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Signal Processing

Signal Processing 94 (2014) 583–594

0165-16http://d

n CorrE-m

asannam♣ E

journal homepage: www.elsevier.com/locate/sigpro

Multiple wideband source detection and tracking usinga distributed acoustic vector sensor array:A random finite set approach

Xionghu Zhong a,n,♣, A.B. Premkumar b

a Centre for Multimedia and Network Technology, The School of Computer Engineering, College of Engineering, Nanyang TechnologicalUniversity, Singapore 639798b Electrical Engineering Department, Faculty of Engineering University of Malaya, Kuala Lumpur, Malaysia

a r t i c l e i n f o

Article history:Received 27 December 2012Received in revised form4 July 2013Accepted 5 July 2013Available online 18 July 2013

Keywords:Acoustic vector sensorDistributed sensor arrayParticle filteringDetection and trackingRandom finite set

84/$ - see front matter & 2013 Elsevier B.V.x.doi.org/10.1016/j.sigpro.2013.07.008

esponding author. Tel.: +65 67904745.ail addresses: xhzhong@ntu.edu.sg (X. Zhongalai@ntu.edu.sg (A.B. Premkumar).

URASIP.

a b s t r a c t

In the past, distributed acoustic vector sensor (AVS) arrays have been employed to localizethe source in a three dimensional space. Least-squares approaches were introducedto triangulate the source position by using the direction of arrival (DOA) measurementsextracted at each AVS. However, such approaches: (1) cannot detect and localize multiplesources; and (2) can be seriously degraded due to inaccurate DOA estimates. In this paper,a practical scenario that the source existence and the number of sources are assumed to beunknown is considered. A random finite set (RFS) approach is developed to jointly detectand track multiple wideband acoustic sources. RFS is able to characterize the randomnessof the state process (i.e., the source dynamics and the number of active sources) as well asthe measurement process (i.e., DOA measurements generated by real sources and falsealarms). Since the relationship between DOAs and source position is highly nonlinear, aparticle filtering approach is employed to arrive at a computationally tractable approx-imation of the RFS densities. Simulations under different acoustic environments demon-strate the performance of the proposed approach and show a significant improvement onposition estimation over the least squares approaches.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Detection, localization and tracking of acoustic sourcesin a three dimensional (3-D, x-, y-, z-coordinate) space areimportant topics both in civilian and military missions suchas sonar and radar array signal processing [1], autonomousunderwater vehicle positioning and navigation [2], and roomspeech enhancement [3]. The tasks are traditionally per-formed by using a number of hybrid vertical and horizontalarrays equipped with acoustic pressure sensors [1]. In recentyears, a new technology namely acoustic vector sensor (AVS)

All rights reserved.

),

[4,5] has beenwidely employed for acoustic source detectionand localization, and different signal processing algorithmshave been developed accordingly [6]. Acoustic vector sensoremploys a co-located sensor structure and measures acousticpressure as well as particle velocity at sensor position.Compared to the traditional pressure sensors, it has follow-ing advantages: (1) it enables 2-D (azimuth and elevation)DOA estimation with a single AVS; (2) it allows elevationangle estimation unambiguously; and (3) its manifold vectoris independent of the source's signal frequency and thus AVSis suitable for wideband source signal processing or scenarioswhere the source signal's frequency is unknown a priori.A full description of AVS in signal processing problems canbe found in [6].

Due to its advantages described above, both the theo-retical aspects and the applications of AVS array have been

X. Zhong, A.B. Premkumar / Signal Processing 94 (2014) 583–594584

widely studied. DOA estimation using a single AVS hasbeen investigated in [7–12]. For linear AVS array config-uration (as shown in Fig. 1(a)), different DOA estimationapproaches such as Capon beamforming [13], MUSIC [14,15],ESPRIT [14,16–18], and subspace intersection based approach[19,20,21] have been investigated. A circular AVS arrayconfiguration (as shown in Fig. 1(b)) is studied in [22],which conducts beamforming using decomposition in the

Fig. 1. Illustration of different AVS array configurations. (a) linear array;(b) circular array; and (c) distributed array.

acoustic mode domain. Tracking the DOA of a singleacoustic source has been recently studied in [9–12] bydeveloping different particle filtering approaches. Also,applications of AVS in room acoustic environments andunderwater acoustic communication are investigated in[23,24] respectively. However, most existing localizationapproaches focus on the 2-D DOA estimation rather thanthe 3-D position estimation.

Recently, advances in distributed sensor arrays inproviding unprecedented capabilities for target detectionand localization have motivated the deployment of dis-tributed sensor arrays for acoustic source detection andlocalization [18,25]. Such an array is configured by using anumber of randomly distributed but with known positionAVSs, as shown in Fig. 1(c). In [18], least square approacheshave been developed for 3-D source position estimation.At each AVS, Capon beamforming is employed to estimatethe 2-D DOAs of the source. These DOA estimates are thenemployed to triangulate a 3-D location by using weightedleast-squares (WL) and re-weighted least-squares (RWL)based approaches. The advantage of this array configura-tion is that each AVS needs only to transmit the DOAestimates to the central processor. Hence, the communica-tion cost is very low. However, such approaches assumethe existence of a single source and cannot be applied formultiple source position estimation. Also, the accuracy ofthe triangulation can be seriously degraded by undesiredDOA estimates at each AVS.

In this paper, the problem of joint detection andtracking of multiple wideband acoustic sources using adistributed AVS array is considered. Such a considerationis more practical since, usually the source existence isunknown and the number of sources may be time-varyingin the tracking scene. Hence, more advanced approachesshould be employed to fuse the DOA information extractedfrom distributed AVSs and estimate the number of sourcesas well as the state of each source. Recently, random finiteset (RFS) approaches have been widely employed formultitarget detection and tracking problem [26–28]. Inessence, an RFS is a random process that is random incardinality as well as in values of each element. It is thusable to naturally characterize the randomness of the stateand measurement processes when target appearance andmeasurement origination are unknown. Generally, RFSframework neglects the intrinsic data association betweensources and measurements, and has been found promisingfor multi-object tracking problem [26–30]. For rigorousmathematical discipline of RFS framework and its appli-cation in multi-object tracking problem, the reader isreferred to [26–30].

An RFS approach is developed to detect and track atime-varying number of acoustic sources in this paper.In the state model, each element of an RFS is employed todescribe the motion dynamics of acoustic sources, andits cardinality is used to model the time-varying numberof sources. At each time step, the DOA measurement isextracted by using Capon beamforming that is exactly thesame as that in [18]. When the signal to noise ratio (SNR)is low and the number of snapshots is small, the Caponbeamforming response may be distorted and spuriouspeaks may be present. Also, acoustic source signals may

X. Zhong, A.B. Premkumar / Signal Processing 94 (2014) 583–594 585

interrupt each other when they are simultaneously active.This can lead to serious false alarms and miss detection inthe DOA measurements, and therefore causes problemsfor the source position estimation. Hence, in the mea-surement space, an RFS presentation is also constructedto describe the source detection, false alarms and misseddetection. Since the relationship between DOAs and sourceposition is highly nonlinear, a particle filtering approach[31–33] is employed to arrive at a computationally tract-able approximation of the RFS densities. Simulations indifferent tracking scenarios are organized to demonstratethe performance of the proposed approaches in detectionand tracking of multiple sources.

The core contribution of this work is that by using adistributed AVS array, joint wideband acoustic sourcedetection and tracking problem has been considered andaccordingly, an RFS based approach has been derived toestimate the number of sources as well as the sourcepositions. It is worth mentioning that since each AVS isonly capable of identifying up to two sources [34], themaximum number of sources at each time step is limitedto two in this paper. A conference precursor of thismanuscript has been published in the 2012 internationalconference on information fusion [35]. In [36], probabilityhypothesis density (PHD) filter approaches are introducedfor multiple (more than two) acoustic source trackingby using acoustic amplitude sensors. However, the signalamplitudes emitted by sources are assumed to be equaland known constants while in our work, the source signalsare assumed to be random and unknown a priori. The restof this paper is organized as follows. In Section 2, the AVSsignal model is introduced. Section 3 presents the trackingalgorithm developed for unknown and time-varyingnumber of acoustic sources. Simulations are organized inSection 4 and conclusions are drawn in Section 5.

2. Signal model and DOA measurement

This section provides a brief introduction of AVS acous-tic signal and distributed AVS array. Capon beamformingbased DOA measurement extraction is also presented. Notethat the general assumption for distributed AVS array andeach individual source signal is the same as those in [18]. Itis also worth pointing out that the proposed approach (aswell as approaches in [18]) can be applied equally well towideband source as well as narrowband sources.

2.1. Distributed AVS array signal model

Assume thatMwideband acoustic source signals smðtÞ∈C,for m¼ 1;…;M impinge on an array equipped with N spat-ially distributed AVSs at discrete time t. The DOA of the mthsource relative to the nth sensor (n¼ 1;…;N) is character-ized by a random vector θmn ðtÞ given by

θmn ðtÞ ¼ ½ϕmn ðtÞ;ψm

n ðtÞ�T ; ð1Þ

where ϕmn ðtÞ∈ð�π; π� and ψm

n ðtÞ∈½�π=2; π=2� represent theazimuth and the elevation angles respectively and super-script T denotes the transpose. Some general assumptions

for the AVS array and the source signal are made as follows[18]:

(A.1)

The distance between the source and the nth sensorrmn ðtÞbλmax, where λmax is the maximum wavelengthof the acoustic signal. This assumption implies a planewave model at each AVS given that the separationsamong co-located sensors inside an AVS are smallcompared with the minimum wavelength.

(A.2)

Each source signal has an independent and identi-cally distributed (i.i.d.) random amplitude εðtÞ andrandom phase ζðtÞ, i.e., sðtÞ ¼ εðtÞejζðtÞ. The phase isassumed to be uniformly distributed over ð0;2π�.This means that s(t) is a wide-band signal and isuncorrelated from one snapshot to the next.

Acoustic vector sensor measures the acoustic pressure aswell as three component particle velocities. Let um

n ðtÞ bethe unit direction vector pointing from the sensor towardto the source and given as

umn ðtÞ ¼

cos ψmn ðtÞ cos ϕm

n ðtÞcos ψm

n ðtÞ sin ϕmn ðtÞ

sin ψmn ðtÞ

264

375: ð2Þ

The received signal model for nth AVS can be written as [6]

ynðtÞ ¼ ∑M

m ¼ 1

1umn ðtÞ

" #smðtÞ þ ϵnðtÞ; ð3Þ

where ϵnðtÞ∈C4�1 represent the channel noise includingthe pressure and velocity noise terms. Note that we havenormalized the particle velocity terms by multiplying bya constant term �ρ0c0, where ρ0 and c0 represent theambient density and the propagation speed of the acousticwave in the medium respectively. Further assumption ismade for noise process as follows:

(A.3)

The noise process ϵnðtÞ is a sequence of complex-valued i.i.d. circular Gaussian random variableswith zero mean and covariance matrix Γ, given asϵnðtÞ∼CN ð0;ΓÞ, where CN ð�jμ;ΣÞ stands for the circu-lar complex Gaussian distribution with mean μ andcovariance Σ.

Usually, a number of snapshots are employed to estimatethe DOA. Assume that T0 snapshots are employed at eachtime step k. That is, each single time step k containsthe discrete time t ¼ ðk�1ÞT0 þ 1;…; kT0, and k¼ 1;…;K .When T0 is small, the source can be assumed to bestationary and θmn ðkÞ can be used to replace θmn ðtÞ duringa measurement frame. Eq. (3) can thus be written in amatrix form as

ynðkÞ ¼AðθnðkÞÞSðkÞ þ ϵnðkÞ; ð4Þ

where

AðθnðkÞÞ ¼ ½aðθ1nðkÞÞ;…;aðθMn ðkÞÞ�; ð5Þ

SðkÞ ¼ ½s1ðkÞ;…; sMðkÞ�T ; ð6Þ

Fig. 2. Illustration of DOA intersection method for position triangulation.

−1000

100

−500

50

2

4

6

8

10

12

x 104

azimuth

φ = − 43.2 ψ = −14.4

elevation

resp

onse

−1000

100

−500

50

5

5.5

6

6.5

x 105

azimuth

φ = −54 ψ = −10.8

elevation

resp

onse

Fig. 3. Response of the Capon beamforming under the environments(a) SNR¼ 10 dB; (b) SNR¼�10 dB. The source signal is located atð�44:4○;�14:4○Þ. The estimated DOA is labeled at the top of each figure.

X. Zhong, A.B. Premkumar / Signal Processing 94 (2014) 583–594586

and

aðθmn ðkÞÞ ¼ ½1; ðumn ðkÞÞT �T ; ð7Þ

where θnðkÞ is a collection of the DOAs of all sources withrespect to the nth sensor.

2.2. DOA measurements extraction

A distributed array contains several randomly distrib-uted AVSs. We assume that the positions of AVSs areknown. Further, assume that the nth AVS is deployed atarbitrary locations x0

n ¼ ½x0n; y0n; z0n�T and the mth sourceis located at xm;k ¼ ½xm;k; ym;k; zm;k�T . According to the arraygeometry, the 2-D DOA θmn ðkÞ is related to the sourceposition by

ϕmn kð Þ ¼ tan �1 xm;k�x0n

ym;k�y0n

!;

ψmn kð Þ ¼ tan�1 zm;k�z0nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðxm;k�x0nÞ2 þ ðym;k�y0nÞ2q

0B@

1CA: ð8Þ

The steering vector (2) contains both the azimuth andelevation information. Hence, the 2-D DOA can be esti-mated by using a single AVS. The Capon spectra estimationfor signal model (4) is [13]

Pnk ðθÞ ¼ fAHðθÞðRn

k Þ�1AðθÞg�1; ð9Þ

where Rnk is the covariance matrix given as

Rnk ¼ E yn kð ÞynðkÞH

n o≈1Nyn kð ÞynðkÞH ; ð10Þ

where E is the expectation operation, and the superscriptH denotes the conjugate transpose. The DOA estimationcan easily be obtained by implementing a 2-D search overθ which can maximize the output of Capon beamformer

θ̂nðkÞ ¼ arg maxθ∈ð�ππ��½�π=2π=2�

∥Pnk ðθÞ∥; ð11Þ

where ∥ � ∥ denotes the amplitude of a complex value. It isobserved in [34] that for a single AVS up to two sourcescan be uniquely identified.

In [18], indirect approaches have been developed for3-D localization. The DOA measurements at each AVSare estimated first by using (11). These DOAs are thenregarded as measurements and employed to triangulatethe 3-D source position by using weighted least-squares(WL) and re-weighted least-squares (RWL) approaches, asshown in Fig. 2. However, such approaches tend to beerroneous by inaccurate DOA estimates. In noisy environ-ments where the SNR is relatively high, Capon spectra isable to present the source DOA by a sharp peak as shownin Fig. 3(a). However, when the SNR is low, the peak maybe distorted and the estimated DOA may diverge from theground truth as shown in Fig. 3(b). Also, the approachescan be used only for single source localization. The authorssuggest a possible extension to multiple source localiza-tion, e.g., obtaining DOAs of multiple sources by enumer-ating multiple peaks. To achieve 3-D localization is not atrivial task since a sophisticated data association techniqueis required to identify the DOAs generated by each source.

In next section, an RFS approach will be developed tojointly detect and track multiple sources based on theextracted DOA measurements (Fig. 4).

−100−50

050

100

−1000

100

−150

−100

−50

0

50

100

150

200

x (m)y (m)

z (m

)

sensorssource 1source 2

Fig. 4. Illustration of sensor positions and source trajectories.

X. Zhong, A.B. Premkumar / Signal Processing 94 (2014) 583–594 587

3. Random finite set approach for joint detection andtracking

This section presents our solution toward to jointdetection and tracking problem. Essentially, an RFS frame-work is formulated to characterize the uncertainties fromsource appearance, source dynamics and DOA measure-ments. The RFS state process is introduced first.

3.1. RFS state model formulation

Since dynamic sources are considered, we assume that themth source moves with a velocity _xm;k ¼ ½ _xm;k; _ym;k; _zm;k�T .The source state Xm;k is thus constructed by the position andvelocity parts, i.e., Xm;k ¼ ½xT

m;k; _xTm;k�T . The constant velocity

(CV) model [37] is employed here to model the sourcedynamics, given as

Xm;k ¼ FXm;k�1 þ Gvk; ð12Þ

where the coefficient matrix F and G are defined respectivelyas

F¼ 1 ΔT0 1

� �⊗I3; G¼ Δ T2

2

ΔT

" #⊗I3; ð13Þ

and ΔT represents the time period in seconds between theprevious and current time step, and ⊗ denotes the Kroneckerproduct, and vk is a zero-mean real Gaussian process, i.e.,vk∼N ð0;ΣvÞ.

For simultaneously detecting and tracking of unknownnumber of multiple acoustic sources, the parameters ofinterest will be the number of sources as well as the 3-Dposition of each source. The state of a single source atcurrent time step k is Xm;k, for m¼ 1;…;mk. All theparameters of interest can be characterized by using asingle finite set, given as

Xk ¼ fX1;k;…;Xmk ;kg; ð14Þ

where mk ¼ jXkj is the number of sources, with j � jrepresenting the cardinality of a set. Given a realizationXk�1 of the RFS state at previous time step k�1, the sourcestate Xk at current step k is modeled by

Xk ¼ BkðbkÞ∪SkðXk�1Þ; ð15Þ

where BkðbkÞ is the state vector of sources born at timestep k, and SkðXk�1Þ denotes the RFS of states that havesurvived at time step k. In this paper, we use birth anddeath processes to describe the source appearance anddisappearance in the tracking scene, and accordingly theterminology ‘born’ is employed to represent that a newsource is appearing in the surveillance area, and ‘die’ refersto existing source disappearing from the surveillance area.For the birth process, we assume that:

�

at most one source is born at a time step; � mkoMmax where Mmax is the maximum number of

sources at each time step.

The first assumption is employed to simplify the problem.Also it is plausible to make such an assumption since thenumber of sources we have considered here is relativelysmall. In practice, it is possible that multiple sourcesturn up simultaneously. Further, the maximum numberof sources in the surveillance area is bounded at Mmax, i.e.,mk ≤Mmax. This means that when jXk�1j ¼Mmax, the newborn state vector is an empty set, i.e., Bk ¼ |. Since a singleAVS is able to identify up to two sources, Mmax ¼ 2 ischosen in this work. The source birth process can thus beformulated as

BkðbkÞ ¼|; ℏbirth;

fbkg; ℏbirth;

|; jXk�1j41;

8><>: ð16Þ

where ℏbirth and ℏbirth are the hypotheses for birth processand non-birth processes respectively, and bk is the initialstate vector under the birth hypothesis. For the initializa-tion of the new source, the position part is assumed to beuniformly distributed over the possible position range, andthe velocity part is assumed to be a Gaussian distributionaround the ground truth velocity. The initial state bk isthus given as

bk ¼X0∼Uðx0Þ �N ð _x0;Σ0Þ; ð17Þ

where Uðx0Þ is a uniform distribution over the range ofpossible x and Σ0 characterize the uncertainties in thevelocity part. The survived state set SkðXk�1Þ can beformulated by considering a death process. Assume thatℏdeath and ℏdeath are the hypotheses for death process andnon-death processes respectively. When a death processhappens, the corresponding state will be set as empty, andthe remaining states will evolve following the motiondynamic equation (12). The survival process after consid-ering death state SkðXk�1Þ can thus be given as

SkðXk�1Þ ¼

Xk�1\fXm;kg; ℏdeath formth source;

⋃jXk�1j

m ¼ 1fFXm;k�1 þ Gvkg; ℏdeath;

8>>>><>>>>:

ð18Þ

with \ denoting the set difference. The RFS state transitiondensity can thus be expressed by the product of birth PDFand survival PDF given as

pðXkjXk�1Þ ¼ pðBkjXk�1ÞpðSkjXk�1Þ: ð19Þ

X. Zhong, A.B. Premkumar / Signal Processing 94 (2014) 583–594588

Assume that the birth and death processes happen withprior probability Pbirth and Pdeath respectively. The PDF ofbirth process can be given as

pðBkjXk�1Þ ¼1�Pbirth; Bk ¼ |;

PbirthpðX0Þ; Bk ¼ fX0g;0; otherwise:

8><>: ð20Þ

To formulate the PDF of death process pðSkjXk�1Þ, wefirstly consider a single source death process. Source dieswith following PDF:

pðSkðXm;k�1ÞjXk�1Þ

¼Pdeath; SkðXm;k�1Þ ¼ |;

ð1�PdeathÞpðXm;kjXi;k�1Þ; SkðXm;k�1Þ ¼ fXm;kg;0; otherwise:

8><>: ð21Þ

Note that for the death process, we also assume that atmost one source can die at a time step. The total PDF ofsurvival process after considering the death process can bewritten as

pðSkjXk�1Þ ¼ Pmk�mk�1death ð1�PdeathÞmk�1 ;

� ∑1 ≤i1≠⋯im ≤mk�1

∏mk

j ¼ 1pðXm;kjXij ;k�1Þ; ð22Þ

where

∑1 ≤i1≠im⋯≤mk�1

¼ ∑n

i1 ¼ 1∑n

i2 ¼ 1;i2≠i1⋯¼ ∑

n

i1 ¼ 1⋯ ∑

n

im ¼ 1;im≠im�1≠⋯≠i1:

ð23ÞHence the RFS state transition PDF is formulated. Inpractice, Pbirth and Pdeath are unknown and are obtainedbased on experimental studies. Usually, increasing Pbirth orPdeath will make the algorithm easier detect the sourcebirth or death respectively. However, overly large Pbirth orPdeath will lead to an overestimation or underestimation onthe number of sources.

3.2. RFS model for 2-D DOA measurement

In [18], linear intersection methods have been devel-oped to localize a single static source. It is desired thatthe largest peaks from Capon response (9) of each AVScorresponds to the source DOAs. However, due to inter-ference of noise and interaction among source signals,spurious peaks may present and some of them maybe even higher than the peaks corresponding to thereal source. Hence, the peaks larger than a threshold (say1=20 of the height of the largest peak) are collectedto obtain the DOA estimates. This results in a finite setmeasurement model for the nth AVS at time step k

Znk ¼ fθ̂n1;k;…; θ̂

nκnk;kg; ð24Þ

where κnk ¼ jZnk j denotes the number of measured DOAs.

From this measurement set, following remarks can be made:

1.

It is desirable that the DOAs generated by real sourcesare included in the measurement set. However, it isunknown a priori that which ones are generated bysources and the associations between the DOAs andeach source Xm;k.

2.

Due to interference from noise and multiple sourcesignals themselves, spurious peaks may be collectedand Zn

k may contain false DOA estimates;

3. Source may be undetected and no DOA measurement is

related to the source. This case occurs when the twosources are closely spaced that the Capon beamformingcannot differentiate them due to a resolution problem.

Based on these observations, the measurement set can bedivided into two separate sets which contain false DOAmeasurements generated by spurious peaks Cnk and mea-surements generated by sources Gn

k respectively. Hence themeasurement set is modeled by

Znk ¼ fθ̂n1;k;…; θ̂

nκnk ;k

g|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}generated by sources Gn

k

∪fθ̂n1;k;…; θ̂nκ̂nk ;k

g|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}false alarms Cnk

; ð25Þ

where κnk and κ̂nk are the number of source generatedmeasurements and the number of false alarms respectively,i.e., jGn

k j ¼ κnk and jCnk j ¼ κ̂nk , and κnk þ κ̂nk ¼ κnk . The source isdetected with a probability of detection Pd. The number offalse alarms is usually assumed to follow a Poisson distribu-tion. Hence the probability of false alarms is

Pf κ̂nk� �¼ e�λf λ

κ̂nkf

κ̂nk !; ð26Þ

where λf is a Poisson factor which denotes the average rateof false alarms.

Given a DOA measurement which is generated byclutter, the likelihood is assumed to be a uniform distribu-tion within the possible 2-D DOA range θ given by,

pðθ̂ni;k Xk ¼ |�� �¼ Uθ½ð�π;�π=2�; ½π; π=2�� ¼ 1

2π2; ð27Þ

where Ud½a; b� is a uniform distribution over the possiblerange ½a; b� for variable d. If the measurement is generatedby a real source, the likelihood is assumed to be the trueDOA corrupted by an additive Gaussian noise. The like-lihood can then be written as,

pðθ̂ni;kjXk ¼ fXm;kgÞ ¼N ðθ̂ni;k;hðXm;kÞ;ΣθÞ; ð28Þ

where hðXm;kÞ is the measurement function (8) and Σθ is thevariance which describes DOA estimation error. Hence, whenthe source state is an empty set, the set likelihood is this

pðZnk jXk ¼ |Þ ¼ Pf ðκnk Þðpðθ̂

ni;kjXk ¼ |ÞÞκnk : ð29Þ

The total likelihood is thus

pðZnk jXkÞ ¼ ∑

~Z nk DZn

k

gð ~Znk jXk ¼ |ÞPmk

d ð1�PdÞj ~Znk j�mk

� ∑1≤i1≠im ≤j ~Z n

k j∏j ~Z n

k j

j ¼ 1pðθ̂nj;kjXij ;kÞ: ð30Þ

Hence, the RFS representation of the time-varying numberof sources and measurements processes is achieved. Inthe following section, the particle filtering approach will beintroduced to obtain the source position as well as sourcenumber estimates.

X. Zhong, A.B. Premkumar / Signal Processing 94 (2014) 583–594 589

3.3. Sequential Monte Carlo tracking algorithm

Above description gives an RFS formulation for AVSsignal based detection and tracking problem. The Bayesianrecursive estimation of the posterior distribution of theRFS state pðXkjZkÞ can be written as

pðXkjZ1:k�1Þ ¼ZFpðXkjXk�1ÞpðXk�1jZ1:k�1Þμð dXk�1Þ; ð31Þ

pðXkjZ1:kÞ∝pðZkjXkÞpðXkjZ1:k�1Þ; ð32Þwhere pðXkjXk�1Þ characterizes the birth, death and survivalprocesses of the state dynamics. The subscript F is the collec-tion of all finite subsets of the state space, and μð dXk�1Þ is ameasure of F . Note that in the case that μ is the Lebesguemeasure, μð dXk�1Þ is the same as dXk�1. Since this work isfrom an application point of view, mathematical concepts onthe set integration and these PDF constructions are beyondthe scope of this paper. Readers are referred to [27,28] for adetailed RFS definition and derivation of these PDFs.

Since the source position is nonlinearly related to themeasurements, closed-form solution for the PDF of DOAestimates is not available. In this work, a particle filteringapproach is employed to approximate the PDFs. Assume thatwe have particles X ðℓÞ

k�1 for ℓ¼ 1;…; L at previous time stepk�1, and the corresponding importance weight wðℓÞ

k�1. Theparticles at current time step k are generated according tothe state dynamic process described in Section 3.1 given as

X ðℓÞk ∼pðX ðℓÞ

k jX ðℓÞk�1Þ: ð33Þ

Since a prior distribution is employed as the importancefunction, the particles are weighted by

wðℓÞk ¼wðℓÞ

k�1pðZkjX ðℓÞk Þ; ð34Þ

where

pðZkjX ðℓÞk Þ ¼ ∏

N

n ¼ 1pðZn

k jX ðℓÞk Þ: ð35Þ

After resampling, the posterior distribution is thus approxi-mated by

pðX ðℓÞk jZkÞ≈ ∑

L

ℓ ¼ 1~wðℓÞk δX ðℓÞ

kðXkÞ; ð36Þ

where ~wðℓÞk is the normalized weight. δX ðYÞ is a set-valued

Dirac delta function. For brevity, δX ðYÞ is defined such thatδX ðYÞ ¼ 1 if X ¼ Y and 0 otherwise.

Due to an RFS presentation, each particle may differfrom others in dimension and the elements in it have anarbitrary order. Extracting the state estimation is thus notas straightforward as that in the known and fixed sourcenumber scenario. Based on the particles and the corre-sponding weights fX ðℓÞ

k ;wðℓÞk gL

ℓ ¼ 1, the number of sourcescan be approximated by

mk≈ ∑L

ℓ ¼ 1wðℓÞ

k jX ðℓÞk j: ð37Þ

Since the number of sources should be an integer, we obtainthe estimation of source number by using a rounding opera-tion, i.e., m̂k ¼ ⌈mk⌋. A K-means algorithm is then employedto cluster all the RFS particles. The centroids of these clustersfX̂m;kg

m̂k

m ¼ 1 are taken as the final state estimates.

The complete steps of implementing RFS particle filteringtracking algorithm are summarized in Algorithm 1. At eachtime step k, particles are sampled according to the sourcedeath, survival and birth processes. This is different fromsingle source particle filtering in which each particle is avector with fixed dimension and is sampled according to thesource dynamics. Here each particle is a random set withvariable dimension. When drawing the particle, the sourcedeath, survival and birth processes are considered and theparticles may contain different number of source states. Thelikelihood for each particle is then calculated and theparticles are resampled based on the high/low importanceweights. It is worth mentioning that when setting the birthand death priors as zero and the number of sources as one,the RFS particle filtering approach is a sequential importancesampling based particle filtering, that is widely employed forsingle source tracking problem. General sequential impor-tance sampling based PF has been developed for DOAtracking of an acoustic source in [10,11].

Algorithm 1. RFS-PF for multiple acoustic source detec-tion and tracking.

3.4. Performance metric

The performance evaluation of multiple source track-ing is fundamentally different from that of single sourcetracking in that the state estimation is a finite set in which

X. Zhong, A.B. Premkumar / Signal Processing 94 (2014) 583–594590

the estimated source number may be different with theground truth. Since the dimension of the state estimationcan be different from the ground truth, the error shouldtake both the state divergence and the dimension mis-match into account. Defining a metric which is able tocharacterize all these errors is a problem on its own. In[38], the authors have proposed an optimal subpatternassignment (OSPA) metric for such a problem. A penaltyvalue is employed in OSPA to transfer the cardinalityerror into the state error and then OSPA is able to presentthe performance on source number estimation as wellas source position estimation. Assume that X̂ k ¼ fX̂1;k;…;

X̂m̂k ;kg is an estimation of the ground truth state setXk ¼ fX1;k;…;Xmk ;kg. Note that here the state cardinalityestimation m̂k may not equal to the ground truth mk. TheOSPA error metric is defined as [38]

eOSPA X̂ k;Xk

¼min

s

ffiffi½

pp� 1mk

� ∑m̂k

i ¼ 1dðcÞðX̂i;k;Xsi ;kÞp þ cpðmk�m̂kÞ

!; ð38Þ

if m̂k ≤mk, and eOSPAðX̂ k;XkÞ ¼ eOSPAðXk; X̂ kÞ if m̂k4mk.Here, c40 is a penalty value which determines therelative weighting of the cardinality error component.The function dðcÞð�Þ is defined as minðc;dð�ÞÞ to guaranteethat the distance error is cut off at c. In OSPA metric,

−2 0 2

−1

0

1

azimuth φ (sen. 1)

elev

atio

n ψ

−1

0

1

azimuth φ (sen. 3)

elev

atio

n ψ

−2 0 2

−1

0

1

azimuth φ (sen. 5)

elev

atio

n ψ

−2 0 2

Fig. 5. Capon beamforming based DOA measurements from six sensors under SN(right). The solid black lines are the ground truth DOAs and the red dots arereferences to color in this figure caption, the reader is referred to the web vers

an appropriate selection of c is able to transfer thecardinality error component into localization error as partof the total error, and p∈ð0;1Þ is a real positive number. In[38], the authors suggest that p¼2 is a more practicalchoice since it yields smooth distance curves. Also, sucha choice makes the localization error similar to the errormetric in the single source scenario, i.e., root-mean-squareerror (RMSE). OSPA metric can also be interpreted as twoerror parts contributed by state error and cardinality errorthat can be given as

eðcÞloc X̂ k;Xk

¼min

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1mk

∑m̂k

i ¼ 1∥CX̂ i;k�CXsi ;k∥

2

s; ð39Þ

and

ecard X̂ k;Xk

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2

mkm̂k�mk� �s

; ð40Þ

respectively, where the minimization is taken over allpossible permutation s, and C¼ ½I;0� such that CXk outputsthe 3-D position part of the state. Generally, a large valueof c means a significant penalty on cardinality error andvice versa. The distance estimation error is cut off at c tomake sure that the state error is smaller than the errormade by cardinality mismatch. In next section, Eq. (38)will be used to evaluate the performance of time-varyingnumber of sources tracking. For the parameters in the

−2 0 2

−1

0

1

azimuth φ (sen. 2)

elev

atio

n ψ

−1

0

1

azimuth φ (sen. 4)

elev

atio

n ψ

−2 0 2

−1

0

1

azimuth φ (sen. 6)

elev

atio

n ψ

−2 0 2

R¼ 4 dB and T0 ¼ 32. From top to bottom: ♯1, ♯3 and ♯5 (left), ♯2, ♯4 and ♯6the DOAs extracted from Capon beamformer. (For interpretation of theion of this paper.)

5 10 15 20 25 30 35 40−100

−50

0

50

100

time step

x−co

ordi

nate

(m)

−150−100−50

050

100150

y−co

ordi

nate

(m)

−150−100−50

050

100150

z−co

ordi

nate

(m)

0

0.5

1

1.5

2

2.5

sour

ce n

um.

5 10 15 20 25 30 35 40

time step

5 10 15 20 25 30 35 40

time step

5 10 15 20 25 30 35 40

time step

Fig. 6. Single source estimation results under SNR¼ 4 dB and T0 ¼ 32 for(a) x-; (b) y-; (c) z-coordinate; and (d) source number.

X. Zhong, A.B. Premkumar / Signal Processing 94 (2014) 583–594 591

error metrics, p¼2 and a moderate penalty value c¼100will be employed.

4. Simulations

Six sensors are deployed to formulate a distributedAVS array. The sensor locations are: (30,�26,40.39) m,(60,�21,169.95) m, (0,0,0) m, (40,38,�10.57) m, (�65,40,�5.43) m, and (�100,�10,51.80) m for sensor ♯1 to sensor♯6 respectively. Such a sensor deployment is exactly thesame as that in [18]. A time-varying number of sources isconsidered: one (S1) is active from ð�10;�40;�120Þmto ð�100;�140;�20Þm, and the other (S2) from ð10;40;120Þm to ð100;140;20Þm with 30 time steps. S1 is activefrom time step 1 to 30, and S2 from time step 10 to 40. Fig. 4describes the sensor deployment and source motion tra-jectories. Such motion results in a velocity of 73.5 m/s.roughly. Fig. 4 describes the sensor deployment and sourcemotion trajectories. Since this is the first time the jointdetection and tracking problem for distributed AVS array isbeing considered, it is difficult to compare the performanceof the proposed approach with that of existing methods.Here, the tracking performance is comparedwith that usinglocalization approaches in [18], i.e., WL and RWL methods.In fact, WL and RWL methods cannot be employed formultiple source localization scenario since neither the DOAmeasurements and source association nor the source exis-tence is known. In our implementation, we assume that theground truthDOAs of each source are known for theWL andRWL approaches. The OSPA metric is then employedto assign the DOA measurements to each source. Themeasurement–source association is achieved by selectingthe combination that minimizes the OSPA metric. Hence,the association between the measured DOAs and sourcescan be perfectly achieved. Further, we assume that thesource power is known for the WL and RWL approaches.Under these assumptions, the WL and RWL approaches arethen applied to multiple source position estimation andthus these approaches are comparable with the proposedone. The wideband source signals are uncorrelated fromeach other and from one snapshot to the next so that eachsignal has a flat spectral density and its bandwidth equalsto the sampling frequency. The simulations are imple-mented under different noisy environments and withdifferent numbers of snapshots. The background noiselevel is evaluated by SNR, and is simulated by addingthe complex circular i.i.d. Gaussian noise into the receivedsignal. The parameters for RFS approach are set as:L¼1000, Pbirth ¼ 0:8, Pdeath ¼ 0:15, λf ¼ 0:1, Σθ ¼ 0:01I2,and Σ0 ¼Σv ¼ 0:01I3M . This parameter setup is foundadequate for all following simulations. The source velo-cities are initialized around the ground truth. The initialpositions are uniformly distributed over the possible sourcelocation area.

4.1. Results from a single trial

In the first simulation, the tracking results from a singletrial are presented. The proposed approach is implementedunder SNR¼ 4 dB and T0 ¼ 32. Fig. 5 shows the DOAmeasurements extracted using Capon beamforming from

six sensors. Due to source dynamics and signal inter-ference between sources, desired DOA measurements canhardly be achieved at all sensors. For some sensors,e.g., sensor ♯1 and ♯5, heavy false alarm can appear. Also,serious miss detection can happen at some time steps, e.g.,some time steps at sensor ♯2. The tracking results from asingle trial are shown in Fig. 6. The proposed RFS trackingapproach is able to track the number of sources and thecorresponding trajectories accurately. Generally, it is able todiscover and lock on to the new source trajectory quickly,and therefore consistently track the source positions. Also,better tracking accuracy can be achieved than using RWLbased localization approach even though optimum assump-tions (known source appearance/disappearance and perfectmeasurement–source associations) are made for the later.

Another implementation under SNR¼�4 dB and T0 ¼128 is also presented. Fig. 7 shows the DOA measure-ments from six sensors. Since the SNR is relatively low,the DOA measurements are degraded more seriously.For some sensors, Capon beamforming can completely

−2 0 2

−1

0

1

azimuth φ (sen. 1)

elev

atio

n ψ

−2 0 2

−1

0

1

azimuth φ (sen. 2)

−2 0 2

−1

0

1

azimuth φ (sen. 3)−2 0 2

−1

0

1

azimuth φ (sen. 4)

−2 0 2

−1

0

1

azimuth φ (sen. 5)−2 0 2

−1

0

1

azimuth φ (sen. 6)

elev

atio

n ψ

elev

atio

n ψ

elev

atio

n ψ

elev

atio

n ψ

elev

atio

n ψ

Fig. 7. Capon beamforming based DOA measurements from six sensors under SNR¼�4 dB and T0 ¼ 128. From top to bottom: ♯1, ♯3 and ♯5 (left), ♯2, ♯4and ♯6 (right). The solid black lines are the ground truth DOAs and the red dots are the DOAs extracted from Capon beamformer. (For interpretation of thereferences to color in this figure caption, the reader is referred to the web version of this paper.)

X. Zhong, A.B. Premkumar / Signal Processing 94 (2014) 583–594592

miss the sources. This results in a significant challenge forthe tracking algorithm. Using the proposed RFS trackingapproach, the estimation result is shown in Fig. 8.The proposed tracking approach is able to track thenumber and the trajectories of the sources accurately eventhough the DOA measurements are seriously distortedand large miss detection presents across some of thesensors. Also, it performs much better than WL and RWLbased localization methods. Although ideal measurement–source assignment is provided for WL and RWL methods,the position estimates are seriously deteriorated whentwo sources are simultaneously active. The reason is thatWL and RWL localization approaches are not as robustas RFS tracking approach when DOA measurements areinaccurate.

4.2. Average performance under different acousticenvironments

To obtain the average detection and tracking perfor-mance, the proposed algorithm is also implemented underdifferent noise environments and using different numbersof snapshots. The OSPA error over 100 Monte Carlo (MC)is presented to illustrate the estimation performance.Fig. 9 presents the error under different SNRs, i.e., SNR¼�4 dB;0 dB, and 2 dB. The number of snapshots is fixed

at T0 ¼ 128. It shows that the proposed RFS trackingapproach is able to provide much better estimation accu-racy for 3-D positions. Also, it can provide accurate sourcenumber estimation. In most time steps, the sources can bedetected and tracked. The degradation only happens at thesource birth and death steps since the algorithm usuallyneeds burn-in period to converge to the ground truth.However, such a burn-in period is very short in allsimulations. Also, due to the interference between thesources, the position estimation performance is degradedwhen two sources are simultaneously active. However, theperformance of proposed RFS approach is much betterthan that of WL and RWL methods, particularly whenmultiple sources are simultaneously active. For all estima-tions methods, larger OSPA error is presented as the SNRdecreases.

Fig. 10 presents the error under different number ofsnapshots, i.e., T0 ¼ 32, 64 and 256. The SNR is fixed atSNR¼ 2 dB. Similar to that in different SNRs, the proposedRFS tracking approach is able to provide better estima-tion accuracy in position estimation than WL and RWLmethods. It also presents good source number estimation.The position estimation performance is degraded whentwo sources are simultaneously active and the number ofsnapshots becomes smaller. In most cases, the proposedRFS approach is able to discover the source birth and death

5 10 15 20 25 30 35 40

−100

−50

0

50

100

time step

x−co

ordi

nate

(m)

−150−100−50

050

100150

y−co

ordi

nate

(m)

−150−100−50

050

100150

z−co

ordi

nate

(m)

0

0.5

1

1.5

2

2.5

sour

ce n

um.

5 10 15 20 25 30 35 40

time step

5 10 15 20 25 30 35 40

time step

5 10 15 20 25 30 35 40

time step

Fig. 8. Single source estimation results under SNR¼�4 dB and T0 ¼ 128for (a) x-; (b) y-; (c) z-coordinate; and (d) source number.

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

90

100

time step

OS

PA

met

ric

Fig. 9. OSPA error for 100 MC runs under different SNRs. The number ofsnapshots T0 is fixed to 128.

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

90

100

time step

OS

PA

met

ric

Fig. 10. OSPA error for 100 MC runs under different number of snapshots.SNR is fixed to 2 dB.

X. Zhong, A.B. Premkumar / Signal Processing 94 (2014) 583–594 593

quickly under all simulated acoustic environments. It isworth pointing out that for WL and RWL methods, perfectDOA measurements to source assignment is given. How-ever, the position estimation is still much worse than thatof RFS approach.

5. Conclusions

This paper considers a practical wideband acousticsource localization problem: jointly detect and track atime-varying number of sources in noisy environments.Traditionally, least-squares approaches are employed totriangulate the DOA measurements extracted from a dis-tributed AVS array to obtain a 3-D position estimation.Such an approach can be rendered erroneous by inaccurateDOA estimates, and also multiple source localization isimpossible. In this work, an RFS framework is employedto formulate the source birth and death processes aswell as false alarms and source detections. Hence, thetime-varying nature of the source state process and DOA

measurement process can be characterized. A particlefiltering approach is then introduced to arrive at a com-putationally tractable approximation of the RFS densities.Simulations in different tracking scenarios demonstratethe ability of the proposed approaches in detecting andtracking multiple acoustic sources.

In our future work, a distributed array that consists of anumber of uniform linear AVS arrays will be consideredand tracking a larger number of sources (Mmax42) willbe studied. When the source number is relatively large,the cardinalized probability hypothesis density (CPHD)recursion [39] and the cardinality balanced multi-targetmulti-Bernoulli filter [40] will be more promising. They areusually able to provide more accurate source numberestimation. Also, the applications of the proposed approachin real acoustic environments such as room acoustics andunderwater acoustics will be interesting directions.

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