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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 8, AUGUST 2009 3041

Circular Acoustic Vector-Sensor Array forMode BeamformingNan Zou and Arye Nehorai, Fellow, IEEE

Abstract—Undersea warfare relies heavily on acoustic meansto detect a submerged vessel. The frequency of the acousticsignal radiated by the vessel is typically very low, thus requiresa large array aperture to achieve acceptable angular resolution.In this paper, we present a novel approach for low-frequencydirection-of-arrival (DOA) estimation using miniature circularvector-sensor array mounted on the perimeter of a cylinder. Underthis approach, we conduct beamforming using decomposition inthe acoustic mode domain rather than frequency domain, to avoidthe long wavelength constraints. We first introduce a multi-layeracoustic gradient scattering model to provide a guideline and per-formance predication tool for the mode beamformer design andalgorithm. We optimize the array gain and frequency responsewith this model. We further develop the adaptive DOA estimationalgorithm based on this model. We formulate the Capon spectraof the mode beamformer which is independent of the frequencyband after the mode decomposition. Numerical simulations areconducted to quantify the performance and evaluate the theoret-ical results developed in this study.

Index Terms—Acoustic signal processing, acoustic transducers,array signal processing, direction-of-arrival estimation.

I. INTRODUCTION

T RADITIONALLY, the passive approach for underwateracoustic surveillance consists of installing a long linear

array in the ocean, with omnidirectional hydrophones as thesensing elements. For an array of pressure sensors, the arraygain in directivity is fundamentally limited by the size of thearray. In littoral or shallow water environment, especially areaswith heavy shipping density, the complex ambient noise charac-teristics and varying ocean bottom conditions often impair theability to detect the acoustic signal produced by the underwatertarget with the long array [1], [2].

The radiated noise produced by the vessel’s machinery andits motion in sea consists of broadband and narrowband com-ponents. The propeller and the hydrodynamic turbulence pro-duce a broadband component whereas the propeller, propulsionsystem, and auxiliary machinery generate the narrowband com-ponent. Generally, the frequency of both narrowband and broad-

Manuscript received August 12, 2008; accepted February 05, 2009. First pub-lished March 24, 2009; current version published July 15, 2009. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Prof. Olivier Besson. The work of Arye Nehorai was supported by theONR Grant N000140810849.

N. Zou is with the DSO National Laboratories, Republic of Singapore (e-mail:[email protected]).

A. Nehorai is with the Department of Electrical and System Engineering,Washington University in St. Louis, St. Louis, MO 63130 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2009.2019174

band components fall within a few tens to a few hundreds ofHertz [3], [4], [37]. Such a low-frequency band requires a largearray aperture to achieve the desired angular resolution.

Circular and cylindrical hydrophone arrays have been studiedover the past three decades and several practical systems havebeen built and demonstrated [6], [7]. The circular arrays canavoid the left-right ambiguity on bearing estimation that lineararrays normally encounter. However, the diameter of an under-water surveillance circular array usually reaches a few metersas it is normally used to sense the low frequency componentsof the acoustic sources. The huge array frame makes its deploy-ment, maintenance, and recovery difficult.

A novel acoustic vector sensor, which measures the acousticpressure and acoustic particle velocity field simultaneously,has been proven to be very effective for shallow water surveil-lance [1]. The acoustic vector sensors have been used indirectional sonobuoy for the past half-century [35]. Nehoraiand Paldi [22] first developed the measurement model of theacoustic vector sensor (AVS) array and analyzed the theoret-ical performance bounds on the direction finding. The DOAestimation using AVS array was further investigated by manyresearchers [23]–[25]. New application areas such as near fieldholography [26] and air acoustical signal extraction [27] havebeen explored by both academic and industrial communities.

A conventional acoustic vector sensor consists of an omnidi-rectional pressure sensor and a particle velocity sensor. It simul-taneously measures the acoustic pressure and three orthogonalcomponents of the particle velocity at single point in an acousticfield by single snapshot [28], [29]. Cray proved a single vectorsensor can provide 6 dB of directivity gain [30]. One of our goalsis to propose a miniaturized cylindric vector array that can pro-vide much better DOA estimation ability.

In this paper, we introduce a novel acoustic vector-sensorarray architecture and its associated DOA estimation algorithm.In Section II we discuss the physical acoustic model of our ap-proach. We introduce the acoustic signal model of a circularvector-sensor array which senses both the radial and tangentialcomponents of the acoustic field gradient. The miniature array ismounted in the perimeter of an acoustically coated cylinder. Weexplore the mode domain decomposition of the acoustic fieldthat is the basis for the proposed DOA estimation algorithm.In Section III we develop the adaptive DOA estimation algo-rithm. We show that for the novel algorithm, the array mani-fold is no longer a function of wavelength. Instead, the arrayDOA estimation performance is mainly decided by the acous-tical and physical characteristics of the cylinders. We illustrateand numerically prove that by coating the cylinder, the requiredmode number and frequency response curve can be created to fitthe incident acoustic waves. This enables our approach to pro-

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3042 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 8, AUGUST 2009

Fig. 1. Geometry of the cylindrical mode beamformer.

vide low-frequency broadband beamforming with miniaturizedarray aperture comparing with conventional methods. Numer-ical simulations are presented in Section IV. The impact of themodal coefficients on the beamforming bandwidth is numericalassessed. The mode beamformer array gain and beam patternare computed and a design example based on practical materialacoustic characteristics are illustrated. Section V concludes ourfindings and results of this paper.

II. PHYSICAL MODEL OF THE CYLINDRICAL MODE ARRAY

We consider a harmonic, plane acoustic wave impinging ona coated cylinder shown in Fig. 1. The cylinder is positionedvertically at the origin of the cylindrical coordinate system. Theaxis of the cylinder is taken as the -axis. This cylindrical sensornode is submerged in water. Denote the water density and soundvelocity in the water by and , respectively. The outer layer ofradius is made of viscoelastic material with density of . Itselastic and viscous Lamé constants [15] are , , , and

, respectively. The cylindrical array frame is made of elasticmaterial of radius with elastic Lamé constants of and

, respectively. The inner cylinder compartment of radiusis filled with air of density and sound velocity .

Assume a unit-magnitude steady-state plane acoustic wavewith radial frequency traveling in the direction of azimuthangle . Without loss of generality, we assume the incidentelevation angle is . This assumption is normally true whenthe underwater source is far enough comparing with the waterdepth. The pressure of the incident wave can be written as [9],[18], [19]

(1)

where and is the scattering angle

(2)

where is the distance from the observation to the origin pointof the coordinate system

(3)

where is the wave number in water, is the thorder Bessel function of the first kind, and is the Neumannfunction

.(4)

The scattered acoustic wave can be written as [10]–[17]

(5)

where is the Hankel function of the first kind. The reflec-tion coefficient is determined by the boundary conditionsthat the pressure and radial component of the particle velocitymust be continuous at the boundaries [20], [21].

Governed by the Helmholtz equations, the scalar acoustic po-tentials in the viscoelastic coating and the node frame can berepresented as

(6)

where the is the th order Neumann function. For theviscoelastic outer layer material

(7)

where

(8)


ZOU AND NEHORAI: CIRCULAR ACOUSTIC VECTOR-SENSOR ARRAY FOR MODE BEAMFORMING 3043

The acoustic field in the cylinder is

(9)

where

(10)

The coefficients of (5), (6), and (9) can be solved as

(11)

where [see (12) and (13) shown at bottom of page] and

(14)

The details of the expressions of the elements of (12) to (13) aregiven in the Appendix.

The total complex acoustic pressure in the perimeter of thecylinder is

(15)

Denote

(16)

then combining (1) and (5) yields

(17)

where

(18)

Placing the acoustic vector sensor in a circular array form assketched in Fig. 2, the radial and tangential components of theacoustic field gradient can be denoted as

(19)

and

(20)

Equations (19) and (20) can be further denoted as

(21)

where

(22)

It can be observed from (21) that both the radial and tangentialgradient of the acoustic field can be decomposed into a series ofcosine and sine components. We define asthe mode component of the radial gradient with mode coef-ficient and as the mode component ofthe tangential gradient with mode coefficient , respectively.

III. CIRCULAR VECTOR-SENSOR ARRAY DOA ESTIMATION

In this section, we establish the rationale for DOA estima-tion based upon the physical model we introduced in Section II.The outputs of the circular vector-sensor array are transformedinto mode domain subspace and spatially filtered to produce theDOA estimation.

(12)

(13)



Fig. 2. Top view of the cylindrical vector-sensor array.

Integrating the radial component of (21) over , we have

(23)

Observe that we dropped the time dependence of (21) for easierreadability. Using the following integral identities [31], [32]:

(24)

where is the Kronecker function defined as

.(25)

Taking (24) and (25) into (23) yields

(26)

where is the acoustic source incident angle, as defined inSection II. Since the acoustic sensors can sense the acousticgradient only discretely, by assuming the sensors are placedequally along the perimeter of the cylinder and satisfies theShannon’s sample theorem, namely where

is the highest order of the mode; the discrete form of (26)becomes

(27)

where

(28)

Similarly, by transforming the tangential component of (21)over , we have

(29)

We consider a two-dimensional gradient vector-sensor arrayof elements uniformly placed in the perimeter of the cylinder.Each vector sensor measures both the radial and tangential com-ponents of the incident acoustic field. The received radial andtangential components at snapshot are

(30)

and

(31)



By transforming the radial and tangential particle velocitycomponents following (27) and (29) and taking the linear sumthe output of the transformation at time , the decomposedmode coefficients of the radial are

(32)

where

(33)

[see (34) at the bottom of the page] and for the tangential gra-dients are

(35)

where

(36)

[see (37) shown at the bottom of the page].Define a new acoustic decomposition vector

(38)

By applying a spatial filter to the transformed subspace vector, the output of the filter and sum beamformer yields

(39)

where

(40)

Given temporal samples , theCapon spectra is [5]

(41)

where

(42)

is the covariance matrix of the acoustic gradient field decompo-sition output.

IV. NUMERICAL EXAMPLES

In this section, we illustrate by numerical examples some ofthe points made in Sections II and III.

A. Modal Coefficients

Fig. 3 shows plots of the radial and tangential gradientacoustic field for 1, 5, and 10, respectively, following(21) and (22).

Without loss of generality, we adopt the following parame-ters for our simulations shown in Table I [15]. We assume thatthe far-field acoustic wave impinges on the sonar node with az-imuth angle of . Observation of Fig. 3 shows a few facts. Theradial acoustic gradient varies more significantly with the vari-ation of than with the tangential component. Whenis small, the radial gradient signal energy is distributed on theback scattering region. When increases, strong signal oc-curs in the forward scattering region. The tangential componentis present at both the forward and back scattering regions. Withthe increase of , the tangential gradient energy is narrowedmore to the backscattering area.

Fig. 4 shows the mode coefficients of the decomposed radialand tangential gradient of the acoustic field following (11)–(22).

.... . .

.... . .

(34)

.... . .

.... . .

(37)



Fig. 3. Magnitude of radial (left) and tangential (right) acoustic gradient field of the scattering signal with �� (top), �� (middle) and ��

(bottom).

TABLE ISIMULATION PARAMETERS

The simulation parameters are the same as shown in Table I.Observation of the plot shows that up to 6 modes of the ra-dial gradient component can be activated above 0 dB whenis greater than 2. The modes are usable for broadband beam-forming until reaches about 5. For the tangential compo-nent, Mode 0 is not able to be activated and the magnitude ofthe mode coefficients in most situations is weaker than radialcomponent. The mode 1 decays the fastest for both the radialand tangential components with the increase of . However,the first notch of mode 1 of the radial component has the strengthof about 12 dB at about 3.8 while the counter part of thetangential component drops sharply to below 12 dB. So forhigh signal-to-noise ratio (SNR) condition, adopting both ra-dial and tangential components is more appropriate to increasethe array gain whereas for low SNR, utilizing the radial compo-



Fig. 4. (a) Radial and (b) tangential acoustic gradient field mode decomposi-tion.

nent is more appropriate to ensure the detection capability forthe beamforming.

The flat frequency response area can be controlled by the nodephysical dimension and material characteristics. Fig. 5 shows thecurve of the mode 1 frequency response as a function of the outerlayer thickness. Observations of Fig. 5 show that when

, the mode 1 coefficient increases with the increase of the outerlayer thickness. However, when , the mode 1 coeffi-cient decreases with the increase of the thickness. By controllingthe physical dimension of the frame and material characteristics,the usable frequency bandwidth can be controlled.

B. Beam Pattern and Array Gain

Fig. 6 shows the single mode beam pattern of the decomposedradial and tangential acoustic gradient field following (26) and(29). After the field decomposition, the decomposed mode isno longer a function of frequency, namely the beam pattern isfrequency independent. It forms a very unique feature for ap-plications requiring frequency invariant constant beam steering.Among them are speech capture, battle field acoustic informa-tion collection etc. Theoretically an infinite mode number can beexcited to increase the directionality of the beamformer. How-ever, in practical implementations, higher mode might be buriedin the ambient noise due to its weak amplitude. The array de-signer has to assess the array frame architecture to select the

Fig. 5. Frequency response of mode 1 for radial gradient (top) and tangentialcomponent (bottom) with different outer layer thickness.

appropriate frequency band and mode number, as we discussedin Section IV-A.

The WNG (white noise gain) of an array is defined as

(43)

where denotes the coefficient of the beamformer and is therepresentation which depends on the array geometry and sourcedirection [36].

We compare the mode decomposition vector sensor arraybeamformer with conventional and vector sensor circular arraybeamformers, of the same size and number of sensors. TheWNG of our mode decomposition vector-sensor array is

(44)

where is defined by (40) and

(45)

Similarly, the conventional circular array WNG can be de-noted

(46)

where

(47)

(48)



Fig. 6. Beam patterns of the radial (red) and tangential (blue) gradient components. a–e: beam patterns of the decomposed radial acoustic gradient field with modenumber from 0 to 4. f–i: beam patterns of the decomposed tangential acoustic gradient field with mode number from 1 to 4.

and

(49)

where

(50)The conventional circular vector-sensor array WNG is

(51)

where

(52)

(53)

(54)

and

(55)

where

(56)

(57)

where and the number of sensors is . is themaximum mode number for the proposed beamformer.

The white noise array gain of the conventional, vector sensorand the proposed circular array are shown in Figs. 7, 8, and 9,respectively. It can be clearly observed that under the same arrayphysical dimension, our approach outperforms the conventionalarray significantly on beam width and sidelobe perspective.

C. Design Example

We next design a sonar node using the physical and acousticspecifications listed in Table I. We assume an acoustic sourceimpinging from far field at an incident angle of 60 . The syn-thetic signal is generated following (19) and (20). The maximummode number of 16 is chosen to produce the synthetic signal.



Fig. 7. The white noise array gain of a conventional circular array with � �

��. The bearing of the source is ��, the array radius is .26 m, and the soundspeed in water is 1500 m/s.

Fig. 8. White noise array gain of the conventional circular vector sensor arraywith � � ��. The bearing of the source is ��, the array radius is .26 m, andthe sound speed in water is 1500 m/s.

Fig. 9. White noise array gain of the proposed mode decomposition vectorsensor array with � � �. The bearing of the source is ��, the array radius is.26 m, and the sound speed in water is 1500 m/s.

To fulfil the spatial Nyquist sample role, 12 vector sensors areevenly distributed in the perimeter of the cylinder each mea-suring both the radial and tangential gradient component of theincident acoustic field. The signal is selected to be narrow bandat 600 Hz. The signal is further digitalized at sample rate of8 kHz.

The maximum mode of the beamformer is selected as 5.Every 2048 points of data is used to estimate the covariancematrix of the beamformer. The sensor signal is convertedto mode domain using (27) and (28). Then the 2048 pointstransformed data are used for DOA estimation.

For comparison, we also calculate the beamforming resultsbased upon the conventional beamforming approaches. Underthe conventional far field assumption, the unit-magnitude fieldmeasured at sensor and due to a source at azimuthal DOAis given by [5]

(58)

where

(59)

and

(60)

So the scalar, radial and tangential components are

(61)

where is the sensor number. The measurement of the scalararray output is

(62)

and

(63)

The conventional circular vector array output measurement is

(64)

and

(65)The computation results are shown in Fig. 10. It is obvious

that under the same simulation conditions, although all threebeams peak at 60 , the angle the beam drops 3 dB occurs at 5or mode beamforming whereas the counterpart of conventional



Fig. 10. DOA estimation performance comparison between the conventionaland mode beamforming algorithms.

vector and hydrophone array beam appear at 105 and 175 , re-spectively. Furthermore the conventional hydrophone array hasa much shallow notch comparing with conventional vector andvector mode beamformer. Strong sidelobe is also found for theconventional vector sensor beamforming under the simulationcondition.

V. CONCLUSION

In this paper, we studied a two-layer cylindric acoustic gra-dient scattering model. We applied the physical acoustic modelto a circular vector-sensor array which senses both the radial andtangential components of the acoustic field gradient. The minia-ture array is mounted in the perimeter of an acoustically coatedcylinder. We explored the mode domain decomposition of theacoustic field that is the basis for the proposed DOA estima-tion algorithm. Numerical simulation were conducted to studythe impact of the cylinder to the decomposed acoustic field.We considered the case where the outer layer material is vis-coelastic while the inner layer is elastic material. From (11), weobserved that under certain configurations, the matrix mayturn to be zeros thus the mode coefficients can turn to be higheven without the existence of the external acoustic source (res-onant scattering). We concluded that special attention should bepaid to the choice of beamformer bandwidth and node configu-ration. By more extensive numerical computations for differentconfigurations, the frequency response curve and the maximumdecomposition modes can be further optimized.

We further developed an adaptive DOA estimation algorithmbased on the model we developed. Both radial and tangentialgradient sensors are used for DOA estimation algorithm de-velopment. However, in practical implementation, the user canselect either the radial sensor or radial and tangential compo-nents to form the beamformer. It should be noted that there isno zero-order mode contribution from the tangential sensor sothat is not recommended to use tangential sensor alone for thebeamformer. For mode decomposition from the circular array,it should be noted that the sensor number (both radial and tan-gential) must be greater than two times of the maximum modenumber to avoid aliasing.

We conclude that there are two advantages of our algorithmover the conventional circular array or a single vector-sensorarray. Firstly, our approach processes the signals in mode do-main, the DOA performance and the array gain rely mostly onthe highest mode that can be activated. The model we developedwas used to show how the highest mode and frequency responsecurve can be estimated and are controlled by the coating thick-ness. We have found by numerical simulation that a 26-cm arrayaperture can accurately estimate a narrow band acoustic sourceat 600 Hz.

The second advantage our algorithm produces is that oncethe signal is converted from the sensor domain to the modedomain, the beamformer performance is actually independentof the signal frequency. For practice, we can first separate thesignal into narrow band frequencies and conduct mode decom-position using precalculated mode coefficients and .After that, the signal at each band is beamformed at the modedomain with the same mode number hence same beamwidth.This forms an important feature for certain applications suchas speech capture or acoustic intelligent information collection.Although our current study only covers the narrow band DOAscenario, we will aim to show more benefits of our approach forbroadband beamforming in out future work.

We confined our study to azimuth domain DOA estimationbut the model can be easily extended to both azimuth and eleva-tion DOA estimation. The current research will readily extend totwo-dimensional DOA estimation. Our study is mainly focusedon underwater acoustic sensing. However since the electromag-netic (EM) scattering mechanism is quite similar to acousticscattering, the algorithm is likely to be extended to be imple-mented to electromagnetic wave DOA estimation.

APPENDIX

COEFFICIENTS OF THE SCATTERING MATRIX

The elements of matrix in (12) are given as follows:



(66)

(67)

(68)

The elements of in (13) are given as follows:

(69)

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Nan Zou received the B.S. degree in electronic en-gineering from TsingHua University, China, in 1985and the M.S. degree in electronic engineering fromthe Second Academy of the Ministry of AstronauticalIndustry, China, in 1988.

Since 1993, he has been with DSO National Lab-oratories, Republic of Singapore. He is the PrincipalMember of Technical Staff at DSO since 2003. Hisresearch interests are mainly on underwater signal re-mote sensing and array processing, with applicationsto passive sonar and magnetic signal detection.

Arye Nehorai (S’80–M’83–SM’90–F’94) receivedthe B.Sc. and M.Sc. degrees in electrical engineeringfrom the Technion—Israeli Institute of Technology,Haifa, and the Ph.D. degree in electrical engineeringfrom Stanford University, Stanford, CA.

From 1985 to 1995, he was a faculty member withthe Department of Electrical Engineering at YaleUniversity. In 1995, he joined as Full Professor theDepartment of Electrical Engineering and ComputerScience at the University of Illinois at Chicago(UIC). From 2000 to 2001, he was Chair of the

department’s Electrical and Computer Engineering (ECE) Division, whichthen became a new department. In 2001, he was named University Scholar ofthe University of Illinois. In 2006, he became Chairman of the Departmentof Electrical and Systems Engineering at Washington University in St. Louis.He is the inaugural holder of the Eugene and Martha Lohman Professorshipand the Director of the Center for Sensor Signal and Information Processing(CSSIP) at WUSTL since 2006.

Dr. Nehorai was Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL

PROCESSING during the years 2000 to 2002. From 2003 to 2005, he wasVice-President (Publications) of the IEEE Signal Processing Society, Chair ofthe Publications Board, member of the Board of Governors, and member of theExecutive Committee of this Society. From 2003 to 2006, he was the foundingeditor of the special columns on Leadership Reflections in the IEEE SIGNAL

PROCESSING MAGAZINE. He was co-recipient of the IEEE SPS 1989 SeniorAward for Best Paper with P. Stoica, coauthor of the 2003 Young Author BestPaper Award and co-recipient of the 2004 Magazine Paper Award with A.Dogandzic. He was elected Distinguished Lecturer of the IEEE SPS for theterm 2004 to 2005 and received the 2006 IEEE SPS Technical AchievementAward. He is the Principal Investigator of the new multidisciplinary universityresearch initiative (MURI) project entitled Adaptive Waveform Diversity forFull Spectral Dominance. He has been a Fellow of the Royal Statistical Societysince 1996.


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