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Underdetermined 2D-DOD and 2D-DOA Estimation for Bistatic Coprime EMVS-MIMO Radar: From the Difference Coarray Perspective

In this paper, the underdetermined 2D-DOD and 2D-DOA estimation for bistatic coprime EMVS-MIMO radar is considered. Firstly, a 5-D tensor model was constructed by using the multi-dimensional space-time characteristics of the received data. Then, an 8-D tensor has been obtained by using the auto-correlation calculation. To obtain the difference coarrays of transmit and receive EMVS, the de-coupling process between the spatial response of EMVS and the steering vector is inevitable. Thus, a new 6-D tensor can be constructed via the tensor permutation and the generalized tensorization of the canonical polyadic decomposition. {According} to the theory of the Tensor-Matrix Product operation, the duplicated elements in the difference coarrays can be removed by the utilization of two designed selection matrices. Due to the centrosymmetric geometry of the difference coarrays, two DFT beamspace matrices were subsequently designed to convert the complex steering matrices into the real-valued ones, whose advantage is to improve the estimation accuracy of the 2D-DODs and 2D-DOAs. Afterwards, a third-order tensor with the third-way fixed at 36 was constructed and the Parallel Factor algorithm was deployed, which can yield the closed-form automatically paired 2D-DOD and 2D-DOA estimation. The simulation results show that the proposed algorithm can exhibit superior estimation performance for the underdetermined 2D-DOD and 2D-DOA estimation.


I. INTRODUCTION
Studies about the electromagnetic vector sensors (EMVS) have attracted extensive attention due to their excellent measurement capabilities of angle parameters and polarization parameters [1]- [5]. Compared with the scalar array, EMVS with three orthogonal electric dipoles and three orthogonal magnetic dipoles can obtain the electromagnetic field vector information of the targets.
Recently, in order to measure the polarization information of targets in bistatic MIMO radars, the EMVS has been used as the transmit and receive array in the bistatic MIMO radar system.
For the achievement of multi-dimensional parameter estimation in bistatic EMVS-MIMO radar, the ESPRIT algorithm was firstly proposed in [6]. But, this algorithm has high computational complexity due to the singular value decomposition on the covariance matrix of the array received data. In [7], a computationally fridendly PM estimator was proposed to construct the virtual direction matrix for bistatic EMVS-MIMO radar. In [8], a tensor subspace based algorithm has been developed to improve the estimated performance by exploiting the inherent multidimensional structure of the received data. Additional optimization process in [6]- [8] is inevitable because of the pair matching between 2D-DOD and 2D-DOA. Thus, an improved PM algorithm in [9] has been raised to realize the automatic pair of 2D-DOD and 2D-DOA, and this method can further decrease the computational complexity. The disadvantage of the PM algorithms in [7] and [9] is the relatively poor estimation performance. In [10], the PARAFAC algorithm was utilized to realize the automatically paired 2D-DOD and 2D-DOA estimation. In [11], the bistatic coprime EMVS-MIMO radar was designed to enhance the 2D-DOD and 2D-DOA estimaiton performance.
It is apparent that the above-mentioned algorithms do not utilize the inherent difference coarray of transmit and receive EMVS, when conducting multi-dimensional parameter estimation in bistatic EMVS MIMO radar. According to [12]- [17], the difference coarray of the original array can offer larger array apertures, which means that the underdetermined DOA estimation is obtain-able. For bistatic EMVS-MIMO radar, the challenge for constructing the difference coarray of the transmit and receive EMVS is the coupling in the spatial response alongside the steering vector.
In this paper, the de-coupling process has been completed with the help of tensor operation. To estimate more targets than elements, the difference coarray of transmit and receive EMVS has been attained by the high order tensor permutation along with the generalized tensoriation of the canonical polyadic decomposition. In addition, two designed selection matrices have been used to remove the repeated elements in the difference coarray via the Tensor-Matrix product rule. The geometry of the difference coarray for both transmit and receive EMVS exhibit centrosymmetry.
As discussed in [22]- [26], the DFT beamspace matrix can be exploited to convert the complex array manifold matrix with centro-symmetric structure into the real-value one. This process can enhance the 2D-DOD and 2D-DOA estimation performance in the bistatic EMVS-MIMO radar.
Besides, a third-order tensor has been constructed with the dimension of the third-way fixed at 36. Finally, The automatically paired transmit/receive elevation angle, transmit/receive azimuth angle, transmit/receive polarization angle and transmit/receive polarization phase difference can be accurately derived from the estimated transmit and receive factor matrices acquired by the PARAFAC algorithm. Simulation results have verified its effectiveness.
The paper is organized as follows. Section II describes the nomenclature employed in this publication. Section III formulates the signal model of the bistatic coprime EMVS-MIMO radar.
Section IV reports the detailed derivation process of the proposed algorithm. Section V presents simulations to verify the effectiveness of the proposed algorithm. Section VI provides a summary.

II. NOMENCLATURE
• Some Definations of Coprime Array in [12]- [13] Definition 1 (D, Difference Coarray). For a sparse coprime array specified by an integer set S, its difference coarray D is defined as Definition 2 (U, maximum central contiguous ULA ). Let S denote a coprime array and D be its difference coarray. The maximum central contiguous ULA segment in D is Definition 3 (W, Weight Functions). The weight function w(m) of an array S is defined as the number of sensor pairs that lead to coarray index m, i.e.,|{(n 1 , n 2 ∈ S 2 )|n 1 − n 2 = m}| • Some Definations of Tensor in [18]- [21] Definition 4 (Mode-n Tensor-Matrix Product). The mode-n product of an N -order tensor set X ∈ C I 1 ×I 2 ×···×I N and a matrix A ∈ C Jn×In is defined as Definition 5 (Outer product of two tensors). The outer product of two tensors A ∈ C I 1 ×I 2 ×···×I N and B ∈ C J 1 ×J 2 ×···×J M is defined as whose elements are y i 1 ,i 2 ,··· ,i N j 1 ,j 2 ,··· ,j M = a i 1 ,i 2 ,··· ,i N · b j 1 ,j 2 ,··· ,j M Definition 6 (PARAFAC decomposition). For an N th-order tensor X ∈ C I 1 ×I 2 ×···×I N , it can be expressed as a linear combination of rank-1 tensors as follows: where a kn ∈ C In×1 , k = 1, 2, · · · , K denotes the rank-1 tensor.

III. DATA MODEL
A bistatic coprime EMVS-MIMO radar system with M transmit EMVS and N receive EMVS is demonstrated in Fig. 1. Let the array positions of transmit EMVS and receive EMVS are L t and L r , respectively Assuming the number of targets is K, the transmit steering vector and receive steering vector of the bistatic coprime EMVS-MIMO radar can be expressed as where θ t k , θ r k ∈ [0, π), φ t k , φ r k ∈ [0, 2π), γ t k , γ r k ∈ [0, π/2) and η t k , η r k ∈ [−π, π) denote the transmit/receive elevation angle, transmit/receive azimuth angle, transmit/receive polarization angle and transmit/receive polarization phase difference, respectively. q t k (θ t k , φ t k , γ t k , η t k ) and q r k (θ r k , φ r k , γ r k , η r k ) represent the spatial response of the transmit EMVS and the receive EMVS of the k-th target, respectively. a t k =[e −j2πd t0 sin(θt k ) , · · · , e −j2πd tM sin(θt k ) ] T , a r k =[e −j2πd r0 sin(θr k ) , · · · , e −j2πd rN sin(θr k ) ] T . For an EMVS, the spatial response q can be expressed as where F (θ, φ) ∈ C 6×2 denotes the spatial angular location matrix, g(γ, η) ∈ C 2×1 is representative of the polarization states vector.
According to [1], the elevation angle θ and azimuth angle φ can be extracted from the normalized Poynting vector of q(θ, φ, γ, η) [1,2,3] q [1,2,3] × q * [4,5,6] q [4,5,6] By using the orthogonality of the bistatic coprime EMVS-MIMO radar transmit and receive waveforms, the signal model after matched filtering at the receive front can be denoted as and Q r = q r 1 q r 2 · · · q r K denote the transmit steering matrix, the receive steering matrix, the transmit spatial response matrix and the receive spatial response matrix, respectively. For the total L snapshots, the sample data can be further derived as For the bistatic coprime EMVS-MIMO radar with M -element transmit EMVS and N -element receive EMVS, the matrix dimension of (C t C r ) is C 36M N ×K . In order to achieve the underdetermined 2D-DOD and 2D-DOA, the difference coarray of the transmit EMVS and receive EMVS will be deployed to construct a new third-order tensor model. Moreover, from (14), it can be seen that the received data Y satisfies the space-time multi-dimensional tensor structure.
Here, some assumptions are made in order to effectively estimate the transmit 4-D parameter • The signals are stationary, non-Gaussian random process and statistically independent of each other.
• The number of signals K is konwn or corrected estimated by the existing algorithm.
• The N is zero-mean, complex Gaussian white noise.
• The noise is statistically independent of the signals.

A. Construction of the Difference Coarray
For the estimation of more targets than elements, the difference coarray of transmit and receive EMVS are constructed. From (9) and (10), it can be found that a t k and q t k , a r k and q r k are copuling, respectively. Therefore, in order to obtain the difference coarrays, the de-coupling is invetiable.
To reserve the mutiple space-time structure in Y , a 5-D tensor Y can be constructed as Then, a 8-D tensor R can be obtained via the following correlation operation between Y and where σ 2 k , k = 1, 2, · · · , K denotes the power of the k-th source. And, according to Definition 1, the correspoding difference coarray of transmit EMVS and receive EMVS can be obtained by combining the factor vectors a tk and a * tk , a rk and a * rk , respectively. Thus, the factor vectors can be rearranged as Then, the indeces 1 and 2, 4 and 5, 7 and 8 need to be combined by the use of the generalized tensorization of the canonical polyadic decomposition in Definition 7. As a result, a new 5-D [7,8] can be achieved as whereã tk = a tk ⊗ a * tk ,ã rk = a rk ⊗ a * rk ,q k = q * tk ⊗ q * rk . According to Definition 1, for the M transmit coprime EMVS with the number of the first subarray is M 1 and the second subarray is M 2, the N transmit coprime EMVS with the number of the first subarray is N 1 and the second subarray is N 2, the number of continue virtual array elements in the corresponding difference coarrays can be deonted as However, the difference coarrays of the coprime transmit array and the coprime receive array contain many duplicated elements. To attain the maximum central contiguous ULA in the difference coarrays, it is necessary to remove the repeated elements, which can be realized by the application of tensor matrix product rules. Two selection matrices J 1 and J 2 are designed on the basis of Algorithm 1. So, a new 6-D tensor can be expressd as where It can be found that both the transmit and the receive difference coarray can be achievable via the above-mentioned above process. Compared with (15), it is obvious that the proposed process has significantly improved the degree of freedoms.

B. Formulation of the new third tensor
It can be seen that a tk and a rk in (22) and (23) satisfy the centro-symmetric geometry, so, the DFT beamspace matrices are used to transform the complex value of a tk and a rk into the corresponding real-value ones. This operation can improve the estimation performance of the angle parameters and help to reconstruct the spatial response matrices Q t and Q r easily. Firstly, two DFT beamspace matrices are designed as where w rn = e −j π( N −1)ur (n) N · · · e j π(Ñ −1)ur (n) where u t = [0, 1, · · · , M −1], u r = [0, 1, · · · , N −1]. Then, a 5-D tensor with real-value steering matrices can be obtained by exploiting the tensor-matrix product rules where A t = W t A t = [ a t1 , a t2 , · · · , a tK ] and A r = W r A r = [ a r1 , a r2 , · · · , a rK ] denote the real-value steering vector matrices of the transmit array and the receive array, respectively, and their detailed forms are as (29) and (30).
Afterwards, a new three way tensor R 5 can be attained via combining the indices 1 and 2, 3 and 4 Fig. 2. The three-way tensor R5.
As shown in Fig. 2, the dimension of the newly constructed tensor R 5 is C 6 M ×6 N ×36 , whose third way is fixed at 36. On its basis, the PARAFAC algorithm can be utilized to estimate the factor matrices C t , C r , and Q, respectively. According to Definition 6, the different slices of the third-order tensor R 5 can be further obtained as where D i , D j , D k represent the diagonal matrix operation, and the elements on the diagonal matrices are the kth-rows of the factor matrices C t , C r , and Q, respectively.
In order to realize the estimation of transmit steering matrix C t , the receive steering matrix C r and the signal matrix Q, the trilinear alternating least squares algorithm is deployed as follows Then, letQ,C t andC r denote the estimated factor matrices, respectively For the factor matricesQ,C t andC r obtained by multiple iterations of PARAFAC-TALS, the transmit steering matricesC t and the receive steering matricesC r are in one-to-one correspondence. This means that the 2D-DOD and 2D-DOA inC t andC r are automatically paired.
Therefore, the proposed algorithm need not the construction of an additional pairing optimization function compared to the algorithms in [6]- [8].

C. Estimaiton of the angle and polarization parameters
For the estimatedC t andC r , the corresponding transmit elevation angle and receive elevation angle can be obtained. Firstly, structural characteristics of the steering vectors a tk in A t and a rk in A r are analyzed. The m-th and (m + 1)-th elements in a tk and n-th and (n + 1)-th elements in a rk can be expressed as It can be seen that the adjacent elements in a tk and a rk satisfy the following relationships On the basis of (37) and (38), the following rotation-invariant relationship for A t and A r can be constructed . . .
By exploiting the property of Khatri-Rao product, the transmit spatial response Q t can be reconstructed as whereC t (6m − 5 : 6m, :) denote (6m − 5)-th row to 6m-th row ofC t . Due to the real-value transform in (28), there is no need to multiply an additional diagonal matrix when estimating the Q t , which differs from the reconstruction of Q t in [6]- [11]..
For the reconstructed transmit spatial response Q t , the normalized Poynting vector corresponding to the transmit EMVS array is  Therefore, the estimated transmit azimuth angle φ t k , k = 1, 2, · · · , K can be expressed as After obtaining (θ t k , φ t k ), k = 1, 2, · · · , K, the corresponding transmit polarization state vector g t k (γ t k , η t k ) can be achieved Consequently, the polarization parameters (γ t k , η t k ), k = 1, 2, · · · , K of the transmit EMVS can be obtained asγ Finally, the automatically paired transmit 4-D parameters (θ t , φ t , γ t , η t ), k = 1, 2, · · · , K can be gained by using the mentioned above process, and the receive 4-D parameters (θ r , φ r , γ r , η r ), k = 1, 2, · · · , K can be estimated in the same way. Algorithm 2 offers the framework of the proposed algorithm.

D. Deterministic Cramer-Rao bound
According to [10], for a bistatic EMVS-MIMO radar with M transmit EMVS and N receive EMVS, the Cramer-Rao bound for the transmit 4-D parameters (θ t , φ t , γ t , η t ) and the receive 4-D parameters (θ r , φ r , γ r , η r ) is where C tr = (C t C r ) is the joint transmit-receive matrix of the bistatic EMVS-MIMO radar, (Π) ⊥ Ctr = I 36M N − C tr C † tr denotes the projection matrix of C tr , ⊕ denotes the Hadamard product, R ss represents the signal covariance matrix, 1 8×8 is representative of the all-one ∂Ctr ∂θr ∂A ∂φ r ∂Ctr ∂γ r ∂Ctr ∂η r is the joint derivative matrix of the C tr for (θ t , φ t , γ t , η t ) and (θ r , φ r , γ r , η r ).

E. Identified targets
Assume the maximum number of identified targets is K by using the proposed algorithm in this paper. The value of K is dependent on the PARAFAC algorithm, owing to the characteristic of PARAFAC decomposition, whose uniqueness, as reported in [18], can be guaranteed via the following Kruskal's theorem.
Algorithm 2 Framework of the proposed algorithm Input: array received data matrix Y Output: the transmit 4-D parameters (θ t , φ t , γ t , η t ) and the receive 4-D parameters (θ r , φ r , γ r , η r ) 1: construct the 5-D tensor Y according to (15) 2: compute the covariance tensor R according to (16) 3: calculate the new 5-D tensor R 2 by using the tensor permutation and the generalized tensorization of the canonical polyadic decomposition according to (17) and (18) 4: remove the repeated elements in the difference coarrays in R 2 according to (21) 5: construct the three-way tensor R 5 according to (24)-(31) 6: perform the PARAFAC algorithm on R 5 to obtain the estimated transmit factor matrix C t and the receive factor matrix C r according to (32)-(34) 7: construct the select matrices Γ t1 , Γ t2 , Γ r1 and Γ r2 to computeΦ(θ t ) andΦ(θ r ), respectively.
And, by using SVD to obtain the estimatedθ tk , k = 1, 2, · · · , K andθ rk , k = 1, 2, · · · , K according to (43)-(50) 8: estimate (φ t , γ t , η t ) by using the reconstructed matrix Q t accoding to (53)-(57) 9: estimate the receive 3-D parameters (φ r , γ r , η r ) by using the similar way according to step 8 10: return (θ t , φ t , γ t , η t ) and (θ r , φ r , γ r , η r ) Theorem 1 (Kruskal's theorem) For a three-way tensor model R 5 ∈ C 6 M ×6 N ×36 , the PARAFAC decomposition is unique, if where κ Ct ,κ Cr and κ Q denote the κ-rank of C t , C r and Q, respectively. The max κ-rank of Thus, the maximum number of identified targets K is Resultantly, the maximum number of identified targets is larger than the number of transmit EMVS and receive EMVS. The constructed difference coarrays in this paper are effective for the underdetermined 2D-DODs and 2D-DOAs estimation in the bistatic EMVS-MIMO radar system . Table I compares the main computational complexity of the proposed algorithm, the ESPRIT algorithm in [6], PM algorithm in [7], Tensor subspace-based algorithm in [8] and PARAFAC algorithm in [10]. Here, the CM, SVD and HOSVD denote the covariance matrix calculation, the singular value decomposition and the higher-order singular value decomposition, respectively.

F. Computational complexity
For a bistatic EMVS-MIMO radar with M transmit array and N receive array, the computational complexity of the proposed algorithm is higher than that of the PM algotihm and the PARAFAC algorithm, while lower than that of the ESPRIT algorithm and the HOSVD algorithm.

V. SIMULATION RESULTS
In order to evaluate the estimation performance of angle parameters and polarization parameters in bistatic coprime EMVS-MIMO radar, the following simulation experiments are   Table II. Additionally, the SNR is set to 10dB, the snapshot is 200. The noise is assumed to be independent zero-mean additive Gaussian white noise, and the signal and noise are independent of each other. The scatterplot in Fig. 3 is obtained via 100 trails. Fig.   3(a) and Fig. 3(b) show that the underdetermined 2D-DODs and 2D-DOAs are accurately estimated. In the meantime, Fig. 3(c)- Fig. 3(f) demonstrate that the corresponding transmit 4-D parameters (θ t , φ t , γ t , η t ) and receive 4-D parameters (θ r , φ r , γ r , η r ) are automatically paired via the PARAFAC algorithm. Therefore, the effectiveness of proposed algorithm for the underdetermined 2D-DODs and 2D-DOAs estimation in the bistatic coprime EMVS-MIMO radar has been verified.
Secondly, the RMSEs performance of different algorithms versus SNR are compared. The average root mean square error (RMSE) is calculated as '-d'and '-p' in the legend refer to angle parameter and polarization parameter, respectively. Fig. 4 shows that the proposed algorithm exhibits a better angle and polarization parameters estimation Thirdly, the RMSEs performance of different algorithms versus the number of snapshots are assessed. The snapshots are set to vary from 100 to 1000 with a step of 100 and the SNR is fixed at 10dB. Other simluation parameters are the same as those in the Secondly experiment.
As evident from Fig. 5, the RMSE versus snapshots of the proposed algorithm has the smallest estimation error both for the angle parameters and polarization parameters. As a consequence, the difference coarray of the copirme EMVS used in this paper promote the angle parameters and polarization parameters estimation performance.
Fourthly, the RMSEs performance of different algorithms versus the number of impinging sources K has been compared. The number of sources vary from 2 to 8 and the corresponding transmit 4-D parameter (θ t , φ t , γ t , η t ) and receive 4-D parameters (θ r , φ r , γ r , η r ) are choosed from the Table II. The SNR and snapshots are set to 10dB and 200, respectively. It can be seen from Fig. 6 that the proposed method is able to maintain a reasonable estimation error for different number of sources, while the RMSEs for the other three algorithms become large with the increase of K. This phenomenon indicates that the proposed algorithm is robust to the different number of sources.
Finally, the estimation performance of the two closely-located targets has been evaluated by using the proposed method. Here, the biases between the true angle/polarization parameters and the mean of estimated angle/polarization parameters versus SNR and snapshots are examined.

VI. CONCLUSION
A tensor-based algorithm has been investigated to deal with the underdetermined 2D-DOD and 2D-DOA estimation in the bistatic EMVS-MIMO radar system, and its effectiveness has been well verified via simulations. This algorithm is advantageous to the realization of de-coupling between the spatial response and steering matrix, and it also contributes to the difference coarrays construction of transmit and receive EMVS, which plays an important role to estimate more targets than elements. More meaningfully, a variety of newly-designed sparse arrays in recent research can also be used as the transmit/receive EMVS in the bistatic MIMO radar system and this method can pave a path to obtain the corresponding difference coarrays to enhance the estimation performance.