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 Method for Direction Estimation in Bistatic MIMO Radar With Unknown Spatial Colored Noise FANGQING WEN1,2 , (Member, IEEE), ZIJING ZHANG3 , (Member, IEEE), GONG ZHANG2 , (Member, IEEE), YU ZHANG2 , XINHAI WANG2 , AND XINYU ZHANG4 1 Electronic
and Information School, Yangtze University, Jingzhou 434023, China Laboratory of Radar Imaging and Microwave Photonics, Ministry of Education, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Laboratory of Radar Signal Processing, Xidian University, Xi’an 710071, China 4 School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, China 2 Key
3 National
Corresponding author: Zijing Zhang (
[email protected]) This work was supported in part by China NSF under Grant 61701046, Grant 61471191, Grant 61501233, Grant 61501152, and Grant 61571349, and in part by the Electronic and Information School of Yangtze University Innovation Foundation under Grant 2016-DXCX-05.
ABSTRACT In this paper, we investigate into direction estimation in bistatic multiple-input multipleoutput (MIMO) radar in the presence of unknown spatial colored noise. Taking the stationary property of the spatial colored noise into consideration, a transform-based tensor covariance differencing method is proposed. The spatial colored noise is eliminated by forming the difference of the original and the transformed covariance matrices. To further exploit the inherent multidimensional nature, a fourth-order tensor is constructed, which helps to achieve more accurate subspace estimation. Thereafter, the traditional subspace-based methods are applied for ambiguous direction estimation. Finally, a special matrix is formed to associate the real angles with the targets. The proposed scheme does not bring virtual aperture loss, and it has complexity lower than the existing tensor-based subspace methods. Numerical simulations verify the improvement of our scheme. INDEX TERMS Bistatic MIMO radar, direction estimation, spatial colored noise, covariance differencing, Tucker decomposition.
I. INTRODUCTION
The topic of joint direction-of-departure (DOD) and direction-of-arrival (DOA) estimation in bistatic MIMO radar has 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 resolution estimation performance. As suggested in the literature, the received noise exhibits uniform white Gaussianity or has a known covariance matrix. By exploiting the subspace decomposition technique or the pre-whitening strategy, the signal/noise subspace can be properly determined. However, in practice, the received array noise may not fulfill a uniform white Gaussian distribution. A typical scenario in radar system is that the noise fields can be highly colored and the received array noise are strongly correlated [9]. Moreover, the underlying noise generating mechanism in radar is too
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complicated to allow the predication of the noise covariance. In such cases, the dominate singularvectors do not span the signal subspace owing to the nonuniform of the noise variances, and the existing subspace-based methods are no longer applicable. Several methods have been proposed to deal with the spatial colored noise in MIMO radar [10]–[15]. For instance, by exploiting the non-correlation between the matched noise corresponding to different transmit antenna, the spatial crosscorrelation methods have been addressed in [10]–[12], where the transmit antenna array is divided into non-overlapping subarrays. The spatial cross-correlation matrix is formulated to eliminate the colored noise. Nevertheless, a common weakness these methods share is that the virtual aperture loss, which will degrades the angle estimation performance. In contract with the spatial cross-correlation methods,
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the temporal cross-correlation matrix-based approach has been studied in [14]. The temporal uniform white Gaussian characteristic of the noise in individual receive channel is exploited to suppress the spatial colored noise, which would not hurt the virtual aperture of MIMO radar. More recently, the tensor subspace-based versions of the above mentioned cross-correlation methods have been derived [13], [15], in which the multidimensional structure of the array data is utilized to achieve performance improvement. Nevertheless, both of which are computationally inefficient and the visual aperture loss still exists in [13]. The covariance difference principle has a long history in enhancing detection and estimation performance of a radar or a sonar. Motivated by the stationary property of the received array noise, a new tensor subspace method is proposed. Two array covariance matrices are formed, in which the colored noise filed remains invariant while the signal filed undergoes some changes. By computing the difference of the array covariances, the unknown noise term is removed. To further explore the multidimensional nature of the array data, the de-noising process is expressed in tensor format. Since the covariance difference measurement is a low-rank Herminate tensor, the higher-order singular value decomposition (HOSVD) is applicable for eigenstructure extraction. Thereafter, the existing subspace algorithms (MUSIC and ESPRIT) are utilized for ambiguous angle estimation, and finally some extra calculation is carried out to determine the unique directions. The computational complexities of the proposed methods have been analyzed. Since the uniform white noise is a special case of colored noise, the proposed method can be regarded as a generalized HOSVD method. The paper outline is as follows. The data model for the bistatic MIMO radar with spatial colored noise is presented in section 2. The details of the proposed scheme are given in section 3. The complexity analysis is given in section 4. Simulation results are illustrated in section 5. The paper is ended by a brief concluding in section 6. Notation, bold capital letters, e.g., X, bold lowercase letters, e.g., x, and boldface Euler script letters, e.g., X , denote matrices, vectors, and tensors, respectively. The M × M identity matrix is denoted by IM . The superscript (X)T ,(X)H and (X)−1 stand for the operations of transpose, Hermitian transpose and inverse, respectively; ⊗ and represent, respectively, the Kronecker product and the Khatri-Rao product (column-wise Kronecker product); diag (·) and vec (·) denotes the diagonalization and the vectorization operation, respectively. E (·) returns the expectation of a variable, rank (·) and det (·) denote rank operator and the determinant of a matrix, respectively. II. BASIC TENSOR OPERATIONS AND SIGNAL MODEL A. TENSOR BASES
A tensor is a multidimensional array [16]. Let X ∈ CI1 ×I2 ×···IN denotes an N -th order tensor. The fibers are the higher-order analogue of matrix rows and columns. 18452
A mode-n fiber of X is an In -dimensional column vector obtained from X by varying the index in and keeping the other indices fixed. Some useful definitions concerning tensor operation are listed as follows: Definition 1 (Unfolding or Matricization): The mode-n unfolding of an N -th order tensor X ∈ CI1 ×I2 ×···×IN is denoted by [X ](n) . The (i1 , i2 , · · · , iN )-element of X maps to the (in , j)-th element of [X ](n) , where j = 1 + PN Qk−1 k=1,k6=n (ik − 1)Jk with Jk = m=1,m6=n Im . Definition 2 (Mode-n Tensor-Matrix Product): The mode-n product of an N -order tensorX ∈ CI1 ×I2 ×···×IN and a matrix A ∈ CJn ×In , denoted by X×n A, is a tensor of size I1 × · · · × In−1 × Jn × In+1 × · · · × IN , obtained by taking the inner product between each mode-n fiber and the rows of the matrix A, i.e., Y = X×n A ⇐⇒ [Y](n) = A [X ](n)
(1)
The mode-n product admits the following properties X×n A×m B = X×m B×n A, m 6 = n X×n A×n B = X×n (BA)
(2)
Definition 3 (Tucker Decomposition): The Tucker decomposition of an N -order tensorX ∈ CI1 ×I2 ×···×IN is given by X = G×1 A1×2 A2×··· AN
(3)
which can be regarded as a multilinear transformation of a core tensor G ∈ CJ1 ×J2 ×···×JN represents the core tensor by the factor matrices An ∈ CIn ×Jn (n = 1, 2, · · · N ), and it fulfills [X ](n) = An ·[G](n) ·[An+1 ⊗ · · · ⊗ AN ⊗ A1 · · · ⊗ An−1 ] (4)
FIGURE 1. Bistatic MIMO radar configuration.
B. DATA MODEL
Consider a bistatic MIMO radar system configured with M transmit antennas and N receive antennas, as illustrated in Fig. 1, both of which are uniform linear arrays with halfwavelength spacing. The transmit antennas emit M orthogonal coded waveforms, and the length of symbols per pulse duration is Q. Assume that there are K far-field targets appearing in the same range bin of the radar system, ϕk and θk represent the DOD and DOA of the k-th target, respectively. The transmitted signals are reflected by the targets, and the echoes are collected by the receive antennas. We consider a coherent processing interval consisting of L pulses. VOLUME 5, 2017
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The received array signal at the l-th (l = 1, 2, · · · , L) pluse period takes the form [5], [13] Xl = AR diag (bl ) ATT S + Nl
(5)
where AR , bl , AT , S, Nl denote, respectively, the receive direction matrix, the echo coefficient vector, the transmit direction matrix, the transmit code matrix, the colored noise matrix, and AR = [ar (θ1 ) , ar (θ2 ) , · · · , ar (θK )] ∈ CN ×K h iT ar (θk ) = 1, ejπ sin θk , . . . , ejπ(N −1) sin θk AT = [at (ϕ1 ) , at (ϕ2 ) , · · · , at (ϕK )] ∈ CM ×K h iT at (ϕk ) = 1, ejπ sin ϕk , . . . , ejπ(M −1) sin ϕk h iT bl = α1 ej2πlf1 /fs , α2 ej2πlf2 /fs , . . . , αK ej2πlfK /fs S = [s1 , s2 , · · · , sM ] ∈ CM ×Q where ar (θk ) and at (ϕk ) are the k-th receive steering vector and the k-th transmit steering vector, respectively; αk , fk (k = 1, 2, . . . , K ) and fs represent the radar cross section (RCS) amplitude, the Doppler frequency and the pulse repeat frequency, respectively; sm ∈ C1×Q is the m-th (m = 1, 2, · · · , M ) baseband code and sm sH m = Q. The columns of Nl are independent and identical distribution circularly symmetric complex Gaussian random vectors with zero mean and unknown covariance matrix C, i.e., ( n o 0, p 6= q E vec Np vecH Nq = (6) IQ ⊗ C, p = q The received signals are matched by sm /Q, m = 1, 2, · · · , M . By stacking the output along the pulse direction, we get 1 Y = [AT AR ] BT + W (7) Q where B = [b1 , b2 , . . . , bL ]T , W = [w1 , w2 , . . . , wL] denotes the matched noise matrix with wl = vec Nl SH , (l = 1, 2, . . . , L). The model in Eq. (7) can be viewed as the matrix form of the array measurement. According to Eq. (7), the array data is sampled on an 3-dimensional lattice. However, the multidimensional nature inherent in the lattice is ignored in Y. Actually, Y can be rearranged into a third-order tensor Y ∈ CN ×M ×L as [7] 1 (8) Y = IK ×1 AR×2 AT ×3 B + W Q where IK is the K × K × K identity tensor. The relations between Eq. (7) and Eq. (8) are Y = [Y]T(3) and W = [W]T(3) , respectively. III. THE PROPOSED ALGORITHM A. TENSOR-BASED COVARIANCE DIFFERENCING
In the traditional subspace-based methods, the covariance matrix R is first estimated, which is given by h i R = E YYH 1 = [AT AR ] RB [AT AR ]H + 2 RW (9) Q VOLUME 5, 2017
where RB = E BT B∗ , RW = E WWH . Considering the case targets are uncorrelated, in which the RB = L1 diag ρ12 , ρ22 , · · · , ρK2 is a real diagonal matrix, ρk2 denotes the coefficient variance of the k-th target. In practice, R can be estimated from finite measurement via ˆ = YYH /L. Let p, q ∈ {1, 2, · · · , L}, we get R n o E wp wH q n o = E vec Np SH vecH Nq SH n io h i h T = E S∗ ⊗ IN vec Np vecH Nq S ⊗ IN 0, p 6 = q = E S∗ ⊗ IN IQ ⊗ C ST ⊗ IN , p = q 0, p 6= q = (10) Q IQ ⊗ C , p = q It can be concluded from Eq. (10) that RW is proportional to IM ⊗ C. Since RW is no longer a scaled identity matrix, we are now faced with an unknown noise field. In this case, the noise subspace can not be separated correctly from the signal subspace, and the performances of the traditional subspace-based algorithms would degrade seriously. Fortunately, the noise process is stationary, thus C is a Hermitian symmetric Toeplitz matrix, and it fulfills JN C∗ JN = C
(11)
where JN denotes a N × N exchange matrix with N ones on its anti-diagonal and zeros elsewhere. With the property in Eq. (11), we further get (IM ⊗ JN ) R∗W (IM ⊗ JN ) = RW
(12)
Observably, the noise component keep unchanged after its conjugate linear transform. To eliminate the spatial colored noise, the covariance differencing technique is hereby adopted. Suppose that there is no paired targets that are located symmetrically about the transmit array broadside and 2K < MN , then the following difference matrix can be formed 1R = R − (IM ⊗ JN ) R∗ (IM ⊗ JN ) = [AT AR ] RB [AT AR ]H H − A∗T (AR 9R ) RB A∗T (AR 9R ) RB , 0 ∗ = AT AR , AT (AR 9R ) 0, −RB H ∗ × AT AR , AT (AR 9R ) RB , 0 ∗ = AT , AT [AR , AR ] 0, −RB H ∗ AT , AT [AR , AR ] (13) where 9R = diag([ej(N −1)πsinθ1 , ej(N −1)πsinθ2 , · · · , j(N −1)πsinθ K ]) is a diagonal matrix. Clearly, the unknown e noise covariance has been subtracted from 1R while the signal covariances remains unchanged. Moreover, 1R is singular Hermitian matrix. As a result, R can be approximated 18453
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by its truncated eigenvalue decomposition (EVD), i.e., 1R ≈ Es 6s EH s
(14)
where 6s and Es contain the 2K dominant eigenvalues and the corresponding of 1R, respectively. It is easy eigenvectors to prove Es and AT , A∗T [AR , AR ] span the same subspace. This subspace will henceforth be termed the signal subspace and it contains the angle information of the targets. To further explore the multi-dimensional inherent structure, the tensor covariance model can be utilized. According to [17] and [18], the covariance tensor can be expressed as R = RB×1 AR×2 AT ×3 A∗R×4 A∗T + RW
(15)
The fourth-order tensor decomposition model in Eq. (15) is commonly known as the noisy ‘‘Tucker4’’ decomposition (Definition 3) of 1R, where the associated core tensor is given by RB ∈ CK ×K ×K ×K and the factors matrices are AR , AT , A∗R and A∗T , respectively, RW is the covariance tensor of the noise. It is easy to show that R is a Hermitian tensor [19]. In fact, R can be obtained by letting the first 2 indices of R vary 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 interpreted as the Hermitian unfolding of R ([19, Lemma 1]), which is marked by R = [R](H ) . Similarity, the relations between RB and RB , RW and RW are RB = [RB ](H ) , RW = [RW ](H ) , respectively. Consequently, the difference covariance tensor can be constructed as 1R = R − R×1 JN ×3 JN ˜ B×1 [AR , AR]×2 AT , A∗T =R
∗ ∗ ∗ A , AR ×4 AT , AT ×3 R (16)
˜ ∈ C2K ×2K ×2K×2K denotes the core tensor with where R h i B RB , 0 ˜B R = , [AR , AR ], AT , A∗T , A∗R , A∗R , 0, −R (H ) B ∗ AT , AT are, respectively, the corresponding factor matrices. Also, 1R is a Hermitian tensor and the unknown noise is subtracting out from 1R. B. TENSOR COVARIANCE-BASED SUBSPACE ESTIMATION
Our goal is to estimate the signal subspace from 1R. In this paper, a direct tensor decomposition method is utilized. According to [19], the HOSVD of 1R is given by 1R = G×1 U1×2 U2×3 U3×4 U4
(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-mode matrix unfolding of 1R as [1R](n) = Un 6n VH n . It is easy to find that U1 = U∗3 and U2 = U∗4 . Similar to the traditional eigenstructure method, 1R can be expressed by its truncated HOSVD as [5] 1Rs = Gs×1 U1s×2 U2s×3 U∗1s×4 U∗2s 18454
(18)
where U1s ∈ CN ×K is the K dominant column vectors of U1 , U2s ∈ CM ×2K contains the column vectors of U2 corresponding to the 2K dominant singular values, respectively; Gs denotes for the signal component of G, which is given by H T T Gs = 1R×1 UH 1s×2 U2s×3 U1s×4 U2s
(19)
Combining with Definition 2, we get H T T 1Rs = 1R×1 UH 1s×2 U2s×3 U1s×4 U2s×1 U1s×2 U2s×3
×U∗1s×4 U∗2s = 1R×1 U1s UH U2s UH 1s 2s ×3 ×2 ∗ T ∗ T × U1s U1s U2s U2s
(20)
×4
By the Hermitian unfolding of 1Rs , we can form a new cross-correlation matrix Rs from 1Rs , which is given by Rs = [1Rs ](H ) h i H = U1s UH ⊗ U U 2s 1s 2s h iH H ×1R U1s UH 1s ⊗ U2s U2s
(21)
Worthnoting is that Rs is a Hermitian matrix. Insertion of Eq.(14) into Eq.(21) yields h i T H H H H H Rs = U1s U1s ⊗ U2s U2s Es 6 U1s U1s ⊗ U2s U2s Es (22) H Since U1s UH 1s and U2s U2s are unitary matrices, Rs can be ¯ s 6s E¯ H , approximated by its truncated EVD as Rs ≈ E s where 6s and Es are the 2K (K ≤ MN ) dominant eigenvalues and corresponding eigenvectors, respectively. Obviously, ¯ s and Es span the same signal subspace. As a result, there E exists a full-rank matrix T that E¯ s = AT , A∗T [AR , AR ] T (23)
Remark 1: It is assumed that there is no paired targets located symmetrically about the transmit array broadside, hence AT , A∗T [AR , AR ] has full column rank, otherwise ¯ s can not be estimated correctly. E Remark 2: The HOSVD of an tensor is equivalent to the SVD of all its matrix unfolding. Since U1 = U∗3 and U2 = U∗4 , we only need to calculate SVD of the mode-1 and mode-2 matrix unfolding of 1R, resulting in a significant computational savings. Remark 3: The maximum number of targets that can be uniquely identified by the proposed method is determined by the maximum rank of AT , A∗T [AR , AR ], which is MN . Notably, the identifiability of the proposed method is the same as the methods in [14] and [15]. However, the methods in [11] and [13] can identify at most min {M1 N , M2 N } targets, where M1 and M2 represent the antenna numbers of two subarrays. VOLUME 5, 2017
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C. JOINT DOD AND DOA ESTIMATION
Unlike the traditional signal subspace, Es contains the real target direction (φ, θ) and its mirror direction (−φ, θ). In this paper, a two-step framework is presented to obtain the real ¯ s . In the first step, the traditional subspacedirections from U based algorithms, such as MUSIC and ESPRIT, are utilized for ambiguous DOD and DOA estimation. In the second step, a criterion is given to determine the unique directions. 1) MUSIC METHOD
By exploiting the orthogonality characteristic between the noise subspace and the signal subspace, the spatial spectrum searching method can be exploited. The DODs and DOAs are obtained via maximization the following spectrum function f (φ, θ) =
1 [at (φ) ⊗ ar (θ )] Fn [at (φ) ⊗ ar (θ )] H
(24)
¯ where Fn = IMN − Eo EH o , Eo is the orthogonal basis of Es . Eq.(25) involves two-dimensional peak searches, which is computationally inefficient. Here, the reduced-dimension MUSIC [1] idea can be adopted to lower the computational cost, where directions estimation is linked to the following quadratic optimization problems ˆ θ = argmin f (θ ) = argmin det [I ⊗ a (θ )]H F [I ⊗ a (θ )] M r n M r φˆ = argmin f (φ) = argmin det [at (φ) ⊗ IN ]H Fn [at (φ) ⊗ IN ]
(25)
Since the rank reduction of f (θ ) and f (φ), respectively, will take place on the estimations of θk and ±φk (k = 1, 2, · · · , K ), the DODs and DOAs can be estimated via two one-dimensional spectrum searches. After which extra computation is carried out to pair the estimated DOD and DOA. Remark 4: Since the estimated DODs are symmetrical to zero, the computational load in f (φ) can be further reduced by constraining φ ∈ [0, π). All the possible DODs can be recovered by the union of the estimated DODs and their negative values.
ejπsinθK , ejπsinθ1 , ejπsinθ2 , · · · , ejπsinθK ]) are rotational invariance matrices. The least square solutions for 9t and 9r are ( H ¯ ¯H H ¯ ˆ t = E¯ H 9 s C2 C2 Es Es C2 C1 Es (27) ¯ s E¯ H CH C3 E¯ s ˆ r = E¯ H CH C4 E 9 s
4
s
4
The diagonal elements of 9t and 9r contain the direction information of the targets, which can be easily obtained and ˆ t and 9 ˆ r. paired form 9 To assign K DODs each corresponding to the matrices At , and A∗t , we need to solve for the difference source covariance matrix. Let A be a guess of the form of estimated AT , A∗T [AR , AR ], then we calculate the following matrix −1 −1 5 = AH A AH 1RA AH A (28) If the initial guess is correct, 5 will have positive values along the upper half of the diagonal and negative elements along the lower half of its diagonal. Once the initial guess is incorrect, this result will not be observed, but the signs of diagonal 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 follows step.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 E¯ s ; step.4 Obtain the ambiguous DOD and DOA pairs via MUSIC or ESPRIT. Finally, determine the unique directions via (28). IV. COMPLEXITY ANALYSIS
The rotational invariance properties can be explored to obtain a closed-form solution for the DODs and DOAs. According to Eq.(21), there exists the following rotational invariance properties ( C1 E¯ s = C2 E¯ s 9t (26) C3 E¯ s = C4 E¯ s 9r
The computational complexity of the proposed method is summarized as follows. The estimation of 1R needs M 2 N 2 L complex multiplications. The complexity of HOSVD on 1R is on the order 2O M 3 N 3 . The load of computing Rs in Eq.(19) is 8M 2K + 2N 2 K + 2M 3 N 3 , and its EVD requires O M 3 N 3 complex multiplications. The estimation complexity of the ambiguous directions is l 2M 3 N 2 + M 2 N 3 + O M 3 + 0.5O N 3 (MUSIC-based method) 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 4M 2 N 2 K + 8MNK 2 + 8O K 3 . We summarize the total computational loads of the proposed methods, [11](marked with Chen’s method), [13](marked with Wang’s method), [14](marked with Fu’s method) and [15](marked with Wen’s method) in Table 1. It can be seen that the complexity of the proposed ESPRIT method is lower than Wang’s method and Wen’s method while the proposed MUSIC method may be more complex than them.
where C1 = CM 1 ⊗ IN , C2 = CM 2 ⊗ IN , C3 = IM ⊗ CN 1 , C4 = IM ⊗ CN 2 are selection matrices; 9t = diag([ejπsinϕ1 , ejπsinϕ2 , · · · , ejπsinϕK , e−jπsinϕ1 , e−jπsinϕ2 , · · · , e−jπsinϕK ]), 9r = diag([ejπsinθ1 , ejπsinθ2 , · · · ,
In this section, 200 Monte Carlo trials were performed to verify the improvement of the proposed methods (Matlab code is available on https://pan.baidu.com/s/1gfw3PYz). The bistatic
2) ESPRIT METHOD
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V. SIMULATION RESULTS
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TABLE 1. Comparison of the complexity.
FIGURE 3. RMSE performance comparison versus L in case (1).
MIMO radar is configured with M = 10 transmit elements and N = 12 receive elements. Pulse number Q and pulse repeat frequency fs are set to Q = 256, fs = 20KHz. Assume that there exist K = 3 uncorrelated sources located at the angles (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 conform to the Swerling I model. The following root mean square error (RMSE) is used for performance measure v K u 200 2 u 1 X X 2 1 t RMSE = θˆi,k − θk + ϕˆi,k − ϕk K 200 k=1
de-noising methods provide better RMSE performances than ESPRIT method at low SNR regions (SNR ≤ −5dB), as spatial colored noise has been eliminated. However, due to the virtual aperture loss, Chen’s method and Wang’s method perform worse than ESPRIT at high SNR regions (SNR ≥ 0dB). Additionally, the performance of proposedESPRIT method coincides with Wen’s method while the proposed-MUSIC method provides better RMSE performance than Wen’s method, since no visual aperture loss occurs in our methods and Wen’s method, and the identifiability of MUSIC is better than ESPRIT. Fig. 3 gives the RMSEs in the case of different numbers of pluse with SNR = −15dB. As expected, the proposed-MUSIC method significantly outperform all the compared methods.
i=1
where θˆi,k and ϕˆi,k correspondingly represent the estimates of θk and ϕk for the i th Monte Carlo trial. For comparison, the performances of ESPRIT [2], Chen’s method and Wang’s method are evaluated.
FIGURE 4. RMSE performance comparison versus SNR in case (2).
FIGURE 2. RMSE performance comparison versus SNR in case (1).
In the first simulation, the (p, q)-th element in C is given by C (p, q) = 0.9|p−q| ejπ(p−q)/2 . Fig. 2 depicts the resultant RMSEs at different SNRs with L = 200. It can be observed that RMSE performance of all the methods gradually improves with the growing SNR. Besides, all the 18456
In the second simulation, the spatial colored noise is modeled as a second-order autoregressive (AR) process with the coefficients z = [1, −1, 0.8] [11], [13]. Fig. 4 gives the RMSEs 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 first simulation. It worth noting that for SNR = −15dB, all the denoising mechanisms work. However, Wang’s method, Wen’s method and our methods perform better than Chen’s method, which implies that the multidimensional inherent structure of the array data helps to achieve a more accurate subspace estimation. Furthermore, Wen’s method and our methods offer VOLUME 5, 2017
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FIGURE 5. RMSE performance comparison versus L in case (2).
better RMSE performances than Wang’s method, especially at high SNR regions, because the covariance differencing technique would not decrease the visual array aperture while the spatial cross-correlation method would bring virtual aperture loss. Since the dimension of subspace in our is larger than that in Wen’s method, the accuracy of subspace estimation may not as good as Wen’s method, thus the proposedESPRIT method perform worse than Wen’s method at low SNR regions. As mentioned in the last section, the complexity of tensor decomposition in our methods is lower than Wen’s method, therefore, our methods can obtain a tradeoff between performance and complexity. VI. CONCLUSION
In this paper, we investigate into tensor-based covariance differencing method for angle estimation in bistatic MIMO radar in the presence of unknown spatial colored noise. Unlike the existing methods, both the inherent multidimensional structure and the full virtual aperture are considered in the proposed scheme. A two-step framework is proposed for angle estimation, where an ambiguous angle estimation is first achieved and follows a unique angle determination. The proposed scheme provides more accurate parameters estimation performance while it has less computational complexity than the existing tensor-based methods, which will lead to a brighter prospect in applications. Finally, numerical simulation are given to verify the improvement of the proposed scheme. REFERENCES [1] X. F. Zhang, L. Y. Xu, L. Xu, and D. Z. Xu, ‘‘Direction of departure (DOD) and direction of arrival (DOA) estimation in MIMO radar with reduceddimension MUSIC,’’ IEEE Commun. Lett., vol. 14, no. 12, pp. 1161–1163, Dec. 2010. [2] C. Duofang, C. Baixiao, and Q. Guodong, ‘‘Angle estimation using ESPRIT 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 estimation 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-based transmit angle and receive angle estimation for multiple-input multipleoutput radar,’’ IET Radar Sonar Navigat., vol. 5, no. 6, pp. 626–631, 2011. VOLUME 5, 2017
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FANGQING WEN (M’17) was born in 1988. He received the B.S. degree in electronic engineering from the Hubei University of Automotive Technology, Shiyan, China, in 2011, and the Ph.D. degree from the Nanjing University of Aeronautics and Astronautics, China, in 2016, where he is currently pursuing the Master’s degree with the College of Electronics and Information Engineering. From 2015 to 2016, he was a Visiting Scholar with the 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 interests include MIMO radar, array signal processing, and compressive sensing. He is a member of the Chinese Institute of Electronics. 18457
F. Wen et al.: Tensor-Based Covariance Differencing Method for Direction Estimation in Bistatic MIMO Radar
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 of Technology, Harbin, China, in 1989 and 1992, respectively, and the Ph.D. degree in electrical engineering from Xidian University, Xi’an, China, in 2001. In 2006, he was a Visiting Scholar with The University of Manchester, U.K. In 2016, he was a Visiting Scholar with the University of Delaware, USA. Since 1992, he has been with the National Laboratory of Radar Signal Processing, Xidian University. His current research interests include radar signal processing and multirate filter banks design.
XINHAI WANG was born in 1988. He received the B.S. degree from the School of Electronic Information Engineering, Nanjing University of Information Science and Technology, Nanjing, China, in 2012, and the M.S. degree from the College of Electronics and Information Engineering, Nanjing University of Aeronautics and Astronautics, where he is currently pursuing the Ph.D. degree with the College of Electronics and Information Engineering. His research interests are array signal processing, wireless communication, and signal processing.
GONG ZHANG (M’07) received the Ph.D. degree in electronic engineering from the Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China, in 2002. From 1990 to 1998, he was a Member of Technical Staff with the No724 Institute of China Shipbuilding Industry Corporation, Nanjing. Since 1998, he has been with the College of Electronics and Information Engineering, NUAA, where he is currently a Professor. His research interests include radar signal processing and compressive sensing. He is a member of the Committee of Electromagnetic Information, Chinese Society of Astronautics and a Senior Member of the Chinese Institute of Electronics.
YU ZHANG was born in 1991. He received the B.S. degree from the Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China, in 2013, and the M.S. degree from the College of Electronics and Information Engineering, NUAA, in 2015, where he is currently pursuing the Ph.D. degree with the College of Electronics and Information Engineering. His research interests are array signal processing, statistical signal processing, and compressive sensing.
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XINYU ZHANG was born in 1999. He is studying at Lanzhou University, China. His research interests are artificial intelligence and signal processing.
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