Received April 19, 2018, accepted May 24, 2018, date of publication May 31, 2018, date of current version June 20, 2018.

Digital Object Identifier 10.1109/ACCESS.2018.2842262

Fast DOA Estimation Algorithms for SparseUniform Linear Array With MultipleInteger FrequenciesAIHUA LIU , XIN ZHANG, QIANG YANG, AND WEIBO DENGDepartment of Electronic and Information Engineering, Harbin Institute of Technology, Harbin 150001, China

Corresponding author: Qiang Yang (yq@hit.edu.cn)

This work was supported in part by the National Natural Science Foundation of China under Grants 61171182 and 61032011, and in partby the Fundamental Research Funds for Central Universities under Grants HIT.MKSTISP.2016 13 and HIT.MKSTISP.2016 26.

ABSTRACT The sparse uniform linear array (SULA) with larger array aperture is adopted to improve theaccuracy of direction of arrival (DOA) estimation with a limited number of physical antennas. In the case ofsingle working frequency configuration, spatial aliasing problem occurs due to the spatial under-sampling.However, with a sufficient anti-aliasing condition for multiple frequencies, the unique DOA estimationscan be obtained without ambiguity by using multiple integer frequencies. The multiple equivalent arraystructures of the multiple integer frequencies are analyzed. To perform DOA estimation, in the first part ofthe paper, a pair of fast DOA estimation methods are proposed. Based on the fact that a real DOA and itsambiguous positions are periodic distributed in the sine domain, the two proposed methods are applied toeach equivalent array separately to obtain the real DOA positions as well as their replica positions. Then,an effective matching algorithm is proposed to pick out the real DOAs from the ambiguous ones. Thefirst algorithm is a partial spectral search-based algorithm, and the second algorithm is a search-free-basedmethod. These two algorithms are referred to as matching partial spectral search-based algorithm (matchingPPS) and matching root-multiple signal classification algorithm (matching root-MUSIC), respectively. Ourthird proposed algorithm is a combined root-MUSIC which is applied to the multiple equivalent arraysby taking them as sub-arrays of a common filled ULA. The combined root-MUSIC algorithm utilizesthe MUSIC null-spectrum functions of all equivalent arrays to form a new polynomial, and the DOAestimations can be obtained efficiently without any spatial ambiguity by applying polynomial rooting tothe new polynomial. Simulations are provided to verify the validity of the proposed algorithms.

INDEX TERMS Direction of arrival estimation, sparse uniform linear array, multiple integer frequencies,root-MUSIC.

I. INTRODUCTIONDirection of arrival (DOA) estimation is a very importantapplication of array signal processing. It is well known thatthe angle resolution of an antenna array is heavily dependenton the array aperture. A larger array aperture usually means abetter angular resolution, while the spacing between two adja-cent sensors should no larger than a half wavelength of theworking frequency to avoid the spatial aliasing problem [1].With a given number of sensors, one approach to obtain thetradeoff between the aperture size and the sensor spacing isto use a sparse array. There are many kinds of sparse lineararrays such as minimum redundancy arrays (MRAs) [2], [3],Nested arrays [4], [5] and co-prime linear arrays [6]. Given N

sensors, the co-arrays of these sparse arrays all have O(N 2)degrees of freedom (DOF), which can be exploited to identifyO(N 2) sources by applying an extension of multiple signalclassification (MUSIC) algorithm to the co-array domain [4].For MRAs, there is no closed form expression for the arrayconstruction and achievable DOF, while Nested and co-primearrays can be designed for any specified N .When the number of sources is less than the sensor num-

ber N , multiple signal classification (MUSIC) or some sim-ilar algorithms can be directly applied to the sparse arrayrather than its co-array. For a given number of sensors,the unambiguous identifiability property has been studiedin [7], it also shows that nested and co-prime arrays have

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higher resolvability than the filled uniform linear array (ULA)owing to array sparsity. Since MUSIC on a co-prime arrayrequires total spectral search over the angular field-of-view,it suffers from high computational cost. To reduce the compu-tation complexity, a co-prime array is decomposed into a pairof sparse uniform linear sub-arrays in [8] and [9], then theconventional MUSIC is applied to each sub-array separately,which brings some reduction of computation at the expenseof phase ambiguity in the MUSIC spectrum of each sub-array. To remove the phase ambiguity, common peaks of thespectra of the two sub-arrays are searched and the real DOAscan be uniquely estimated. To further reduce the complexity,a partial spectral search-based DOA estimation method isproposed [10] and [11]. The partial spectral search-basedmethod only searches the peaks of the MUSIC spectrum overa limited angular sector rather than the whole angular field ofview by using the fact that peaks are uniformly distributed inthe transformed sine domain.

On the one hand, since the two sub-arrays in the co-primearrays both have the uniform linear array structure, the mani-fold array of each sub-array has a Vandermonde form, whichallows the search step to be replaced by the polynomialrooting. These are so-called search-free algorithms, whichincludes root-MUSIC [12], ESPRIT [13] and MODE [14].A fast search-free DOA estimation algorithm for co-primearrays based on MODE is proposed in [15], which oper-ates on each sub-array and uses a projection-like method tocombine the estimations from the two different sub-arrays.This method greatly reduces the computational complexity.However, it requires the DOA matching processing, whichis proven to be invalid since multi-target matching errorexists in some extreme scenarios as pointed out in [16].To tackle this problem, an improved DOA estimation algo-rithm based on root-MUSIC algorithm is proposed in [16]which solves the matching error problem by using the dataof co-prime rather than its sub-arrays. However, it uses thesteer arrays of the sub-arrays to perform polynomial root find-ing separately. Like methods that operate on the sub-arraysof the co-prime arrays separately, only the DOFs of thesub-arrays rather than that of the co-prime array are usedto obtain the DOA estimations, thus it can only resolve asmaller number of sources than that of the sensors in bothsub-arrays.

On the other hand, the co-prime array is actually a nonuni-form array, the methods for arbitrary arrays that transformthe arbitrary arrays to equivalent ULAs can be applied tothe co-prime array. These methods include manifold sep-aration [17], array interpolation [18], [19] and Fourier-Domain(FD) root-MUSIC [20], which take advantage ofefficient algorithms for ULAs by approximating the steeringvector of an arbitrary array by that of a virtual ULA. However,they have the following disadvantage, a large number ofvirtual elements is always required to achieve a satisfactoryperformance, which increase the computational complexity.What’s more, they do not coverage to Cramer-Rao Low

Bound (CRLB) when the noise-to-signal ratio (SNR) is high,because the approximating error dominates the performanceat high SNR.

Unlike the co-prime array, which uses two sparse uniformlinear sub-arrays to form a sparse nonuniform array witha large array aperture. The authors in [21]–[23] proposean alternative structure to implement the co-prime arrays,which uses a single sparse uniform linear array with multipleco-prime frequencies to increase the DOF of the proposedsystem in the co-prime domain. This structure mainly aims atthe increased DOF benefiting from the usage of the cross-lagsbetween equivalent sub-arrays of two co-prime frequencies.Themain difference between the co-prime array and co-primefrequency array is that the reflection characteristic in the latteris different due to the differences in the propagation phaseand possible the target reflectivity. Therefore, the equivalentarrays corresponding to different frequencies cannot directlycombined to form a non-uniform liner array. To make fulluse of the informations provided by all co-prime frequen-cies, the DOA estimation problem of co-prime frequencyarray is formulated as a group sparse problem, which can besolved utilizing some algorithms such as the group LASSOalgorithm [24], Block OMP [25] and the CMT-BCS algo-rithm [26]. One common drawback of these methods is thatthey suffer from dictionary mismatches. To solve the thisproblem, the group sparsity based method proposed in [27].Another common disadvantage of these algorithms have highcomputational complexities.

Following the spirit in [23], we use a sparse ULA witha large aperture in this paper to improve angle resolutionwith a limited number of physical array elements. Differentfrom the analysis in [23], our study mainly focuses on theoccasion where the number of source is less than that ofthe physical elements in the array with multiple frequencies.Therefore, we do not need to perform DOA estimation onthe co-prime domain. Such scheme can be taken as a specialcase in [1] and [28]. However, the DOA estimation methodproposed in [1] only takes the single target scenario intoconsideration, which fails in the multi-target scenario dueto the influence of different target reflection characteristicat different frequencies. Meanwhile, the sparse signal rep-resentation (SSR) method used in [28] also suffers fromthe dictionary mismatches and requires large computationalcomplexity.

Different from the SSR method, MUSIC has no dictionarymismatches and it has lower computation complexity.By judiciously choosing the frequencies combinations,the spatial aliasing problem can be completely avoided.On the one hand, we can extend the scheme in [8]and [10] or directly apply the complex strategy in [11] tothe equivalent arrays of the multiple frequencies for the gen-uine DOAs. On the other hand, the combined MUSIC algo-rithm in [29] can be used, however, it needs a total spectralsearch over the angular field-of-view. To reduce the compu-tation complexity, at the first stage, two fast matching-based

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algortihms are proposed, the first one is a partial spec-tral search-based algorithm based on the peaks matchingof the MUSIC spectra (matching PPS), which is similarto the method in [8]. The difference is that the matchingalgorithm in our paper is applicable to multiple sub-arrays,while the matching scheme used in [8] is only applicable toco-prime array with two sub-arrays. The second algorithmis a search-free method which applies root-MUSIC to eachequivalent array separately for the real DOAs and ambiguousDOAs just as the matching PPS method does. Then the samematching algorithm as used in matching PPS is utilized toselect the real DOAs. The third one is also a search-freemethod called combined root-MUSIC which takes averageof the MUSIC spectrum null functions of all the equivalentarrays. This proposed algorithm takes full advantage of theinherent connections between these equivalent arrays and afilled ULA. Unlike the first two algorithms, the combinedroot-MUSIC can uniquely determine the DOAs directly with-out any matching processing.

The rest of the paper is organized as follows. The signalmode of a sparse ULA with multiple integer frequencies isintroduced in Section II. In Section III, The proposed threefast DOA estimation algorithms for multiple integer frequen-cies are presented. In Section IV, Cramer-Rao Lower Boundfor multiple integer frequencies is introduced. Section V pro-vides some discussions on the choice of the multiple integerfrequencies combinations. The computation complexity com-parison for different algorithms are provided in Section VI.Then, simulation results are provided in section VII to com-pare the performances of the three proposedmethods. Finally,some conclusions are provided in Section VIII.Notation: Throughout this paper, bold symbols are used

for vectors and matrices. A−1 A∗, AH and AT are inverse,conjugate, conjugate transpose and transpose of the matrix A,respectively. C denotes the sets of real numbers and complexnumbers. | · | denotes the absolute value of a scalar or car-dinality of a set. ‖·‖2 denotes the 2-norm, and ‖·‖F denotesFrobenius norm. � denotes the Hadamard product, 6 is theangle operator. [A]j,k is the (j, k) th entry of a matrix A.I denotes an identity matrix,< denotes the real part of theexpression. E{·} is the expectation notion.

II. SPARSE UNIFORM LINEAR ARRAY WITH MULTIPLEINTEGER FREQUENCIES SIGNAL MODELConsider a multiple frequencies radar system consisting of asingle transmitting antenna and an N -element receiving arraywhich is a sparse uniform linear array (SULA) with elementspacing d0, which is a half wavelength of the reference fre-quency f0, i.e. d0 = λ0/2 = c/2f0, where c is the lightspeed. The transmitting antennas transmit L continuous-wavesignals with integer frequencies fl, l = 1, . . . ,L simultane-ously. By integer frequencies, we mean frequency fl is integermultiples of the reference frequency f0, i.e. fl = Ml f0, whereMl is an integer. By sparse ULA, we mean that Ml > 1.Suppose that there are K < N far field targets in the fieldof view with DOAs θ = [θ1, θ2, . . . , θK ], the received signal

for the frequency fl can be expressed as

yl(t) =K∑k=1

al(θk )sk,l(t)ej2π fl t + nl(t), l = 1 . . . ,L. (1)

where sk,l(t) is the complex envelop of the echoed signalof the source k corresponding to the frequency fl , whichdepends on the transmitting power, the receiving antennagain, the propagation loss and the radar cross section (RCS)of the target. nl(t) is the additive noise vector whose elementsare spatially and temporally white following the complexGaussian distribution CN (0, σ 2

n IN ) and are independent ofthe target signals. We use the same assumption in [23] thatsk,l(t) is independent with the receive antenna channel, butis essentially frequency-dependent due to the different prop-agation phase delays. sk,l(t) is also uncorrelated for differenttargets over one scan due to target RCSfluctuations (SwerlingII) or target motion. In addition, al(θk ) is the steering vectorcorresponding to θk at the frequency fl , which is expressed as

al(θk ) =[1, e

−j2πd0 sin(θk )λl , . . . , e

−j2π (N−1)d0 sin(θk )λl

]T, (2)

where λl is the wavelength of frequency fl .To obtain the baseband signal, we make demodulation

utilizing the respective frequencies. Filtering through thelow-pass filter, we obtain

yl(t) =K∑k=1

al(θk )sk,l(t)+ nl(t),

= Al(θ )sl + nl(t), l = 1 . . . ,L, (3)

where Al(θ ) = [al(θ1), al(θ2), . . . , al(θK )] and sl(t) =[s1,l(t), . . . , sK ,l(t)]T . We use σ 2

n,l to denote the noise vari-ance at the filter output with frequency fl .

Without loss of generality, we assume that the inte-gers Ml, l = 1, . . . ,L are sorted in a descending order,i.e.,M1 > M2 > · · · > ML . It is noted that our signal model isthe samewith the one in [23] ifMl aremutually co-prime inte-gers.With fl = Ml f0, we have λ0 = Mlλl , then we can rewritethe steering vectors in a reference frequency-independentform as

al(θk ) =[1, e

−j2πMld0 sin(θk )λ0 , . . . , e

−j2πMl (N−1)d0 sin(θk )λ0

]T=

[1, e−jπMl sin(θk ), . . . , e−jπMl (N−1) sin(θk )

]T. (4)

By analyzing the structure of (4), we can take it as a steervector of a virtual sparse ULA with inter-element spacing ofMld0 whose working frequency is the reference frequency f0.As a result, we can obtain L equivalent arrays. These arrayshave the same number of elements, N . In addition, the arrayaperture of the l th equivalent array isMl(N −1)d0. Since thesignals corresponding to different virtual arrays have distinctphases or RCSs, DOA estimation method needs to take thisfact into account.

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The covariance matrix corresponding to the l th equivalentarray can be written as

Rl = E[yl(t)y

Hl (t)

]= Al(θ )Rs,lAHl (θ )+ σ

2n,lIN , (5)

where Rs,l = E[sl(t)sHl (t)

]is the signal covariance matrix.

In practical situations, the exact array covariance matrix Rl isnot available, we can use T snapshots to obtain its estimate

Rl =1T

T∑t=1

yl(t)yHl (t). (6)

The eigenvalue decomposition of Rl yields

Rl = Us,l3s,lUHs,l + Un,l3n,lU

Hn,l, (7)

where Us,l ∈ CN×K and Un,l ∈ CN×(N−K ) are thesignal-subspace and noise-subspace, respectively.

III. SPARSE UNIFORM LINEAR ARRAY WITH MULTIPLEINTEGER FREQUENCIES SIGNAL MODELIn this section, we firstly introduce the spatial aliasing prob-lem of the sparse equivalent arrays, then the anti-aliasingcondition for multiple integer frequencies system is pre-sented. After that, two algorithms are proposed, which are thematching based partial spectral searching algorithm (match-ing PPS) and the matching based root-MUSIC (matchingroot-MUSIC). These two fast DOA estimation algorithmsboth consist of two steps: in the first step, the aliasingpeaks (roots) and the real peaks (roots) are obtained. To pickout the real ones corresponding to the real targets, an efficientmatching algorithm is proposed in the second step. Anothertwo match-free algorithms are also presented here, one is aconventional combined MUSIC based on spectral searching,the other is a combined root-MUSIC which is actually asearch free version of the former.

A. SPATIAL ALIASING PROBLEM FOR MULTIPLEINTEGER FREQUENCIESSince the distance between two adjacent elements in thelth equivalent array isMlλ0/2 withMl > 1, this has the sameeffect as we under-sample a signal by a factor of Ml in thetemporal domain. According to the spatial sampling theorem,if the interval of the elements is larger than λ0/2, the DOAs ofthe targets are not able to resolve uniquely because more thanone different DOAs will produce the same steering vector,which leads to the problem of angular ambiguity. Accordingto [11, Th. 1], for each real DOA, there exist other (Ml-1)distinct replicas that are uniformly distributed in sine domainfrom -1 to 1. The relationship between a real DOA, θk and itsambiguous DOA, θ (k)a,l is expressed as [8], [10], [11]

sin(θk )− sin(θ (k)a,l ) = 2ml/Ml, . (8)

where ml is a nonzero integer between−(Ml−1) and (Ml−1).

B. ALIASING SUPPRESSION FOR MULTIPLEINTEGER FREQUENCIESThe sufficient anti-aliasing condition for multiple integerfrequency with a sparse ULA is given in [1] and [28]: If thereexist at least two frequencies, say fm and fn, satisfying

0 < |fm − fn| ≤c2d0

, (9)

then the unique DOA of a true target can be obtained withoutany aliasing. This condition only aims at avoiding the over-lap of spatial aliasing corresponding to different frequenciesinstead of avoiding spatial aliasing for each frequency. There-fore, it can be view as a relaxed constraint.

Using the sufficient anti-aliasing condition, sinced0 = λ0/2, we have |fm − fn| ≤ f0, which means|Mm −Mn| ≤ 1. Considering the fact that Mm and Mn areintegers, we have |Mm −Mn| = 1. Without loss of generality,we set Mm > Mn, then Mm = Mn + 1. According to theabove analysis, only two integer frequencies can be utilizedto obtain the unique DOAs. If only two frequencies areused, say M1f0 and M2f0, then we have M1 = M2 + 1,which means that M1 and M2 are mutually co-prime. As aconsequence, the two corresponding equivalent arrays can betaken as sparse sub-arrays of a co-prime array as expressedin [9] and [10]. However, unlike a real co-prime array,in which the number of sub-array sensors is different, the twocorresponding equivalent arrays share the same number ofsensors. According to the two propositions in [9], if theinter-element spacings in two sub-arrays are mutually co-prime, the unique DOA estimation in the case of singlesignal can be obtained without aliasing by using MUSICspectrum peaks match processing, which is consistent withthe anti-aliasing condition in (9).

Note that, the sufficient anti-aliasing condition in (9) onlyprevents the alignment of false replicas belonging to the samedirection of arrival. It is still possible that two spatial spectrafor Mm and Mn share the replicas of different directions ofarrivals. This type of false alignment is referred to as cross-nulling, which has been analysed in [16]. To identify cross-nulls, additional spatial spectra are needed. It can be shownthat at least K + 1 frequencies are required to eliminatepotential cross-nulls for K sources.

C. PEAKS MATCHING BASED ALGORITHMS FORMULTIPLE INTEGER FREQUENCIESSince the echoed signal of a target depends on the transmit-ting frequency, unlike a real co-prime array, in which thesub-arrays share the same return signal for each target. As aresult, we cannot directly combine these equivalent arrays toform a nonuniform array. Instead, we apply partial spectralsearch-basedMUSIC or rootMUSIC to each equivalent arrayseparately to obtain aliasing peaks or roots that contain thereal DOAs, then we combine the aliasing results of all arraysto pick the real DOA estimation with an efficient matchingalgorithm.

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1) PEAKS MATCHING BASED ALGORITHMSFOR MULTIPLE INTEGER FREQUENCIESUsing the orthogonality between the signal subspace andthe noise subspace, the conventional MUSIC null-spectrumfunction of frequency fl can be expressed as

Pl(θ ) =1

aHl (θ )Un,lUHn.lal(θ )

=1∥∥∥aHl (θ )Un,l

∥∥∥2 . (10)

The spectral MUSIC technique estimates the DOAs fromthe maxima of this function by searching over θ witha fine grid. Since the total number of spectral pointsS � N , the computational complexity of this spectralsearch step is typically substantially higher than that ofthe eigen-decomposition step. To reduce the computationcomplexity, a partial spectral search-based DOA estima-tion method is proposed for co-prime linear arrays in [10],which can also be used for DOA estimation using multipleinteger frequencies. Using the transformation u = sin(θ ),(−1 ≤ u ≤ 1) the position of a real peak and its (Ml − 1)ambiguous peaks follow uniform distribution in the u domainas described in (8). If we obtain an arbitrary peak, thenwe can recover other peaks immediately without searchthe whole search field. Therefore, we can equally dividethe total search field of the lth equivalent array into Mlintervals without overlaps as in [10]. The i th interval isdenoted as

2i,l =

{u

∣∣∣∣−1+ 2(i− 1)Ml

≤ u ≤ −1+2iMl

}, (11)

for i = 1, 2, . . . ,Ml . For each source, there must existone ambiguous DOA in arbitrary interval 2i,l . Therefore,the positions of the K corresponding peaks of K sourcescan be obtained by searching over the arbitrary inter-val 2i,l . Denote the K peaks in the ith interval as ui,l =[u1i,l, u

2i,l, . . . , u

Ki,l]. Then the corresponding K ambiguous

peaks in other intervals2q,l(q = 1, 2, . . . ,Ml) can be recov-ered using the following equation

uq,l = ui,l + 2(q− i)/Ml . (12)

Generally, there exist K ×Ml peaks for the lth equivalentarray, denoted as

ul = [u1,l,u2,l, . . . ,uMl ,l]. (13)

2) ALIASING ROOTS POSITIONS USING ROOT-MUSICSince each equivalent array is a sparse ULA, the root-MUSICcan be directly applied to the lth equivalent separately byusing (10) as

Fl(zl) = aTl (1/zl)Un,lUHn,lal(zl)

=

N−1∑ml=−(N−1)

cml zlml , 1/Pl(θ ), (14)

where zl = e−jπMl sin(θ )=e−jπMlu, and

cml =∑

ml=p−q

[Un,lU

Hn,l

]p,q

(15)

is the sum of elements ofmatrix Un,lUHn,l on itsml th diagonal.

This is a polynomial of degree of 2(N−1) whose roots appearin conjugate reciprocal pairs. That is, if z0 is a root of Fl(zl),then 1/z∗0 is its root as well. However, the polynomial in (14)has Ml × K roots instead of K , the problem is that alias-ing occurs with the period of 2π/Ml due to spatial under-sampling. If z(k)i,l = e−jπMluki,l is a root of (14) with uki,l ∈ 2i,l ,then z(k)i,l e

j2mπ/Ml is also a root of (14), where m is a non-zerointeger between −(Ml − 1) and (Ml − 1). Like the aliasingproblem in the partial spectral based DOA algorithm, there isno way to distinguish a real root from its replicas by usingeach equivalent array separately. However, we can find Kaliasing roots corresponding to the K targets by finding the Klargest-magnitude roots that are located inside the unit circle,denoted as z(k)i,l , k = 1, 2, . . . ,K , whose DOA positions in theu domain can be estimated as

uki,l = −6 (z(k)i,l )

πMl∈ 2i,l, k = 1, 2, . . . ,K . (16)

For each uki,l , we can find the Ml aliasing DOA estimationpositions ukl = [uk1,l, u

k2,l, . . . , u

kMl ,l] ∈ [−1, 1] in the

u domain by using the spatial aliasing period of 2/Ml asdescribed in (12). Then we can obtain the Ml × K rootpositions ul = [u1l ,u

2l , . . . ,u

Kl ] as the same in (13). The

remaining work is how to peak the K real root positionsfrom ul .

3) EFFECTIVE MATCHING ALGORITHM FORMULTIPLE EQUIVALENT ARRAYSUsing the result of partial spectral based DOA with thelth equivalent array, i.e. ul = [u1,l,u2,l, . . . ,uMl ,l], or theoutput of the root-MUSIC, i.e. ul = [u1l ,u

2l , . . . ,u

Kl ]. Our

purpose is to select the K genuine peak (root) positionsfrom ul . To identify the genuine peak (root) positions andexclude the replicas, a probability-based decision strategy isused in [11], while a binary decision strategy is used in [15].The simple peaks match process used in [10] is successful inthe case of only two equivalent arrays. However, when thenumber of equivalent arrays is larger than 2, we need a newstrategy to get better results. In this section, we present anefficient target matching processing for multiple equivalentarrays.

According to [11], a genuine peak (root) position is inde-pendent on the sparse inter-element spacing, while the repli-cate peak (root) position is highly dependent on it. If apeak (root) position u = sin(θ ) correspond to a real tar-get, then it is a peak (root) in all spectra Pl(θ )(Fl(zl)) forl = 1 . . . ,L, while a replica one is only a peak (root) for partof the spectra Pl(θ )(Fl(zl)) due to the anti-aliasing conditionin (9). Therefore, we can use the following criterion function

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to find the K genuine peak (root) positions.

F(u) =L∑l=1

aHl (u)Un,lUHn,lal(u), (17)

where al(u) =[1, e−jπMlu, . . . , e−jπMl (N−1)u

]T. By finding

the K minimum values in (17), we can find the K gen-uine positions corresponding to the K real targets ul =[u1,l, u2,l, . . . , uK ,l] in the u domain.Without loss of general-ity, we assume that elements in ul are sorted in an ascendingorder, i.e. u1,l < u2,l < . . . < uK ,l . With L equivalent arrays,we can obtain ul = [u1,l, u2,l, . . . , uK ,l], l = 1 . . . ,L foreach equivalent array, by averaging the kth elements of theseL groups of ul, l = 1 . . . ,L across l, we can obtain the finalDOA estimate of the kth target as

θk = arcsin(1L

L∑l=1

uk,l), k = 1, 2, . . . ,K . (18)

Compared with the matching based partial spectralsearching algorithm (Matching PPS), the matching basedroot-MUSIC (Matching root-MUSIC) has lower com-putational complexity because no spectral searching isneeded.

D. COMBINED MUSIC SPECTRAL DOAESTIMATION ALGORITHMIn [29], the combined MUSIC spectrum algorithm for wide-band signals is proposed. Since the multiple integer frequen-cies can be considered as some discrete frequency bins in awideband signal. The method in [29] can be directly appliedhere. This method first measures the orthogonality at eachinteger frequency, and then combines the resulted measuresfor the multiple frequencies. By using the arithmetic meanmetric both for the individual frequency and the combinationover the multiple integer frequencies, the combined MUSICspectrum is expressed as

Pcombine(θ ) =1

1L

∑Ll=1 1/Pl(θ )

=1

1L

∑Ll=1 a

Hl (θ )Un,lU

Hn.lal(θ )

, (19)

then the estimatedDOAs correspond to theK largest values ofthe spectrum Pcombine(θ ). Compared with the extended partialspectral DOA estimation algorithm introduced above, there isno need of peaks match processing. However, it requires totalspectral search over the angular field-of-view, which cause alarger computational complexity.

E. COMBINED ROOT-MUSIC FOR MULTIPLEINTEGER FREQUENCIESTo reduce the computation complexity in equation (19),we propose a combined root-MUSIC algorithm which canbe considered as a search free version of the DOA estimatorfor (19). The method can estimate the true DOAs directly

without peak matching processing by taking all equivalentarray as sub-arrays of a filled ULA.

1) RELATIONSHIPS OF THE EQUIVALENTARRAYS AND A FILLED ULABased on the analysis in section II, the element positions inthe lth equivalent array form the position set

Sl = {nMld0, 0 ≤ n ≤ N − 1}. (20)

All the L equivalent arrays share the first element position atthe zeroth position. Since M1 > M2 > · · · > ML , the firstequivalent array has the largest array aperture asM1(N−1)d0,the L equivalent arrays can be considered as sub-arrays of afilled ULA with an inter-element spacing is d0. By observingthe element position set of each equivalent array, we findthat there are infinitely many such filled ULAs. However,we can find a unique filled ULA with the smallest numberof elements. Obviously, the unique filled ULA has the samearray aperture as the first equivalent array i.e. M1(N − 1)d0.The element position set of this filled ULA is

Sfull = {md0, 0 ≤ m ≤ M1(N − 1)}. (21)

The manifold matrix of the filled ULA is

Afull(θ ) = [afull(θ1), . . . , afull(θK )]. (22)

where

afull(θk ) = [1, e−jπ sin(θk ), . . . , e−jπM1(N−1) sin(θk )]T

(23)

is the steering vector of the filled ULA corresponding to thekth direction, θk .The lth equivalent array takes only a subset of the ele-

ments of the filled ULA. Denote the subset as �l =

[�1,l, . . . , �N ,l], then the lth equivalent array can be fullyrepresented by this element index set. Assume that the subsetis sorted ascending with�1,l = 1 and�N ,l = Ml(N−1)+1,then we have �n,l = Ml(n − 1) + 1, n = 1, . . . ,N .There also exists a corresponding selection matrix 0l ∈{0, 1}N×(M1N−M1+1) whose nth row contains all zeros but asingle one at the �n,l th position. Obviously, we have thefollowing equation

Al(θ ) = 0lAfull(θ ). (24)

Next, we use an example to show how the selection matrix0l is constructed. Given L = 2, M1 = 3, M2 = 2, N = 2,then we have S1 = {0, 3} × d0, S2 = {0, 2} × d0, Sfull ={0, 1,2,3} × d0, the subsets �1 = {1, 4}, and �2 = {1, 3},and their corresponding selected matrixes are

01 =

[1 0 0 00 0 0 1

]and 02 =

[1 0 0 00 0 1 0

],

respectively.

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2) COMBINED ROOT-MUSIC FOR MULTIPLEINTEGER FREQUENCIESUsing (24), the MUSIC spectrum used for peak search for thelth equivalent array in (10) can be rewritten as

Pl(θ ) =1

aHl (θ )Un,lUHn.lal(θ )

=1

aHfull(θ )0Hl Un,lU

Hn,l0lafull(θ )

=1

aHfull(θ )GlGHl afull(θ )

(25)

where Gl = 0Hl Un,l . Therefore, the combined MUSICspectrum in (19) can be rewritten as

Pcombine(θ ) =1

1L

∑Ll=1 a

Hl (θ )Un,lU

Hn,lal(θ )

=1

1L

∑Ll=1(a

Hfull(θ )0

Hl Un,lU

Hn,l0lafull(θ ))

=1

aHfull(θ )(1L

∑Ll=1 GlG

Hl )afull(θ )

. (26)

Instead of searching the combined MUSIC spectrum forpeaks, we can apply the polynomial root finding to (26)to obtain the true DOA estimate directly. Let afull(z) =[1, z, . . . , zM1(N−1)]

T, where z = e−jπ sin(θ ). Then we can

turn (26) a the root finding problem with the followingpolynomial

Fcombine(z) = aTfull(1/z)(1L

∑L

l=1GlGHl )afull(z)

=

M1(N−1)∑m=−M1(N−1)

cmzm , 1/Pcombine(θ ), (27)

where

cm =∑

m=p−q

[1L

∑L

l=1GlGHl

]p,q

(28)

is the sum of elements of matrix 1L

∑Ll=1 GlG

Hl on its

mth diagonal. This is a polynomial of degree of 2M1(N − 1)whose roots appear in conjugate reciprocal pairs, that is, if z0is a root of Fcombine(z), 1/z∗0 is its root as well. By findingthe K largest-magnitude roots that are located inside the unitcircle, denoted as z(k), k = 1, 2, . . . ,K , the K real DOAs canbe estimated as follows

θk = arcsin(−6 (z(k))π

), k = 1, 2, . . . ,K . (29)

F. THE RELATIONSHIP OF COMBINED ROOT-MUSICAND MATCHING ROOT-MUSICSince each equivalent array is a sparse ULA, the root-MUSIC can be directly applied to lth equivalent separately

by using (10). With (24), we can rewrite (14) as

Fl(zl) = aTl (1/zl)Un,lUHn,lal(zl) =

N−1∑ml=−(N−1)

cml zlml

= aTfull(1/z)GlGHl afull(z) =

M1(N−1)∑m=−M1(N−1)

cm,lzm

= Fl(z) , 1/Pl(θ ) (30)

where

cm,l =∑

m=p−q

[GlGHl

]p,q. (31)

From (30), we have zl = zMl , the coefficients cm,l andcoefficients cml meet the following relationship

cm,l =

{cml , m = nml, n = −(N−1), . . . , 0, . . . , (N−1)0 elsewhere

(32)

Obviously, from (27) and (30), we have

Fcombine(z) =1L

∑L

l=1Fl(z). (33)

Furthermore, we have cm = 1L

L∑l=1

cm,l .

The polynomial Fl(zl) and The polynomial Fl(z) are essen-tially the same, except that they have different variablesand polynomial degrees, the polynomial Fl(zl) has degreeof 2(N − 1), while the polynomial Fl(z) has degree of2M1(N − 1). These two polynomials both have the sameMl×K roots instead ofK due to spatial aliasing. According tothe analysis in Section III.C, due to the low polynomial degreeof Fl(zl), we can only obtain K aliasing roots correspondingto the K targets by finding the K largest-magnitude roots thatare located inside the unit circle. However, we can directlyobtain the Ml × K roots of polynomial Fl(z) by finding theMl × K largest-magnitude roots that are located inside theunit circle of at the expense of more computation complexity,which will be discussed in the following section.

According to the analysis in the Section III.B, with the suf-ficient condition formultiple integer frequencies system, onlythe roots z(k) = e−jπ sin(θk ), k = 1, 2, . . . ,K correspondingto the real K targets satisfy equations Fl(z(k)) = 0 for l =1, . . . ,L, thus we have Fcombine(z(k)) = 1

L

∑Ll=1 Fl(z

(k)) = 0.Therefore, we can obtain the K genuine DOAs by findingthe K largest-magnitude roots of polynomial Fcombine(z) thatare located inside the unit circle. The combined root MUSICalgorithm can directly find the K DOAs without positionmatching processing, which is the key point of the combinedroot-MUSIC algorithm. It is noted that we cannot take aver-age of Fl(zl) to construct a similar polynomial as Fcombine(z),because variable zl is different for different equivalent array.That is the reason we consider every equivalent array as asub-array of the same filled full ULA.

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IV. CRAMER-RAO LOWER BOUND FOR MULTIPLEINTEGER FREQUENCIESIn this section, we present the Cramer-Rao LowerBound (CRLB) for the multiple integer frequencies followingthe theoretical analysis in [32] and [33]. Since the multipleinteger frequencies can be parts of discrete frequency binsin a wideband signal, which can be consider as a specialwidebandmodel, we can use the CRLB of widebandmodel asthe CRLB of multiple integer frequencies. Under the assump-tions of the wideband Gaussian model, the CRLB is givenby [32] and [33]

CRLB(θ ) =12T

[L∑l=1

1

σ 2n,l

<

{(DHl (θ )P

⊥AlDl(θ )

)�W l

}]−1,

(34)

where W l =

(Rs,l −

(R−1s,l +

1σ 2n,l

AHl (θ )Al(θ ))−1)T

,

P⊥Al = [IL − Al(θ )(AHl (θ )Al(θ ))−1AHl (θ )] and Dl(θ ) =

[d l(θ1), . . . , d l(θK )], where d l(θ ) = ∂al(θ )/∂θ .Considering the case of single sourceK = 1, (34) becomes

CRLB(θ ) =6

π2cos2(θ )N (N 2 − 1)L∑l=1

(M2l · SNRl)

, (35)

where the SNRl is the signal-to-noise (SNR) of the target atfrequency fl . For simplicity, we assume that the SNRl doesnot vary with fl , i.e. SNRl = SNR, l = 1, 2, . . . ,L, with afixed N , the CRLB of the multiple integer frequencies highlydepends on the combination of frequencies, especially thevalue ofM1. In general, a larger value ofM1 leads to a lowerCRLB. However, with a fixed M1, increasing the numberof multiple integer frequencies L has less influence on theCRLB, which will be demonstrated in the simulation section.

V. DISCUSSION ON THE COMBINATION OF THEMULTIPLE INTEGER FREQUENCIESIn this section, we discuss how to choose the best combina-tion of multiple integer frequencies with a given number ofphysical elements N and the fixed number of frequencies L.The angular resolution of the multiple integer frequencies

is mainly limited by the array aperture of the first equiv-alent array. Given a required angular resolution, we canfind the corresponding M1 and then we use the ant-aliasingcondition (9) and the CRLB to find the other L − 1 inte-ger frequencies. If we want to obtain the lowest CRLBwith the assumption that SNRl = SNR, l = 1, 2, . . . ,L,according to (35), we have Ml = M1 − l + 1, l =1, 2, . . . ,L. Naturally, one may think that we can improvethe angular resolution with arbitrary big M1 with given Nand L. However, as the increasing of M1, the much morenumber of replica peaks or replica roots are generated forMUSIC spectra in (10) or polynomials in (14) and (30),respectively, which lead to cross-nullings in demanding sce-narios such as low SNR and small number of snapshots.

Furthermore, with increase of the number of targets, higherprobability of cross-nulling.

VI. COMPUTATION COMPLEXITY COMPARISONThe computational complexity of the proposed matchingPPS, matching root-MUSIC and combined root-MUSICalgorithms are compared with that of the conventional com-bined MUSIC algorithm in Table 1.

TABLE 1. The orders of computational complexities of matching PPS,matching root-MUSIC, combined MUSIC and combined root-MUSIC.

The computation complexity of the estimation of L covari-ance matrices is O(LN 2T ), and the eigen-decompositionof the L covariance matrices involve O(LN 3) operations.The computational cost of spectral search for the match-

ing PPS algorithm is O(N 2SL∑l=1

1/Ml). The complexity of

the peak match processing takes O(N 2KL∑l=1

Ml) operations,

while the spectral search for the combined MUSIC algo-rithm is O(LN 2S). The complexities to compute the match-ing root-MUSIC and combined root-MUSIC polynomialcoefficients are given by O(LN 3) and O((M1(N − 1))3),respectively.

According to the above analysis, with fixed L, N , T andsearching interval, the complexity of the combined MUSICalgorithm is independent with the combination of multipleinteger frequencies, while that of the matching PPS algo-rithm is highly dependent on the choice of multiple inte-ger frequencies. With the value of the integer frequenciesincrease, the computation cost decreases. The computationcomplexity of the combined root-MUSIC highly depends onthe M1, which increases fast as M1 increases. In the typicalfor spectral search-based estimation case of large S(S � N ,S � M1), the proposed combined root-MUSIC is moreefficient than the combined MUSIC algorithm. Comparedwith the other three algorithms, the proposed match-ing root-MUSIC algorithm has the lowest computationcomplexity.

We use an example to show the comparison of the computa-tion complexity of these algorithms. The number of snapshotsis set to be T = 400, N = 6 elements are used in thesparse ULA, and L = 3 frequencies are used with Ml =

M1− l+ 1, l = 1, 2, . . . ,L, whereM1 is set integers rangingfrom 4 to 40. The search intervals for both the matching PPSalgorithm and the combined MUSIC algorithm are the same,

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we use two sets of searching intervals for comparison, onesearching interval is 0.01◦ (S = 1.8 × 104) and the other is0.001◦ with (S = 1.8× 105). The results are show in Fig.1.

FIGURE 1. Computation complexity of the four algorithms.

From Fig.1, the combined MUSIC algorithm has the high-est computation complexity when the search step is 0.001◦,since it requires the total spectral search over the entire angu-lar field-of-view while the matching root-MUSIC has thelowest computation complexity. The computation complex-ity of the matching PPS algorithm decreases rapidly as M1increases while that of the combined root-MUSIC increaseas rapidly M1 increases. The combined root-MUSIC haslower complexity than the combined MUSIC with searchingstep of 0.001◦. When M1 < 25, its complexity is smallerthan that of the combined MUSIC algorithm with searchingstep of 0.01◦. When M1 < 20, combined root-MUSIChas lower complexity than that of the matching PPS algo-rithm with searching step of 0.001◦, and it is more efficientthan the matching PPS algorithm with search step 0.01◦

when M1 < 11.

VII. SIMULATION RESULTSIn this section, simulations are performed to evaluate the theperformances of the proposed three algorithms including thematching PPS algorithm, the matching root-MUSIC algo-rithm and the combined root-MUSIC algorithm. Throughoutour simulations, 500 independent Monte Carlo runs havebeen used in each example. The sparse ULA consists ofN = 6 physical elements. To obtain a low CRLB with afixed number of frequencies L, we set Ml = M1 − l + 1,l = 1, 2, . . . ,L. For matching PPS, S = 1.8 × 104 has beenchosen with searching interval of 0.01◦. The DOA estimationroot-mean-square-error (RMSE) is used to compare the per-formances of these three algorithms. CRLB in the (34) is alsoused for comparison.

A. IDENTICAL RCS AT DIFFERENT FREQUENCIESIn this section, the source powers at different integer frequen-cies are assumed to be identical and only phase differencesexist between the received signals at different frequencies.Thus the SNRl does not vary with fl , i.e. SNRl = SNR,

l = 1, 2, . . . ,L as in [23]. This assumption is reasonable ifthe frequency separations between these frequencies are rela-tively small such that the absolute value of the RCS of a targetstays unchanged with these multiple integer frequencies.

1) RMSE VERSUS SNRIn the first simulation, we use K = 2 equally powered targetswith DOAs of θ = [−0.9◦, 0.9◦]. A total of T = 400snapshots are used, and the SNR varies from−15dB to 20dB.Firstly, the performances of each algorithm separately withthree different numbers of integer frequencies L are com-pared. For each algorithm, three sets of multiple integer fre-quencies combinations are used. All these three combinationsshare the same M1 = 11. L = 2, 3, 4 frequencies are usedrespectively for each algorithm. When L = 4, the elementpositions of the physical array, the equivalent arrays and thecorresponding filled ULA array are given in Fig.2.

FIGURE 2. Element locations of the physical array, equivalent arrays andthe filled ULA with M1 = 11, L = 4.

The results of the matching PPS algorithm, the match-ing root-MUSIC algorithm and the combined root-MUSICtogether with the CRLB are presented in Fig.3 (a), Fig.3(b),and Fig.3(c), respectively. The results show that, with thesame SNR, a larger L leads to a lower CRLB, but theimprovement in CRLB with increasing of L is relativelynon-significant. For each algorithm, they all reach the corre-sponding CRLB when in the high SNR regime. While in thelow SNR regime, the matching PPS method is not sensitiveto L, since when L increases from 3 to 4, the improve-ment in the RMSE performance is not remarkable. On thecontrary, the matching root-MUSIC method and the com-bined root-MUSIC both have notable improvements whenL changes from 3 to 4. Then, the number of frequen-cies is fixed at L = 3, the performances of the threealgorithms are compared and the results in Fig.3(d) showthat the matching root-MUSIC outperforms the matchingPPS method, while the combined root-MUSIC has the bestperformance when the SNR > −10dB. The matchingPPS method and the matching root-MUSIC method are alittle more robust than the combined root-MUSIC whenSNR < −10dB.

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FIGURE 3. RMSEs versus SNR, T = 400, θ1 = −0.9◦, θ2 = 0.9◦. (a) Matching PPS. (b) Matching root-MUSIC. (c)Combined root-MUSIC. (d) Comparison of the three algorithms with L = 3.

2) RMSE VERSUS NUMBER OF SNAPSHOTSIn the second simulation, the RMSE performances of theproposed methods versus the number of snapshots are com-pared. We fix the SNR at −5dB and vary the number ofsnapshots T from 50 to 1000. All other parameters areidentical to those in the first example. Firstly, we comparethe performance of each algorithm separately with differentnumber of integer frequencies L. The results of the matchingPPS algorithm, the matching root-MUSIC algorithm and thecombined root-MUSIC together with the CRLB are pre-sented in Fig.4(a), Fig.4(b) and Fig.4(c), respectively. Theresults show that with the same number of snapshots T ,a larger L leads to a lower CRLB, but the improvement inCRLB with the increase of the number of frequencies is rela-tively non-significant. For each algorithm, they all approachthe corresponding CRLB when the number of snapshots islarge. When L = 2, both the matching PPS algorithm andthe matching root-MUSIC algorithm approach CRLB whenT > 600, while the combined root-MUSIC requires T > 870snapshots to reach the CRLB. When T is small, the matchingPPS method is not sensitive to L. Since when L increasesfrom 3 to 4, the improvement in the RMSE performance is notremarkable. However, the matching root-MUSICmethod andthe combined root-MUSIC both have notable improvement

whenL changes from 3 to 4. Then, we fixed the number of fre-quencies at L = 3, the performances of the three algorithmsare compared in Fig.4(d). The results show that the matchingroot-MUSIC outperforms the matching PPS method and thematching root-MUSIC method when L = 3. These resultsindicate that the combined root-MUSIC algorithm is moresensitive to the number of frequencies L.

3) RMSE VERSUS ANGULAR SEPARATIONIn the third simulation, we examine the performances ofthe three proposed algorithms versus the angular separationbetween two targets. We fix the SNR at 0dB and the numberof the snapshots T = 400. The DOA of the first target is fixedand equals to θ1 = 2◦, while the DOA of the second target θ2changes from 2.4◦ to 10◦. All other parameters are identicalwith those in the previous two examples. At first, we comparethe performances of each algorithm separately with differ-ent numbers of integer frequencies L. The results of thematching PPS algorithm, the matching root-MUSIC algo-rithm and the combined root-MUSIC together with the CRLBare presented in Fig.5(a), Fig.5(b), and Fig.5(c), respectively.We can see that with the same angular separation, a larger Lleads to a lower CRLB, but the improvement in CRLB withincreasing of the number of frequencies L is relatively small.

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FIGURE 4. RMSEs versus input number of snapshots, SNR = −5dB, θ1 = −0.9◦, θ2 = 0.9◦. (a) Matching PPS.(b) Matching root-MUSIC. (c) Combined root-MUSIC. (d) Comparison of the three algorithms with L = 3.

For each algorithm, they all approach the correspondingCRLB when the angular separation is big enough exceptfor the case of L = 2. When L = 2, all these threeproposed algorithms only approach the CRLB with someangular separations, while big RMSEs occur at other angularseparations. Because the cross-nulling problem occurs whenonly a total of L = 2 frequencies are used, the two spatialspectra of these two frequencies share the replicas of differentdirections of arrivals. Large bias occurs when two replicasof the two real DOAs are closer than that of the real twoDOAs due to low SNR. However, adding one more frequencyimproves the RMSE performance dramatically as can beenseen in Fig.5(a), Fig.5(b) and Fig.5(c). Then, we fixed thenumber of frequencies at L = 3, the performances of the threealgorithms are presented in Fig5.(d). The results show thatthe matching root-MUSIC outperforms the matching PPSmethod when a total number L = 3 frequencies are utilized,while the combined root-MUSIC has the best performancewhen the angular separation is larger than 1◦. These resultsindicate that the combined root-MUSIC algorithm is moresensitive to the number of frequencies L.

B. DISTINCT RCSS AT DIFFERENT FREQUENCIESIn this section, we evaluate the DOA estimation performancesof the three proposed algorithms in the case of distinct RCSs

at different frequencies. We consider the same array configu-ration as in the simulations of Section VII.A with L = 3 andM1 = 11. However, the two sources are now assumed to havedistinct RCSs at different frequencies. The source powers areidentical at each frequency but different across frequencies.More specifically, the source powers at frequency f1 areassumed to be unity, whereas the source powers associatedwith f2 and f3 are assumed to be 1.5625 and 2.25 times ofunity, respectively.

1) RMSE VERSUS SNRIn the first simulation, we consider the same source config-urations used in Section VII.A.1). The results of the threeproposed algorithms are shown in Fig.6. The results show thatRMSE curves of the three algorithms all approach the CRLBin the high SNR regime. When SNR < 12dB, the matchingPPS algorithm and the matching root-MUSIC algorithm havethe comparable good performance, and they both have betteraccuracy than combined root-MUSIC. When SNR > 12dB,matching root-MUSIC and combined root-MUSIC have bet-ter performance than matching PPS.

2) RMSE VERSUS NUMBER OF SNAPSHOTSIn the second simulation, we consider the same source con-figurations used in Section VII.A.2). The results of the three

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FIGURE 5. RMSEs versus angular separation, SNR = 0dB, T = 400, θ1 = 2◦. (a) Matching PPS. (b) Matchingroot-MUSIC. (c) Combined root-MUSIC.(d) Comparison of the three algorithms with L = 3.

FIGURE 6. RMSEs versus SNR, L = 3, T = 400, θ1 = −0.9◦, θ2 = 0.9◦.

proposed methods are shown in Fig.7. The results show thatRMSE curves of the three methods all approach the CRLBwith increase of number of snapshot. When T < 100,the combined root-MUSIC have lower RMSE than matchingPPS and the matching root-MUSIC, while the curves ofmatching PPS and matching root-MUSIC are closer to CRLBthan that of combined root-MUSIC when T > 100. Theresults indicate that combined root-MUSIC is sensitive to thedistinct RCSs at different frequencies.

FIGURE 7. RMSEs versus number of snapshot, L = 3, SNR = −5dB,θ1 = −0.9◦, θ2 = 0.9◦.

3) RMSE VERSUS ANGULAR SEPARATIONIn the second simulation, we consider the same source con-figurations used in Section VII.A.3). The results of the threeproposed algorithms are shown in Fig.8. The results showthat RMSE curves of the three algorithms all approach theCRLB with increase of the angular separation between thetwo sources.When the separation is smaller than 1◦, matchingroot-MUSIC has the best performance, and the combined

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FIGURE 8. RMSEs versus angular separation, L = 3, SNR = −5dB, T = 400.

root-MUSIC have lower RMSE than matching PPS. Whenthe separation is larger than 1◦, the curves of matching PPSand matching root-MUSIC are closer to CRLB than that ofcombined root-MUSIC. The results also indicate that com-bined root-MUSIC is more sensitive to the distinct RCSs atdifferent frequencies.

VIII. CONCLUSIONIn this paper, the problem of fast DOA estimation forsparse ULA with integer frequencies has been addressed.To improve the angular resolution of the system with limitednumber of physical array elements, multiple integer frequen-cies are proposed, the equivalent array structure for each fre-quency is analyzed and the sufficient anti-aliasing conditionis provided here.

Taking the different reflections of a target with differ-ent frequencies into account, we cannot use the multipleequivalent arrays to form a sparse nonuniform array. Instead,we apply two fast algorithms to each equivalent array sep-arately, which gives the real positions of targets as well astheir replicas by using the fact that a real DOA positionand its replica positions are uniformly distributed on thesine domain, to select the real DOA positions, an effectivematching algorithm is proposed.

To avoid the matching processing step, a combinedroot-MUSIC algorithm is proposed, which takes all equiv-alent arrays as sub-arrays of a filled ULA. Then thenull-spectrum MUSIC functions of all equivalent array arecombined to formulate a new polynomial rooting find prob-lem which gives the real DOAs of targets directly.

It has been verified through simulations that the threeproposed algorithms are all effective, and at least threeinteger frequencies are needed to provide sufficiently goodperformance to locate two targets. In addition, in the caseof identical RCS at different frequencies, the combinedroot-MUSIC method provides better performance than theother two proposed algorithms when the number of snapshotsis large or the SNR is high. However, in the case of distinctRCSs at different frequencies, matching PPS and matchingroot-MUSIC both have better DOA estimation accuracy than

combined root-MUSIC in the cases of large number of snap-shots, high SNR and large separation of sources. Taking thecomputational complexity into consideration, the matchingroot-MUSIC is preferred.

It is noted that, throughout this paper, multiple snapshotsare used to develop the signal model. In the case of singlesnapshot, the DOA estimation with multiple integer frequen-cies for sparse array is also important, and a similar algorithmwill be developed in the future.

REFERENCES[1] M. Ulrich and B. Yang, ‘‘Multi-carrier MIMO radar: A concept of

sparse array for improved DOA estimation,’’ in Proc. IEEE Radar Conf.,May 2016, pp. 494–498.

[2] S. U. Pillai, Y. Bar-Ness, and F. Haber, ‘‘A new approach to array geometryfor improved spatial spectrum estimation,’’ Proc. IEEE, vol. 73, no. 10,pp. 1522–1524, Oct. 1985.

[3] Y. I. Abramovich, D. A. Gray, A. Y. Gorokhov, and N. K. Spencer,‘‘Positive-definite Toeplitz completion in DOA estimation for nonuniformlinear antenna arrays. I. Fully augmentable arrays,’’ IEEE Trans. SignalProcess., vol. 46, no. 9, pp. 2458–2471, Sep. 1998.

[4] P. Pal and P. P. Vaidyanathan, ‘‘Nested arrays: A novel approach to arrayprocessing with enhanced degrees of freedom,’’ IEEE Trans. Signal Pro-cess., vol. 58, no. 8, pp. 4167–4181, Aug. 2010.

[5] C.-L. Liu and P. P. Vaidyanathan, ‘‘Super nested arrays: Linear sparsearrays with reduced mutual coupling—Part I: Fundamentals,’’ IEEE Trans.Signal Process., vol. 64, no. 15, pp. 3997–4012, Aug. 2016.

[6] S. Qin, Y. D. Zhang, and M. G. Amin, ‘‘Generalized coprime array config-urations for direction-of-arrival estimation,’’ IEEE Trans. Signal Process.,vol. 63, no. 6, pp. 1377–1390, Mar. 2015.

[7] P. P. Vaidyanathan and P. Pal, ‘‘Direct-MUSIC on sparse arrays,’’ in Proc.Int. Conf. Signal Process. Commun. (SPCOM), Jul. 2012, pp. 1–5.

[8] C. Zhou, Z. Shi, Y. Gu, and X. Shen, ‘‘DECOM: DOA estimation withcombined MUSIC for coprime array,’’ in Proc. Int. Conf. Wireless Com-mun. Signal Process., Oct. 2013, pp. 1–5.

[9] C. Zhou, Y. Gu, S. He, and Z. Shi, ‘‘A robust and efficient algorithm forcoprime array adaptive beamforming,’’ IEEE Trans. Veh. Technol., vol. 67,no. 2, pp. 1099–1112, Feb. 2017.

[10] F. G. Sun, P. Lan, and B. Gao, ‘‘Partial spectral search-based DOA esti-mation method for co-prime linear arrays,’’ Electron. Lett., vol. 51, no. 24,pp. 2053–2055, Nov. 2015.

[11] J. Liu, Y. Zhang, Y. Lu, and W. Wang, ‘‘DOA estimation based on multi-resolution difference co-array perspective,’’Digit. Signal Process., vol. 62,pp. 187–196, Mar. 2017.

[12] B. D. Rao and K. V. S. Hari, ‘‘Performance analysis of root-music,’’ IEEETrans. Acoust., Speech, Signal Process., vol. 37, no. 12, pp. 1939–1949,Dec. 1989.

[13] R. Roy and T. Kailath, ‘‘ESPRIT-estimation of signal parameters viarotational invariance techniques,’’ IEEE Trans. Acoust., Speech, SignalProcess., vol. 37, no. 7, pp. 984–995, Jul. 1989.

[14] P. Stoica and A. Nehorai, ‘‘MUSIC, maximum likelihood, and Cramer-Raobound,’’ IEEE Trans. Acoust., Speech, Signal Process., vol. 37, no. 5,pp. 720–741, May 1989.

[15] Z. Weng and P. Djuric, ‘‘A search-free DOA estimation algorithm forcoprime arrays,’’ Digit. Signal Process., vol. 24, pp. 27–33, Jan. 2014.

[16] D. Zhang, Y. Zhang, G. Zheng, C. Feng, and J. Tang, ‘‘Improved DOA esti-mation algorithm for co-prime linear arrays using root-MUSIC algorithm,’’Electron. Lett., vol. 53, no. 18, pp. 1277–1279, Aug. 2017.

[17] F. Belloni, A. Richter, and V. Koivunen, ‘‘DoA estimation via manifoldseparation for arbitrary array structures,’’ IEEE Trans. Signal Process.,vol. 55, no. 10, pp. 4800–4810, Oct. 2007.

[18] B. Friedlander, ‘‘The root-MUSIC algorithm for direction finding withinterpolated arrays,’’ Signal Process., vol. 30, no. 1, pp. 15–29, 1993.

[19] B. Friedlander and A. J. Weiss, ‘‘Direction finding using spatial smoothingwith interpolated arrays,’’ IEEE Trans. Aerosp. Electron. Syst., vol. 28,no. 2, pp. 574–587, Apr. 1992.

[20] M. Rubsamen and A. B. Gershman, ‘‘Direction-of-arrival estimationfor nonuniform sensor arrays: From manifold separation to Fourierdomain MUSIC methods,’’ IEEE Trans. Signal Process., vol. 57, no. 2,pp. 588–599, Feb. 2009.

29964 VOLUME 6, 2018

A. Liu et al.: Fast DOA Estimation Algorithms for SULA With Multiple Integer Frequencies

[21] Y. D. Zhang, M. G. Amin, F. Ahmad, and B. Himed, ‘‘DOA estimationusing a sparse uniform linear array with two CW signals of co-primefrequencies,’’ in Proc. 5th IEEE Int. Workshop Comput. Adv. Multi-SensorAdapt. Process., Dec. 2013, pp. 404–407.

[22] S. Qin, Y. D. Zhang, andM.G.Amin, ‘‘DOA estimation exploiting coprimefrequencies,’’ Proc. SPIE, vol. 9103, p. 91030E, May 2014.

[23] S. Qin, Y. D. Zhang, M. G. Amin, and B. Himed, ‘‘DOA estimationexploiting a uniform linear array with multiple co-prime frequencies,’’Signal Process., vol. 130, pp. 37–46, Jan. 2017.

[24] L.Meier, S. van de Geer, and P. Bühlmann, ‘‘The group LASSO for logisticregression,’’ J. Roy. Statist. Soc. B, vol. 70, no. 1, pp. 53–71, 2008.

[25] Y. C. Eldar, P. Kuppinger, and H. Bolcskei, ‘‘Block-sparse signals: Uncer-tainty relations and efficient recovery,’’ IEEE Trans. Signal Process.,vol. 58, no. 6, pp. 3042–3054, Jun. 2010.

[26] Q. Wu, Y. D. Zhang, M. G. Amin, and B. Himed, ‘‘Complex multitaskBayesian compressive sensing,’’ in Proc. IEEE Int. Conf. Acoust., SpeechSignal Process., May 2014, pp. 3375–3379.

[27] Q. Shen, W. Cui, W. Liu, S. Wu, Y. D. Zhang, and M. G. Amin, ‘‘Under-determined wideband DOA estimation of off-grid sources employing thedifference co-array concept,’’ Signal Process., vol. 130, pp. 299–304,Jan. 2017.

[28] Z. Tang, G. Blacquiere, and G. Leus, ‘‘Aliasing-free wideband beam-forming using sparse signal representation,’’ IEEE Trans. Signal Process.,vol. 59, no. 7, pp. 3464–3469, Jul. 2011.

[29] K. Han and A. Nehorai, ‘‘Wideband Gaussian source processing usinga linear nested array,’’ IEEE Signal Process. Lett., vol. 20, no. 11,pp. 1110–1113, Nov. 2013.

[30] P. Stoica and A. Nehorai, ‘‘Performance study of conditional and uncondi-tional direction-of-arrival estimation,’’ IEEE Trans. Acoust., Speech SignalProcess., vol. 38, no. 10, pp. 1783–1795, Oct. 1990.

[31] A. J. Weiss and B. Friedlander, ‘‘On the Cramer-Rao bound for directionfinding of correlated signals,’’ IEEE Trans. Signal Process., vol. 41, no. 1,p. 495, Jan. 1993.

AIHUA LIU received the B.Sc. degree in elec-tronic information engineering from the HarbinInstitute of Technology, Harbin, China, in 2013,where he is currently pursuing the Ph.D. degree ininformation and communication engineering. Hisresearch interests include array signal processing,compressed sensing, direction of arrival estimate,and radar signal processing.

XIN ZHANG received the B.Sc., M.Sc., and Ph.D.degrees in information and communication engi-neering from the Harbin Institute of Technology,Harbin, China, in 2009, 2011, and 2016, respec-tively. He is currently a Lecturer with the School ofElectronic Engineering, Harbin Institute of Tech-nology. His research interests are in radar signalprocessing, clutter suppression, space–time adap-tive processing, and compressed sensing.

QIANG YANG received the B.Sc., M.Sc., andPh.D. degrees in information and communicationengineering from the Harbin Institute of Tech-nology, Harbin, China, in 1992, 1996, and 2002,respectively. He is currently a Professor with theSchool of Electronic Engineering, Harbin Instituteof Technology. He has published over 30 papers invarious journals and conferences. His research hasbeen supported by the National Natural ScienceCouncil. He is currently involved in weak target

detection, real-time processing, and information extraction.

WEIBO DENG received the B.Sc. degree in radiotechnology from Tsinghua University, Beijing,China, in 1984, and theM.Sc. and Ph.D. degrees ininformation and communication engineering fromthe Harbin Institute of Technology, Harbin, China,in 1992 and 2002, respectively. He is currentlythe Director and a Professor with the School ofElectronic Engineering, Harbin Institute of Tech-nology. He has over 60 papers published by inter-national conferences and journals. His research

interests include radar signal processing, clutter suppression, radar targetscattering characteristics, antenna design, and compressed sensing. He is aSenior Member of the Chinese Institute of Electronics.

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