Robust Angle Estimation for MIMO Radar with the Coexistence of Mutual Coupling and Colored Noise

This paper deals with joint estimation of direction-of-departure (DOD) and direction-of- arrival (DOA) in bistatic multiple-input multiple-output (MIMO) radar with the coexistence of unknown mutual coupling and spatial colored noise by developing a novel robust covariance tensor-based angle estimation method. In the proposed method, a third-order tensor is firstly formulated for capturing the multidimensional nature of the received data. Then taking advantage of the temporal uncorrelated characteristic of colored noise and the banded complex symmetric Toeplitz structure of the mutual coupling matrices, a novel fourth-order covariance tensor is constructed for eliminating the influence of both spatial colored noise and mutual coupling. After a robust signal subspace estimation is obtained by using the higher-order singular value decomposition (HOSVD) technique, the rotational invariance technique is applied to achieve the DODs and DOAs. Compared with the existing HOSVD-based subspace methods, the proposed method can provide superior angle estimation performance and automatically jointly perform the DODs and DOAs. Results from numerical experiments are presented to verify the effectiveness of the proposed method.


Introduction
As an innovative radar framework, multiple-input multiple-output (MIMO) radar system plays an important role in improving detection and estimation performance. MIMO radar uses multiple antennas to emit mutual orthogonal waveforms, and attempts to detect the locations of interest targets from the echoes received by multiple antennas. By exploiting matched filters, the information from an individual transmitter-to-receiver path can be extracted, and a virtual array with a large aperture is formed. Thus, MIMO radar can achieve superior performance compared with the traditional phase-array radar [1]. In terms of the antennas configuration, MIMO radar can be generally divided into two classes: statistical MIMO radar and collocated MIMO radar [2,3]. The former can deal with the scintillation problem by taking advantage of widely separated antennas and the latter can achieve unambiguous angle estimation with closely placed antenna arrays. In this paper, we focus on the bistatic MIMO radar, which is a kind of the collocated MIMO radar.
In bistatic MIMO radar, joint estimation of direction-of-departure (DOD) and direction-of-arrival (DOA) is a key issue. Many excellent algorithms, such as spatial spectrum searching methods [4,5], estimate of signal parameter via rotational invariance techniques (ESPRIT) based methods [6,7], Toeplitz matrix constructed by the vector r; diag(r) denotes the diagonalization operation; angle(·) returns the phase angles of an element in radians.

Tensor Preliminaries
Let us briefly review some necessary operations in tensor domain. For more details about tensor algebra, the readers can refer to the review article [35]. Definition 1 (Unfolding or Matricization). An (I 1 × I 2 × · · · × I N )-dimensional tensor has N indices. The mode-n unfolding of a tensor X ∈ C I 1 ×I 2 ×···×I N is denoted by [X ] (n) , where the (i 1 , i 2 , . . . , i N )-element of X maps to the (i n , j)-th element of [X ] (n) , where j = 1 + ∑ N k=1,k =n (i k − 1)J k with J k = ∏ k−1 m=1,m =n I m .
Definition 2 (Mode-n Tensor-Matrix Product). The mode-n product of an N-order tensor X ∈ C I 1 ×I 2 ×···×I N and a matrix A ∈ C J n ×I n , is denoted by Y = X ×n A. In the matrix form, it can be expressed as Moreover, the mode-n tensor-matrix product satisfies the following properties Definition 3 (Tensor Decomposition). The HOSVD of a tensor X ∈ C I 1 ×I 2 ×···×I N is given by where G ∈ C I 1 ×I 2 ×···×I N is the core tensor, and U n ∈ C I n ×I n (n = 1, 2, 3, . . . , N) is a unitary matrix, which is consist of the left singular vectors of [X ] (n) .

Signal Model
Now we consider a general bistatic MIMO radar system, as shown in Figure 1. There are M transmitters and N receivers, and both of which are uniform linear arrays (ULAs) with half-wavelength spacing. The transmitters emit M mutual orthogonal waveforms and illuminate a given area. Suppose that there are K point targets located in the far field. The signals are reflected by the slow-moving targets and the echoes are collected by the receivers. With the assumtion that each pulse period of the transmitted waveform is consist of Q coded symbols, let us consider a coherent processing interval (CPI) consisting of L pulses. Then the received data during the l-th (l = 1, 2, . . . , L) pulse period can be expressed as where A r ∈ C N×K is the receive direction matrix, A t ∈ C M×K is the transmit direction matrix, s l ∈ C K×1 is the echo coefficient vector, W ∈ C M×Q is the baseband code matrix, and F l ∈ C N×Q is the received spatial colored noise. The details of the above mentioned matrices are given below where a r (θ k ) and a t (ϕ k ) denote the receive steering vector and the transmit steering vector, respectively; α k , f k and f s represent the radar cross section (RCS) amplitude, the Doppler frequency and the pulse repeat frequency, respectively and w m ∈ C Q×1 is the m-th transmitted baseband code with w H m w n = 0, m = n P, m = n m, n ∈ {1, 2, . . . , M}. The received noise is spatially colored, which indicates that the columns of F l are independently and identically distributed complex Gaussian random vectors with zero mean and unknown covariance matrix C, i.e., for any a, b ∈ {1, 2, . . . , L}, there exist 0.98,0.00,0.00 where F a and F b are the columns of F l . At the receive array, the received data X l is matched by w m /Q, m = 1, 2, . . . , M, respectively. After this operation, the matched data can be stacked in to a matrixỸ according to the spatial-temporal order. Similar to [24], we havẽ where S = [s 1 , s 2 , . . . , s L ] T is the echo coefficient matrix and N = [n 1 , n 2 , . . . , n L ] denotes the matched noise matrix with n l = vec F l W H , (l = 1, 2, . . . , L). Generally speaking, mutual coupling effect would exist in the antenna array due to the radiation effects of the antenna elements [16]. The mutual coupling between antenna elements of a ULA can be described as a banded symmetric Toeplitz matrix, known as the mutual coupling matrix. The mutual coupling coefficient between two antennas in a ULA is opposite to their distance and is approximated as zeros when the distance is large enough [24]. Assuming that the number of nonzero mutual coupling coefficients is P + 1 for both transmit and receive arrays, and P is satisfied with min{M, N} > 2P. Taking the mutual coupling effect into consideration, the data model in Equation (7) becomes where C t = toeplitz c T t , 0 1×(M−P−1) ∈ C M×M and C r = toeplitz c T r , 0 1×(N−P−1) ∈ C N×N are the mutual coupling matrices with c t = [c t0 , c t1 , ..., c tP ] and c r = [c r0 , c r1 , ..., c rP ], and c ip (i = r, t; p = 0, 1, 2, . . . , P) is the P + 1 nonzero mutual coupling coefficients, which Since each element in Y is formed by multiplying the unique entity inÃ t ,Ã r and S, the matched data Y exhibits three diversities. However, these diversities have been ignored by the traditional matrix-based estimation methods. To further exploiting the inherent multidimensional structure, Y is rearranged into a third-order tensor Y shown as Figure 2. Then the (m, n, l)-th (m = 1, . . . , M; n = 1, . . . , N; l = 1, . . . , L) element is given by where Y (m, n, l) denotes the (m, n, l)-th element in Y and similar to others, and N ∈ C M×N×L is the rearranged noise tensor. According to Definition 1, it is obvious that . In HOSVD-wise format, Equation (9) can be expressed as The tensor diagram of received data.

Robust Covariance Tensor-based Angle Estimation Method
Because the conventional subspace-based methods or tensor decomposition technique [24][25][26][27][28][29]31,33,34] considers the influence of mutual coupling and spatial colored noise independently, they unavoidably have performance degradation or fail to work with the coexistence of unknown mutual coupling and spatial colored noise. Let us formulate a novel robust covariance tensor-based angle estimation method to remove the influence of both mutual coupling and spatial colored noise in tensor domain.

Spatial Colored Noise Suppression and Mutual Coupling Elimination
In Equation (10), the mutual coupling has an impacts on the transmit direction matrixÃ t and receive direction matrixÃ r , i.e,Ã t andÃ r are no longer Vandermonde matrices. Fortunately, the mutual coupling matrices are banded symmetric Toeplitz, and two sub-matrices can be extracted from transmit and receive direction matrices for decoupling. By defining two selection matrices as where are column vectors composed of the first N − 2P and M − 2P elements of a r (θ k ) and a t (ϕ k ), respectively. From Equation (11), it is easy to know that for each target, β tk and β rk are constant, which indicates that the direction matricesÂ have the Vandermonde structure. Thus, the influence of mutual coupling is eliminated after the decoupling operation. The decoupling operation in Equation (11) can be extended to Equation (10) in tensor domain, which showsŶ whereN = N ×1 J 3×2 J 4 is a part ofN , which corresponds to the spatial colored noise of (M − 2P)(N − 2P) elements after the decoupling operation in matrix domain. According to Equation (12), it can be indicated that the effect of mutual coupling has been eliminated in tensor domain, and the tensor noiseN also holds the same characteristic with N . Thus, the decoupling in Equation (12) also hold the influence of spatial colored noise, which makes the performance degradation. In order to further eliminate the spatial colored noise, it is necessary to analysis the covariance of the colored noise. Let a, b ∈ {1, 2, . . . , L}, we get Based on Equation (13), it can be concluded that the covariance matrix of the spatial colored noise with different pulse period is 0, i.e., the spatial colored noise is temporal uncorrelated. Motivated by this feature, the tensorŶ is divided into two sub-tensors Z 1 and Z 2 , which are shown as According to the characteristic of the spatial colored noise in Equation (13), we have where RŜ = E S T 1 S * 2 , and according to Equation (13), = E [n 1 ,n 2 , . . . ,n L−1 ] [n 2 ,n 3 , . . . ,n L ] H = 0, which indicates that the spatial colored noise is eliminated.
Thus, similar to [26], the de-noising method in Equation (15) can be extended in tensor format. Using the characteristic of the spatial colored noise, a new fourth-order tensor R ∈ C N×M×N×M is constructed with the (n, m, q, p)-th (n, q ∈ {1, 2, . . . , N − 2P} ; m, p ∈ {1, 2, . . . , M − 2P}) element as According to the structure of R, the relationship between R Z and R can be written as 0.98,0.00,0.00 From Equation (17), it is clearly seen that the spatial colored noise is suppressed in R, which indicates that the covariance tensor R is without the influence of both mutual coupling and spatial colored noise. Similar to the model in Equation (10), R can be rewritten as where R S ∈ C K×K×K×K is the Hermitian folding of RS in the same way as Equation (17), i.e., R S (:, k 1 , :, k 2 ) = 0, k 1 = k 2 RS (k 1 , k 2 ) , k 1 = k 2 , k 1 , k 2 ∈ {1, 2, . . . , K}. The decomposition of fourth-order tensor in Equation (18) is commonly known as the 'Tucker4' decomposition, where the corresponding core tensor is R S .

Tensor-Based Signal Subspace Estimation and Angle Estimation
In order to exploit the subspace technique for angle estimation, the signal/noise subspace is required to estimate firstly. Although the traditional SVD technique is easy to implement for estimating the signal/noise subspace, the multidimensional structure inherent in R is ignored. Fortunately, the HOSVD technique is an effective way to estimate the signal subspace with capturing the multidimensional structure information in tensor. The HOSVD of the measurement R is given by [31] R where G ∈ C (N−2P)×(M−2P)×(N−2P)×(M−2P) is the associated core tensor, By Hermitian unfolding of R s , a new cross-correlation matrixR is formed to bê Since the rank of R Z is K, it can be approximated by its truncated EVD as R Z ≈ U s Λ s U H s , where U s contains the singular vectors corresponding to the K-dominate singular values. Obviously, U s spans the same subspace by the columns of Ã t Ã r . Inserting RỸ into Equation (22), we get After truncating EVD of R s , a new signal subspaceÛ s is obtained aŝ Equation (24) describes the relationship between the signal subspace obtained via matrix-based method and that estimated via tensor-based method, from which we can observe that U s is weighted by U 1s U H 1s ⊗ U 2s U H 2s . Due to the weighting effect, the noise in the tensor measurement is suppressed to obtain an enhanced signal subspace. Obviously,Û s spans the same space as U s . Therefore, there exists a full-rank matrix T that

Joint DOD and DOA Estimation
After obtaining the tensor-based signal subspace without the influence of both mutual coupling and spatial colored noise, the conventional ESPRIT method can be applied to estimate the DODs and DOAs [6,7]. Letâ =â t (ϕ k ) â r (θ k ), and define J t1 = I ( Let Φ t = diag e −jπ sin ϕ 1 ande −jπ sin ϕ 2 , . . . , e −jπ sin ϕ K , Φ r = diag e −jπ sin θ 1 , e −jπ sin θ 2 , . . . , e −jπ sin θ K . Combined with Equation (25) and Equation (26), we can get the least squares (LS) estimation of Φ t and Φ r via where the relations betweenΦ t , Φ t andΦ r , Φ r areΦ t = TΦ t T −1 , andΦ r = TΦ r T −1 , respectively. OnceΦ t has been estimated, the estimation e −jπ sinφ k can be obtained via the eigenvalue ofΦ t . LetT be the estimated eigenvectors ofΦ t , then the estimation e −jπ sinθ k can be obtained via the diagonal element ofTΦ rP −1 . Finally, the k-th (k = 1, 2, . . . , K) DOD and DOA are given by BecauseΦ t andΦ r share the same eigenvectors, the estimated DODs and DOAs are paired automatically.
The proposed method is summarized as follows.

Related Remarks
Remark 1. The estimation algorithms in [24][25][26] are only effective with uniform white noise, while the denoising methods in [28][29][30][31][32][33][34] may suffer from mutual coupling. Consequently, their performances would degrade to deal with the co-existence mutual coupling and spatial colored noise. However, the proposed method can circumvent the limitations above to achieve better performance than these early reported algorithms.

Remark 2.
The main drawback of the MUSIC-Like algorithm is the ambiguity [23], since the columns in the mutual coupling matrices may be linear dependent. From Equation (12) we can observe that the direction submatrices exhibit a Vandermonde-Like structure to remove the ambiguity in the proposed method.

Remark 3.
The HOSVD of a tensor is calculated by SVDs of all the unfolded matrices. However, since U 1s = U * 3s , U 2s = U * 4s , we only need to compute the truncated SVDs of the mode-1 and mode-2 unfolding of R, resulting in a significant computational saving.

Computation Complexity
Now we analysis the complexity of the proposed method in terms of the number of complex multiplications. The complexity of forming R is M 2 N 2 (L − 1) and performing truncated HOSVD of R requires 2(M − 2P) 3 (N − 2P) 3 . The computation load of calculatingR is 2(M − 2P) 2 Estimating DOD and DOA throughΦ t andΦ r needs K 3 complex multiplications. I summary, the total complexities of the proposed method, the ESPRIT-Like method [24], the PM-Like method [25] and the HOSVD method [26] are presented in Table 1. Clearly, the complexity of the proposed method is a slight heaver than the HOSVD method, the ESPRIT-Like method and the PM-Like method. However, it provides much better estimation performance than all these other methods, which will be shown in the simulation section.

Simulation Results
In the simulation, the bistatic MIMO radar is configured with M = 10 transmit elements and N = 12 receive elements. The transmit code length is Q = 256 and the pulse repeat frequency is f s = 20 KHz. Suppose that there are K = 3 uncorrelated targets located at the directions (θ 1 , ϕ 1 ) = (−30 • , 10 • ), (θ 2 , ϕ 2 ) = (20 • , 0 • ), (θ 3 , ϕ 3 ) = (0 • , and−20 • ). The corresponding RCS are α 1 = α 2 = α 3 = 1 and the Doppler shifts are { f k } 3 k=1 = {200, 400, 850}Hz, respectively. The spatial colored noise is modeled as a second-order autoregressive (AR) process with the coefficient z = [1, −1, 0.8] [31,33]. The SNR in the simulation is defined by SNR = 10log 10 where X l and F l are the matrices in Equation (5). In the following simulations, two measures are applied for performance assessment [33]. The first one is the root mean square error (RMSE), defined as whereθ i,k andφ i,k represent the estimations of θ k and ϕ k for the i th Monte Carlo trial, respectively. ζ is the total number of Monte Carlo trials. The other one is the probability of the successful detection (PSD) defined as PSD = D ζ × 100%, where D denotes the total numbers of successful trial and a successful trial is recognized if the absolute errors of all the estimated angles are smaller than 0.1 • .
In the first step simulation results from Figures 3-5, we consider the transmit and receive arrays with a weak mutual coupling defined by the mutual coefficients c t = [1, 0.1174 + j0.0577] and c r = [1, −0.0121 − j0.1029], respectively. The number of snapshots is 50. Figure 3 illustrates the estimation results of the proposed method with SNR = 0dB, from which we can observe that the DODs and DOAs are accurately obtained and correctly paired, which proves the effectiveness of the proposed method. Figures 4 and 5 show the RMSE and the PSD performance comparisons of the proposed method with the ESPRIT-Like method [24], the PM-Like method [25] and the HOSVD method [26], respectively. It can be concluded from Figure 4 that the proposed method provides much better estimation perfprmance than the ESPRIT-Like method, the PM-Like method and the HOSVD method in the low SNR region. The reason is that the proposed method can eliminate the effect of the mutual coupling and spatially colored noise simultaneously, and the other methods only consider the effect of the mutual coupling. It is shown in Figure 5 that all the methods exhibit a 100% successful detection in the high SNR region. As the SNR decreases, the PSD of each method starts to drop at a certain point, which is defined as the SNR threshold [33]. The proposed method provides lower PSD threshold and better detection performance than all the compared methods.  In the second step simulation results shown in Figure 6 to Figure 7, we consider the transmit and receive arrays with a strong mutual coupling, and the mutual coupling coefficients are fixed at c t = [1, 0.8 + j0.5, 0.2 + j0.1], and c r = [1, 0.6 + j0.4, 0.1 − j0.3]. The number of snapshots is 50. Figures 6 and 7 illustrate the RMSE and PSD curves of various methods versus SNR. According to the results, the RMSE and PSD performances of all the methods gradually improved with the increasing SNR. However, the ESPRIT-Like method, the PM-Like method and the HOSVD method provide higher RMSE than the proposed method in low SNR region. On the other hand, the proposed method has estimation performance similar to that achieved by the HOSVD method in high SNR region. This is because that the spatially colored noise is not the important factor to effect the estimation performance when the SNR is high enough. From Figure 7, it is obvious that the proposed method has better PSD than the other methods, i.e., it outperforming all the compared methods.

Conclusions
In this paper, a robust covariance tensor-based angle estimation method is developed for bistatic MIMO radar with the co-existence mutual coupling and spatial colored noise. The proposed method can capture the multidimensional nature of the matched array data. At the same time, it can eliminate the effect of mutual coupling and colored noise in the tensor domain. As a result, the proposed algorithm provides better estimation performance than the existing estimation algorithms. The robustness and superiority of the proposed method are clearly verified by simulation results.