a tensor-based covariance differencing method for...

Received August 3, 2017, accepted September 1, 2017, date of publication September 6, 2017,date of current version September 27, 2017.

Digital Object Identifier 10.1109/ACCESS.2017.2749404

A Tensor-Based Covariance Differencing Methodfor Direction Estimation in Bistatic MIMO RadarWith Unknown Spatial Colored NoiseFANGQING WEN1,2, (Member, IEEE), ZIJING ZHANG3, (Member, IEEE),GONG ZHANG2, (Member, IEEE), YU ZHANG2, XINHAI WANG2,AND XINYU ZHANG41Electronic and Information School, Yangtze University, Jingzhou 434023, China2Key Laboratory of Radar Imaging and Microwave Photonics, Ministry of Education, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China3National Laboratory of Radar Signal Processing, Xidian University, Xi’an 710071, China4School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, China

Corresponding author: Zijing Zhang ([email protected])

This work was supported in part by China NSF under Grant 61701046, Grant 61471191, Grant 61501233, Grant 61501152, andGrant 61571349, and in part by the Electronic and Information School of Yangtze University Innovation Foundation underGrant 2016-DXCX-05.

ABSTRACT In this paper, we investigate into direction estimation in bistatic multiple-input multiple-output (MIMO) radar in the presence of unknown spatial colored noise. Taking the stationary propertyof the spatial colored noise into consideration, a transform-based tensor covariance differencing methodis proposed. The spatial colored noise is eliminated by forming the difference of the original and thetransformed covariance matrices. To further exploit the inherent multidimensional nature, a fourth-ordertensor is constructed, which helps to achieve more accurate subspace estimation. Thereafter, the traditionalsubspace-based methods are applied for ambiguous direction estimation. Finally, a special matrix is formedto associate the real angles with the targets. The proposed scheme does not bring virtual aperture loss, andit has complexity lower than the existing tensor-based subspace methods. Numerical simulations verify theimprovement of our scheme.

INDEX TERMS Bistatic MIMO radar, direction estimation, spatial colored noise, covariance differencing,Tucker decomposition.

I. INTRODUCTIONThe topic of joint direction-of-departure (DOD) anddirection-of-arrival (DOA) estimation in bistaticMIMO radarhas aroused extensive attention in the past decade [1]–[8].It has been shown that typical subspace-based methods,such as MUSIC and ESPRIT [1], [2], provide super resolu-tion estimation performance. As suggested in the literature,the received noise exhibits uniformwhite Gaussianity or has aknown covariance matrix. By exploiting the subspace decom-position technique or the pre-whitening strategy, the sig-nal/noise subspace can be properly determined. However,in practice, the received array noise may not fulfill a uni-form white Gaussian distribution. A typical scenario in radarsystem is that the noise fields can be highly colored and thereceived array noise are strongly correlated [9]. Moreover,the underlying noise generating mechanism in radar is too

complicated to allow the predication of the noise covariance.In such cases, the dominate singularvectors do not spanthe signal subspace owing to the nonuniform of the noisevariances, and the existing subspace-based methods are nolonger applicable.

Several methods have been proposed to deal with the spa-tial colored noise in MIMO radar [10]–[15]. For instance,by exploiting the non-correlation between the matched noisecorresponding to different transmit antenna, the spatial cross-correlation methods have been addressed in [10]–[12], wherethe transmit antenna array is divided into non-overlappingsubarrays. The spatial cross-correlation matrix is formulatedto eliminate the colored noise. Nevertheless, a commonweakness these methods share is that the virtual apertureloss, which will degrades the angle estimation perfor-mance. In contract with the spatial cross-correlation methods,

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F. Wen et al.: Tensor-Based Covariance Differencing Method for Direction Estimation in Bistatic MIMO Radar

the temporal cross-correlation matrix-based approach hasbeen studied in [14]. The temporal uniform white Gaussiancharacteristic of the noise in individual receive channel isexploited to suppress the spatial colored noise, which wouldnot hurt the virtual aperture of MIMO radar. More recently,the tensor subspace-based versions of the above mentionedcross-correlation methods have been derived [13], [15],in which the multidimensional structure of the array data isutilized to achieve performance improvement. Nevertheless,both of which are computationally inefficient and the visualaperture loss still exists in [13].

The covariance difference principle has a long historyin enhancing detection and estimation performance of aradar or a sonar. Motivated by the stationary property ofthe received array noise, a new tensor subspace methodis proposed. Two array covariance matrices are formed,in which the colored noise filed remains invariant whilethe signal filed undergoes some changes. By computing thedifference of the array covariances, the unknown noise termis removed. To further explore the multidimensional natureof the array data, the de-noising process is expressed intensor format. Since the covariance difference measurementis a low-rank Herminate tensor, the higher-order singularvalue decomposition (HOSVD) is applicable for eigenstruc-ture extraction. Thereafter, the existing subspace algorithms(MUSIC and ESPRIT) are utilized for ambiguous angle esti-mation, and finally some extra calculation is carried out todetermine the unique directions. The computational complex-ities of the proposed methods have been analyzed. Since theuniformwhite noise is a special case of colored noise, the pro-posed method can be regarded as a generalized HOSVDmethod.

The paper outline is as follows. The data model for thebistatic MIMO radar with spatial colored noise is presentedin section 2. The details of the proposed scheme are givenin section 3. The complexity analysis is given in section 4.Simulation results are illustrated in section 5. The paper isended by a brief concluding in section 6.

Notation, bold capital letters, e.g., X, bold lowercase let-ters, e.g., x, and boldface Euler script letters, e.g., X , denotematrices, vectors, and tensors, respectively. The M × Midentity matrix is denoted by IM . The superscript (X)T ,(X)H

and (X)−1 stand for the operations of transpose, Hermitiantranspose and inverse, respectively; ⊗ and � represent,respectively, the Kronecker product and the Khatri-Rao prod-uct (column-wise Kronecker product); diag (·) and vec (·)denotes the diagonalization and the vectorization opera-tion, respectively. E (·) returns the expectation of a variable,rank (·) and det (·) denote rank operator and the determinantof a matrix, respectively.

II. BASIC TENSOR OPERATIONS AND SIGNAL MODELA. TENSOR BASESA tensor is a multidimensional array [16]. Let X ∈

CI1×I2×···IN denotes an N -th order tensor. The fibers arethe higher-order analogue of matrix rows and columns.

A mode-n fiber of X is an In-dimensional column vectorobtained from X by varying the index in and keeping theother indices fixed. Some useful definitions concerning ten-sor operation are listed as follows:Definition 1 (Unfolding or Matricization): The mode-n

unfolding of an N -th order tensor X ∈ CI1×I2×···×INis denoted by [X ](n). The (i1, i2, · · · , iN )-element of Xmaps to the (in, j)-th element of [X ](n), where j = 1 +∑N

k=1,k 6=n (ik − 1)Jk with Jk =∏k−1

m=1,m 6=n Im.Definition 2 (Mode-n Tensor-Matrix Product):Themode-n

product of an N -order tensorX ∈ CI1×I2×···×IN and a matrixA ∈ CJn×In , denoted by X×nA, is a tensor of size I1 ×· · · × In−1 × Jn × In+1 × · · · × IN , obtained by taking theinner product between each mode-n fiber and the rows of thematrix A, i.e.,

Y = X×nA⇐⇒ [Y](n) = A [X ](n) (1)

The mode-n product admits the following properties{X×nA×mB = X×mB×nA, m 6= nX×nA×nB = X×n (BA)

(2)

Definition 3 (Tucker Decomposition): The Tucker decompo-sition of an N -order tensorX ∈ CI1×I2×···×IN is given by

X = G×1A1×2A2×···AN (3)

which can be regarded as a multilinear transformation of acore tensor G ∈ CJ1×J2×···×JN represents the core tensor bythe factor matrices An ∈ CIn×Jn (n = 1, 2, · · ·N ), and itfulfills

[X ](n) = An ·[G](n) ·[An+1 ⊗ · · · ⊗ AN ⊗ A1 · · · ⊗ An−1]

(4)

FIGURE 1. Bistatic MIMO radar configuration.

B. DATA MODELConsider a bistatic MIMO radar system configured withM transmit antennas and N receive antennas, as illustratedin Fig. 1, both of which are uniform linear arrays with half-wavelength spacing. The transmit antennas emit M orthogo-nal coded waveforms, and the length of symbols per pulseduration is Q. Assume that there are K far-field targetsappearing in the same range bin of the radar system, ϕk andθk represent the DOD and DOA of the k-th target, respec-tively. The transmitted signals are reflected by the targets,and the echoes are collected by the receive antennas. We con-sider a coherent processing interval consisting of L pulses.

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The received array signal at the l-th (l = 1, 2, · · · ,L) pluseperiod takes the form [5], [13]

Xl = ARdiag (bl)ATTS+ Nl (5)

where AR, bl , AT , S, Nl denote, respectively, the receivedirection matrix, the echo coefficient vector, the transmitdirection matrix, the transmit code matrix, the colored noisematrix, and

AR = [ar (θ1) , ar (θ2) , · · · , ar (θK )] ∈ CN×K

ar (θk ) =[1, ejπ sin θk , . . . , ejπ(N−1) sin θk

]TAT = [at (ϕ1) , at (ϕ2) , · · · , at (ϕK )] ∈ CM×K

at (ϕk ) =[1, ejπ sinϕk , . . . , ejπ(M−1) sinϕk

]Tbl =

[α1ej2π lf1/fs , α2ej2π lf2/fs , . . . , αK ej2π lfK /fs

]TS = [s1, s2, · · · , sM ] ∈ CM×Q

where ar (θk ) and at (ϕk ) are the k-th receive steering vec-tor and the k-th transmit steering vector, respectively; αk ,fk (k = 1, 2, . . . ,K ) and fs represent the radar crosssection (RCS) amplitude, the Doppler frequency and thepulse repeat frequency, respectively; sm ∈ C1×Q is them-th (m = 1, 2, · · · ,M ) baseband code and smsHm = Q.The columns of Nl are independent and identical distributioncircularly symmetric complex Gaussian random vectors withzero mean and unknown covariance matrix C, i.e.,

E{vec

(Np)vecH

(Nq)}=

{0, p 6= qIQ ⊗ C, p = q

(6)

The received signals are matched by sm/Q,m = 1, 2, · · · ,M .By stacking the output along the pulse direction, we get

Y = [AT � AR]BT +1QW (7)

where B = [b1,b2, . . . ,bL]T , W = [w1,w2, . . . ,wL]denotes the matched noise matrix with wl = vec

(NlSH

),

(l = 1, 2, . . . ,L). The model in Eq. (7) can be viewedas the matrix form of the array measurement. Accordingto Eq. (7), the array data is sampled on an 3-dimensionallattice. However, the multidimensional nature inherent in thelattice is ignored in Y. Actually, Y can be rearranged into athird-order tensor Y ∈ CN×M×L as [7]

Y = IK×1AR×2AT×3B+1QW (8)

where IK is the K × K × K identity tensor. The relationsbetween Eq. (7) and Eq. (8) are Y = [Y]T(3) andW = [W]T(3),respectively.

III. THE PROPOSED ALGORITHMA. TENSOR-BASED COVARIANCE DIFFERENCINGIn the traditional subspace-based methods, the covariancematrix R is first estimated, which is given by

R = E[YYH

]= [AT � AR]RB [AT � AR]H +

1Q2RW (9)

where RB = E[BTB∗

], RW = E

[WWH

]. Con-

sidering the case in which the targets are uncorrelated,RB =

1L diag

([ρ21 , ρ

22 , · · · , ρ

2K

])is a real diagonal matrix,

ρ2k denotes the coefficient variance of the k-th target. In prac-tice, R can be estimated from finite measurement viaR = YYH/L. Let p, q ∈ {1, 2, · · · ,L}, we get

E{wpwH

q

}= E

{vec

(NpSH

)vecH

(NqSH

)}= E

{[S∗ ⊗ IN

] [vec

(Np)vecH

(Nq)] [

ST ⊗ IN]}

=

{0, p 6= qE{[S∗ ⊗ IN

] [IQ ⊗ C

] [ST ⊗ IN

]}, p = q

=

{0, p 6= qQ(IQ ⊗ C

), p = q

(10)

It can be concluded from Eq. (10) that RW is proportionalto IM ⊗ C. Since RW is no longer a scaled identity matrix,we are now faced with an unknown noise field. In thiscase, the noise subspace can not be separated correctlyfrom the signal subspace, and the performances of the tra-ditional subspace-based algorithms would degrade seriously.Fortunately, the noise process is stationary, thus C is aHermitian symmetric Toeplitz matrix, and it fulfills

JNC∗JN = C (11)

where JN denotes a N × N exchange matrix with N oneson its anti-diagonal and zeros elsewhere. With the propertyin Eq. (11), we further get

(IM ⊗ JN )R∗W (IM ⊗ JN ) = RW (12)

Observably, the noise component keep unchanged after itsconjugate linear transform. To eliminate the spatial col-ored noise, the covariance differencing technique is herebyadopted. Suppose that there is no paired targets that arelocated symmetrically about the transmit array broadside and2K < MN , then the following difference matrix can beformed

1R = R− (IM ⊗ JN )R∗ (IM ⊗ JN )

= [AT � AR]RB [AT � AR]H

−[A∗T � (AR9R)

]RB

[A∗T � (AR9R)

]H=[AT � AR,A∗T � (AR9R)

] [RB, 00, −RB

]×[AT � AR,A∗T � (AR9R)

]H={[AT ,A∗T

]� [AR,AR]

} [RB, 00, −RB

]{[AT ,A∗T

]� [AR,AR]

}H (13)

where 9R = diag([ej(N−1)πsinθ1 , ej(N−1)πsinθ2 , · · · ,ej(N−1)πsinθK ]) is a diagonal matrix. Clearly, the unknownnoise covariance has been subtracted from 1R while thesignal covariances remains unchanged. Moreover,1R is sin-gular Hermitian matrix. As a result, R can be approximated

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by its truncated eigenvalue decomposition (EVD), i.e.,

1R ≈ Es6sEHs (14)

where 6s and Es contain the 2K dominant eigenvalues andthe corresponding eigenvectors of1R, respectively. It is easyto proveEs and

[AT ,A∗T

]�[AR,AR] span the same subspace.

This subspace will henceforth be termed the signal subspaceand it contains the angle information of the targets. To furtherexplore the multi-dimensional inherent structure, the tensorcovariance model can be utilized. According to [17] and [18],the covariance tensor can be expressed as

R = RB×1AR×2AT×3A∗R×4A∗T +RW (15)

The fourth-order tensor decomposition model in Eq. (15) iscommonly known as the noisy ‘‘Tucker4’’ decomposition(Definition 3) of 1R, where the associated core tensor isgiven by RB ∈ CK×K×K×K and the factors matrices are AR,AT , A∗R and A∗T , respectively,RW is the covariance tensor ofthe noise. It is easy to show thatR is a Hermitian tensor [19].In fact, R can be obtained by letting the first 2 indices of Rvary along the columns (the 1th index is fasten, then index 2)and the last 2 indices along the rows (in the same order)of R [5], [13]. The above arrangement can be interpretedas the Hermitian unfolding of R ([19, Lemma 1]), whichis marked by R = [R](H). Similarity, the relationsbetween RB and RB, RW and RW are RB = [RB](H),RW = [RW](H), respectively. Consequently, the differencecovariance tensor can be constructed as

1R = R−R×1JN×3JN= RB×1[AR,AR]×2

[AT ,A∗T

]×3

[A∗R,A

∗R]×4

[A∗T ,AT

](16)

where RB ∈ C2K×2K×2K×2K denotes the core tensor with[RB

](H)=

[RB, 00, −RB

], [AR,AR],

[AT ,A∗T

],[A∗R,A

∗R

],[

A∗T ,AT]

are, respectively, the corresponding factormatrices. Also, 1R is a Hermitian tensor and the unknownnoise is subtracting out from 1R.

B. TENSOR COVARIANCE-BASED SUBSPACE ESTIMATIONOur goal is to estimate the signal subspace from 1R. In thispaper, a direct tensor decomposition method is utilized.According to [19], the HOSVD of 1R is given by

1R = G×1U1×2U2×3U3×4U4 (17)

where G ∈ CN×M×N×M is the core tensor, Un(n ∈ {1, 2, 3, 4}) are the left singular vectors of the n-modematrix unfolding of 1R as [1R](n) = Un6nVH

n . It is easyto find that U1 = U∗3 and U2 = U∗4. Similar to the traditionaleigenstructure method,1R can be expressed by its truncatedHOSVD as [5]

1Rs = Gs×1U1s×2U2s×3U∗1s×4U∗

2s (18)

where U1s ∈ CN×K is the K dominant column vectorsof U1, U2s ∈ CM×2K contains the column vectors of U2 cor-responding to the 2K dominant singular values, respectively;Gs denotes for the signal component of G, which is given by

Gs = 1R×1UH1s×2U

H2s×3U

T1s×4U

T2s (19)

Combining with Definition 2, we get

1Rs = 1R×1UH1s×2U

H2s×3U

T1s×4U

T2s×1U1s×2U2s×3

×U∗1s×4U∗

2s

= 1R×1(U1sUH

1s

)×2

(U2sUH

2s

)×3

×

(U∗1sU

T1s

)×4

(U∗2sU

T2s

)(20)

By the Hermitian unfolding of 1Rs, we can form a newcross-correlation matrix Rs from 1Rs, which is given by

Rs = [1Rs](H)

=

[(U1sUH

1s

)⊗

(U2sUH

2s

)]×1R

[(U1sUH

1s

)⊗

(U2sUH

2s

)]H(21)

Worthnoting is that Rs is a Hermitian matrix. Insertion ofEq.(14) into Eq.(21) yields

Rs=

[(U1sUH

1s

)⊗

(U2sUH

2s

)Es]6

[(U1sUH

1s

)⊗

(U2sUH

2s

)TEs

]H(22)

Since U1sUH1s and U2sUH

2s are unitary matrices, Rs can beapproximated by its truncated EVD as Rs ≈ Es6sEHs ,where 6s and Es are the 2K (K ≤ MN ) dominant eigenval-ues and corresponding eigenvectors, respectively. Obviously,Es and Es span the same signal subspace. As a result, thereexists a full-rank matrix T that

Es =([AT ,A∗T

]� [AR,AR]

)T (23)

Remark 1: It is assumed that there is no paired targetslocated symmetrically about the transmit array broadside,hence

[AT ,A∗T

]� [AR,AR] has full column rank, otherwise

Es can not be estimated correctly.Remark 2: The HOSVD of an tensor is equivalent to

the SVD of all its matrix unfolding. Since U1 = U∗3 andU2 = U∗4, we only need to calculate SVD of the mode-1 andmode-2 matrix unfolding of 1R, resulting in a significantcomputational savings.Remark 3: The maximum number of targets that can be

uniquely identified by the proposed method is determined bythe maximum rank of

[AT ,A∗T

]� [AR,AR], which is MN .

Notably, the identifiability of the proposed method is thesame as the methods in [14] and [15]. However, the methodsin [11] and [13] can identify atmostmin {M1N ,M2N } targets,where M1 and M2 represent the antenna numbers of twosubarrays.

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C. JOINT DOD AND DOA ESTIMATIONUnlike the traditional signal subspace, Es contains the realtarget direction (φ, θ) and its mirror direction (−φ, θ). In thispaper, a two-step framework is presented to obtain the realdirections from Us. In the first step, the traditional subspace-based algorithms, such as MUSIC and ESPRIT, are utilizedfor ambiguous DOD and DOA estimation. In the second step,a criterion is given to determine the unique directions.

1) MUSIC METHODBy exploiting the orthogonality characteristic between thenoise subspace and the signal subspace, the spatial spectrumsearching method can be exploited. The DODs and DOAs areobtained via maximization the following spectrum function

f (φ, θ) =1

[at (φ)⊗ ar (θ)]H Fn [at (φ)⊗ ar (θ)](24)

where Fn = IMN − EoEHo , Eo is the orthogonal basis of Es.Eq.(25) involves two-dimensional peak searches, which iscomputationally inefficient. Here, the reduced-dimensionMUSIC [1] idea can be adopted to lower the computationalcost, where directions estimation is linked to the followingquadratic optimization problemsθ = argmin f (θ)= argmin det

{[IM ⊗ ar (θ)]H Fn [IM ⊗ ar (θ)]

}φ = argmin f (φ)= argmin det

{[at (φ)⊗ IN ]H Fn [at (φ)⊗ IN ]

} (25)

Since the rank reduction of f (θ) and f (φ), respec-tively, will take place on the estimations of θk and ±φk(k = 1, 2, · · · ,K ), the DODs and DOAs can be esti-mated via two one-dimensional spectrum searches. Afterwhich extra computation is carried out to pair the estimatedDOD and DOA.Remark 4: Since the estimated DODs are symmetrical to

zero, the computational load in f (φ) can be further reducedby constraining φ ∈ [0, π). All the possible DODs can berecovered by the union of the estimated DODs and theirnegative values.

2) ESPRIT METHODThe rotational invariance properties can be explored to obtaina closed-form solution for the DODs and DOAs. Accordingto Eq.(21), there exists the following rotational invarianceproperties {

C1Es = C2Es9t

C3Es = C4Es9r(26)

where C1 = CM1 ⊗ IN , C2 = CM2 ⊗ IN , C3 =

IM ⊗ CN1, C4 = IM ⊗ CN2 are selection matri-ces; 9t = diag([ejπsinϕ1 , ejπsinϕ2 , · · · , ejπsinϕK , e−jπsinϕ1 ,e−jπsinϕ2 , · · · , e−jπsinϕK ]), 9r = diag([ejπsinθ1 , ejπsinθ2 , · · · ,

ejπsinθK , ejπsinθ1 , ejπsinθ2 , · · · , ejπsinθK ]) are rotational invari-ance matrices. The least square solutions for 9t and 9r are{

9t =(EHs C

H2 C2Es

)EHs C

H2 C1Es

9r =(EHs C

H4 C4Es

)EHs C

H4 C3Es

(27)

The diagonal elements of 9t and 9r contain the directioninformation of the targets, which can be easily obtained andpaired form 9t and 9r .To assign K DODs each corresponding to the matrices At ,

andA∗t , we need to solve for the difference source covariancematrix. LetA be a guess of the form of estimated

[AT ,A∗T

]�

[AR,AR], then we calculate the following matrix

5 =(AHA

)−1AH1RA

(AHA

)−1(28)

If the initial guess is correct, 5 will have positive valuesalong the upper half of the diagonal and negative elementsalong the lower half of its diagonal. Once the initial guessis incorrect, this result will not be observed, but the signs ofdiagonal elements in 5 still suggest the appropriate form.Till now, we have achieved the proposal of our scheme. The

detailed steps for the proposed method are shown as followsstep.1 Stack the matched data into a third-order tensor

as Eq. (8);step.2 Estimate the covariance tensor R, and step further

to get 1R according to Eq. (16);step.3 Perform HOSVD on 1R, and get Rs through

Eq.(21). Perform EVD of Rs to get the signal subspace Es;step.4 Obtain the ambiguous DOD and DOA pairs via

MUSIC or ESPRIT. Finally, determine the unique directionsvia (28).

IV. COMPLEXITY ANALYSISThe computational complexity of the proposed methodis summarized as follows. The estimation of 1R needsM2N 2L complexmultiplications. The complexity of HOSVDon 1R is on the order 2O

(M3N 3

). The load of com-

puting Rs in Eq.(19) is 8M2K + 2N 2K + 2M3N 3,and its EVD requires O

(M3N 3

)complex multiplications.

The estimation complexity of the ambiguous directions isl(2M3N 2

+M2N 3+ O

(M3)+ 0.5O

(N 3))

(MUSIC-basedmethod) or 8 (M − 1)NK 2

+ 8 (N − 1)MK 2+ 16O

(K 3)

(ESPRIT-based method). The complexity of unique direc-tion determine is 4M2N 2K + 8MNK 2

+ 8O(K 3). We sum-

marize the total computational loads of the proposedmethods, [11](marked with Chen’s method), [13](markedwith Wang’s method), [14](marked with Fu’s method)and [15](marked with Wen’s method) in Table 1. It can beseen that the complexity of the proposed ESPRIT methodis lower than Wang’s method and Wen’s method while theproposed MUSIC method may be more complex than them.

V. SIMULATION RESULTSIn this section, 200Monte Carlo trials were performed to ver-ify the improvement of the proposed methods (Matlab code isavailable on https://pan.baidu.com/s/1gfw3PYz). The bistatic

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TABLE 1. Comparison of the complexity.

MIMO radar is configured with M = 10 transmit elementsand N = 12 receive elements. Pulse number Q and pulserepeat frequency fs are set to Q = 256, fs = 20KHz. Assumethat there exist K = 3 uncorrelated sources located at theangles (θ1, ϕ1) = (30◦,−30◦), (θ2, ϕ2) = (−45◦, 10◦),(θ3, ϕ3) = (10◦, 20◦), and Doppler frequency shifts are{fk}3k=1 = {200, 400, 850}Hz. The RCS coefficients conformto the Swerling I model. The following root mean squareerror (RMSE) is used for performance measure

RMSE =1K

K∑k=1

√√√√ 1200

200∑i=1

{(θi,k − θk

)2+(ϕi,k − ϕk

)2}where θi,k and ϕi,k correspondingly represent the estimatesof θk and ϕk for the i th Monte Carlo trial. For comparison,the performances of ESPRIT [2], Chen’s method and Wang’smethod are evaluated.

FIGURE 2. RMSE performance comparison versus SNR in case (1).

In the first simulation, the (p, q)-th element in C is givenby C (p, q) = 0.9|p−q|ejπ (p−q)/2. Fig. 2 depicts the resul-tant RMSEs at different SNRs with L = 200. It canbe observed that RMSE performance of all the methodsgradually improves with the growing SNR. Besides, all the

FIGURE 3. RMSE performance comparison versus L in case (1).

de-noising methods provide better RMSE performances thanESPRIT method at low SNR regions (SNR ≤ −5dB),as spatial colored noise has been eliminated. However, dueto the virtual aperture loss, Chen’s method and Wang’smethod perform worse than ESPRIT at high SNR regions(SNR ≥ 0dB). Additionally, the performance of proposed-ESPRIT method coincides with Wen’s method while theproposed-MUSIC method provides better RMSE perfor-mance than Wen’s method, since no visual aperture lossoccurs in our methods and Wen’s method, and the iden-tifiability of MUSIC is better than ESPRIT. Fig. 3 givesthe RMSEs in the case of different numbers of pluse withSNR = −15dB. As expected, the proposed-MUSIC methodsignificantly outperform all the compared methods.

FIGURE 4. RMSE performance comparison versus SNR in case (2).

In the second simulation, the spatial colored noise is mod-eled as a second-order autoregressive (AR) process with thecoefficients z = [1,−1, 0.8] [11], [13]. Fig. 4 gives theRMSEs of different methods versus the SNR with L = 200.Fig. 5 illustrates the RMSEs versus L with SNR = −15dB.It can be found that the results are very similar that in the firstsimulation. It worth noting that for SNR = −15dB, all the de-noising mechanisms work. However, Wang’s method, Wen’smethod and our methods perform better than Chen’s method,which implies that the multidimensional inherent structure ofthe array data helps to achieve a more accurate subspace esti-mation. Furthermore, Wen’s method and our methods offer

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FIGURE 5. RMSE performance comparison versus L in case (2).

better RMSE performances than Wang’s method, especiallyat high SNR regions, because the covariance differencingtechnique would not decrease the visual array aperture whilethe spatial cross-correlation method would bring virtual aper-ture loss. Since the dimension of subspace in our is largerthan that in Wen’s method, the accuracy of subspace estima-tion may not as good as Wen’s method, thus the proposed-ESPRIT method perform worse than Wen’s method at lowSNR regions. Asmentioned in the last section, the complexityof tensor decomposition in our methods is lower than Wen’smethod, therefore, our methods can obtain a tradeoff betweenperformance and complexity.

VI. CONCLUSIONIn this paper, we investigate into tensor-based covariance dif-ferencingmethod for angle estimation in bistaticMIMO radarin the presence of unknown spatial colored noise. Unlikethe existing methods, both the inherent multidimensionalstructure and the full virtual aperture are considered in theproposed scheme. A two-step framework is proposed forangle estimation, where an ambiguous angle estimation isfirst achieved and follows a unique angle determination. Theproposed scheme provides more accurate parameters estima-tion performance while it has less computational complexitythan the existing tensor-based methods, which will lead toa brighter prospect in applications. Finally, numerical simu-lation are given to verify the improvement of the proposedscheme.

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[2] C. Duofang, C. Baixiao, and Q. Guodong, ‘‘Angle estimation usingESPRIT in MIMO radar,’’ Electron. Lett., vol. 44, no. 12, pp. 770–771,Jun. 2008.

[3] B. Tang, J. Tang, Y. Zhang, and Z. Zheng, ‘‘Maximum likelihood estima-tion of DOD and DOA for bistatic MIMO radar,’’ Signal Process., vol. 93,no. 5, pp. 1349–1357, 2013.

[4] X. Zhang, Z. Xu, L. Xu, and D. Xu, ‘‘Trilinear decomposition-basedtransmit angle and receive angle estimation for multiple-input multiple-output radar,’’ IET Radar Sonar Navigat., vol. 5, no. 6, pp. 626–631,2011.

[5] Y. Cheng, R. Yu, H.Gu, andW. Su, ‘‘Multi-SVDbased subspace estimationto improve angle estimation accuracy in bistatic MIMO radar,’’ SignalProcess., vol. 93, no. 7, pp. 2003–2009, 2013.

[6] X. Wang, W. Wang, J. Liu, Q. Liu, and B. Wang, ‘‘Tensor-based real-valued subspace approach for angle estimation in bistatic MIMO radarwith unknown mutual coupling,’’ Signal Process., vol. 116, pp. 152–158,Nov. 2015.

[7] B. Xu, Y. Zhao, Z. Cheng, and H. Li, ‘‘A novel unitary PARAFAC methodfor DOD and DOA estimation in bistatic MIMO radar,’’ Signal Process.,vol. 138, pp. 273–279, Sep. 2017.

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[10] M. Jin, G. Liao, and J. Li, ‘‘Joint DOD and DOA estimation for bistaticMIMO radar,’’ Signal Process., vol. 89, no. 2, pp. 244–251, 2009.

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[12] H. Jiang, J.-K. Zhang, and K. M. Wong, ‘‘Joint DOD and DOA estimationfor bistatic MIMO radar in unknown correlated noise,’’ IEEE Trans. Veh.Technol., vol. 64, no. 11, pp. 5113–5125, Nov. 2015.

[13] X.Wang,W.Wang, X. Li, and J.Wang, ‘‘A tensor-based subspace approachfor bistatic MIMO radar in spatial colored noise,’’ Sensors, vol. 14, no. 3,pp. 3897–3907, 2014.

[14] W.-B. Fu, T. Su, Y.-B. Zhao, and X.-H. He, ‘‘Joint estimation of angle anddoppler frequency for bistatic MIMO radar in spatial colored noise basedon temporal-spatial structure,’’ J. Electron. Inf. Technol., vol. 33, no. 7,pp. 1649–1654, 2011.

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FANGQING WEN (M’17) was born in 1988.He received the B.S. degree in electronic engi-neering from the Hubei University of AutomotiveTechnology, Shiyan, China, in 2011, and the Ph.D.degree from the Nanjing University of Aeronauticsand Astronautics, China, in 2016, where he is cur-rently pursuing the Master’s degree with the Col-lege of Electronics and Information Engineering.From 2015 to 2016, he was a Visiting Scholar withthe University of Delaware, USA. Since 2016,

he has been with the Electronic and Information School, Yangtze University,China, where he is currently an Assistant Professor. His research interestsincludeMIMO radar, array signal processing, and compressive sensing. He isa member of the Chinese Institute of Electronics.

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ZIJING ZHANG (M’11) was born in Beijing,China, in 1967. He received the B.S. and M.S.degrees in dynamics from the Harbin Institute ofTechnology, Harbin, China, in 1989 and 1992,respectively, and the Ph.D. degree in electricalengineering from Xidian University, Xi’an, China,in 2001. In 2006, he was a Visiting Scholar withThe University of Manchester, U.K. In 2016, hewas a Visiting Scholar with the University ofDelaware, USA. Since 1992, he has been with the

National Laboratory of Radar Signal Processing, Xidian University. Hiscurrent research interests include radar signal processing and multirate filterbanks design.

GONG ZHANG (M’07) received the Ph.D. degreein electronic engineering from the Nanjing Uni-versity of Aeronautics and Astronautics (NUAA),Nanjing, China, in 2002. From 1990 to 1998, hewas a Member of Technical Staff with the No724Institute of China Shipbuilding Industry Corpora-tion, Nanjing. Since 1998, he has been with theCollege of Electronics and Information Engineer-ing, NUAA, where he is currently a Professor. Hisresearch interests include radar signal processing

and compressive sensing. He is a member of the Committee of Electromag-netic Information, Chinese Society of Astronautics and a Senior Member ofthe Chinese Institute of Electronics.

YU ZHANG was born in 1991. He receivedthe B.S. degree from the Nanjing University ofAeronautics and Astronautics (NUAA), Nanjing,China, in 2013, and the M.S. degree from the Col-lege of Electronics and Information Engineering,NUAA, in 2015, where he is currently pursuingthe Ph.D. degree with the College of Electronicsand Information Engineering. His research inter-ests are array signal processing, statistical signalprocessing, and compressive sensing.

XINHAI WANG was born in 1988. He received theB.S. degree from the School of Electronic Infor-mation Engineering, Nanjing University of Infor-mation Science and Technology, Nanjing, China,in 2012, and the M.S. degree from the Collegeof Electronics and Information Engineering,Nanjing University of Aeronautics and Astro-nautics, where he is currently pursuing thePh.D. degree with the College of Electronics andInformation Engineering. His research interests

are array signal processing, wireless communication, and signal processing.

XINYU ZHANG was born in 1999. He is study-ing at Lanzhou University, China. His researchinterests are artificial intelligence and signalprocessing.

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