A Compressive Sensing-Based Bistatic MIMO Radar Imaging Method in the Presence of Array Errors

A robust transmit-receive angle imagingmethod for bistatic MIMO radar based on compressed sensing is proposed. A new imaging model with array gain and phase error is established. The array gain error and phase error were modeled as a random interference for observation matrix by mathematical derivation. A constraint of observationmatrix error is constructed in optimization problem of sparse recovery to reduce the effect of the interference of observation matrix. Then, the iterative algorithm of the optimization problems is derived. The proposed recovery method is more robust than the existed method in small samples, especially in the case of one snapshot. It is applicable in the case of relatively small array gain and phase errors. Simulation results confirm the effectiveness of the proposed method.


Introduction
Bistatic MIMO radar has the potential advantages of both bistatic radar and MIMO radar.Particularly, bistatic MIMO radar has the capability of obtaining the transmit angle information by processing the received data.As a new radar system, bistatic MIMO radar has been applied to target localization [1][2][3][4], clutter cancellation [5,6], and imaging [7,8].Compressed sensing (CS) is attracted technique used in the field of radar imaging.Compressive sensing theory can achieve high-resolution imaging in small samples.Introduction of compressive sensing to bistatic MIMO radar can achieve more accurate imaging results.However, when the gain and phase errors exist in the array elements, the detection accuracy and imaging quality of the radar will deteriorate seriously.Many literatures have studied robust compression sensing recovery methods [7,[9][10][11][12].A sparse recovery-based transmit-receive angle imaging scheme is proposed for bistatic multiple-input multiple-output (MIMO) radar in [8].The method is robust under large error conditions.However, the method does not take into account the array phase error.Furthermore, it is not robust in the case of one snapshot.Therefore, it is necessary to further study the robust recovery method of bistatic MIMO radar in the nonideal systems.Based on the above work, a new constraint is added to the sparse recovery method, and the corresponding algorithm is deduced.The simulation results show that in the case of one snapshot, the method proposed in [8] cannot accurately reconstruct the sparse signal, whereas the proposed method can still recover the sparse signal robustly.
This paper is organized as follows.The sparse signal model for bistatic MIMO radar with array errors is derived in Section 2. In Section 3, a robust CS algorithm is proposed to achieve target imaging in the presence of array errors.The proposed method is tested via simulations and analysis, which appear in Section 4. Finally, Section 5 concludes the paper.

Signal Model of Bistatic MIMO Radar with Array Errors
The configuration of a bistatic MIMO radar is shown in Figure 1.Consider the radar has M-transmit and N-receive uniform linear array.The transmitting signals are where L is the length of transmitting signals.Assuming that the p th pixel point of the target is at the angle θ tp , θ rp .The received signal can be written as where A t = a tp M×P and a tp = 1 e j 2π/λ d t sin θ tp e j 2π/λ 2d t sin θ tp … e j 2π/λ M−1 d t sin θ tp .A r = a rp N×P , where a rp = 1 e j 2π/λ d r sin θ rp e j 2π/λ 2d r sin θ rp … e j 2π/λ N−1 d r sin θ rp .A t is transmit steering matrix and A r is receive steering matrix of P pixel points.λ represents the transmit signal wavelength.d t and d r are the ideal element space at the transmitter and receiver.D q = diag d 1 , … , d p denotes the scattering coefficient of P pixel points of the target in the qth pulse period.
The noise E q is assumed to be independent, and zero-mean complex Gaussian distribution withE q ~Nc 0, σ 2 n I N .As shown in Figure 1, the region of interest is divided into two dimensional discrete set of angular positions Ω = θ k , θ l : k, l ∈ 1, … , G × 1, … , G .The target pixel distribution is assumed to be X q ∈ ℂ G×G and the transmit signal is orthogonal waveform.After the matched filtering, the received signal can be expressed as Y q = Α r X q Α t + E q , the vector form of it is where A = Α r ⊗ Α t , x q = vec X q , and e q = vec E q .When the gain and phase errors exist for both the transmit and receive array elements, the transmit and receive steering matrix can be written as where Γ t = diag ρ t1 , … , ρ tM and Γ r = diag ρ r1 , … , ρ rN are the diagonal matrices with array gain and phase errors at diagonal elements.ρ ti = a ti e jφ ti , where a ti is gain error and φ ti is phase error in the transmit array elements.ρ ri = a ri e jφ ri , where a ri is gain error and φ ri is phase error in the receive array elements.The receiving data with the gain and phase errors of the array can be written as Y ~q = Α ~rX q Α ~t + E q .
The vector form of it is where It can be observed from ( 4) and ( 5) that the array error can be modeled as an additive error matrix for ideal observation matrix A. The next task is to construct the sparse recovery problem under this nonideal model.

Robust CS-Based Sparse Imaging Algorithm
Considering more general situation, the E matrix in (4) can be relaxed as a random perturbation matrix which added on the ideal observation matrix A. Inspired by the robust beamforming method [12], the norm of the observation matrix is constrained to reduce the effect of matrix E. A quadratic optimization problem with sparse constraint can be constructed as In this optimization problem, matrix Β is an unknown matrix of the same size as the observation matrix A. We use it to approximate the actual observation matrix.Since the actual observation matrix is derived from the sum of the ideal observation matrix A and an unknown perturbation matrix E, we obtain more accurate values of the actual observation matrix by the above optimization methods.
The iterative method is derived to solve the optimization problem.In the k th iteration, the algorithm performs two steps.In the first step, we solve the familiar convex optimization problem holding B matrix fixed.This is a convex optimization problem that is known to yield unique sparse solutions.The next step is to hold the coefficients x fixed and find a better observation matrix solution by using the Lagrange multiplier method.The updating formula of the observation matrix is The above two steps are executed iteratively until an ideal solution is obtained.In the end, we introduce the derivation of the updating formula for observation matrix.We construct the Lagrange function according to the optimization problem By getting the partial derivative of the Lagrange function to B and making it equal to zero, the optimal estimation of the actual observation matrix can be obtained.
The first item is The second item is The third item is Then, we get the estimation of the actual observation matrix.
The algorithm is summarized as Algorithm 1.
For the error measurement parameter ε, we take the value according to the empirical data.Usually we set ε as one tenth of the error value of the first iteration.That is to say, our algorithm stops when the estimation error of sparse signal to one tenth of the initial estimation error.This rule leads to relative good results in practice.More appropriate selection methods remain to be further studied.
Furthermore, the method we propose is a more generally applied method.As long as the signal model can be expressed Step 4: Judge the condition of convergence x k+1 − x k 2 2 ≤ ε If the condition is satisfied, the iteration is stopped, and Step 2 is returned if it is not satisfied.
Algorithm 1: Algorithm Flow.as a form of random interference added to the observation matrix, the proposed method can be used for sparse reconstruction.So algorithm is still effective in the case of many nonideal situations, such as array location uncertainties, and CS grid error.

Simulation Results
In this section, we simulate the method proposed in this article and compare it with the method proposed in [8].It is assumed that the transmit array element and the receive array element are uniform linear arrays with halfwavelength space between adjacent elements, and the number of transmit and receive array elements is 15.Both of the transmit angular region and receive angular region range from 1 °to 10 °.Assume that there are two pixels of a target at angle θ t1 , θ r1 = 2 °, 9 °and θ t2 , θ r2 = 9 °, 2 °.We use only one snapshot for the sparse recovery.
The form of the transmit and receive array gain and phase error is where σ t and σ r are the parameters governing the array errors.N 0, σ 2 r denotes the Gaussian distribution.The single snapshot data is used in the simulations.
Figure 2 shows the MSE of the recovery results varying with iteration time.In the statistical sense, the algorithm converges at about 20 iteration times.The convergence rate of the algorithm is related to the size of the observation matrix and the selection of Lagrange multipliers.
Figure 3 shows the results of the image recovery using the proposed method compared with existing sparse methods.It   International Journal of Antennas and Propagation can be observed that the image recovery of the direct CS method and the algorithm proposed in [8] have relatively higher sidelobes.It is obvious that the performance of the proposed method is prior to both of them.The running time of the examples in Figure 3 by MATLAB in the same computer is listed in Table 1.It can be observed that the computational complexity of the proposed method increases while it improves the performance of the sparse recovery.
Figure 4 plots the mean square error of recovery results of the proposed method, the method in [8], and the direct CS method.The governing parameters of the array errors are equal for all the elements.The array error parameter changes from 0 to 0.6 with the interval 0.01.For each simulation, 500 Monte Carlo trails are run.It is shown that the performance of the method in [8] is very poor, even worse than the direct CS method, in the case of one snapshot.The proposed method has the lowest MSE among three methods.The results confirm that the proposed method is robust for array errors and suitable to one snapshot case.It can be observed from Figure 4 that the performance of the proposed method is degraded when the array gain and phase errors are greater than 0.4.The result implies that the proposed method is applicable in the case of relatively small array gain and phase errors.Array gain and phase error MSE of the recovery results Method proposed in [12] Proposed method Direct CS method shows the effect of γ on the convergence of the algorithm.The parameter μ is set as 0.001 and γ increases from 0.0001 to 0.01 with the interval 0.00005.For each γ, 100 simulations are performed to get the average number of iterations required for convergence of the algorithm.The same simulations are done to evaluate the effect of μ on the convergence of the algorithm in Figure 5 (b).It is shown that the iteration number will be small with the increase of γ and μ.However, too large γ and μ will result to the divergence of the algorithm.

Conclusions
A robust transmit-receive angle imaging method for bistatic MIMO radar based on CS had been proposed in this paper.The method can be used to enhance the performance of CS in the case of both gain and phase errors of the array.In addition, only one snapshot is required in the proposed method.Simulation and analysis demonstrated the effectiveness of the proposed method.

2 2 + x 1 Step 3 :
Input: y, A, λ, μ Output: x, B Step 1: Initialization: x 0 = 0 n×1 , B 0 = A Step 2: Fix B and solve the optimization problem x k = arg min x y ~q − B k x Fix x and calculate the matrix B B k+1 = y x k H + γA x k x k H + γ + μ I −1

Figure 4 :
Figure 4: Comparison of the performance of sparse recovery methods.

Table 1 :
The running time of each method.