Sparsity-Based Robust Bistatic MIMO Radar Imaging in the Presence of Array Errors

A sparse recovery method for robust transmit-receive angle imaging in a bistatic MIMO radar is proposed to deal with the effect of array gain-phase errors. .e impact of multiplicative array gain-phase errors is changed to be additive through model reformulation, and transmit-receive angle imaging is formulated to a sparse total least square signal problem. .en, an iterative algorithm is proposed to solve the optimization problem. Compared with existing methods, the proposed method can achieve a significant performance gain in the case that the number of snapshots is small. Simulation results verify the effectiveness of the proposed method.


Introduction
Multiple-input multiple-output (MIMO) radars use multiple channels to transmit orthogonal waveforms and multiple channels to receive echo signals, where the transmitting aperture can be fully exploited [1][2][3][4][5][6]. Compared to a conventional phased-array radar, an MIMO radar can enhance spatial resolution, improve target detection performance, etc. e performance of the MIMO radar can usually be improved by increasing the number of channels. According to the transmitting and receiving antenna configurations, there are two main classes of MIMO radars. e first class is the statistical MIMO radar [7,8], where transmitting and receiving antennas are widely separated. By exploiting the spatial diversity, the statistical MIMO radar can resist the performance degradations caused by target scintillations. e second class is the colocated MIMO radar [2], where the transmitting antennas and receiving antennas are closely spaced, and performance gain can be achieved by multichannel coherent processing. A colocated MIMO radar can obtain a virtual aperture larger than its real aperture, resulting in lower sidelobes and a narrower beam width.
A bistatic MIMO radar, firstly proposed in [3], has the advantages of both the bistatic radar and the MIMO radar.
In the bistatic MIMO radar, the direction of arrival (DOA) and the direction of departure (DOD) of targets can be obtained at the same time by processing the received signals, and it has been employed to identify and locate multiple targets [9,10], clutter cancellation [11,12], and imaging [13][14][15]. Nevertheless, due to the presence of the gain and phase errors of the transmitting array and receiving array, the imaging quality and detection accuracy of these techniques can deteriorate seriously. Many methods have been investigated to mitigate the impact of array gain-phase errors in the bistatic MIMO radar [16,17]. However, these methods work well with a large number of snapshots, but they do not work properly when the number of snapshots is small.
Sparse recovery techniques [18][19][20] can be employed to achieve high-resolution imaging with a small number of samples, and they have been applied to bistatic MIMO radar imaging. A sparse recovery-based imaging method is proposed in [14], which is robust to large gain errors. However, the method does not consider the array phase errors and does not work well with a single snapshot. In [15], a transmit-receive angle imaging method is proposed, but its complexity is high and its performance degrades severely when the gain-phase errors are relatively large. In this work, we propose a robust sparse recovery method for transmit-receive angle imaging in the bistatic MIMO radar. Signals are transformed into a sparse domain by discrete wavelet transform.
en, the reconstruction is formulated as an optimization problem which is solved iteratively. e proposed method is more robust in dealing with the imperfection of array gain-phase errors and noise in the bistatic MIMO radar system. We analyse the key parameters affecting the performance of recovery results. We also analyse the influence of the number of iterations on the convergence of the algorithm. Simulation results show that compared with existing methods, the proposed method exhibits better robustness.
is paper is organized as follows: e bistatic MIMO radar sparse signal model with array gain-phase errors is derived in Section 2. In Section 3, a robust iterative algorithm is proposed to achieve target image reconstruction in the presence of array gain-phase errors. In Section 4, the effectiveness of the proposed method is verified by simulations. Some conclusions are drawn in Section 5.

Signal Model with Array Gain-Phase Errors
We consider a bistatic MIMO radar system with M transmitting antennas and N receiving antennas, where both the transmitter and the receiver are equipped with a uniform linear array (ULA). e transmitted pulses are denoted by S ∈ C M×L , where M is the number of coded periodic signals and L is the length of the coding sequence in one pulse period.
As shown in Figure 1, the location of a target can be determined by its angle pair (φ t , θ r ), where φ t and θ r are the angles of the target with respect to the transmitting array and the receiving array, respectively. e received signal can be expressed as where A T � [a t1 , a t2 , . . . , a tg , . . . , a tG ] M×G and A R � [a r1 , a r2 , . . . , a rg , . . . , a rG ] N×G are the steering matrices of the transmitting array and receiving array with G pixel points, (·) T denotes the transpose operator, and D q � diag(d 1 , . . . , d g ) is a diagonal matrix with diagonal elements being the target scattering coefficients for the qth pulse period. It is assumed that the transmitted signals are orthogonal, i.e., SS H � I, and E q denotes complex Gaussian distributed noise with zero mean and σ 2 n I N covariance. e steering vectors of the receiving array a rg and the transmitting array a tg can be expressed as a rg � 1, e j(2π/λ)d r sinθ rg , e j(2π/λ)2d r sinθ rg , . . . , e j(2π/λ)(N− 1)d r sinθ rg T , where λ is the signal wavelength and d t and d r denote the transmitting antenna spacing and receiving antenna spacing, respectively. Consider the region of interest Ω, as shown in Figure 1, and we divide it into two-dimensional grids consisting of After match filtering by the transmitted waveforms, the received signals turn into where X q ∈ C G×G is the matrix of target pixel distribution and is nonzero only when it is the pixel of the target. When we process the data of one range cell, only surface of the ellipse has the pixels of the target, and any other grid points in Ω are zeros. We can recover the image range by range. e vectorized Y q can be expressed as , and ⊗ denotes the Kronecker product. With the presence of gainphase errors in the transmitting and receiving arrays, the steering matrices can be represented as where are the diagonal matrices containing gain-phase errors. e diagonal elements σ Ri � a Ri e jφ Ri and σ Ti � a Ti e jφ Ti , where a Ri and a Ti denote the gain errors of the receiving array and transmitting array elements and φ Ri and φ Ti denote the phase errors of the receiving array and transmitting array elements, respectively. e received signal with the gain-phase errors can be expressed as Hence, the vector form of equation (6) can be written as where Δν ′ � ΔS R ⊗ A T + A R ⊗ ΔS T + ΔS R ⊗ ΔS T can be modeled as an additive random perturbation matrix on the ideal measurement matrix Φ ′ . We use the symlet wavelet in discrete wavelet transform (DWT), to transform the image into a sparse domain, and then reconstruct the image with our robust algorithms in the sparse domain. We assume that the wavelet transform matrix is W ∈ C G×G and then Wx q ′ � x q , so x q is a representation of x q ′ in the sparse domain. Accordingly, equation (7) can be rewritten as where Φ � Φ ′ W − 1 and Δν � Δν ′ W − 1 . e objective is to recover the sparse vector x q , which is elaborated in Section 3. (8) considers both gain-phase errors and noise. e optimization problem with the sparsity constraint can be formulated as [20] x q , Δν, e q � arg min

Iterative Sparse Recovery Imaging Algorithm
e cost function includes two parts: the error term ‖Δν, e q ‖ 2 F and the regularization term c‖x q ‖ 1 , where c is a regularization parameter to control the sparsity of the solution. In the problem formulated in (9), the observation vector z q and measurement matrix Φ are given and Δν and e q are the unknown perturbation matrix and noise vector, respectively. Our aim is to recover the unknown signal vector x q , which is a nonconvex optimization problem.
We use a coordinate descent method to solve the optimization problem. In the ith iteration, the method performs two steps. In the first step, x q is updated, which can be formulated as where the matrix Δν(i − 1) is obtained in the last iteration, and the optimization problem above is convex. e second step is to update the perturbation matrix Δν with x q (i) obtained in the first step, i.e.,

Δν(i) � arg min
e method fixes a parameter between x q and Δν while optimizing the other one, until a stop criterion is satisfied. In each iteration, x q can be solved by the following optimization problem [21]: In the above, we add the penalty term ρ/2‖x q − w‖ 2 2 to bring robustness to the dual ascent method, where ρ is a penalty parameter. en, we can form the augmented Lagrangian: where the constraint s.t. x q − w � 0 can be written as the term L T (x q − w) and L is a Lagrangian multiplier vector. We use the alternating iteration method to solve (13). In the k + 1th iteration, the method performs three steps. In the first step, x q can be updated as By calculating the partial derivative of Λ(x q , w k , L k ) with respect to x q and making it equal to zero, we can update x q as where w and L are fixed and (Φ + Δν) T (Φ + Δν) + ρI is always invertible since ρ > 0. en, w can be updated by Similarly, we can get w k+1 by letting the partial derivative of Λ(x k+1 q (i), w, L k ) with respect to w equal to zero, i.e., where x k+1 q (i) and L k are fixed and ⊙ denotes point multiplication. e updated estimates of x k+1 q (i) and w k+1 are then employed to update the current estimate of L k+1 : with x q (i) available, the above formula is quadratic. en, letting the first-order derivative of (11) with respect to Δν to be zero, we have e proposed algorithm is given in Algorithm 1.

International Journal of Antennas and Propagation
We set the regularization parameter c as 15 and penalty parameter ρ as 0.2. e algorithm stops when the difference of the squared l 2 error of two consecutive iterations is less than δ � 1e − 10 . After obtaining x q , we use x q ′ � W − 1 x q to transform x q to x q ′ to obtain the reconstructed image. e computational complexity of the proposed algorithm is O ((N 3 )G), where N is the number of receiving antennas and G � mn denotes the product of two iteration numbers. Generally, G � 200 will lead to satisfactory quality of the reconstructed image.

Simulation Results
In this section, we examine the performance of the proposed method in comparison with that of the Lasso method and the methods proposed in [14] and [15]. We use the mean square error (MSE) and performance recovery coefficient (PRC) with the following definitions to evaluate the reconstruction quality: where x t represents the true image and x q ′ denotes the reconstructed image. e PRC measures the similarity of the true target coefficient and the estimated target coefficient. Generally, the larger the PRC, the smaller the MSE and the better the quality of the reconstructed image. In our simulations, both the transmitting array and receiving array are ULAs with 15 elements and half-wavelength spacing. ere are 64 pixels in each range. In each Input: z q , Φ, c, ρ Output: x q , Δν Initialization with Δν(0) � 0 M×N , w � 0 N×1 , L � 0 1×N , and err(0) � 0 for i � 1, 2, . . ., m do for k � 0, . . ., n do Update the iterate x k+1  range is a one-dimensional two-slit image. e number of Monte Carlo trails is 500 in all simulations. All the simulations are run using MATLAB 2014a in a computer with the following configuration: Intel(R) Core (TM) i7-7700CPU 3.6 GHz and 16 GB memory. e convergence of our algorithm is shown in Figure 2. It illustrates the MSE of the reconstruction results with iteration numbers i and k. Normally, the algorithm converges within i � 60 and k � 150. Figure 3 shows the computation time of each algorithm with different numbers of antennas. It can be seen that the proposed method takes less time than the iterative method in [15] but takes longer time than the constrained optimization method [14] and Lasso method. Figure 4 shows the results of the image recovery by using different methods. It can be seen in Figure 4 that the iterative method [15] and Lasso method work slightly better than the constrained optimization method [14] and the proposed method achieves much better performance than the iterative method [15] and Lasso method. Simulations show that the MSEs of the constrained optimization method [14], the iterative method [15], the Lasso method, and the proposed method are 14.93, 11.07, 12.40, and 4.78, respectively. Figure 5 shows the MSE and PRC of the four methods, where the error parameter ε changes from 0.2 to 0.8 with the interval 0.02. We can see that the MSE of the proposed method is much smaller than that of the other three methods International Journal of Antennas and Propagation 5 and the PRC is much higher than that of the other three methods when the array gain-phase error is greater than 0.35. However, the iterative method [15] and Lasso method are better than the proposed method with small errors. e proposed method achieves better sparse recovery results than others with one snapshot. As we can see from Figure 5, the proposed method is robust to array errors in the case of one snapshot.
It is indicated in Figure 6 that the performance of the proposed method is much better than that of the other three methods. As the SNR increases, better sparse recovery results can be obtained. e SNR changes from 10 dB to 30 dB with the interval 1 dB. Figure 7 displays the effect of the regularization parameter c and penalty parameter ρ on the MSE of the recovery results. e penalty parameter ρ is set as 0.2 in   Figure 7(b) and ρ changes from 0.1 to 0.5. It can be observed that the MSE decreases with the increase of c and ρ, but when c is too large, the algorithm will diverge.

Conclusions
In this work, we have investigated a robust sparse recoverybased transmit-receive angle imaging method for the bistatic MIMO radar. Signals are transformed into a sparse domain by discrete wavelet transform, and the reconstruction is achieved by a robust iterative algorithm. e proposed method is more robust to deal with the imperfection of array gain-phase errors and noise in the bistatic MIMO radar system. Simulation results have been provided to show the superiority of the proposed method, which can achieve significantly better PRC and MSE than existing methods.

Data Availability
e simulation data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest. International Journal of Antennas and Propagation 7