Low Complexity Direction and Doppler Frequency Estimation for Bistatic MIMO Radar in Spatial Colored Noise

We investigate the algorithm of direction and Doppler frequency estimation for bistatic multiple-input multiple-output (MIMO) radar in spatial colored noise. A novel method of joint estimation of direction and Doppler frequency in spatial colored noise based on propagator method (PM) for bistatic MIMO radar is discussed. Utilizing the cross-correlation matrix which is formed by the adjacent outputs of match filter in the time domain, the special matrix is constructed to eliminate the influence of spatial colored noise. The proposed algorithm provides lower computational complexity and has very close parameters estimation compared to estimation of signal parameters via rotational invariance technique (ESPRIT) algorithm in high signal-to-noise ratio (SNR). It is applicable even if the transmitted waveforms are not orthogonal. The estimated parameters can be paired automatically and the Cramér-Rao Bound (CRB) is given in spatial colored noise. Simulation results confirm the effectiveness of the proposed method.

Target direction estimation is a basic function of a radar system.Many advanced direction estimation algorithms for MIMO radar have been extensively discussed in the current literature which include ESPRIT algorithm, Capon algorithm, parallel factor (PARAFAC) algorithm, multiple signal classification (MUSIC) algorithm, and PM algorithm [25][26][27][28][29][30][31][32][33].In [25,29], ESPRIT algorithm exploited the invariance property of both the transmit array and the receive array for direction estimation in MIMO radar systems.Reference [30] derived a reduced-dimension multiple signal classification (MUSIC) algorithm for direction of departure (DOD) and direction of arrival (DOA) estimation.The algorithm, which only requires one-dimensional search, can avoid the high computational cost of the two-dimensional MUSIC algorithm.However, the mentioned algorithm above did not consider the Doppler frequency estimation, and the noises were assumed to be the Gaussian white noise.In [31], the ESPRIT method was used for DOD-DOA and Doppler frequency estimation which necessitates eigen decomposition of the sample covariance matrix.Huge computation will be involved where the large array size is required in applications.Yunhe [32] proposed the DOA matrix algorithm to estimate the DOD-DOA and Doppler frequency, but it cannot eliminate the influence of spatial colored noise.In this paper, we propose a low-complexity angle and Doppler frequency estimation algorithm which can reduce computational cost.It has very close parameters estimation performance compared to ESRPIT and DOA matrix algorithm in high SNR.And the proposed algorithm pairs the parameters automatically and eliminates the influence of the spatial colored noise.Simulation results illustrate performance of the proposed algorithm.
The remainder of this paper is structured as follows.Section 2 develops the data model for a bistatic MIMO radar system, and Section 3 proposes the proposed algorithm for angle and Doppler frequency estimation in MIMO radar.In Section 4, simulation results are presented to verify improvement for the proposed algorithm, while the conclusions are shown in Section 5.

Data Model
We consider a narrowband bistatic MIMO radar system with -element transmit antennas and -element receive antennas, both of which are half-wavelength spaced uniform linear arrays.The transmit antennas transmit  orthogonal coded signals.s  = [  (1),   (2), . . .,   ()]  ∈ C ×1 ,  = 1, 2, . . .,  denotes the sampled baseband coded signal of the th transmit antenna with one repetition interval with  being the length of the transmitted code sequence, so the transmit signals can be expressed as S = [s 1 , s 2 , . . ., s  ]  .We also assume that there are a number of  far-field independent targets; a  (  ) and a  (  ) are the receive steering vector and transmit steering vector for   (DOA) and   (DOD) of the th target, so the arrival signal of the th target is a   (  )S.The received array through reflections of the target can be expressed as where   and   denote the radar cross-section (RCS) fading coefficient and Doppler frequency of the th target.  is the pulse repeat frequency.Due to the steering vector of ULA, we have a  (  ) = [1,  − sin   , . . .,  −(−1) sin   ]  , a  (  ) = [1,  − sin   , . . .,  −(−1) sin   ]  .w() ∈ C × denotes a Gaussian noise of zeros mean with unknown covariance matrix Q  .Matching the received data with the signal (1/ √ )S  , we obtain where A = [ ā 1 , ā 2 , . . ., ā  ] is an  ×  matrix composed of the  steering vectors, and ā  = a  (  )⊗ ā  (  ) is the Kronecker product of the receive and the transmit steering vectors for the th target.
The covariance matrix of n() is as follows:

Direction and Doppler Frequency Estimation Algorithm for MIMO Radar
3.1.The Proposed Algorithm Description.We assume that a  (  ) and a  (  ) are constant for  samples and define X as , . . ., x()], and we assume that the number of snapshots is .Let where  3) shows that the crosscovariance matrix of noises is 0. This characteristic will be utilized in this paper to improve the estimate performance.
This indicates that the rotation factor Φ is generated by adjacent outputs of match filters.
The covariance matrix of X 1 and X 2 can be written as follows: where . For the independent targets, R  should be a diagonal matrix; then we have the relationship The propagator method relies on the partition of the matrix B [34].B can be denoted by International Journal of Antennas and Propagation 3 is the full rank matrix; B 2 ∈ C (2−)× .The propagator V is a unique linear operator which can be written as Similarly partitioning received data matrix R into two submatrices R 1 and R 2 with dimensions  ×  and (2 − ) × , respectively.Then the unique linear operation holds between R 1 and R 2 : An estimation matrix V can be obtained by minimizing the cost function: The optimal solution is given by V where I  is the identity matrix.Combining Ṽ and ( 7), we obtain Rewrite (8) as where R † 1 is the pseudoinverse of R 1 and T is a nonsingular matrix.From ( 9), the columns in Ṽ span the same signal subspace as the column vectors in B. So the signal subspace can be obtained by avoiding the estimation and eigen decomposition of the sample covariance matrix.The matrix Ṽ can be partitioned into two submatrices Ṽ = [ Ṽ1

Ṽ2
].According to (9), we can get Notice that Φ = Φ −1 Φ 2 ; from (10) we can obtain Equation (11) shows that the main diagonal elements of Φ 2 are equal to the eigen values obtained via the eigen decomposition of Ṽ † 1 Ṽ2 with T the corresponding eigenvectors.Thus the Doppler frequency of the th target can be calculated as where   is the th diagonal element of Φ 2 .The Â can be calculated from (12), because the Φ is the diagonal matrix, and it cannot affect the estimation of DOD and DOA using the least square method from the matrix Â.
We note that the pairing is automatically obtained because the DOA-DODs and Doppler frequencies are given through the corresponding eigenvectors.The matrix Â can be also denoted by where Â = [â  ( 1 ), â ( 2 ), . . ., â (  )] and Â = [â  ( 1 ), â ( 2 ), . . ., â (  )] are the transmit and receive direction matrices, respectively: There exists an  ×  transformation matrix C corresponding to the finite number of row interchanged operations such that where We define Owing to P  Φ = P  , Φ = P  † P  , then we get the estimation of DOA: where   is the th diagonal element of the matrix Φ .We also define Owing to B  Φ = B  , Φ = B  † B  , we can also get the estimation of DOD: where   is the th diagonal element of the matrix Φ .Now we show the major steps of the proposed algorithm as follows.
International Journal of Antennas and Propagation (1) Compute the covariance matrix of the received data through (5).

Discussion
(1) From Figure 1 we find that our algorithm has much lower computational load than ESPRIT algorithm and DOA matrix algorithm.ESPRIT algorithm employs either eigen-value decomposition (EVD) of cross-correlation matrix or singular value.Using the techniques, the computational complexity is very high.Reference [34] has shown the propagator method (PM) for array signal processing to estimate DOA of incident signals without eigen-value decomposition of cross-correlation matrix of the received data.In our proposed algorithm, propagator V is a linear operator which can easily be extracted from the data matrix R.But the construction of matrix Ṽ leads

L/snapshots
The proposed algorithm ESPRIT algorithm [31] DOA matrix algorithm [32]  the proposed algorithm's performance to degrade in low SNR.So the proposed algorithm has very close parameters estimation to ESPRIT algorithm and DOA matrix algorithm in high SNR.
(2) Since the DOA-DODs and Doppler frequencies are given through the corresponding eigenvectors, it can achieve automatically paired estimation of angles and Doppler frequencies.
(3) The proposed algorithm can eliminate the effect of the spatial colored noise since the new matrix is constructed by ( 5) and (6).

The Cramér-Rao Bound (CRB).
In this section, we derive CRB of parameter estimation for MIMO radar and rewrite the received data as where , . . .,   ] can be calculated as follows [31,35]: International Journal of Antennas and Propagation where and Γ = diag() denotes the diagonal matrix constructed by the vector : Then the CRB matrix is

Simulation Results
We present the Monte Carlo simulations to assess the parameter estimation performance of our algorithm., where f, is the estimate of Doppler frequency   of the th Monte Carlo trial.Note that , , , and  are the number of transmit antennas, the receive antennas, the snapshots of the targets, and the number of the targets, respectively.We assume that International Journal of Antennas and Propagation
The pulse repeat frequency   is 10 KHz for a  = 9 and  = 9 bistatic MIMO radar.The (, )th element of the unknown noise covariance matrix Q  is 0.9 |−|  (−)/2 .Figure 2 shows the estimation results with 100 Monte Carlo trials at  = 100.As seen in Figure 2, the DODs, DOAs, and Doppler frequencies of all three targets are correctly paired and well localised.Figures 3 and 4 show the angle and Doppler frequency estimation performance comparison with  = 9,  = 9,    ).The Doppler frequencies of the three targets are 1200 Hz, 1500 Hz, and 1800 Hz, respectively.From Figure 7, we find that our algorithm works well in the case of closely spaced targets.

Conclusion
We have presented a low-complexity angle and Doppler frequency estimation based on propagator method for MIMO radar in spatial colored noise.The proposed algorithm can obtain automatically paired transmit and receive angle estimations in the MIMO radar and eliminate the influence of the spatial colored noise.Furthermore, it provides lower computational complexity and has close parameters estimation compared to ESPRIT algorithm and DOA matrix algorithm in high SNR.It is applicable even if the transmitted waveforms are not orthogonal.

Figure 5 :Figure 6 :
Figure 5: Angle estimation with different values of .