IEEE SIGNAL PROCESSING LETTERS, VOL. 21, NO. 9, SEPTEMBER 2014 1111

Nested Array Processing for Distributed SourcesKeyong Han, Student Member, IEEE, and Arye Nehorai, Fellow, IEEE

Abstract—We consider the problem of using linear nested arraysto estimate the directions of arrival (DOAs) of distributed sourcesand to detect the source number, where we have more sources thanactual physical sensors. Angular spread, caused by the multipathnature of the distributed sources, makes the commonly used point-source assumption challenging. We establish the signal model fordistributed sources, using a nested array. Due to the characteris-tics of distributed sources, the regular spatial smoothing technique,which is used to exploit the increased degrees of freedom providedby the co-array, no longer works. We thus propose a novel spa-tial smoothing approach to circumvent this problem. Based on theanalytical results, we construct the corresponding DOA estimationand source number detectionmethods. The effectiveness of the pro-posed methods is verified through numerical examples.

Index Terms—Direction of arrival estimation, distributedsource, nested array, source number detection.

I. INTRODUCTION

D IRECTION OF ARRIVAL (DOA) estimation and sourcenumber detection are two major applications of antenna

arrays [1]–[3]. However, most existing results are based onsignal and noise models which assume that the signals are prop-agated from point sources. However, in practice, signal sourcesmay often be transmitted by reflection, causing angular spread.Besides, the reflective medium may often be dispersive, thusmaking the point-source assumption questionable. Distributedsources have received a considerable amount of attention in thelast two decades [4]–[6], and are the focus of this letter.Most existing strategies, for both point and distributed

sources, are confined to the case of the uniform linear array(ULA) [7]. Using conventional subspace-based methods suchas MUSIC, a ULA with scalar sensors can resolve at most

sources. Based on a nested array, a systematic approachto achieve degrees of freedom (DOFs) usingsensors was recently proposed in [8], where DOA estimationand beamforming were studied. The jackknifing strategy wasapplied to nested arrays in [9], improving the accuracy ofsource number detection and DOA estimation. Extension towideband sources using nested arrays was investigated in [10],and extension to nested vector-sensor arrays in [11]. Note that

Manuscript received March 31, 2014; accepted May 14, 2014. Date of publi-cation May 16, 2014; date of current version May 23, 2014. This work was sup-ported by the Air Force Office of Scientific Research under Grant FA9550-11-1-0210 and by the Office of Naval Research under Grant N000141310050. Theassociate editor coordinating the review of this manuscript and approving it forpublication was Prof. Muhammad Zubair Ikram.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: kyhannah@ese.wustl.edu; nehorai@ese.wustl.edu).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/LSP.2014.2325000

the nested-array strategy used in [9] and [10] is based on thepoint-source model as in [8].In this letter, we establish the signal model for distributed

sources using a nested array. To exploit the increased DOFsprovided by the difference co-array, we propose a novel spa-tial smoothing approach with a priori knowledge of the angularspreading parameters. Based on the analytical results, we willconstruct corresponding source number detection and DOA es-timation methods.The rest of the letter is organized as follows. In Section II, we

establish the signal models of the nested array for distributedsources. Then we propose a new spatial smoothing techniquein Section III. In Section IV, we propose corresponding sourcenumber detection and DOA estimation methods based on ouranalytical results. In Section V, we use numerical results toshow the effectiveness of our proposed strategies. Our conclu-sions and directions for possible future work are contained inSection VI.

II. SIGNAL MODEL

We consider a nonuniform linear nested array [8] withscalar sensors along the -axis, consisting of two concatenatedULAs. Suppose the inner ULA has sensors with inter-sensor spacing , and the outer ULA has sensors withintersensor spacing . Further suppose thesensor positions are , where

is an integervector containing the sensors’ position information.First, consider the 1-dimensional impinging source directions

. Then, the signal model can be written as

(1)

where is the received signalvector at the sensors at time . Let be the steeringvector, , wheredenotes the carrier wavelength. Then the manifold matrix canbe expressed as

(2)

is the source vector. We sup-pose the source signals follow Gaussian distributions,

, and they are all independent of each other. The noisesignal is assumed to be whiteGaussian with power , and uncorrelated with the sources.Next, we consider the case of the distributed source, which is

a generalization of the collection of -point sources [4]. Sucha source is usually described by a distributed source densitythat indicates the amount of source power coming from each di-rection. Denoting the distributed source density by , wehave , where is a function ofthe signal envelope and unknown parameters. We consider oneclass of distributed sources employed in [4], for which

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1112 IEEE SIGNAL PROCESSING LETTERS, VOL. 21, NO. 9, SEPTEMBER 2014

, with and .Then the output can be expressed as

(3)

If we write the steering vector as

(4)

with , and , following the analysis in [4]we can obtain

(5)

where , and the symbol ‘ ’ denotesthe Hadamard product between two vectors.Therefore, for distributed sources, the output of the nested

array can be written as

(6)

with . The covari-ance matrix of is

(7)

Vectorizing in (7), we get a long vector

(8)

where , and ,with being a vector of all zeros except for a 1 at the th po-sition. We can view the vector as representing new longer re-ceived signals with the new manifold matrix , and newsource signals . The notation means conjugate without trans-pose, and denotes the Khatri-Rao product. The distinct rowsof behave like the manifold of a longer array whosesensors are located at positions given by distinct values in theset .We write as

, with

. To obtain the expression of, we first define two operations between two vectors.

Employing the integer vector , we define Khatri-Raoaddition as , withelement , and Khatri-Rao differenceas , with element

. Based on and , we can get.

III. SPATIAL SMOOTHING

To exploit the increased DOF offered by the differenceco-array, we propose to apply the spatial smoothing techniquein a new fashion based on what is presented in [8], wherethe point source model is employed. Since the strategy is notsuitable for distributed sources, as presented in this section, wethus propose a new fashion of spatial smoothing by exploitinga priori knowledge of the spreading parameters.Considering in model (8), we remove the repeated

rows from and also sort them so that the th row cor-responds to the sensor location in the difference

co-array of the 2-level nested array, with ,giving a new vector:

(9)

where is a vector of all zerosexcept for a 1 at the center position. Denote

and. Then

we can obtain . The integer set, and is the

corresponding integer exponent of when is fixed. Wedenote .As for point sources, according to the analysis in [8], we can

get the corresponding vector by vectorizing the covariancematrix: , with . We cansee that the point source model is a special case with .Additionally, the point source model has a Vandermonde arraymanifold with unit circle entries, whereas the distributed sourcemodel produces a Vandermonde array manifold with non-unitcircle entries. This makes the typical point-source strategy un-suitable for distributed sources. Thus, we will propose a novelstrategy by exploiting a priori knowledge of the spreading pa-rameters. First we consider a special case when all the sourceshave the same distribution parameter, denoted as . This as-sumption is reasonable when the sources are similar.Observing the structures of and , we can verify

that , where .Thus we have . The difference co-arrayof this 2-level nested array has sensors located at

. We now divide thesesensors into overlapping subarrays, each with

elements, where the th subarray has sensors located at. The th subarray

corresponds to the th to th rows of ,denoted as

(10)

where ,and is the corre-

sponding th submatrix of . Specifically, the first submatrix.

Provided a priori knowledge about the spreading parameter, we can conduct the following transformation:

(11)

Recall that for point sources, we have the th subarray vector. Comparing with ,

we can see that the difference is the noise term, from to. We will show that the resulting noise term, which

contains the distributed source parameters, would not affect theestimation and detection performance of nested arrays.Based on (11), we can obtain the spatially smoothed matrix

for distributed sources:

(12)

HAN AND NEHORAI: NESTED ARRAY PROCESSING FOR DISTRIBUTED SOURCES 1113

where . enables us to perform DOA estimationof distributed sources with sensors, as proved by thefollowing theorem:Theorem 1: The spatially smoothed matrix in (12) can

be expressed as where

(13)

Proof: First, we consider

Note that is a real diagonal matrix, so .Since is a vector with all zeros except a 1 at the th position,

and , we can calculatethat

(14)

where is the th element of . Therefore, we have

(15)

Then, we can calculate [8]:

(16)

with . Finally we can get

(17)

with .As mentioned before, the effect of the distributed source

model is to produce a Vandermonde array manifold withnon-unit circle entries, which contributes to the noise variancein with spreading parameters. has the same form as theconventional covariance matrix of the signal received by alonger ULA consisting of sensors. The equivalent arraymanifold is represented by . Thus, we can apply subspacebased methods like MUSIC to identify up to sources.The above analysis is based on the same spreading parameterfor all sources. When the parameters are different, we cannot

easily find the equivalent matrix in (10). Thus, we will notbe able to obtain the simple form (11), and further we cannotachieve results similar to those in Theorem 1. Nevertheless, toinvestigate its performance with different , we propose to usethe average of to replace the in .

IV. SOURCE NUMBER DETECTION AND DOA ESTIMATION

We will use the nested array mentioned above to per-form source number detection using SORTE [3] and performDOA estimation using MUSIC [2]. For both, we use thesample covariance, calculated from the measurements: .In our case, for distributed sources using nested arrays, wewill use for estimation and detection. Based on ,we do eigenvalue decomposition: ,where are the eigenvalues and

is the corresponding eigenvector ma-trix. We assume that the eigenvalues are sorted decreasingly:

.

A. Source Number Detection

SORTE is an eigenvalue-based strategy. To detect the sourcenumber, we define a gap measure:

where , and

Then the source number is .

B. DOA Estimation

Suppose the source number is known or has been detected.Then the noise subspace is formed by a matrix containing thenoise eigenvectors, . Based onthe observation that the steering vectors corresponding to signalcomponents are orthogonal to the noise subspace eigenvectors,we have . There-fore, for , corresponding to the thincoming signal. We define the MUSIC spectrum as

. Then, to obtain the DOA estimates, we conductan exhaustive search over the impinging direction space, com-pute the MUSIC spectrum for all direction angles, and find thelargest peaks.

V. NUMERICAL EXAMPLES

In this section, we use numerical examples to show the ef-fectiveness of our proposed strategy. The nested array we usecontains sensors, with and .

A. MUSIC Spectral

We consider sources using a nested array. Twoclasses of distributed sources are investigated: one with thesame spreading parameter , the other with differentvalues . Note that wehave more sources than sensors. Fig. 1 shows the MUSIC spec-trum after applying the proposed spatial smoothing technique.As can be seen, the proposed strategy can resolve the sevendistributed sources for both cases. The case with the sameperforms better, as expected. Note that it is not always the casethat all the sources can be resolved. With smaller SNRs, fewersamples, or smaller spreading parameters, the probability offalse estimation becomes larger.

1114 IEEE SIGNAL PROCESSING LETTERS, VOL. 21, NO. 9, SEPTEMBER 2014

Fig. 1. MUSIC spectrum of the proposedmethod for two sets of , as a functionof , using a 6-sensor nested array.

Fig. 2. RMSE of estimates of versus SNR, using both the PP-MUSIC andWPP-MUSIC with a 6-sensor nested array.

B. RMSE versus SNR

In this section, we consider one source withspreading parameter . We compare our proposed “priorprocessing” MUSIC (PP-MUSIC) with the regular “withoutprior processing” MUSIC (WPP-MUSIC) [8], by studyingthe root mean squared error (RMSE) of the DOA estimatesversus the signal-to-noise ratio (SNR), which is defined as

. Fig. 2 shows the RMSE of both methodsas a function of SNR for snapshots, averaged over1000 Monte Carlo simulations. We can see that the perfor-mance of both methods improves with increasing SNR, and ourproposed method performs better than the regular method.Note that we fix the spreading parameter at in the

above example. With different , the estimation performancewill improve with increasing . In addition, the PP-MUSIC andWPP-MUSIC will merge at , which is identical to thepoint source model. This is also true for source detection.

C. Source Number Detection

In this section, suppose that we have sources, butthis number is unknown and we need to detect it. We comparethe detection performance of a 6-sensor ULA, a 6-sensor nestedarray, and a 12-sensor ULA. The detection probability versusSNR is depicted in Fig. 3. The detection probability is definedas , where is the trial number and is the number of

Fig. 3. Detection probability of the proposed method versus SNR, witha 6-sensor ULA, a 6-sensor nested array, and a 12-sensor ULA: ,

, and .

times that is detected. We can see that the two-level nestedarray outperforms the corresponding ULAwith same number ofsensors and performs close to the much longer ULA.

VI. CONCLUSION

We considered distributed source processing using linearnested arrays. We established the nested array model for dis-tributed sources. We proposed an improved spatial smoothingstrategy by exploiting a priori knowledge of the spreading pa-rameter, which enables a nested array with sensors to detect

distributed sources. The results were verified throughnumerical examples. In our future work, we will consider moreefficient strategies for the general case with different spreadingparameters.

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