2542 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, …nehorai/paper/Han_Tensor_TSP_2014.pdf ·...

2542 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 10, MAY 15, 2014

Nested Vector-Sensor Array Processing viaTensor Modeling

Keyong Han, Student Member, IEEE, and Arye Nehorai, Fellow, IEEE

Abstract—We propose a new class of nested vector-sensor arrayswhich is capable of significantly increasing the degrees of freedom(DOF). This is not a simple extension of the nested scalar-sensorarray, but a novel signal model. The structure is obtained by sys-tematically nesting two or more uniform linear arrays with vectorsensors. By using one component’s information of the interspec-tral tensor, which is equivalent to the higher-dimensional second-order statistics of the received data, the proposed nested vector-sensor array can provide DOF with only physical sen-sors. To utilize the increased DOF, a novel spatial smoothing ap-proach is proposed, which needs multilinear algebra in order topreserve the data structure and avoid reorganization. Thus, thedata is stored in a higher-order tensor. Both the signal model ofthe nested vector-sensor array and the signal processing strategies,which include spatial smoothing, source number detection, anddirection of arrival (DOA) estimation, are developed in the mul-tidimensional sense. Based on the analytical results, we considertwo main applications: electromagnetic (EM) vector sensors andacoustic vector sensors. The effectiveness of the proposed methodsis verified through numerical examples.

Index Terms—Acoustic vector sensors, direction of arrivalestimation, electromagnetic vector sensors, multilinear algebra,nested array, source number detection, tensor.

I. INTRODUCTION

V ECTOR sensors, which measure multiple physicalcomponents, have proven useful in electromagnetic,sonar, and seismological applications. Many array processingtechniques have been developed for source localization andpolarization estimation using vector sensors. An electromag-netic (EM) vector-sensor array, which consists of six spatiallycollocated antennas, measures the complete electric and mag-netic fields induced by EM signals. The EM vector-sensorarray was first introduced by Nehorai and Paldi in [1], wherea cross-product based DOA estimation method applicableto single-source scenarios was proposed. MUSIC-based al-gorithms were proposed by Wong and Zoltowski [2], [3].Different ESPRIT-based methods for direction of arrival

Manuscript received August 30, 2013; revised December 05, 2013 andMarch06, 2014; acceptedMarch 25, 2014. Date of publication March 31, 2014; date ofcurrent version April 23, 2014. The associate editor coordinating the review ofthis manuscript and approving it for publication was Prof. Martin Haardt. Thiswork was supported by the AFOSR Grant FA9550-11-1-0210 and ONR GrantN000141310050.The authors are with the Preston M. Green Department of Electrical and Sys-

tems Engineering, Washington University in St. Louis, St. Louis, MO 63130USA (e-mail: [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2014.2314437

(DOA) estimation have been developed separately in [4]–[6].[7] and [8] investigated identifiability issues, providing someupper bounds for the number of sources identifiable. Anotherimportant vector-sensor array is the acoustic vector-sensorarray, first proposed by Nehorai and Paldi [9]. The idea of usingvector sensors that measure both pressure and velocity hasbeen widely used to solve the passive DOA estimation problem[10]–[14]. In this paper, we will mainly focus on applicationsof these two vector-sensor arrays.Most of the previous work on DOA estimation with

vector-sensor arrays uses matrix techniques directly derivedfrom scalar-sensor array processing. Such a method is based ona long vector, which is concatenated with all components of thevector-sensor array. A method that keeps the multidimensionalstructures for data organization and processing was proposedin [15] using vector sensors for seismic sources, where thereceived measurements are represented as a multidimensionaltensor. A version of the MUSIC algorithm adapted to themultilinear structure was proposed based on the higher-ordereigenvalue decomposition (HOEVD) of a fourth-order tensor.A tensor is a multidimensional array [16], [17], for which

multilinear algebra provides a good framework to conserve themultidimensional structure of the information. Decompositionsof higher-order tensors have been shown to be of great interestin signal processing [18]–[23]. Two main decompositionsare CANDECOMP/PARAFAC (CP) [24], [25] and HigherOrder Singular Value Decomposition (HOSVD) [26], both ofwhich can be considered to be higher-order generalizations ofthe matrix singular value decomposition (SVD). One DOAestimation strategy based on HOSVD was proposed in [23],where the tensor structure of the data was well exploited, andthe HOSVD was applied to the covariance tensor. Anotherestimation strategy based on HOSVD was proposed in [27],where the HOSVD was applied to the measurement tensor.Though both approaches use HOSVD, they are different strate-gies. Another decomposition, namely HOEVD, was defined in[17]. It uses the concept of simple orthogonality, and allowsdetection of an increased number of sources [15]. However,this approach was shown to be equivalent to using matrixformulism on a “long-vector” in [28]. Additionally, it has beenshown that the HOSVD method in [23] is more effective thanthe HOEVD method. Therefore, in this paper, we will use theHOSVD-based strategy proposed in [23].Source number detection and DOA estimation are two major

applications of antenna arrays. Both applications are oftenconfined to the case of the uniform linear array (ULA) [29].A ULA with scalar sensors can resolve at mostsources using conventional subspace-based methods such as

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HAN AND NEHORAI: NESTED VECTOR-SENSOR ARRAY PROCESSING VIA TENSOR MODELING 2543

MUSIC. A systematic approach to achieve degrees offreedom (DOF) using sensors based on a nested arraywas recently proposed in [30], where DOA estimation andbeamforming were studied. The nested arrays are obtained bycombining two or more ULAs with increasing spacing. Owingto the property of nonuniformity, the resulting differenceco-array has significantly more DOF than the original sparsearray, which makes it possible for the nested array to detectmore sources than the number of sensors. Pal and Vaidyanathan[31], [32] extended the one-dimensional nested array to thetwo-dimensional case, assuming the sensors to be present onlattices, and providing a thorough analysis of the geometricalconsiderations and applications. Another similar nonuniformarray, called the co-prime array, was proposed and developedin [33]–[35], using sensors to obtain DOFfor DOA estimation, where and are co-prime. Han andNehorai [36] applied the jackknifing strategy to nested arrays,improving the accuracy of source number detection and DOAestimation. Extension to wideband sources using nested arrayswas investigated in [37]. The advantages of the nested arraystrategy are obvious, but we need to pay attention to its draw-backs. The nested array approach requires that the impingingsignals be uncorrelated. Additionally, it needs relatively moreobservations than required for a ULA-based approach.In the existing literature, the nested-array strategy was ap-

plied only to scalar-sensor arrays, including one-dimensionaland two-dimensional spatial cases. However, it is of great ana-lytical and practical interest to consider the vector-sensor arraymodel employing the idea of the nested array. In this paper, wewill apply the nested-array concept to the vector-sensor array.More specifically, we will provide a detailed analysis for theconstruction of the signal model of the nested vector-sensorarray. We will see that multilinear algebra plays an importantrole in the signal processing of the proposed array. Similar to thecase of the nested scalar-sensor array, the nested vector-sensorarray shows superior performance in terms of DOF and estima-tion resolution.The rest of the paper is organized as follows. In Section II, we

list the notations used in this paper. In Section III, we presentthe tensor-based signal model of the proposed nested vector-sensor array, also the models of two applications: EM vector-sensor array and acoustic vector-sensor array. We provide a pre-processing strategy, spatial smoothing, for the proposed nestedvector-sensor array, and introduce source number detection andDOA estimation based on HOSVD in Section IV. In Section V,we use numerical examples to show the effectiveness of ourproposed strategy. Our conclusions and directions for possiblefuture work are contained in Section VI.

II. NOTATIONS

In this paper, vectors (tensors of order one) are denoted byitalic boldface lowercase letters, e.g., . Matrices (tensors oforder two) are denoted by italic boldface capital letters, e.g.,. Higher-order tensors (order three or higher) are denoted byitalic boldface “calligraphic” font, e.g., . Scalars are denotedby lowercase italic letters, e.g., . We list some notational con-ventions that will be used in this paper.

• : complex conjugate of• : transpose of• : complex conjugate of• : mode-2 matrix unfolding of tensor

, with dimension , defined as

• : mode-3 matrix unfolding of tensor, with dimension , defined as

• : Kronecker product of and• : Khatri-Rao product of and• : outer product of and ,defined as

• : 3-mode product of and, defined as

• : mode-3 inner product of and, defined as

• : extended Khatri-Rao product of and, with dimension , defined as

III. SIGNAL MODELIn this section, we construct the signal model of the proposed

nested vector-sensor array.

A. Matrix-Based Vector-Sensor ArrayWe assume there is a linear array with vector sensors. The

output of each vector sensor is an -dimensional vector whichcontains all the components. We assume far-field sourcesare in the surveillance region, impinging on this linear arrayfrom directions , where andrepresent the azimuth and the elevation angles of the th signalrespectively. We assume and .The measurement received at the array at time can be modeledas [5]

(1)

where and are complex vectors, respectively,and is the source vector. The array man-ifold can be expressed as

(2)

where , with

Here, is the steering vector of the array associ-ated with a signal coming from the direction . de-notes the phase delay of the th signal at the sensors withrespect to the origin, represents the wavelength of the signals,and denotes the coordinates of the th sensor. The vector

is the unit vectorat the sensor pointing towards the th signal. , which variesfor different kinds of vector sensors, is the steering vector of a


single vector sensor located at the origin. Each element ofcorresponds to one component of a vector sensor. Next, we willconsider applications to both EM and acoustic vector sensors.When we consider EM or acoustic vector sensors, the differ-ences from the original signal model are the steering vectors.Thus, we will present the array steering vectors for both EMand acoustic vector-sensor arrays.1) Electromagnetic Vector Sensors: Electromagnetic vector

sensorsmeasure the complete electromagnetic field [1].We con-sider a linear array with EM vector sensors, each having

components. Here, we consider polarized signals.The array steering vector can be written as ,

with

(3)

where

(4)

and

(5)

Here, is the steering vector of the array associatedwith a polarized signal coming from the direction withpolarization , where and arepolarization parameters referred to as the auxiliary polarizationangle and polarization phase difference, respectively. is thesteering matrix of one EM vector sensor associated with the thsignal. is the polarization vector for the th signal.2) Acoustic Vector Sensors: We assume there is a linear array

with acoustic vector sensors. The output of each vector sensoris an -dimensional vector which contains compo-nents: the acoustic pressure and the acoustic particle-velocityvector [10].The array steering vector can be written as ,

with

(6)

which is the steering vector of a single vector sensor located atthe origin.

B. Tensor-Based Vector-Sensor ArrayIn this section, we propose a tensorial model for sources im-

pinging on a vector-sensor array based on model (1).First, we consider only one source signal . We set thearray manifold matrix for the th source as the outer

product of the phase delay vector and the steering vector :

(7)

Then, we can get the measurement matrix at time :

(8)

Fig. 1. The structure of the tensor .

and are the correspondingmeasurements and noiseat all the components of all the sensors.Considering sources in the surveillance region, we can get

the summed measurement matrix as

(9)

We further transform model (9) to the tensor form:

(10)

where is a tensor with element ,shown in Fig. 1. While is a matrix, it is decomposed in atensor form to explicitly bring out the additional dimension.Comparing models (1) and (10), we can see they contain the

same amount of statistical information. In order to find the DOAof sources and the source number, we will consider the second-order statistics through a “spectral tensor”.

C. Nested Array

Let us put aside the vector-sensor array in this section. In-stead, we consider a nonuniform linear nested array [30] withscalar sensors, consisting of two concatenated ULAs. Sup-

pose the inner ULA has sensors with intersensor spacing, and the outer ULA has sensors with intersensor spacing

, as shown in Fig. 2. We consider the 1-dimen-sional impinging source directions . Then,we can get the signal model:

(11)

where the matrix. The difference

from in the -sensor ULA is that the th element of thesteering vector is , with being theinteger multiple of the basic spacing or . We suppose thesource signals follow Gaussian distributions, ,and they are all independent of each other. The noise signal

is assumed to be whiteGaussian, and uncorrelated with the sources. Based on our


Fig. 2. A 2-level nested array with sensors in the inner ULA andsensors in the outer ULA, with intersensor spacings and respectively.

assumptions, the source autocorrelation matrix is diagonal:. Then, the autocorrelation matrix

of the received signal for the nested array is

(12)

Vectorizing , we get the vector

(13)

where , and ,with being a vector of all zeros except for a 1 at the th posi-tion. We can view vector in (13) as some new longer receivedsignals with the new manifold matrix , andthe new source signal variance vector .Model (13) is used to conduct source number detection and

DOA estimation for a nested array.

D. Tensor-Based Nested Vector-Sensor ArrayNow instead of scalar vectors as in Fig. 2, we consider

now vector sensors, where again each vector sensor haselements. Suppose the sensors are located along the -axis:

where . Then the phase delay vector for the th signalis

(14)

Based on the phase delay vector, we can write the manifoldmatrix in (7):

.... . .

...

.... . .

...

We suppose the signals and noise follow the assumptions insection . Based on model (10), we get the interspectral tensor, which is the fourth-order complex tensor of size

, defined as the second-order automoments and cross-moments between all the components on all sensors, as follows:

(15)

where the element of is given by

(16)

represents the expectation operator. Note that is a tenso-rial version of the covariance matrix .Substituting (10) into (15) yields:

(17)(18)

where is the covariancematrix of the sources. Equation (17) is due to the assumptionsthat sources and noise are independent, sources are zero meanGaussian, and noise is white Gaussian. We show the derivationof (18) in Appendix A.Now, we do the mode-2 matricization of tensor :

(19)

where is a matrix. Here andis a matrix defined as

. . .(20)

where with being a columnvector of all zeros except for a 1 at the th position. Followingthe definition of the extended Khatri-Rao product, we can seethe tensor is of dimension . Thus,after multiplying the third dimension by , the tensor becomesa matrix. The derivation of (19) is shown in Appendix B.Comparing (19) with (10), we can say that in (19) behaves

like the signal received at a longer vector-sensor array whosemanifold is given by . The equivalent source signalvector is represented by , and the noise becomes a deterministicmatrix given by .Looking deeply through the structure of tensor ,

we can find there are sets of horizontal slices, each corre-sponding to one component and containing slices. We pro-vide the internal analysis of in Appendix C. Withineach set, the distinct horizontal slices behave like the manifoldof a longer vector-sensor array whose sensor locations are givenby the distinct values in the set . Thisarray is precisely the difference co-array of the original array[38]. Hence, instead of model (10), we will apply DOA esti-mation and source number detection to the data in model (19),which provides more DOF than a ULA.


IV. SOURCE DETECTION AND DOA ESTIMATION

In this section, we will conduct source number detectionand DOA estimation based on the nested vector-sensor arraysignal model (19). To exploit the increased DOF offered by theco-array, we propose to apply the spatial smoothing techniquein a new fashion, as presented in [30]. Before conductingsource number detection and DOA estimation, we presentthe higher-Order singular value decomposition (HOSVD) oftensors [23].

A. Spatial Smoothing

Considering one set of horizontal slices in tensor ,with , we construct a newtensor , following the procedure in Appendix D. Equivalently,we can get a new model by removing the corresponding rowsfrom the observation matrix and sorting them accordingly:

(21)

where is a matrix with all zeros except for a1 at the position . The difference co-array of this 2-levelnested array has sensors located at

We now divide these sensors into overlapping sub-arrays, where the th subarray has sensors located at

We can see that each subarray has sensors. The th subarraycorresponds to the th to th rows of ,denoted as

(22)

where is a matrix, with all zeros except for a 1 atposition ( , 1). In Appendix E, we show that the th subarray isrelated to the first subarray by

(23)

where

(24)

Further we get

(25)

We define that

(26)

Taking the average of over all , we get

(27)

We call the tensor the spatially smoothed in-terspectral tensor. Note that the tensor is quite different from afourth order cumulant. We will use it to conduct source numberdetection and DOA estimation. We would like to clarify thatthe operations defined in this section for spatial smoothing areused to exploit the increased DOF provided by the nested array.They themselves do not contribute to the DOF. Additionally, theVandermonde structure of the array manifold guarantees theunique source localization.Note that we have set of horizontal slices. Each of them

corresponds to one component, and can be used to derive a spa-tially smoothed interspectral tensor . Without loss of gener-ality, we will consider using the first set in the following sec-tions. It has been shown that a 2-level nested array with scalarsensors can provide DOF. Therefore, based on , we expectto estimate up to sources as well, by using the HOSVDmethod presented in the next section.

B. Higher-Order Singular Value Decomposition

HOSVD, as stated in [23], efficiently exploits the tensorstructure of the multidimensional data. The HOSVD of thespatially smoothed interspectral tensor can be written as

(28)

where , and are or-thonormal matrices, provided by the singular value decompo-sition of the -dimension unfolding of tensor :

(29)

is the core tensor. Since is an Hermitiantensor, i.e., , the HOSVDof can be written as

(30)

C. Source Number Detection Using SORTE

For source number detection, we use the sample interspectraltensor, calculated from the measurements:

(31)

where is the number of snapshots. Based on , following(19), (21), and (25)–(27), we will obtain the sample spatiallysmoothed interspectral tensor . Further, we get the sample ma-trices and , and we write

(32)


TABLE IALGORITHM FOR SOURCE NUMBER DETECTION USING SORTE WITH A

2-LEVEL NESTED VECTOR-SENSOR ARRAY

Suppose the eigenvalues are sorted decreasingly:

Researchers have developed numerous detection methodsbased on different techniques, including eigenvalues, eigenvec-tors, and information theory [39]–[43]. Here we use the secondorder statistic of eigenvalues (SORTE) [43] method to considerthe source detection problem.We define a gap measure:

,

(33)where , and

(34)Then the source number is

(35)

The algorithm for source number detection using SORTEbased on a 2-level nested vector-sensor array is shown inTable I.

D. DOA Estimation Using Tensor-MUSICDOA estimation is based on the condition that we already

know, or have already estimated, the source number. MUSICis one of the earliest proposed subspace-based algorithms forDOA estimation.Suppose we know the source number is . Based on and, we obtain the approximation matrices and by trun-

cating the first and columns respectively. Here andare the number of important values in the SVD of and ,where is equal to the source number .Thus, we can construct the tensor-MUSIC (TM) estimator as

(36)

with the steering matrix

(37)

TABLE IIALGORITHM FOR DOA ESTIMATION USING TENSOR MUSIC WITH A 2-LEVEL

NESTED VECTOR-SENSOR ARRAY

where

(38)

Then, to obtain the DOA estimates, we conduct an exhaustivesearch over the impinging direction space, compute the MUSICspectrum for all direction angles, and find the largest peaks.As for the polarized sources using EM vector sensors, thesteering matrix in (37) will also be related to the polarizationparameters. We can use similar strategies to estimate them.The algorithm is shown in Table II.

V. NUMERICAL EXAMPLES

In this section, we use numerical examples for both EM andacoustic cases to show the effectiveness of our strategies basedon the proposed nested vector-sensor array signal model. Thenested array we use contains vector sensors, with, . As mentioned in the former sections, we havesets of horizontal slices. Without loss of generality, we use thefirst set of horizontal slices. Note that any set of the horizontalslices is the manipulated results of all the original compo-nents’ received information. Since the 2-level nested array has12 DOF, we also consider the corresponding performance of aULA with EM or acoustic vector sensors, with sensorpositions [0 1 2 3 4 5 6 7 8 9 10 11] . We will use the 12-sensorULA as a benchmark at high SNRs.Note that, in this paper, the ULA-based method exploits the

interspectral tensor which is similar to (15), rather thanthe spatially smoothed interspectral tensor in (27). Note thatthe tensor is achieved by using a ULA of vector sen-sors. Based on , we can conduct the estimation or de-tection using the proposed strategy in Section IV-D. Since theNA-based strategy increases the degrees of freedom by con-sidering the difference co-array, the NA-based approach canresolve more sources than the ULA-based approach when thenumber of sensors is the same. We will verify this through thefollowing numerical examples.

A. MUSIC Spectral for the EM Case

We provide numerical examples for MUSIC spectral analysiscorresponding to the following three cases.• Case 1: sources, with impinging directions

, , for, and polarization parameters ,

, for .


Fig. 3. MUSIC spectrum using a nested EM vector-sensor array with 6 sensors,as a function of elevation angle , , , . Thehorizontal axis is the elevation angle, whereas the vertical axis is the MUSICspectrum. (a) 6-sensor nested array, (b) 6-sensor ULA.

We use this scenario to illustrate the superior performanceof the nested EM vector-sensor array in terms of degrees offreedom for 1-dimensional DOA estimation.Suppose we know the azimuth angles of all the six sources.

Fig. 3 shows the MUSIC spectrum with respect to different ele-vation angles using both a 6-sensor nested array and a 6-sensorULA. In this example, we use snapshots at an SNRof 21.97 dB, where SNR is the signal to noise ratio, defined as

(39)

Here, we assume all sources are of equal power. As can be seenfrom Fig. 3(a), our method resolves the 6 sources well with thenested array. However, for the given number of sources, since , the presented tensor-MUSIC method could

not have been applied on a ULA having EM vectorsensors. This is verified by Fig. 3(b). Note that we assume allsources have the same polarization parameters in this example,but our algorithm also works for cases with unequal polarizationparameters.• Case 2: sources, with impinging directions

, , and polarization pa-rameters and .

Now, we consider 2-dimensional DOA estimation with twoclose sources in the surveillance region. For the purpose of in-tuitive demonstration, the polarization parameters are assumedto be known. The 2-dimensional MUSIC spectrum with respectto azimuth and elevation angles using the nested array is shownin Fig. 4. We can see that the two sources are well estimated.One thing to note is that the peaks are a little sharper alongthe direction of than along . This is reasonable because thesensors are aligned along the -axis. As a comparison, we alsoplot the case of the ULA with 6 vector sensors in Fig. 5. Wecan see that the estimation performance is poor, and we can not

Fig. 4. MUSIC spectrum using a nested EM vector-sensor array with 6 sensorsusing the proposed algorithm, as a function of azimuth and elevation angles ,

, , , and true directions ,.

Fig. 5. MUSIC spectrum using a ULA with 6 EM vector sensors, as a functionof azimuth and elevation angles , , , ,and true directions , .

tell where the sources are located. To show the superiority ofour proposed algorithm, we also consider the estimation perfor-mance of the HOEVD-based strategy proposed in [15], whichis plotted in Fig. 6. We can see that our algorithm outperformsthe HOEVD-based method.• Case 3: source, with impinging directions

, , and polarization parametersand .

We consider only one source in this example, but here westudy the estimation performance of the polarization parame-ters. We represent the estimator values in the polarizationplane in Fig. 7, from which we can see the polarization param-eters are estimated well.

B. Detection Performance for the EM CaseIn the previous examples, we assumed the number of sources

to be known. However, in practical situations, we need to de-termine the source number first, before conducting estimation.Using the SORTE method presented in Section III, we inves-tigate the detection performance of the proposed nested EMvector-sensor array.


Fig. 6. MUSIC spectrum using a nested EM vector-sensor array with 6 sensorsusing the HOEVD-based algorithm, as a function of azimuth and elevationangles , , , , and true directions

, .

Fig. 7. MUSIC spectrum using a nested EM vector-sensor array with 6 sensors,as a function of polarization parameters and , , ,

, and true polarization parameters , .

We consider sources, with impinging directions, , and polarization parameters

, , for , 2. The probability of detec-tion of the proposed method using as a function ofSNRs is plotted in Fig. 8. For comparison, we also plot the cor-responding performance of the 6-sensor and 12-sensor ULAs.We define the probability of detection as , where is thetrial number, and is the number of times that is detected.In this example, .We can see that the detection perfor-mance of all the three arrays improves with increasing SNRs. Inaddition, the nested array outperforms the corresponding ULAwith same number of sensors and performs close to the muchlonger ULA.

C. Nested Acoustic Vector-Sensor Array

The performance results of the nested acoustic vector-sensorarray, including DOA estimation and source number detection,are similar to the case of the EM vector-sensor array.

Fig. 8. Probability of detection versus SNR using a nested array with 6 EMvector sensors and ULAs with 6 and 12 EM vector sensors, , ,and true directions , .

Fig. 9. MUSIC spectrum using a nested vector-sensor array with 6 acousticsensors, as a function of elevation angle , , , .

We first consider sources, with impinging directions, and , for

. Fig. 9 shows the MUSIC spectrum with respectto different elevation angles, using snapshots at anSNR of 0 dB. We can see all the six sources are resolved.Next, considering two close sources with

and , weinvestigate the estimation resolution using both a 6-sensornested array and a 6-sensor ULA. The estimation results areshown in Fig. 10 and Fig. 11. We can see that the nested arrayresolves the two sources well, but the ULA with the samesensor number fails.In the end, we consider the source number detection using

acoustic sensors. Suppose we have sources, with im-pinging directions and .The probability of detection of the proposed method using

as a function of SNRs is plotted in Fig. 12, where we con-sider three arrays. We can see that the detection performance issimilar to that of the EM case.


Fig. 10. MUSIC spectrum using a nested acoustic vector-sensor array with 6sensors, as a function of azimuth and elevation angles , , ,

, and true directions , .

Fig. 11. MUSIC spectrum using a ULA with 6 acoustic vector sensors, as afunction of azimuth and elevation angles , , ,

, and true directions , .

Fig. 12. Probability of detection versus SNR using a nested array with 6acoustic vector sensors and ULAs with 6 and 12 acoustic vector sensors,

, , and true directions , .

VI. CONCLUSIONIn this paper, we proposed a novel sensor array model: a

nested vector-sensor array. By exploiting multilinear algebra,we constructed the analytical foundation of the proposed modelfor signal processing. The number of elements in the co-array,namely the DOF, was increased to with only sen-sors. Based on one set of horizontal slices of the matricized in-terspectral tensor, which corresponds to one component of thevector-sensor array, we proposed a novel spatial smoothing al-gorithm to exploit the increased DOF. HOSVD was used toconduct the tensor decomposition, based on which we detectedthe source number and estimated the DOAs of sources. Numer-ical examples demonstrated the effectiveness of the proposedstrategy. The nested array with vector sensors also outperformsthe ULA with vector sensors in terms of estimation resolution.In future work, we will consider using other decomposition

methods instead of HOSVD, and compare their performancewhen used for source number detection and DOA estimation.We will compare different source number detection schemesusing our proposed strategies. In addition, we will find a wayto combine the information of all the components to further im-prove the performance.

APPENDIX ADERIVATION OF (18)

We define two new tensors and as

(A.40)

and(A.41)

They are both tensors. We consider eachelement of :

(A.42)

The second to the last step is due to the independence assump-tion between sources, and to the zero mean assumption. Simi-larly, we get the elements of :

(A.43)

Obviously, , so .


APPENDIX BDERIVATION OF (19)

We use the following notations:

where , .We derive (19) through two steps. First we show that the

mode-2 matricization of , denoted as ,is equal to , denoted as :

Considering any element in tensor , there is a cor-responding element in such that

(B.44)

The second step is shown in Appendix A. Now, we consider thecorresponding element in :

(B.45)

The last two steps are due to the definitions of extendedKhatri-Rao product and matricization. From (B.44) and (B.45),it is obvious that

(B.46)So now we have .

Next, we show that the mode-2 matricization of ,denoted as , is equal to , denoted as :

Based on the white Gaussian noise assumption, we have

if ,otherwise.

(B.47)

Since

(B.48)

we have

if ,otherwise.

(B.49)According to the definition of , we can see that

if ,otherwise.

(B.50)Therefore, we have .Based on the above analysis, we can conclude that

(B.51)

APPENDIX CINTERNAL ANALYSIS OF

The mode-3 unfolding of tensor , can be written as

(C.52)

where

.... . .

...(C.53)

with

(C.54)

Then we can write

(C.55)


where we have sets of horizontal slices, each correspondingto one component. Let the th set be

(C.56)

with element

We can see that there are slices in . Following the defini-tion of the extended Khatri-Rao product, we have

(C.57)

with

(C.58)

where we define the Khatri-Rao minus as. Now, we have provided the closed form of each element of

, and can easily see that there are parallel sets ofhorizontal slices in . Over the sets, the exponentialterms of the corresponding elements are the same.

APPENDIX DGENERATING

Following the analysis in Appendix C, we can see that thereare parallel sets of horizontal slices in (C.55). We considerthe dimensional th set , as defined in (C.56).Looking at the element in (C.57), we can see that the

first dimensional index affects only in the exponential term.Since there are many repeated values in the virtual sensor posi-tion vector in (C.58), we get the corresponding repeated hor-izontal slices in .By removing the repeated horizontal slices and sorting the

remaining ones so that the th slice corresponds to the virtualsensor position in the difference co-array of the2-level nested array, we can construct a newtensor .To make it clearer, we consider the case by fixing the second

and third indexes in :

(D.59)After we remove the repeated elements and sort them accordingto the above strategy, we will have

(D.60)with

(D.61)

Based on (D.60), we can easily obtain by extending theand indexes.

APPENDIX EDERIVATION OF EQUATION (23)

We consider the th set in (C.56). Following (D.60) we canwrite any element of the first subarray tensor as

(E.62)

where , , and . Sim-ilarly, for the th subarray tensor , its element can be writtenas

(E.63)

According to (D.61), we can easily get that

(E.64)

Thus, in the tensor form, we can get

(E.65)

where

(E.66)

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Keyong Han (S’12) received the B.Sc. degree inelectrical engineering from University of Scienceand Technology of China in 2010, and M.Sc. degreein electrical engineering from Washington Univer-sity in St. Louis, MO, in 2012.Currently, he is a Ph.D. degree candidate in

electrical engineering at Washington University inSt. Louis, under the guidance of Dr. Arye Nehorai.His research interests include statistical signal pro-cessing, radar systems, array processing, and tensordecomposition.

Arye Nehorai (S’80–M’83–SM’90–F’94) receivedthe B.Sc. andM.Sc. degrees from the Technion, Israeland the Ph.D. from Stanford University, California.He is the Eugene and Martha Lohman Professor

and Chair of the Preston M. Green Departmentof Electrical and Systems Engineering (ESE) atWashington University in St. Louis (WUSTL). Heis also Professor in the Division of Biology andBiomedical Studies (DBBS), Professor in the De-partment of Biomedical Engineering, and Directorof the Center for Sensor Signal and Information

Processing at WUSTL. Earlier, he was a faculty member at Yale Universityand the University of Illinois at Chicago. Under his leadership as departmentchair, the undergraduate enrollment has more than tripled in the last four years.Dr. Nehorai served as Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL

PROCESSING from 2000 to 2002. From 2003 to 2005 he was the Vice President(Publications) of the IEEE Signal Processing Society (SPS), the Chair of thePublications Board, and a member of the Executive Committee of this Society.He was the founding editor of the special columns on Leadership Reflectionsin IEEE Signal Processing Magazine from 2003 to 2006. He received the 2006IEEE SPS Technical Achievement Award and the 2010 IEEE SPS MeritoriousService Award. He was elected Distinguished Lecturer of the IEEE SPS for aterm lasting from 2004 to 2005. He received a number of Best Paper awards inIEEE journals and conferences. In 2001 he was named University Scholar of theUniversity of Illinois. He was the Principal Investigator of the MultidisciplinaryUniversity Research Initiative (MURI) project titled AdaptiveWaveformDiver-sity for Full Spectral Dominance from 2005 to 2010. He is a Fellow of the RoyalStatistical Society since 1996 and Fellow of AAAS since 2012.

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