γn)fq (γn)

= U2D,n · f2D,n (32)

where the symbol X denotes an aribitrary entry and has noconsequence to the outcome if the first parameter of the vectorf2D,n is constrained to 0. Applying the constraining row vectorp0 = (1,0,0) and together with (37) and (32), one can obtain

(EH

W,n ·U2D,n

p0

)f2D,n =

(−EH

W,n · v2D,n

0

)(33)

Denoting, respectively, for nth vector sensor(EH

W,n ·U2D,n

p0

)= Hn (34)

and (−EW,n · v2D,n

0

)= qn (35)

the least square estimate for the rotational parameters forsingle vector sensor based on subspace criteria is

f2D,n =(HH

n Hn

)−1HH

n qn (36)

2) Vector Sensor Array: Define Ew as the noise subspaceof a vector sensor array, the subspace property states that

(Ew)H · a2D + (Ew)

H ·∆Φ2D · a2D = 0

(37)

where ∆Φ2D was defined in (20). Define, respectively,

U2D =

U2D,0 0 0

0. . . 0

0 0 U2D,N−1

·diag

h⊗

111

(38)

and

f2D =

f2D,0f2D,1

...f2D,N−1

(39)

where h was defined in (5). As shown in (33), the parametersfor all pressure channels are required to be constrained to 0,resulting in the following expression

EHw ·U2D

p0

p1

...pN−1

f2D =

−EW · a2D

00...0

(40)

where the constraining row vectors pn denote a vector with’1’ at the position of nth pressure channel and zero otherwise.These row vectors are to ensure that an arbitrary value canbe inserted into the respective U2D,n so that U2D is non-singular. It is important to note that the parameter of pressurechannels must be constrained individually to ensure a propersolution. Similarly, one can obtain a least square estimate ofarray rotational parameter f2D as follow:

f2D =(HH

a Ha

)−1HH

a qa (41)

where Ha and qa are respectively defined as

Ha =

EH

w ·U2D

p0

p1

...pN−1

(42)

and

qa =

−Ew · a2D

00...0

(43)

The expressions from (40) to (43) are valid when only 1source is present. For multiple sources, the nominal signalposition vectors for all sources (or at least their estimates)will need to be included to obtain the correct estimates forrotational perturbations. Assuming K sources are present,where K < N , and located at (θ1, φ1), (θ2, φ2) · · · (θk, φk)· · · (θK , φK), with respective U2D (θk, φk) and a2D (θk, φk),one can write the following

EHw ·U2D(θ1, φ1)

EHw ·U2D(θ2, φ2)

...EH

w ·U2D(θK, φK)p0

p1

...pN−1

f2D =

−EW · a2D(θ1, φ1)−EW · a2D(θ2, φ2)

...−EW · a2D(θK, φK)

00...0

(44)

and solve for f2D using (40) to (43).

V. COMPUTER SIMULATIONS

A. Variance of Estimator

A computer simulation was conducted to assess the perfor-mance of estimators shown in (30) and (41) under the influ-ence of zero mean white gaussian noise. The sample size usedis 20000 and the source is positioned at θ = 90o and φ = 45o.The variance of the estimator, MSE(f) = E

{(f − f)2

}, is

plotted as shown in Fig. 3 with its CRLB. From the plot, it canbe observed that the performance of both estimators are closeto the bound and are consistent along the different signal-to-noise ratio.

B. Beamforming

In this subsection, preliminary test of the 2 algorithms de-rived in Section IV-B (Direct Least Square Method) and IV-C(Subspace Least Square Method) are conducted in computersimulations with 2 high resolution beamforming algorithmsof interest, i.e, Minimum Variance Distortionless Response(MVDR) and MUliple SIgnal Classification (MUSIC), and theresults are presented in this section. Denoting R as the spatialcovariance matrix and Ew as the noise subspace, the MVDRbeamforming algorithm with calibration matrix (for rotationalperturbations) is given by

Fig. 3. CRLB vs Variance of Estimation by Direct Least Square and SubspaceLease Square Method. Sample size = 40000, 500 trials, φ = 45o, θ = 90o.

Pmvdr (θ, φ) =1(

Φ2Da2D (θ, φ))H

R−1(

Φ2Da2D (θ, φ))

(45)

and the MUSIC beamforming algorithm with calibration ma-trix is as follows:

Pmusic (θ, φ) =1(

Φ2Da2D (θ, φ))H

EwEHw

(Φ2Da2D (θ, φ)

)(46)

where the parameters of Φ2D are estimated with bothalgorithms. An array of 4 vector sensor elements spaced at0.5m apart is used as the subject under study, and narrowbandcontinuous wave signals of 1kHz and 2 kHz are used. Thesignal-to-noise ratio at sensor level is 40dB and the rollperturbation of each sensor is modelled as gaussian randomvariable with standard deviation of 35 degrees. The source isplaced at polar angle of 90 degrees and at several azimuthangles. One source and two sources cases are included inthis short simulation studies and they are placed at thebroadside of the array. The sampling frequency used in thissimulation studies is 20kHz and the sample size generatedfor each simulation is 40000. The simulation results areshown in Fig. 4 to Fig. 11. In the figures, the spatial spectraalong azimuth angle of the array beamformer is plotted tohighlight the impact between with and without compensationof roll perturbations. The ideal spectra assuming the case ofperfectly known perturbations, are also plotted to benchmarkthe performance of the algorithms under test.

As rotational perturbations are consistent under staticsituation, it is important to verify that the estimated solutionis indeed consistent on all simultaneous sources and at disjointtime frame. Hence four scenarios are used in this simulation,(i)1 source 1 measurement, (ii) 1 source 2 measurements, (iii)

2 sources 1 measurement and (iv) 2 sources 2 measurements.The objective of 2-source scenario is to check the consistencyof the solution on all current sources while the objective of2 measurements is to check the consistency of the solutionacross time. The summary of results are as follows:

1) Scenario 1: 1 Source 1 Measurement: Fig. 4 and Fig. 5show the result of single source MVDR and MUSIC directionfinding respectively on scenario (i) where the source is locatedat (θ = 90o, φ = 80o). It can be observed that the functionalityof both estimation algorithms are verified as the performanceof calibrated roll-perturbed array outperforms the uncalibratedarray by a large margin and also corrected the bias in DOAestimation.

2) Scenario 2: 1 Source 2 Measurements: In scenario (ii),two temporal disjoint measurements are simulated with the1kHz source first at (θ = 90o, φ = 80o) and then shifted to(θ = 90o, φ = 130o) in the second measurement. The algo-rithms are first applied to the first measurement to obtain anestimate of roll-perturbations. The calibration matrix is thenformulated based on the estimates and subsequently appliedto the second measurement. The spatial spectra of the secondmeasurement are shown in Fig 6 and Fig 7. The results showthat the estimates are consistent as they correct the bias andrestore the level of the peak after compensation on the secondmeasurement.

3) Scenario 3: 2 Sources 1 Measurement: The simulationresults of scenario (iii) are shown in Fig 8 and Fig 9. Inthis scenario, two sources at 1kHz and 2 kHz are generatedsimultaneously. The estimation algorithm is used to estimatethe roll-perturbations of the first source at 1kHz and thenapplied to the MVDR and MUSIC beamforming of the arraydata. The results show that for both spatial spectra, the DOAestimation bias is corrected and the two sources can be seenmuch more distinctly compared to no calibration. Again, theresults show that the algorithms are able to obtain consistentestimates of roll-perturbations.

4) Scenario 4: 2 Sources 2 Measurements: The last sce-nario (iv) is to check if the roll-perturbation estimations areconsistent for 2 sources at different temporal measurement.The results are shown in Fig 10 and Fig 11. Similar tothe results of scenario (ii), the azimuth spatial spectra forthe second measurement are plotted for both MVDR andMUSIC algorithm. The results again demonstrate that thealgorithms are able to obtain consistent and good estimates ofroll perturbations under the influence of 2 sources at differenttime.

VI. CONCLUSION

This paper presented 2 algorithms that are capable of esti-mating and compensate for rotational perturbations from datameasurement in the presence of one and multiple sources. Thederivation of both algorithms are based on direct measurementof steering vector and subspace criteria and no assumptions ofsmall perturbation angles are made during the formulation.Hence the algorithms can perform well under both small and

Fig. 4. MVDR Spatial Spectrum, 4-element 2D vector sensor array, 1 Source,1 Measurement, sample size = 40000, frequency 1kHz, azimuth angle φ =80o, elevation angle θ = 90o.

Fig. 5. MUSIC Spatial Spectrum, 4-element 2D vector sensor array, 1 Source,1 Measurement, sample size = 40000, frequency 1kHz, azimuth angle φ =80o, elevation angle θ = 90o.

Fig. 6. MVDR Spatial Spectrum for 2nd Measurement, 4-element 2D vectorsensor array, 1 Source, 2 Measurements, sample size = 40000, frequency1kHz, azimuth angle φ = 130o, elevation angle θ = 90o.

Fig. 7. MUSIC Spatial Spectrum for 2nd Measurement, 4-element 2D vectorsensor array, 1 Source, 2 Measurements, sample size = 40000, frequency1kHz, azimuth angle φ = 130o, elevation angle θ = 90o.

Fig. 8. MVDR Spatial Spectrum, 4-element 2D vector sensor array, 2Sources, 1 Measurement, sample size = 40000, frequency 1kHz and 2kHz,azimuth angles φ1 = 45o and φ2 = 120o, elevation angle θ = 90o.

Fig. 9. MUSIC Spatial Spectrum, 4-element 2D vector sensor array, 2Sources, 1 Measurement, sample size = 40000, frequency 1kHz and 2kHz,azimuth angles φ1 = 45o and φ2 = 120o, elevation angle θ = 90o.

Fig. 10. MVDR Spatial Spectrum for 2nd Measurement, 4-element 2D vectorsensor array, 2 Sources, 2 Measurement, sample size = 40000, frequency1kHz and 2kHz, azimuth angles φ1 = 200o and φ2 = 250o, elevation angleθ = 75o.

Fig. 11. MUSIC Spatial Spectrum for 2nd Measurement, 4-element 2D vectorsensor array, 2 Sources, 2 Measurement, sample size = 40000, frequency1kHz and 2kHz, azimuth angles φ1 = 200o and φ2 = 250o, elevation angleθ = 75o.

large roll-perturbation angles. Computer simulations show thatthe performance of both algorithms approach the theoreticalbound and are able to produce consistent estimates in thepresence of single source and multiple sources. The augmentedbeamformer with the estimated rotational parameters is ableto correct bias in direction of arrive estimation and restorespectra level.

REFERENCES

[1] B. P. Ng and M. H. Er and C. Kot, “Array gain/phase calibrationtechniques for adaptive beamforming and direction finding”’, IEE Pro-ceedings on Radar, Sonar and Navigation, Vol.141, No.1, pp 25-29, Feb1994.

[2] A. Nehorai and E. Paldi, “Acoustic Vector Sensor Array Processing”,IEEE Trans on Signal Processing, Vol.42, No.4, pp 2481-2491, April1994.

[3] M. Hawkes and A. Nehorai, “Acoutic Vector Sensor Beamforming andCapon Direction Estimation”, IEEE Trans on Signal Processing, Vol.46,No.9, pp 2291-2304, Sept 1998.

[4] H. Chen and J. W. Zhao, “Wideband MVDR Beamforming for AcousticVector Sensor Linear Array”, IEE Proceedings on Radar, Sonar andNavigation, Vol.151, No.3, pp 158-162, 2004.

[5] S. Miron and N. Le Bihan and J. I Mars, “Quaternion-MUSIC for VectorSensor Array Processing”, IEEE Trans on Signal Processing, pp 1218-1229 Vol.45, N0.4, April 2006.

[6] N. Le Bihan and S. Miron and J. I Mars, “MUSIC Algorithm for VectorSensors Array Using Biquaternions”, IEEE Trans on Signal Processing,Vol. 55, No.9, pp 4523-4533, Sept 2009.

[7] H. S. Mir and J. D. Sahr, “Passive Direction Finding Using AirborneVector Sensors in the Presence of Manifold Perturbations”, IEEE Transon Signal Processing, Vol.55 No.1, pp 156-164, Jan 2007.

[8] H. Cox and H. Lai, “Performance of Line Arrays of Vector and HigherOrder Sensors”, 41st Asilomar Conference on Signals, Systems andComputers, 2007.

[9] J. P. Kitchens, “Acoustic Vector Sensor Array Performance”, Master ofScience Thesis, Massachusetts Institute of Technology, June 2008.

[10] H. Lai and K. Bell and H. Cox, “DOA Estimation using Vector SensorArrays”, 42nd Asilomar Conference on Signals, Systems and Computers,2009.

[11] Andrew J. Poulsen and Raj R. Nadakuditi and Arthur B. Baggeroer,“Robust Adaptive Vector Sensor Processing in the Presence of Mismatchand Finite Sample Support”, 5th IEEE Sensor Array and MultichannelSignal Processing Workshop, 2008.

[12] A. J. Poulsen, “Robust Vector Sensor Array Processing and PerformanceAnalysis”, Doctor of Philosophy Thesis, Massachusetts Institute ofTechnology, Feb 2009.

[13] P. K. Tam and K. T. Wong, “Cramer-Rao Bounds for Direction Findingsby an Acoustic Vector Sensor Under Nonideal Gain-Phase Responses,Noncollocation or Nonorthogonal Orientation”, IEEE Sensors Journal,Vol.9, No.8, pp 969-982, August 2009.

[14] J. C. Osler and D. M. F. Chapman and P. C. Hines and G. P. Dooleyand A. P. Lyons, “Measurement and Modeling of Seabed Particle MotionUsing Buries Vector Sensors”, IEEE Journal of Oceanic Engineering,Vol 35, No.3, pp 533-534, July 2010.

[15] J. P. Kitches, “Acoustic Vector Sensor Array Processing”, Doctor ofPhilosophy Thesis, Massachusetts Institute of Technology, June 2010.

[16] P. Wang and G. J. Zhang and C. Y. Xue and W. D. Zhang, “Root-MUSICAlgorithm with Real Values Eigen-Decomposition for Acoustic VectorSensor Array”, 1st International Conference on Pervasive Computing,Signal Processing and Application, 2010.