6118 ieee transactions on signal processing, vol. 61, …nehorai/paper/han... · and...

6118 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 23, DECEMBER 1, 2013

Improved Source Number Detection and DirectionEstimation With Nested Arrays and ULAs

Using JackknifingKeyong Han, Student Member, IEEE, and Arye Nehorai, Fellow, IEEE

Abstract—We consider the problem of source number detectionand direction-of-arrival (DOA) estimation, based on uniformlinear arrays (ULAs) and the newly proposed nested arrays. AULA with sensors can detect at most sources, whereasa nested array provides degrees of freedom withsensors, enabling us to detect sources with sensors.In order to make full use of the available limited valuable data,we propose a novel strategy, which is inspired by the jackknifingresampling method. Exploiting numerous iterations of subsetsof the whole data set, this strategy greatly improves the resultsof the existing source number detection and DOA estimationmethods. With the assumption that the subsets of the data setcontain enough information, we theoretically prove that theimprovement of detection or estimation performance, comparedwith the original performance without jackknifing, is guaranteedwhen the detection or estimation accuracy is greater than or equalto 50%. Numerical simulations demonstrate the superiority ofour strategy when applied to source number detection and DOAestimation, both for ULAs and nested arrays.

Index Terms—Direction of arrival estimation, jackknifing,nested array, source number detection, uniform linear array.

I. INTRODUCTION

S OURCE number detection and DOA estimation are twomain applications of linear arrays. Source number detec-

tion is often a prerequisite for DOA estimation. The use of aULA for source number detection has received a considerableamount of attention in the last three decades [1]–[9]. Variousmethods have been proposed according to different mathemat-ical criteria. The most commonly used techniques are based oninformation theoretic criteria, such as the Akaike informationcriterion (AIC) [4], the Kullback-Leibler information criterion(KIC) [5], and Rissanen’s minimum description length (MDL)[6] principle. These methods conduct detection by combiningeigenvalue decomposition, the maximum likelihood function,and penalty functions. Another eigenvalue-based method,called the second order statistic of eigenvalues (SORTE) [7],is based on a gap measure of the eigenvalues. A predicted

Manuscript received February 07, 2013; revised May 29, 2013; acceptedSeptember 15, 2013. Date of publication September 25, 2013; date of currentversion November 05, 2013. The associate editor coordinating the review ofthis manuscript and approving it for publication was Prof. Ana Perez-Neira.This work was supported by the AFOSR Grant FA9550-11-1-0210 and ONRGrant N000141310050.The authors are with the Preston M. Green Department of Electrical and Sys-

tems Engineering, Washington University, St. Louis, MO 63130 USA (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.2013.2283462

eigen-threshod (ET) approach was proposed by Chen [8],which detects the number of sources by setting an upper boundon the eigenvalues and then implementing a hypothesis testingprocedure. All the aforementioned methods are based on eigen-values of the sample covariance matrix. Eigenvectors can alsobe used for the determination of sources. Jiang and Ingram[9] proposed an eigenvector-based method by exploiting theproperty of the variance of the rotational submatrix (VTRS).The DOA of a signal is basically estimated by using sen-

sors or antenna arrays [10]. Various theories and techniqueshave been developed for array signal processing related toDOA estimation [11]. Generally, DOA techniques can bebroadly classified into two categories: spectral-based methodsand parametric methods. The spectral-based methods canbe further classified into beamforming techniques [12] andsubspace-based methods, including the multiple signal classi-fication (MUSIC) algorithm [13] and the estimation of signalparameters via the rotational invariance technique (ESPRIT)[14]. However, their performances are generally not satisfac-tory under high-resolution scenarios. To address these issues,[15]–[18] introduce and develop the concept of sparse opti-mization in DOA estimation.Source number detection and DOA estimation are two major

applications of antenna arrays. Both applications are mostlyconfined to the case of ULAs [19]. A ULA with sensorscan resolve at most sources using conventional sub-space-based methods such as MUSIC. A systematic approachto achieve degrees of freedom (DOF) usingsensors based on a nested array was recently proposed in [20],where DOA estimation and beamforming were studied. Thenested arrays are obtained by combining two or more ULAswith increasing spacing. Owing to the property of nonunifor-mity, the resulting difference co-array has significantly moreDOF than the original sparse array, which makes it possiblefor the nested array to detect more sources than the numberof sensors. Pal et al. [21], [22] extended the one-dimensionalnested array to the two-dimensional case, assuming the sensorsto be present on lattices, and providing thorough analysisabout the geometrical considerations and applications. Hanand Nehorai extended the narrowband source case to widebandsource case, especially for Gaussian sources [23]. Anothersimilar nonuniform array, called the co-prime array, was pro-posed and developed in [24]–[26], using sensors toobtain freedom for DOA estimation, where andare co-prime. Both the nested array and co-prime array are

nonuniform linear arrays, and we will transform their signalmodels to equivalent ULAs’ when conducting source numberdetection and DOA estimation.All the existing strategies, for both source number detection

and DOA estimation, exploit all the available data together to

1053-587X © 2013 IEEE

HAN AND NEHORAI: IMPROVED SOURCE NUMBER DETECTION 6119

calculate the whole sample covariance matrix. However, thisfails to make full use of the available limited information. Jack-knifing is a general data-resampling method used in statisticalanalysis. We can consider applying it to the measurement dataunder any scenario. The resampling method replaces theoret-ical derivations in statistical analysis by repeatedly resamplingthe original data and making inferences from the resamples.Quenouille [27] invented this method with the intention of re-ducing the bias of the sample estimate. Tukey [28] extended thismethod to construct variance estimators. Without resting on atheoretical formula that is derived under some model assump-tion, jackknifing shows better robustness, making it less suscep-tible to violation of the model assumptions. The performance ofjackknifing is dependent on the independence of the data. How-ever, extensions of jackknifing to allow for dependence of thedata have been proposed as well [29].The basic idea behind jackknifing lies in systematically re-

computing the statistics, leaving out one or more observationsat a time from the sample set. From this newly generated setof replicates of the statistics, more accurate estimates of thevariables can be calculated. Jackknifing helps fully exploit thereceived data to improve the detection and estimation perfor-mance. Since we exploit numbers of data subsets, and conductthe detection and estimation for each of them, the extra compu-tation would be the cost.In this paper, we will apply the idea of jackknifing to source

number detection and DOA estimation for both ULAs andnested arrays. Specifically, rather than employing various de-tection and estimation algorithms on the covariance matrix ofthe whole data set, we choose to operate on a series of subsets,generated randomly from the whole set of measurements.Combining the results of all the subsets, we choose the valuethat occurs most frequently as the final estimated value. We canshow that this strategy helps improve the accuracy of detectionand estimation. As far as we know, our work here is the firstattempt to apply jackknifing to source number detection andDOA estimation. In this paper, we will propose a sufficientcondition for the improvement of jackknifing.The rest of the paper is organized as follows. In Section II,

we present the signal models for both ULAs and nested arrays.We introduce the pre-processing strategy, spatial smoothing, fornested arrays, and present several source number detection andDOA estimation methods in Section III. In Section IV, we willpropose and analyze the novel strategy algorithm, jackknifing,for array signal processing. In Section V, we use numerical re-sults to show the advantage of our proposed strategy. Our con-clusions and directions for possible future work are containedin Section VI.

II. SIGNAL MODEL

A. Uniform Linear Array

Suppose we have sensors, with equal spacing betweensensors, as shown in Fig. 1(a). We assume narrowbandsources are in the surveillance region, impinging on this lineararray from directions . We can obtain thereceived signal as

(1)

where is the received signalvector at the sensors at time . Let be thesteering vector with the th element . is the

Fig. 1. (a) An N-sensor ULA with equal spacing . (b) A 2-level nested arraywith sensors in the inner ULA, and sensors in the outer ULA, withintersensor spacings and respectively.

position of th sensor, which is an integer multiple of the basicspacing . denotes the carrier wavelength. Then the mani-fold matrix can be expressed as .

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, and uncorrelated with the sources.Based on our assumptions, the source autocorrelation matrixis diagonal: . Then the autocor-

relation matrix of the received signal is

(2)

where is the noise power, and is the identity matrix.Suppose we have snapshots. Stacking all the measurements

together, we rewrite (1) as

(3)

where• , an matrix,• , an matrix, and• , an matrix.

B. Nested Array

We assume there is a nonuniform linear nested array withsensors, consisting of two concatenated ULAs. Suppose the

inner ULA has sensors with intersensor spacing and theouter ULA has sensors with intersensor spacing

, as shown in Fig. 1(b). With the same assumptions as forthe N-sensor ULA, we can get the similar signal model as (1):

(4)

where the matrix . Thedifference from in the -sensor ULA is that the th elementof the steering vector is , with beingthe integer multiple of the basic spacing or . Thus, theautocorrelation matrix of the received signal for nested array is

(5)

Vectorizing , we get

(6)


where , and ,with being a vector of all zeros except a 1 at the th position.We can view vector in (6) as some new longer received signalswith the new manifold matrix , and the new sourcesignals . Here, denotes conjugation without transpose, anddenotes the Khatri-Rao product.We will conduct detection and estimation based on the con-

structed signal model (6) for nested arrays.

III. SOURCE DETECTION AND DOA ESTIMATION

We will use the ULA and nested array mentioned above toperform source detection and DOA estimation. First, we willbriefly introduce spatial smoothing [20], which is used to exploitthe increased DOF generated by the nested array. Then, we willpresent four source detection methods and the well known DOAestimation approach, MUSIC.

A. Spatial Smoothing

In order to exploit the increased DOF provided by theco-array, we need to apply spatial smoothing. We remove therepeated rows from and also sort them so that the throw corresponds to the sensor locationin the difference co-array of the 2-level nested array, giving anew vector:

where is a vector of all zeros except a 1at the center position.The difference co-array of this 2-level nested array has sen-

sors located at

We now divide these sensors intooverlapping subarrays, each with elements, wherethe th subarray has sensors located at

. The th subarray corresponds to theth to th rows of , denoted

as

(7)

We can check that , where.

Viewing as a newly received vector, we can get the equiva-lent covariance matrix . Taking the average of ,we get

(8)

The spatially smoothed matrix enables us to perform de-tection of sources with sensors.We can further show that , where

, with

.... . .

...

and . Here, , and. When applied on a ULA

with sensors whose manifold matrix is , thematrix has the same form as the conventional covariancematrix used in source detection techniques. We will employ

when conducting source number detection and DOAestimation.

B. Source Number Detection

As mentioned in the introduction, the sample covariance ma-trix is a key element for source detection. Considering a uniformlinear array with sensors, based on model (3), the sample co-variance matrix is

We do eigenvalue decomposition:

where

(9)

are the eigenvalues and

(10)

is the corresponding eigenvector matrix. Suppose the eigen-values are sorted decreasingly:

Researchers have been developing numerous detection methodsbased on different techniques, including eigenvalues, eigenvec-tors, and information theory. Four popular detection methodsare presented as follows:• SORTEA gap measure is defined:

where , and

Then the source number is .• VTRSSuppose is the combined signal eigenvectors of ,and and are the first rows and lastrows of respectively. Solving basedon the least square criterion, we get matrix . Define

. Then the source number is

where is the Frobenius norm.


• ETDefine the eigen-threshold

(11)where is a pre-set parameter, and

Based on this, we keep testing the binary hypothesis:and . Accept or

according to . If is accepted,then we set , and continue. Otherwise, if isaccepted, stop testing, and assign .

• AICDefine

and . Then the sourcenumber is determined as

C. MUSIC

DOA estimation is based on the condition that we alreadyknow, 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 . Then the

noise subspace is formed by a matrix containing the noiseeigenvectors:

The cornerstone of MUSIC is the remarkable observation thatthe steering vectors corresponding to signal components are or-thogonal to the noise subspace eigenvectors:

Therefore, for , corresponding tothe th incoming signal. We define the MUSIC spectrum as

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.

IV. JACKKNIFING ARRAY PROCESSING

All the existing methods for array processing, includingsource number detection and DOA estimation, are based onthe eigenvalues or eigenvectors of the sample covariancematrix , which is calculated over the entire sample data set.However, the received data can tell us more.Researchers have been using all the measurements as a whole

to get the sample covariance, then proceeding further based on

this covariance matrix. Here, we will make full use of the re-ceived data, achieving more accurate detection and estimation.Jackknifing is an effective strategy used in the statistical areato estimate sample statistics [29]. The idea is to use subsets ofavailable data to improve the estimation performance. For bothsource number detection and DOA estimation, we propose anovel array signal processing strategy based on the idea of jack-knifing. Our basic belief is that a large proportion of the avail-able data contains approximately the same amount of informa-tion as the whole available data set does.

A. Source Number Detection Using Jackknifing

Suppose we have snapshots in total:

First, we take a subset of size from the snapshotsmatrix :

where , , and , with expressed asa percentage and satisfying . The low constraintfor helps to guarantee our basic belief that the subsets containenough information, whereas the high constraint guarantees thatthe subsets will not make exactly the same decision as the wholedata set does. Specifically, we randomly pick elements from, without replacement, to form . The sample covariance

based on is

Then we do eigenvalue decomposition for :

(12)

where , sorted non-increasingly.Using and , we conduct source detection using theexisting methods. Suppose we obtain the source number . Wecontinue the above two procedures for iterations, obtainingestimated source numbers, . Before making

the decision of the final source number, we need one more step,counting the occurrence of each estimated number, denotedas , with summation . The final sourcenumber is chosen as the one that occurs most frequently:

The algorithm is shown in Table I.When the detection accuracy is greater than 50%, the im-

provement using jackknifing is guaranteed by the followingtheorem.Theorem 1: If the source number detection accuracy using

the whole data set is , then the detection accuracy ,after applied jackknifing, will be greater than or equal to :.Proof: Recall our basic assumption that the jackknifing

subset contains almost the same information as the originalwhole data set. Therefore the detection accuracy based on ajackknifing subset is assumed to be , satisfying .


TABLE IALGORITHM FOR SOURCE DETECTION USING JACKKNIFING

Suppose we conduct iterations, and the detected sourcenumber is . Then the probability of correct detection for anyindependent iteration is

The false detection probability is denoted as

Let denote the occurrence times of the number , anddenote the occurrence times of other numbers

except . We consider the proof through two cases: or; namely, the iteration number is even or odd.

Case 1: : According to the jackknifing algorithmin Table I, we can obtain the detection accuracy after applyingjackknifing:

(13)

Considering

(14)

we get

Applying

we get

Since , namely , we have

(15)

Case 2: : Similarly, we can calculate the detectionaccuracy after applying jackknifing as

(16)

The above two cases together prove the theorem: for anynumber of iterations , after applied jackknifing, the source de-tection accuracy .


Remarks:• When , there are terms in (13), which canbe split into terms according to (14). Note that thelast term is transformed into one element. However theexpansion of just has elements, thus thereis an extra term, resulting in the strict ‘ ’ in (15). As forthe case , we have elements in (16), whichcan be split into terms. This is exactly the numberof the expansion terms of , leading to ‘ ’ for odditeration numbers.

• Equation (15) provides a lower bound for the improvementof jackknifing when the iteration number is even:

However, when the iteration number is odd, it is not ob-vious to find a lower bound because of the result “ ”.

• Theoretically, when the detection accuracy is higher than50%, the detection performance with jackknifing will def-initely be improved. With over 50% accuracy, the cor-rect number should be detected more frequently than othernumbers. This assumption is based on the condition that thejackknifing sample subset contains enough information toguarantee over 50% accuracy. Otherwise, the jackknifingwill lose its power.

• For methods that are sensitive to the sample number, weneed to increase the sample number to guarantee the ef-ficiency of jackknifing. For example, one method mightperform well with the whole samples. However, whenapplying jackknifing, we just use samples, in whichcase this method may achieve an accuracy lower than 50%.Consequently, jackknifing provides no improvement. Al-ternatively, we can adjust the value of to guarantee theaccuracy.

• When there is a low SNR, namely a high noise level, thedetection methods may fail to detect the source numbercorrectly, with accuracy lower than 50%. This will causejackknifing to perform badly, as discussed in the secondremark.

• One thing to note is that we choose the value that occursmost frequently as the final source number. One problem inthis process is that ties may exist. However, the greater theSNR or the larger the sample number, the less frequentlyties happen. Equivalently, when the detection accuracy isgreater than 0.5, the probability that ties happen will berelatively low. Therefore, based on the assumption that

, we expect the ties rarely happen, and will verifythis through our numerical examples. When ties happen,we just choose the first one after arranging them in a de-scending order. Even if the choice is wrong, this can beignored because of our assumption that .

B. DOA Estimation Using Jackknifing

Similar to the previous discussion for source number detec-tion using jackknifing, we choose the subset of size fromthe snapshots matrix , and obtain the sample covariance ma-trix . According to (12), we get the sample noise subspace

which consists of the last eigenvectors corresponding tothe smallest eigenvalues.

TABLE IIMUSIC FOR DOA ESTIMATION USING JACKKNIFING

The impinging direction of the signal is a continuous variable,so it is impossible for us to conduct an exhaustive search overall the direction space for the sample spectrum:

(17)

Consequently, we will not be able to apply jackknifing to DOAestimation. To circumvent this problem, we discretize the direc-tion space into grid points:

Then the estimated DOA is

(18)

Since we can have at most different estimated DOAs, jack-knifing is suitable for DOA estimation. We iteratively choosesubsets from the whole data set, and employ MUSIC based

on the sub-covariance matrices. Based on the estimated di-rections, we count the occurrence of different entries, and con-sider the DOA to be the one that has the largest frequency.The MUSIC algorithm, applied with jackknifing, is shown inTable II.Similar to source number detection, we have the comparative

theorem for DOA estimation.Theorem 2: If the DOA estimation accuracy using the whole

data set is , then the DOA estimation accuracy , afterapplied jackknifing, will be greater than or equal to : .

Proof: The proof is similar to that for Theorem 1. Just notethat the DOA estimation accuracy is defined as the probabilityof the event that the estimated direction is equal to the truedirection over the direction space .

C. Nested Array Using Jackknifing

As for array processing for source number detection andDOA estimation, the difference between a nested array anda ULA is the covariance matrix we are using. For ULAs, wesimply apply the sample covariance matrix, whereas for nestedarrays, we need to construct the spatially smoothed matrixin (8) for each iteration when using jackknifing. The algorithmfor a nested array using jackknifing is shown in Table III.


TABLE IIINESTED ARRAY PROCESSING USING JACKKNIFING

V. NUMERICAL EXAMPLES

In this section, we use numerical examples to show the su-periority of our proposed strategy, considering source numberdetection and DOA estimation, for both ULAs and nested ar-rays. To show the effectiveness of jackknifing, we also considerthe coprime arrays [24] for source number detection.We consider the following four scenarios:Scenario 1: We consider a ULA with sensors and a

nested array with sensors.• For the 8-sensor ULA:— Sensor position , with spacing .— Source number .— Impinging directions .— Impinging directions .— Source power .

• For the 6-sensor nested array:— Sensor position , with spacing ,and .

— Source number .— Impinging directions

.— Source power .

Scenario 2: We consider a ULA with sensors and anested array with sensors. For both arrays, we supposethere is only one source, with impinging direction , andpower .• For the 6-sensor ULA:— Sensor position , with spacing .


Scenario 3: We consider a ULAwith sensors, a nestedarray with sensors, and a ULA with sensors.For all the three arrays, we suppose there are two sources, withimpinging direction or , andpower .• For the 6-sensor ULA:— Sensor position , with spacing .


• For the 12-sensor ULA:

— Sensor position , with spacing.

Scenario 4: We consider a coprime array with coprime num-bers 3 and 5. Therefore, we have 10 sensors.• Sensor position [0 3 5 6 9 10 12 15 20 25]d, with spacing

.• Source number .• Impinging directions

.

A. Source Number Detection

We consider four cases for source number detection: twocases for scenario 1, one case for scenario 3, and one case forscenario 4.• The ULA in scenario 1.We choose a jackknifing iteration number , the per-

centage of , and the Monte Carlo simulation number. Fig. 2 shows the detection results of the aforemen-

tioned four different methods: SORTE, VTRS, ET, and AIC,with impinging direction . It describes the detection accuracywith respect to different SNRs. We take the SNR as

and the detection accuracy as

where is the trial number, and is the number of times thatis detected. In this example, we use trials.In Fig. 2, all four methods achieve different levels of im-

provement by applying jackknifing. SORTE improves the most,and performs even better when the SNR is low. Note that the de-tection accuracy is always above 0.5 without jackknifing, whichguarantees the improvement of jackknifing. For ET, the perfor-mance is highly related to the appropriate choice of parameterin (11). The decision of this value depends on a priori infor-mation, such as the probability density function of false alarm,SNR level, etc. In many applications some of the information isnot available, in which case the parameter must be chosen basedon empirical decision. In our example, .The computation time with jackknifing is highly related to

the number of iterations . The larger is, the longer it takesusing jackknifing. When we use jackknifing, another part thatconsumes time is randomly picking a subset from the wholedata set iteratively. Therefore, it may take more than timesthe computation cost of the original algorithm. The computa-tion burdens of the four methods are shown in Table IV, with

and . We can see that, as a computer-in-tensive method, jackknifing does cost much more time than thecase without jackknifing. However, because of the availabilityof inexpensive and fast computing, jackknifing is still appreci-ated by current researchers.We also calculated the detection accuracy for impinging di-

rection , which has smaller source spacing. Fig. 3 shows theresults of the four methods using samples. We cansee that jackknifing improves the detection performance muchas in Fig. 2 for SORTE, ET, and AIC. However, VTRS loses itsdetection ability at small SNRs. One thing to note is that, onceVTRS works, namely the detection accuracy is greater than 0.5,jackknifing works as well. We can see this through the high SNRpoints.• The 2-level nested array in scenario 1.


Fig. 2. Performance comparison of four methods with ULA using 1000 sam-ples for : the blue-star line is the performance with jack-knifing, and the red-circle line without jackknifing. The vertical axis representsthe detection accuracy, while the horizontal axis represents the SNR.

TABLE IVCOMPUTATION TIME FOR VARIOUS METHODSBASED ON AN 8-SENSOR ULA WITH OR WITHOUT

JACKKNIFING, WHERE ,

The spacings are and . It is impossiblefor us to use a 6-element ULA to detect 8 sources. Howeverthe spatial matrix in (8) helps a nested array obtain thisgoal. We choose jackknifing iteration number , and thepercentage . We use trials. From Fig. 2, wecan see that SORTE and VTRS perform a little better, thus weconsider only these two methods for the nested array.Fig. 4 shows the performance of SORTE and VTRS, with

and without jackknifing, using Monte Carlo simula-tions. We can see that at high SNR both methods can detect thesource number correctly with high probability. Moreover, withjackknifing, both methods’ detection accuracy increases. In thisexample, we can see that, at high SNRs, the improvement isgreater than that at low SNRs. That jackknifing’s performancedegrades at low SNRs is in accordance with our previous anal-ysis. Additionally, SORTE slightly outperforms VTRS.In Table V, we tabulate the computation time for SORTE and

VTRS with and without jackknifing, where and. The results are similar to the case of ULA.Note that, for jackknifing, we choose the percentage. However, this may not be the best value. In Fig. 5, we

plot the detection accuracy with respect to different percentagevalues for both SORTE and VTRS, at an SNR of 24 dB using

snapshots. We can see is the best choicefor both VTRS and SORTE, which confirms our statement thatshould be moderate, neither too big nor too small. To have aclearer picture of how the percentage value affects the detectionperformance, we list the best for different numbers of snap-shots in Table VI, where we consider just SORTE. We can seethat when the snapshot number increases, the best decreases,which means that larger size of subset may result in worse per-formance. Therefore, to achieve good performance using jack-knifing given a certain number of samples, we should be carefulwhen choosing the percentage parameter.

Fig. 3. Performance comparison of four methods with ULA using 1000 sam-ples for : the blue-star line is the performance with jack-knifing, and the red-circle line without jackknifing. The vertical axis representsthe detection accuracy, while the horizontal axis represents the SNR.

Fig. 4. Performance comparison of SORTE and VTRS with a nested arrayusing 2000 samples.

TABLE VCOMPUTATION TIME FOR SORTE AND VTRSBASED ON A 6-SENSOR NESTED ARRAY WITH OR WITHOUT

JACKKNIFING, WHERE ,

• Scenario 3.Since a nested array takes advantage of the increased DOF

provided by the co-array, we expect its detection accuracy toimprove greatly. Similar to the aforementioned examples, weplot the detection accuracy versus SNR for , shown in Fig. 6using snapshots. The trial number is 1000, the per-centage , and the jackknifing iteration number is setas . Clearly, the two-level nested array outperforms thecorresponding ULA with same number of sensors and performs


Fig. 5. Detection accuracy of SORTE and VTRS for different percentagevalues, with a nested array at an SNR of 24 dB using snapshots.

Fig. 6. SORTE performance comparison of a 6-sensor nested array, a 6-sensorULA, and a 12-sensor ULA using 1000 samples for , with andwithout jackknifing.

TABLE VIBEST PERCENTAGE VALUES FOR DIFFERENT NUMBER OF SNAPSHOTSUSING SORTE WITH A NESTED ARRAY, AT AN SNR OF 24 dB

close to the much longer ULA. Moreover, jackknifing helps allthree arrays achieve substantial improvement.We also calculated the detection accuracy for impinging di-

rection , which has smaller source spacing. Fig. 7shows the results using samples. We can see that the6-sensor nested array and 12-sensor ULA work the same as thewide spacing case. As for the 6-sensor ULA, the performancedegrades at low SNRs. However, once the detection accuracywithout jackknifing is greater than 0.5, jackknifing works well.

Fig. 7. SORTE performance comparison of a 6-sensor nested array, a 6-sensorULA, and a 12-sensor ULA using 1000 samples for , with andwithout jackknifing.

Fig. 8. Performance comparison of SORTE and VTRS with a coprime arrayusing 1000 snapshots.

The result is similar to the aforementioned VTRS in the case ofULA.• Scenario 4.The coprime array is a nonuniform linear array, similar to the

nested array.Wewant to use 10 sensors to detect 12 sources. Justas for the nested array, we employ SORTE and VTRS to conductdetection. Fig. 8 shows the performance of SORTE and VTRSover different SNRs, with and without jackknifing, using

snapshots. The results are similar to those of nested array:jackknifing effectively improves the detection performance.Similar to Table IV and Table V, we tabulate the computation

time in Table VII for SORTE and VTRS based on the coprimearray, with and without jackknifing. We take and

. We can see that the jackknifing strategy needs morethan 50 times the computation time than without jackknifing.To investigate the effect of the percentage value , in Fig. 9 we

plot the detection accuracy with respect to different percentagevalues for both SORTE and VTRS, at an SNR of 24 dB using


TABLE VIICOMPUTATION TIME FOR SORTE AND VTRS

BASED ON A 10-SENSOR COPRIME ARRAY WITH OR WITHOUTJACKKNIFING, WHERE ,

TABLE VIIIBEST PERCENTAGE VALUES FOR DIFFERENT NUMBERS OF SNAPSHOTSUSING VTRS WITH A COPRIME ARRAY, AT AN SNR OF 24 dB

Fig. 9. Detection accuracy of SORTE and VTRS for different percentagevalues, with a coprime array at an SNR of 24 dB using snapshots.

snapshots. The results are similar to the case of thenested array. From the figure, we can see the best percentagefor SORTE is 0.65, while it is 0.6 for VTRS. In addition, we listthe best for different numbers of snapshots in Table VIII forVTRS, from which we can see the trend is the same as the caseof the nested array: as the snapshot number increases, the bestdecreases.

B. DOA Estimation

Considering scenario 2, we split the direction space bychoosing , namely . Byapplying the algorithm in Table II, we get the results for botha ULA and a nested array in Fig. 10 and Fig. 11, respectively.Besides the estimation accuracy, we also plot the root meansquare error (RMSE) versus SNR. We can see that the perfor-mance is slightly improved with jackknifing. Furthermore, thenested array performs better than the ULA, especially at highSNRs, with higher estimation accuracy and a smaller RMSE.The improvements in DOA estimation using jackknifing are

less apparent than those for source number detection. The rea-sons, we believe, are mainly owing to the following issues. First,the size of the result space of DOA estimation is much largerthan that of source number detection, with for DOAestimation versus, at most, for source number detec-tion. Second, the DOA estimation may be more sensitive to

Fig. 10. DOA estimation using a ULA with 6 sensors: the top figure is theestimation accuracy versus SNR, and the bottom figure shows the RMSE versusSNR.

Fig. 11. DOA estimation using a nested array with 6 sensors: the top figure isthe estimation accuracy versus SNR, and the bottom figure shows the RMSEversus SNR.

the sample number: with snapshots, the estimation accuracymay degrade more than the source number detection does. An-other possible explanation is that the the nested or ULA arraymay provide small DOA estimation errors which are alreadyclose to the CRB. Thus the improvement obtained from jack-knifing cannot be very significant.

VI. CONCLUSION

In this paper, by applying the resampling strategy jackknifing,we proposed a novel strategy for source number detection and


DOA estimation. Iteratively employing subsets of the wholedata set, the strategy greatly improves the performance of the ex-isting detection and estimation by making full use of the limitedavailable data. Using jackknifing, we investigated four sourcenumber detection approaches based on different principles, aswell as the MUSIC algorithm for DOA estimation. All achievedifferent levels of improvement. With the assumption that thesubsets of the data set contain enough information, we analyt-ically proved that the improvement is guaranteed when the de-tection or estimation accuracy is greater than or equal to 50%.Both ULAs and the newly developed nested arrays are consid-ered. The advantage of our strategy was verified through simu-lations. The expense for using jackknifing is the higher compu-tation burden, more than times that of the case without jack-knifing. Additionally, we investigated the performance effect ofthe percentage parameter we choose when doing jackknifing,finding that a moderate value is the best choice, and either alarger or smaller subset will degrade the performance.In future work, we will work on the bound on the jackknifing

improvement. In addition, we will find a better way to improveDOA estimation using jackknifing, and investigate this problemfrom the CRB viewpoint. Further, we will consider the DOAestimation for multiple sources, where we will consider thesources pair-matching problem for different subsets when usingjackknifing. We will also consider applying this strategy toother statistical problems.

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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 the M.Sc.degree in electrical engineering from WashingtonUniversity, St. Louis, MO, in 2012.Currently, he is a Ph.D. degree candidate in

electrical engineering at Washington University, St.Louis, under the guidance of Dr. A. Nehorai. His re-search interests include statistical signal processing,radar systems, and array processing.

Arye Nehorai (S’80–M’83–SM’90–F’94) receivedthe B.Sc. and M.Sc. degrees from the Technion,Israel, and the Ph.D. degree from Stanford Univer-sity, CA.He is the Eugene and Martha Lohman Professor

and Chair of the Preston M. Green Departmentof Electrical and Systems Engineering (ESE),Washington University in St. Louis (WUSTL). Heis also Professor in the Division of Biology andBiomedical Studies (DBBS) and Director of theCenter for Sensor Signal and Information Processing

at WUSTL. Earlier, he was a faculty member at Yale University and theUniversity of Illinois at Chicago. Under his leadership as department chair, theundergraduate 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 Reflections inIEEE SIGNAL PROCESSINGMAGAZINE 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 Best Paper awards in IEEE journalsand conferences. In 2001, he was named University Scholar of the Universityof Illinois. He was the Principal Investigator of the Multidisciplinary UniversityResearch Initiative (MURI) project titled Adaptive Waveform Diversity for FullSpectral Dominance from 2005 to 2010. He has been a Fellow of the RoyalStatistical Society since 1996 and Fellow of AAAS since 2012.

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