Abstract
For radar imaging, a target usually has only a few strong scatterers which are sparsely distributed. In this paper, we propose a compressive sensing MIMO radar imaging algorithm based on smoothed norm. An approximate hyperbolic tangent function is proposed as the smoothed function to measure the sparsity. A revised Newton method is used to solve the optimization problem by deriving the new revised Newton directions for the sequence of approximate hyperbolic tangent functions. In order to improve robustness of the imaging algorithm, main value weighted method is proposed. Simulation results show that the proposed algorithm is superior to Orthogonal Matching Pursuit (OMP), smoothed method (SL0), and Bayesian method with Laplace prior in performance of sparse signal reconstruction. Two-dimensional image quality of MIMO radar using the new method has great improvement comparing with aforementioned reconstruction algorithm.
1. Introduction
Multiple Input Multiple Output (MIMO) radar has been widely concerned in recent years. Unlike the conventional radar system which transmits correlated signals, a MIMO radar system transmits multiple independent signals and receives the scattered signals via its antennas [1–4]. A MIMO radar system has many advantages in both distributed MIMO radar scenario and collocated MIMO radar scenario. The distributed MIMO radar takes advantage of diversity of the receive antennas to improve target recognition [5–8]. For collocated MIMO radar, the element spacing of transmit antennas and receive antennas are sufficiently small so that the radar returns from a given scatterer are fully correlated across the array, which can improve the spatial resolution [9–11]. We adopt the latter scenario in this paper. The Bipolar Phase Shift Keying (BPSK) signals and step frequency signals are usually used as the transmit signal. However, the defect of these signals is high sidelobes and it is difficult to eliminate sidelobes and improve imaging quality by conventional methods such as windowing methods. Combining MIMO radar with ISAR technology is discussed in [12], but it cannot realize one snapshot imaging. A single snapshot imaging method is proposed in [13]; however, it needs too many antennas.
The recent developed new field, known as sparse learning, is a technique proposed recently to recovery sparse signal by optimization theory [14–16]. Sparsity can usually be measured by () norm. The sparse learning reconstruction algorithms based on norm are intractable because they require a combinatorial search, and they are sensitive to noise. The computational complexity of the reconstruction algorithms based on norm is high enough, which makes them impractical for some practical applications. Hence many simpler algorithms, such as orthogonal matching pursuit (OMP) [17, 18], were proposed, but they are iteratively greedy algorithms and do not give good estimation of the sources. Mohimani et al. proposed a smoothed function to approximate norm; then the problem of minimum norm optimization can be transferred to an optimization problem for smoothed functions, called smoothed norm (SL0) [19]. The method based on smoothed norm is about two orders of magnitude faster than -magic method, while providing better estimation of the source than -magic.
The targets in sky are usually sparse and can be viewed as ideal point targets for DOA estimation. The signal model in this case fits the requirement of sparse learning. Angle-Doppler estimation of targets using sparse learning of MIMO radar was studied in [20], where narrow band signal was used. A Sparse Learning via Iterative Minimization (SLIM) algorithm is proposed and the application on range-angle-Doppler estimation of MIMO radar is discussed in [21]. For radar imaging, a target usually has only a few strong scatterers which are sparsely distributed. Then sparse learning reconstruction methods can be used in radar imaging. A high resolution imaging method of ground-based radar with sparse learning is proposed in [22]. We will discuss the performance of MIMO radar imaging based on sparse signal recovery algorithm. In order to solve the optimization problem effectively, we utilize a revised Newton method to derive the new revised Newton directions for the approximate hyperbolic tangent function. Because the condition number of the matrix is very large in MIMO radar imaging, the matrix can be ill-conditioned and the algorithm will lose its robustness. We use main value weighted method to improve the robustness of this algorithm.
This paper is organized as follows. Section 2 introduces the MIMO radar signal model. Section 3 introduces the proposed reconstruction algorithm. Simulation results are presented in Section 4. Finally, Section 5 provides the conclusion.
2. MIMO Radar Signal Model
In this section, we describe a signal model for the MIMO radar. Considering a monostatic MIMO radar imaging system with only one snapshot signal, it has a -element transmit array and a -element receive array, both of which are closely spaced uniform linear arrays (ULA). We assume that the targets appear in far field. Therefore, the directions of a target relative to the transmit antennas and the received antennas are the same, and the RCS of a target corresponding to different antennas are also the same. The MIMO radar geometry is shown in Figure 1.
Let and express the positions of the th transmit array and the th receive array, respectively. expresses the transmit code of the th transmit antenna. The baseband transmit waveform can be modeled as where where is the subpulse duration and is the code length. At the transmit site, the signal transmitted by the th transmitter is modeled as , where is the carrier frequency of the signal. The backscattered signal from point observed at the th receiver is given by where , , is the scatterer amplitude. Denote ; the received array signal can be expressed as
The above equation can be rewritten as a simple matrix form where , , , and
Through time domain sampling, the received signal can be expressed as a matrix . is the vectoring operator. Let , ; we have . The imaging region is divided as grids. Basis matrix and scatter amplitudes vector are formed for all grids. If a scatterer is not zero, the received discrete signal can be expressed as where is noise. The aim of imaging is to solve for the vector in (7).
For sparse learning, the optimization algorithms have been developed for real valued signals. We divide the signal into its real and imaginary parts as follows: where and express the real and imaginary part of the complex vector, respectively. So (7) becomes
Because the strong scatterers are sparsely distributed for imaging, is sparse. To solve , we can use the following optimization sparse optimization strategy: where is a small positive number associated with . indicates the norm. The imaging quality depends largely on reconstruction algorithms and an improved SL0 algorithm is proposed and applied to MIMO radar imaging. We give a detailed description about the algorithm in the next section.
3. Sparse Signal Recovery Imaging Algorithm
3.1. Using Hyperbolic Tangent Function Approximate to Norm
In order to obtain an approximate norm solution, a smoothed function was used to replace the norm in [19]. By letting varies from a larger value to zero, the minimum norm solution is obtained. In order to further improve the approximation performance of the smoothed function, we propose a function which is approximate hyperbolic tangent function to approach the norm. The function can be expressed as follows: where is an auxiliary variable; it is obvious that Denote ; expresses the number of nonzero elements of vector . According to the definition of norm, we can obtain . In order to further show the advantages of the approximate hyperbolic tangent function, we compare the performance of approximate hyperbolic tangent function and gauss function at . From Figure 2, one can see that the approximate hyperbolic tangent function is steeper than gauss function, so the performance of approaching to minimum norm is more excellent.
Therefore sparse signal recovery algorithm based on function minimum can be described as
3.2. Reconstruction Algorithm Based on Revised Newton Method (ASL0 Algorithm)
There are many methods to solve (13). The most representative one is the steepest descent method in [19]. However, the search path will be a “sawtooth” shape in the process of searching the optimal value, which will affect the estimation accuracy of minimum norm. In order to better approximate to the norm, we choose the revised Newton method to solve the problem.
First, we compute the Newton direction of approximate hyperbolic tangent function
where From the results above, we can see that the Hessen matrix () in the Newton direction of is not the positive definite matrix, which does not ensure Newton direction of is a descent direction. So in the second step, we will revise the Hessen matrix above and obtain a revised Newton direction. We construct a new matrix , where is a properly chosen positive elements and is the identity matrix, which makes the diagonal elements of all positive.
To meet the requirement, is chosen as Thus the revised diagonal element of is , which meets the requirement of positive definite matrix.
Finally, the Newton direction in this paper is simplified as According to the above expression, the algorithm structure of ASL0 can be described as follows:(i)initialization:(1)let the minimum norm solution of be the initial value, ;(2)choose a proper decreasing sequence of , ;(ii)for :(1)let ;(2)minimize the function on the feasible set ①②for :(a)compute the revised Newton direction (b)use the revised Newton method and obtain ;(c)project back into the feasible set : ;(3)set ;(iii)final answer is .
Now let us explain the choosing of some parameters. is the minimum norm solution. In [19] is chosen as , . In this paper, when we choose , should be chosen as . The choice of is also important. When , the norm approximates to norm. However, norm is not suited to express a vector with many small elements. Because, for radar imaging, there are many small scatterers, the choice of should not be too small. For , we choose , where expresses expectation. Then can be estimated by selecting a few noise samples, computing , selecting the maximum value of , and taking the average value.
3.3. Robust Improvement
In practical application, can be in ill condition because its condition number is very large. In our imaging, the condition number is up to . It cannot be convergent if using directly. The main weighted method can solve the ill condition problem. We add a diagonal matrix to ; then can take the place of . If weighted factor is too small, the improved effect of ill condition matrix is not obvious. If choosing the larger , the convergence speed will slow down, even the solution will be distorted. There is no good way about how to choose the value of . A suggested way is to compute the largest eigenvalue of and then choose .
4. Simulation Results
4.1. Simulation 1: One-Dimensional Synthetic Signals
In simulation 1, we compare the proposed sparse signal recovery algorithm with OMP, SL0 method, and Bayesian method with Laplace prior. The signal model with noise is and the dimension of is . is the matrix with Gauss distribution. is the sparse signal, whose nonzero coefficients are uniform random spikes signal. is Gaussian random vector with standard variance . The SNR is defined as . We consider SNR = 15, 20 dB conditions. For ANSL0 method and SL0 method, the numbers of outer loop and inner loop are 25 and 20, respectively, . We choose the step size adjustment . The MSE is defined as , where is the true solution and is the estimation value. We assume that if the MSE is less than , the positions are estimated correctly. The experiment was then repeated 200 times; the values of MSE and correct position estimation frequencies are averaged. Figure 3 shows the average computational cost for 15 dB SNR case. For other SNR cases, the computational costs are similar to this case and are not shown. From this figure, we can see that OMP and ANSL0 method in this paper are more efficient than smoothed l0 norm based method and Bayesian method.
When there is no noise, the reconstruction probability for different methods with different and different number of measurements are shown in Figures 4 and 5. The MSE and reconstruction probability for different methods with different and different SNR are shown in Figures 6 and 7. We can see that the performance of ANSL0 method is better than other methods.
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4.2. Simulation 2: Imaging of Scatterers Located on the Grid Points
In simulation 2 and simulation 3, we consider imaging of a target in two-dimension. The center frequency GHz. We use the random BPSK codes as the transmitting signal and the length of the signal is 150. The bandwidth is MHz. The transmitter and the receiver . The distances between transmitters and receivers are m and m, respectively. The transmitting and receiving array are located on a line. The distance of the target to the radar is 200 km. We assume that the scatterers are ideal isolated scatterers and located on the grid points. The target contains 19 discrete scatterers. The amplitudes and positions of the original target are shown in Figure 8(a). The SNR of the smallest scatterer is 15 dB. We choose the factor . The numbers of outer loop and inner loop are 25 and 20, respectively, for SL0 and ANSL0. Images using OMP, SL0, Bayesian, and ANSL0 are shown in Figures 8(b), 8(c), 8(d), and 8(e). We can see that the performance of OMP is poor. ANSL0 and Bayesian are more superior than SL0.
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4.3. Simulation 3: Imaging of Scatterers Not Located on the Grid Points
In simulation 3, some scatterers of the target are not on the grid points; then the scenario is a more realistic situation. The parameters of MIMO radar are the same as simulation 2. The target consists of 9 scatterers. The distance from the target to the radar is 200 km. The relative positions of the scatterers are (1, 1), (0.5, 5), (1.5, 7), (5.5, 4.5), (5, 6.5), (5.5, 3), (9, 1.25), (9.25, 5), and (9.25, 8.25) (m, m). The sampling distance of grid points in two dimensions is all 1 m. is chosen as the factor. The SNR is 15 dB. The contour plots of the reconstructed images using different methods are shown in Figure 9. One can see that ANSL0 has better reconstruction performance than other methods.
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4.4. Simulation 4: Robustness of the Proposed Method
In the simulation 4, the parameters of MIMO radar are the same as simulation 2.
There are three target scatterers in the imaging region. The distance from the target to the radar is 150 km. Three target scatterers locate at (14, 11), (13, 14.2), (16, 11.2) (m, m).
The targets locate at close range. The complex amplitudes are 0.2, 0.5, and 0.9, respectively. is chosen as the factor. Assume and are the estimation value and real value of MIMO radar in the image region, so the root mean square error (RMSE) can be expressed as . The experiment was repeated 200 times. The mesh plots of the reconstructed images using different methods when the SNR is 15 dB are shown in Figure 10. The RMSE of images versus SNR are shown in Figure 11. It can be seen from Figure 11 that the ANSL0 method has low RMSE for the reconstructed images.
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5. Conclusion
Sparse learning reconstruction algorithm can improve the image quality of a target including scatterers discrete distribution. We propose one new approximate hyperbolic tangent function and use revised Newton method to reconstruct MIMO radar imaging algorithm. In practical imaging, the coefficient matrix may be ill conditioned. By using the main value weighted method in this paper, the robustness of this algorithm is strengthened consumedly compared with the traditional algorithm described in [19]. Simulation results show that the ANSL0 method can improve the image quality of a target.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Chinese National Natural Science Foundation under Contracts nos. 61471191, 61201367, and 61271327 and Funding of Jiangsu Innovation Program for Graduate Education and the Fundamental Research Funds for the Central Universities (CXZZ12_0155) and partly funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PADA).