MIMO Radar Imaging Based on Smoothed l 0 Norm

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 [subscript] l 0 [/subscript] 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 [subscript] l 0 [/subscript] 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.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Jun-Jie Feng 1,2 and Gong Zhang 1 and Fang-Qing Wen 1

Academic Editor:Jian Li

1, Key Laboratory of Radar Imaging and Microwave Photonics, Ministry of Education, College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
2, Department of Physics and Information Technology, Liupanshui Normal University, Liupanshui, Guizhou 553004, China

Received 23 October 2014; Revised 22 December 2014; Accepted 23 December 2014; 3 February 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) norm. The sparse learning reconstruction algorithms based on [figure omitted; refer to PDF] norm are intractable because they require a combinatorial search, and they are sensitive to noise. The computational complexity of the reconstruction algorithms based on [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] norm; then the problem of minimum [figure omitted; refer to PDF] norm optimization can be transferred to an optimization problem for smoothed functions, called smoothed [figure omitted; refer to PDF] norm (SL0) [19]. The method based on smoothed [figure omitted; refer to PDF] norm is about two orders of magnitude faster than [figure omitted; refer to PDF] -magic method, while providing better estimation of the source than [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] -element transmit array and a [figure omitted; refer to PDF] -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.

Figure 1: Geometry of MIMO radar imaging.

[figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] express the positions of the [figure omitted; refer to PDF] th transmit array and the [figure omitted; refer to PDF] th receive array, respectively. [figure omitted; refer to PDF] expresses the transmit code of the [figure omitted; refer to PDF] th transmit antenna. The baseband transmit waveform can be modeled as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the subpulse duration and [figure omitted; refer to PDF] is the code length. At the transmit site, the signal transmitted by the [figure omitted; refer to PDF] th transmitter is modeled as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the carrier frequency of the signal. The backscattered signal from point [figure omitted; refer to PDF] observed at the [figure omitted; refer to PDF] th receiver is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the scatterer amplitude. Denote [figure omitted; refer to PDF] ; the received array signal can be expressed as [figure omitted; refer to PDF]

The above equation can be rewritten as a simple matrix form [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF]

Through time domain sampling, the received signal can be expressed as a matrix [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is the vectoring operator. Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; we have [figure omitted; refer to PDF] . The imaging region is divided as grids. Basis matrix [figure omitted; refer to PDF] and scatter amplitudes vector [figure omitted; refer to PDF] are formed for all grids. If a scatterer is not zero, the received discrete signal can be expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is noise. The aim of imaging is to solve for the vector [figure omitted; refer to PDF] 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: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] express the real and imaginary part of the complex vector, respectively. So (7) becomes [figure omitted; refer to PDF]

Because the strong scatterers are sparsely distributed for imaging, [figure omitted; refer to PDF] is sparse. To solve [figure omitted; refer to PDF] , we can use the following optimization sparse optimization strategy: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a small positive number associated with [figure omitted; refer to PDF] . [figure omitted; refer to PDF] indicates the [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] Norm

In order to obtain an approximate [figure omitted; refer to PDF] norm solution, a smoothed function [figure omitted; refer to PDF] was used to replace the [figure omitted; refer to PDF] norm in [19]. By letting [figure omitted; refer to PDF] varies from a larger value to zero, the minimum [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] norm. The function can be expressed as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is an auxiliary variable; it is obvious that [figure omitted; refer to PDF] Denote [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] expresses the number of nonzero elements of vector [figure omitted; refer to PDF] . According to the definition of [figure omitted; refer to PDF] norm, we can obtain [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] . From Figure 2, one can see that the approximate hyperbolic tangent function is steeper than gauss function, so the performance of approaching to minimum [figure omitted; refer to PDF] norm is more excellent.

Figure 2: The comparison of Gauss function and the approximate hyperbolic tangent function.

[figure omitted; refer to PDF]

Therefore sparse signal recovery algorithm based on [figure omitted; refer to PDF] function minimum can be described as [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] norm. In order to better approximate to the [figure omitted; refer to PDF] norm, we choose the revised Newton method to solve the problem.

First, we compute the Newton direction of approximate hyperbolic tangent function

[figure omitted; refer to PDF] where [figure omitted; refer to PDF] From the results above, we can see that the Hessen matrix ( [figure omitted; refer to PDF] ) in the Newton direction of [figure omitted; refer to PDF] is not the positive definite matrix, which does not ensure Newton direction of [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a properly chosen positive elements and [figure omitted; refer to PDF] is the identity matrix, which makes the diagonal elements of [figure omitted; refer to PDF] all positive.

To meet the requirement, [figure omitted; refer to PDF] is chosen as [figure omitted; refer to PDF] Thus the revised diagonal element of [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , which meets the requirement of positive definite matrix.

Finally, the Newton direction in this paper is simplified as [figure omitted; refer to PDF] According to the above expression, the algorithm structure of ASL0 can be described as follows:

(i) initialization:

(1) let the minimum [figure omitted; refer to PDF] norm solution of [figure omitted; refer to PDF] be the initial value, [figure omitted; refer to PDF] ;

(2) choose a proper decreasing sequence of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ;

(ii) for [figure omitted; refer to PDF] :

(1) let [figure omitted; refer to PDF] ;

(2) minimize the function [figure omitted; refer to PDF] on the feasible set [figure omitted; refer to PDF]

[...]: [figure omitted; refer to PDF]

[...]: for [figure omitted; refer to PDF] :

(a) compute the revised Newton direction [figure omitted; refer to PDF]

(b) use the revised Newton method and obtain [figure omitted; refer to PDF] ;

(c) project [figure omitted; refer to PDF] back into the feasible set [figure omitted; refer to PDF] : [figure omitted; refer to PDF] ;

(3) set [figure omitted; refer to PDF] ;

(iii): final answer is [figure omitted; refer to PDF] .

Now let us explain the choosing of some parameters. [figure omitted; refer to PDF] is the minimum [figure omitted; refer to PDF] norm solution. In [19] [figure omitted; refer to PDF] is chosen as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . In this paper, when we choose [figure omitted; refer to PDF] , [figure omitted; refer to PDF] should be chosen as [figure omitted; refer to PDF] . The choice of [figure omitted; refer to PDF] is also important. When [figure omitted; refer to PDF] , the norm [figure omitted; refer to PDF] approximates to [figure omitted; refer to PDF] norm. However, [figure omitted; refer to PDF] norm is not suited to express a vector with many small elements. Because, for radar imaging, there are many small scatterers, the choice of [figure omitted; refer to PDF] should not be too small. For [figure omitted; refer to PDF] , we choose [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] expresses expectation. Then [figure omitted; refer to PDF] can be estimated by selecting a few noise samples, computing [figure omitted; refer to PDF] , selecting the maximum value of [figure omitted; refer to PDF] , and taking the average value.

3.3. Robust Improvement

In practical application, [figure omitted; refer to PDF] can be in ill condition because its condition number is very large. In our imaging, the condition number is up to [figure omitted; refer to PDF] . It cannot be convergent if using [figure omitted; refer to PDF] directly. The main weighted method can solve the ill condition problem. We add a diagonal matrix [figure omitted; refer to PDF] to [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] can take the place of [figure omitted; refer to PDF] . If weighted factor [figure omitted; refer to PDF] is too small, the improved effect of ill condition matrix is not obvious. If choosing the larger [figure omitted; refer to PDF] , the convergence speed will slow down, even the solution will be distorted. There is no good way about how to choose the value of [figure omitted; refer to PDF] . A suggested way is to compute the largest eigenvalue [figure omitted; refer to PDF] of [figure omitted; refer to PDF] and then choose [figure omitted; refer to PDF] .

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 [figure omitted; refer to PDF] and the dimension of [figure omitted; refer to PDF] is [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is the matrix with Gauss distribution. [figure omitted; refer to PDF] is the sparse signal, whose nonzero coefficients are uniform [figure omitted; refer to PDF] random spikes signal. [figure omitted; refer to PDF] is Gaussian random vector with standard variance [figure omitted; refer to PDF] . The SNR is defined as [figure omitted; refer to PDF] . 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, [figure omitted; refer to PDF] . We choose the step size adjustment [figure omitted; refer to PDF] . The MSE is defined as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the true solution and [figure omitted; refer to PDF] is the estimation value. We assume that if the MSE is less than [figure omitted; refer to PDF] , 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.

Figure 3: Computation costs of different methods.

[figure omitted; refer to PDF]

When there is no noise, the reconstruction probability for different methods with different [figure omitted; refer to PDF] and different number of measurements are shown in Figures 4 and 5. The MSE and reconstruction probability for different methods with different [figure omitted; refer to PDF] and different SNR are shown in Figures 6 and 7. We can see that the performance of ANSL0 method is better than other methods.

Figure 4: Correct position estimation for different [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 5: Correct position estimation for different number of measurement.

[figure omitted; refer to PDF]

MSE for different [figure omitted; refer to PDF] and SNR: (a) SNR = 15 dB and (b) SNR = 20 dB.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Correct position estimation frequencies for different [figure omitted; refer to PDF] and SNR: (a) SNR = 15 dB and (b) SNR = 20 dB.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] GHz. We use the random BPSK codes as the transmitting signal and the length of the signal is 150. The bandwidth is [figure omitted; refer to PDF] MHz. The transmitter [figure omitted; refer to PDF] and the receiver [figure omitted; refer to PDF] . The distances between transmitters and receivers are [figure omitted; refer to PDF] m and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] . 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.

(a) Original image of the target, (b) reconstructed image of OMP, (c) reconstructed image of Bayesian method with Laplace prior, (d) reconstructed image of SL0, and (e) reconstructed image of ANSL0.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

(e) [figure omitted; refer to PDF]

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. [figure omitted; refer to PDF] 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.

Contour plot of reconstructed images using (a) OMP, (b) Bayesian method with Laplace prior, (c) SL0, and (d) ANSL0.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

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. [figure omitted; refer to PDF] is chosen as the factor. Assume [figure omitted; refer to PDF] and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] . 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.

(a) Original image of the targets, (b) reconstructed image of OMP, (c) reconstructed image of Bayesian method with Laplace prior, (d) reconstructed image of SL0, and (e) reconstructed image of ANSL0.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

(e) [figure omitted; refer to PDF]

Figure 11: RMSE of images versus SNR.

[figure omitted; refer to PDF]

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.

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).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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