Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 841986, 10 pages

http://dx.doi.org/10.1155/2015/841986

## MIMO Radar Imaging Based on Smoothed Norm

^{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

Academic Editor: Jian Li

Copyright © 2015 Jun-Jie Feng et al. 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.

#### 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 (SL_{0}), 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.