Separate DOD and DOA Estimation for Bistatic MIMO Radar

A novel MUSIC-type algorithm is derived in this paper for the direction of departure (DOD) and direction of arrival (DOA) estimation in a bistatic MIMO radar. Through rearranging the received signal matrix, we illustrate that the DOD and the DOA can be separately estimated. Compared with conventional MUSIC-type algorithms, the proposed separate MUSIC algorithm can avoid the interference between DOD and DOA estimations effectively. Therefore, it is expected to give a better angle estimation performance and have a much lower computational complexity. Meanwhile, we demonstrate that our method is also effective for coherent targets in MIMO radar. Simulation results verify the efficiency of the proposed method, particularly when the signal-tonoise ratio (SNR) is low and/or the number of snapshots is small.


Introduction
Multiple-input multiple-output (MIMO) radar, which utilizes multiple antennas to simultaneously transmit diverse waveforms and receive reflected signals, has many potential advantages over the conventional phased-array radar [1][2][3][4].Direction of departure (DOD) and direction of arrival (DOA) estimation [5][6][7][8] is a key issue in MIMO radar signal processing, and it has attracted a lot of attention.Many algorithms for DOD and DOA estimation have been established in the literatures [9][10][11].By exploiting the invariance property of both the transmit array and the receive array, [12] developed a subspace method based on the classical rotational invariance techniques (ESPRIT) algorithm, but additional pair matching is required.To avoid pair matching, an improved ESPRIT algorithm was presented in [13], whose complexity is lower than the algorithm in [12].A real-valued ESPRIT algorithm was proposed in [14], where all the complex computations are transformed into real-valued ones.As a consequence, it can further reduce the computational complexity and ameliorate the performance for DOD and DOA estimation.Moreover, a propagator method (PM) algorithm for DOD and DOA estimation for MIMO radar was investigated in [15], which can construct the signal subspace without the eigenvalue decomposition of covariance matrix.So, the PM algorithm has lower complexity than ESPRIT-type methods [12-14, 16, 17], but it has a low performance of DOD and DOA estimation.
It is well known that multiple signal classification (MUSIC) algorithms have better performance than ESPRITtype and PM-type algorithms.It has been proved that twodimension MUSIC (2D-MUSIC) algorithm can be used for DOD and DOA estimation in MIMO radar and has a good angle estimation accuracy; however, it requires high computation complexity.The method in [18] combines ESPRIT and root-MUSIC to achieve the compromise between the complexity and estimation performance.In [19], a reduceddimension MUSIC (RD-MUSIC) algorithm was proposed for DOD and DOA estimation, which can reduce the computational cost by replacing the two-dimensional searching with one-dimensional searching.All these MUSIC-type algorithms [19][20][21][22] can pair DOD and DOA estimation automatically.However, none of them can avoid the interference between DOD and DOA estimations.For example, if DOA is first estimated in these methods, then the estimation of DOD will be influenced by the estimation error of DOA.

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International Journal of Antennas and Propagation Therefore, the performance of these MUSIC-type algorithms might seriously degrade.
To solve the aforementioned problem, in this paper, a separate MUSIC algorithm for DOD and DOA estimation is presented.The algorithm first addresses the DOD estimation and then rearranges the received signal matrix to estimate DOA.Compared with conventional MUSIC-type algorithms, the proposed algorithm, due to the utilization of separate DOD and DOA estimation, avoids the interference between DOD and DOA estimations effectively; therefore, the algorithm gives a better angle estimation accuracy and has a much lower computational complexity, when signal-to-noise ratio (SNR) is low and/or the number of snapshots is small.Meanwhile, this paper guarantees that the algorithm is also effective for coherent targets in MIMO radar.
This paper is organized as follows.Section 2 addresses the data model for bistatic MIMO radar.Section 3 reviews the existing MUSIC-type algorithms for DOD and DOA estimation.Section 4 proposes our separate MUSIC algorithm for DOD and DOA estimation.Finally, simulation results and conclusions are given in Sections 5 and 6, respectively.

Data Model
Consider a bistatic MIMO radar system with  transmit antennas and  receive antennas, both of which are halfwavelength spaced uniform linear arrays (ULAs).At the transmit site,  different narrowband waveforms are emitted simultaneously, which have identical bandwidth and central frequency but are temporally orthogonal.In each receiver, the echoes are processed for all of the transmitted waveforms.Assume that there are  uncorrelated targets located at the same range, and the DOD and DOA of the th target relative to the transmitter and the receiver are denoted by   and   , respectively.
The output of the matched filters at the receiver can be expressed as where A = [a  ( 1 ) ⊗ a  ( being the receive steering vector and the transmit steering vector, respectively.⊗ is the Kronecker product and [⋅]  is the transpose operation.s() = [ 1 (),  2 (), . . .,   ()]  is a column vector consisting of the amplitudes and phases of the  targets at time .n() represents an  × 1 complex Gaussian white noise vector of zeros mean and covariance matrix  2 I  .

Review of MUSIC-Type Algorithms
Let X be denoted as the data matrix composed of  snapshots of x(  ), 1 ≤  ≤ ; then the  ×  matrix X can be expressed as where The covariance matrix of X is defined as where R  = (1/)SS  is the source covariance matrix and [⋅]  is the Hermitian transpose.Let the eigenvalue decomposition of R be where Σ  denotes a  ×  diagonal matrix formed by  largest eigenvalues and Σ  denotes a diagonal matrix formed by the rest of the  −  smaller eigenvalues.U  and U  represent the signal subspace and noise subspace, respectively, of which U  contains the eigenvectors corresponding to the  largest eigenvalues and U  consists of the rest of the eigenvectors.

Separate MUSIC Algorithm for DOD and DOA Estimation
Obviously, the DOD estimation of RD-MUSIC algorithm is based on the estimate of the DOA.It is likely to degrade the DOD estimation performance due to the interference caused by the DOA estimation.In this section, we will propose a separate MUSIC algorithm to handle the problem.To this end, we first address the DOD estimation and then rearrange the received signal matrix X to estimate DOA.Finally, we will address the pair matching.
. . . where and diag[v] denotes a diagonal matrix constructed by the vector v.
In order to independently estimate DOD, we construct a new matrix containing the information of DOD; that is, According to the structure of X in (14), X  can be rewritten by International Journal of Antennas and Propagation Note that S  and N  can be seen as the signal matrix and noise matrix corresponding to the new measurement matrix X  , respectively.
The covariance matrix of X  is defined as ] where )  is the source covariance matrix corresponding to X  .Performing the eigenvalue decomposition of R  , we can get the noise subspace U  corresponding to X  .Then, we construct the following MUSIC spatial spectrum function for DOD estimation: Searching  ∈ [−90 ∘ , 90 ∘ ], we can obtain  largest peaks of   ().The corresponding ( θ1 , θ2 , . . ., θ ) are taken as the estimates of the DODs.Actually, our method is effective for coherent DOD estimation.Considering a completely coherent environment [23][24][25], we have where   (  ̸ = 0) represents the complex attenuation of the th signal with respect to the first signal  1 ().Using (20) with [ 2  1 (  )] = 1,  = 1, 2, . . ., , it is easy to see that where  = [ 1 ,  1 , . . .,   ]  .So the source covariance matrix R  takes the form [26] where Clearly, the rank of R  is equal to the rank of B. Since B = DV and the diagonal matrix D is of full rank, the rank of B is the same as that of V. Now, the rank of the  ×  Vandermonde matrix V is rank(V) = min(, ), and, hence, rank(V) =  if  ≥ .We can find that the rank of R  is the same as the number of the targets.Therefore, the proposed algorithm can be applied to solve multiple coherent targets for DOD estimation.

DOA Estimation. Define a new 𝑀𝑁 × 𝑃 matrix
, where a   = a  (  ) ⊗ a  (  ).In order to simplify the notation, we rewrite a  () = [1,   ,  2  , . . .,  −1  ] with   = exp( sin   ); then, we have Obviously, there exists an  ×  transformation matrix B corresponding to the finite number of row interchanged operations such that A  = BA, where Using the structure of the matrix A  , we introduce a virtual  ×  data matrix X  ≜ BX, or, equivalently, where N  ≜ BN.Divide the matrix X  into  submatrices: . . . where In order to estimate DOA independently, we construct a new  ×  matrix X  : which contains the information of DOA.According to the structure of X  in ( 27), X  can be expressed as Note that S  and N  can be seen as the virtual signal matrix and noise matrix corresponding to the measurement X  , respectively.The covariance matrix of X  can be calculated by where )  is the source covariance matrix corresponding to X  .Performing the eigenvalue decomposition of R  , we obtain the noise subspace matrix U  of X  .Thus, the MUSIC spatial spectrum function for DOA estimation can be expressed as Searching  ∈ [−90 ∘ , 90 ∘ ], we obtain the  largest peaks of   () corresponding to the estimated DOAs (φ 1 , φ2 , . . ., φ ).Moreover, since R  has similar structure characteristics to R  , the rank of R  is also the same as the number of the targets.Therefore, the proposed algorithm can also be applied to solve multiple coherent targets for DOA estimation.
Step 2. Calculate the sample covariance matrix of X  and X  , respectively, and obtain the noise subspace matrices U  and U  correspondingly.Step 3. Acquire the estimates of DODs and DOAs by searching  through (19) and  through (32), respectively.
Step 4. Pair the estimates of DODs and DOAs with (33).

Simulation Results
In this section, we introduce the simulation results of the proposed algorithm in comparison with the standard ESPRIT method [13], the Unitary ESPRIT method [14], the RD-MUSIC method [19], and the PM method [15].The root mean squared error (RMSE) of the DOD or DOA estimation is defined as where β, is the estimate of DOD or DOA corresponding to true value   of DOD or DOA of the th Monte Carlo trial.
Simulation 2 shows the performance of RMSE versus SNR for the DOD and DOA estimation of the proposed algorithm in comparison with the standard ESPRIT method, the Unitary ESPRIT method, the RD-MUSIC method, and the PM method.In this simulation, the number of snapshots is  = 10.We assume a ULA composed of  = 10 transmit antennas and  = 10 receive antennas, and we consider  = 4 uncorrelated targets located at angles ( 1 ,  1 ) = (−15 ∘ , −15 ∘ ), ( 2 ,  2 ) = (6 ∘ , 10 ∘ ), ( 3 ,  3 ) = (50 ∘ , 50 ∘ ), and ( 4 ,  4 ) = (−45 ∘ , 25 ∘ ), respectively.Figure 2(a) depicts DOD estimation performance of RMSE versus SNR. Figure 2(b) presents DOA estimation performance of RMSE versus SNR.As shown in the figure, the proposed algorithm has much better DOD and DOA estimation performance than other algorithms, particularly at the low SNR.Furthermore, we use the Cramer-Rao bound (CRB), calculated in the Appendix, as the performance benchmark.Obviously, the performance of the proposed algorithm is closer to the CRB than others.
Simulation 3 is used to verify whether our method can improve the interference between DOD and DOA estimations, where we compare it with the RD-MUSIC method and the CRB.In this simulation, we use the same parameters as in Simulation 2. As shown in Figure 3, obviously, the DOA estimation has a little influence on the DOD estimation of the proposed algorithm; however, the RD-MUSIC algorithm has large interference between DOD and DOA estimations.Hence, we have a conclusion that the proposed algorithm, owing to utilizing the separate DOD and DOA estimation, reduces the interference between DOD and DOA estimations and thus improves the accuracy of DOD and DOA estimation.Furthermore, the performance of the proposed algorithm is closer to the CRB than the RD-MUSIC method.
Simulation 4 shows the DOD and DOA estimation performance of the proposed algorithm with different snapshots .We consider a ULA composed of  = 12 transmit antennas and  = 12 receive antennas, and we assume  = 3 uncorrelated targets located at angles ( 1 ,  1 ) = (−25 ∘ , −20 ∘ ), ( 2 ,  2 ) = (0 ∘ , 45 ∘ ), and ( 3 ,  3 ) = (45 ∘ , 5 ∘ ), respectively.Figure 4(a) depicts DOD estimation performance of RMSE versus different number of snapshots . Figure 4(b) presents DOA estimation performance of RMSE versus different number of snapshots .As indicated in the figure, the performance of the proposed algorithm for DOD and DOA estimation is improved with  increasing, and we also draw a conclusion that the proposed algorithm works well in the case of small sampling sizes (e.g.,  = 10).

( 10 ∘Figure 1 :
Figure 1: Paired results of the proposed algorithm for all five targets. ) and 5(b) depict DOD and DOA estimation performance of RMSE versus different number of transmit antennas  under  = 10 condition, respectively.Figures 6(a) and 6(b) present DOD and DOA estimation performance of RMSE versus different number of transmit antennas  under  = 10 condition, respectively.It is clearly shown in Figures 5 and 6 that the angle estimation performance of the proposed algorithm is gradually improved