International Journal of Antennas and Propagation

Volume 2016 (2016), Article ID 8523143, 16 pages

http://dx.doi.org/10.1155/2016/8523143

## Radar Coincidence Imaging for Off-Grid Target Using Frequency-Hopping Waveforms

School of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China

Received 8 November 2015; Accepted 14 March 2016

Academic Editor: Wen-Qin Wang

Copyright © 2016 Xiaoli Zhou 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

Radar coincidence imaging (RCI) is a high-resolution staring imaging technique without the limitation of the target relative motion. To achieve better imaging performance, sparse reconstruction is commonly used. While its performance is based on the assumption that the scatterers are located at the prediscretized grid-cell centers, otherwise, off-grid emerges and the performance of RCI degrades significantly. In this paper, RCI using frequency-hopping (FH) waveforms is considered. The off-grid effects are analyzed, and the corresponding constrained Cramér-Rao bound (CCRB) is derived based on the mean square error (MSE) of the “oracle” estimator. For off-grid RCI, the process is composed of two stages: grid matching and off-grid error (OGE) calibration, where two-dimension (2D) band-excluded locally optimized orthogonal matching pursuit (BLOOMP) and alternating iteration minimization (AIM) algorithms are proposed, respectively. Unlike traditional sparse recovery methods, BLOOMP realizes the recovery in the refinement grids by overwhelming the shortages of coherent dictionary and is robust to noise and OGE. AIM calibration algorithm adaptively adjusts the OGE and, meanwhile, seeks the optimal target reconstruction result.

#### 1. Introduction

Radar coincidence imaging (RCI), originated from the optical coincidence imaging, is a staring imaging technique which can obtain focused high-resolution image without the limitation of the target relative motion [1–3]. RCI can operate under the nonideal observing geometry of forward-looking/staring, with significant potentials for resolution enhancement, interference, and jamming suppression. The essential principle of RCI is to produce time-space independent signals in the imaging area, while the frequency-hopping (FH) waveforms are good candidates because they are easily generated and have constant modulus [4, 5]. Besides, comparing with the linear frequency modulated (LFM) waveforms which are often used in the traditional radar systems, FH waveforms can suppress the range ambiguity, decouple the range and Doppler, and are also attractive for their merits on electronic counter-countermeasures (ECCM) and reducing interference between adjacent radar systems for sharing the frequency spectrum [6]. Hence, we focus on RCI using FH waveforms (FH-RCI) in this paper.

In RCI, the continuous target space needs to be discretized to a fine grid and the target-scattering centers are assumed to be exactly located at these prediscretized grid-cell centers [3]. Then, the detecting signals at different grid-cell centers can be formed as the atoms of sparse representation dictionary. Meanwhile, the scatterers of target are often distributed sparsely in most radar imaging applications. Thus, sparse recovery approaches and compressive sensing (CS) [7, 8] are suitable for RCI by exploiting the sparsity of target in the target space. In the sparse reconstruction theory, signal reconstruction depends on presetting an appropriate sparsifying dictionary which is supported on the assumed grids and defines the signal sparsity. However, as the scatterers are distributed in a continuous scene, the scattering centers are generally located off the grid-cell centers, no matter how fine the grid is; therefore, off-grid yields [9]. Then, the performance of RCI would be severely affected.

Off-grid would lead to the mismatch between the assumed and actual sparsifying dictionaries directly, which causes the performance of conventional sparse recovery methods to degrade considerably [10–13]. Intuitively speaking, the sparse elements in the signal may not lie on the assumed grids and not perfectly match the dictionary; thus, the true signal is not exactly supported on the assumed dictionary. Moreover, the signal recovery is robust to the mismatch in the sense that the recovery error grows with the mismatch level and is independent of the sparsity of the original signal. Thus, the sparse reconstruction performance of radar imaging degrades severely [3, 9, 14–17].

Considering the off-grid, several algorithms have been proposed. One simple approach is to use multiresolution refinement strategy and decrease the grid size iteratively [18]. Nevertheless, a finer grid may enhance the coherence between the columns of dictionary and increase the computational complexity and numerical instability of reconstruction [17]. Modeling the off-grid as a multiplicative perturbation, the sparse total least squares (S-TLS) [19] and joint correlation-parameterization (CP) [3] algorithms are proposed. However, the algorithms are inefficient without considering possibly available a priori information and the performance degrades rapidly when the off-grid is significant. To explore the structure of dictionary mismatch, the support-constrained orthogonal matching pursuit (SCOMP) [16] and joint sparse signal recovery methods [20] are proposed based on the first-order Taylor expansion to utilize the support constraint, while the methods eventually break down when the gridding error dominates the data. Lately, from the sparse Bayesian learning (SBL) perspective, several approaches are proposed, such as off-grid sparse Bayesian inference (OGSBI) [21], sparse adaptive calibration recovery via iterative maximum a posteriori (SACR-iMAP) [9], and variational expectation-maximization [10, 22] algorithms, to achieve joint sparse recovery. In the SBL framework, the sparsity is exploited in the signal of interest. The merit of SBL is its flexibility in modeling sparse signals that can not only promote the sparsity of its solution but also exploit the possible structure of the signal to be recovered [21], whereas it offers few guarantees on the signal recovery accuracy.

Another way to sidestep the off-grid is to work directly on the continuous parameter space. An atomic norm minimization approach [23], which yields an infinite dictionary of continuous atoms and arbitrarily high coherence, is proposed to exactly identify the unknown parameters directly. In [24], the continuous basis pursuit (CBP), which uses a dictionary with an auxiliary interpolation function to overcome the off-grid, is proposed to overcome the limitation of BP. However, any noise will make the exact results unidentifiable, and the computational burden and numerical instability are significant.

In conclusion, there are variable methods to solve the off-grid. However, most algorithms provide no performance guarantees on signal recovery and the performance deteriorates significantly when the off-grid error (OGE) increases. Furthermore, most approaches cannot be applied to off-grid RCI directly, as they are proposed for DOA or spectral estimation applications without considering the specific problems existing in RCI, such as the coupling among the parameters in 2D/3D imaging. Moreover, the computational complexity increases significantly in 2D/3D case.

Thus, we investigate the off-grid FH-RCI in this paper and present an off-grid imaging approach. The main contributions of this paper are as follows.

(a) The off-grid effect is investigated. The off-grid FH-RCI model is derived and the methodology to analyze the off-grid is established. The relative imaging error (RIE), gridding error, and signal-error-ratio (SER) are introduced to model the OGE. The gridding error is seriously sensitive to the OGE and induces the imaging quality to degrade drastically. Furthermore, the corresponding constrained Cramér-Rao bound (CCRB) is also derived to analyze the off-grid effect.

(b) A novel sparse recovery approach for off-grid RCI is presented. Off-grid RCI is a nonconvex optimization problem and can be solved by two stages. The first is grid matching; that is, the scatterers are captured by the closest grid-cells. The band-excluded locally optimized orthogonal matching pursuit (BLOOMP) approach [25] is introduced and extended to 2D version which is operated on range-azimuth space for RCI. OGE calibration, namely, estimate the OGE between the actual scatterer location and its closest grid-cell center. The sparse alternating iteration minimization (AIM) approach is used. Numerical experiments show that the proposed method realizes the target reconstruction robustly and achieves both high-resolution and outstanding imaging quality and is also simple to implement.

The rest of the paper is organized as follows. Section 2 presents the off-grid FH-RCI model in the range-azimuth space. Section 3 investigate**s** the off-grid effects by both numerical simulations and theoretical derivations. Then, the image reconstruction method is proposed in Section 4. In Section 5, some numerical examples are given to verify the performance of the presented method. Finally, some comments and conclusions are shown in Section 6.

A comment on notation: we use boldface lowercase letters for vectors and boldface uppercase letters for matrices. , , and denote the transpose, inverse, and pseudoinverse of a matrix, respectively. , , and are the diagonalization, Hadamard product, and vectorization operation, separately. Finally, denotes the Euclidean norm of a vector.

#### 2. Problem Formulation

##### 2.1. Signal Model

RCI can be realized by a multiple-input multiple-output (MIMO) radar system to transmit time-independent and group-orthogonal signals [1]. Thus, the spatial variety of wavefront increases, and the scatterers within a beam then reflect different detecting signals according to their respective locations. So the superresolution within the beam emerges. Compared with conventional MIMO radar which focuses on multiple paths or multiple observation angles, RCI needs the interference of transmitted waveforms to make the wavefront show spatial fluctuation and increase the spatial variety of detecting signals. Besides, the components of each path are separated utilizing the waveform orthogonality in conventional MIMO radar, while the components are not separated in the whole RCI procedure.

In this paper, a RCI system with transmitters and receivers is considered; each transmitter emits an independent FH waveform. Assume that both transmitters and receivers are configured as a uniform linear array (ULA) and the interelement spacings of the transmit and receive antennas is and , respectively. The th transmitter emits constant modulus FH waveform . Assuming pulses comprise a waveform, the signal from the th transmitter is [4]where andwhere , , and denote the pulse repetition interval (PRI), FH duration, and FH interval, respectively. is the FH code which specifies the transmitted frequency during each hopping interval, where is a positive integer. is the length of the code. Thus, the PRI is .

To ensure the orthogonality of waveforms, for each hopping interval, the codes are assumed to be constrained to satisfy [5] can be arranged into an dimensional code matrix specifying the transmitted frequencies.

Without loss of generality, the target is assumed to consist of several ideal point scattering centers for an enough high carrier frequency, which is widely used in the imaging radar system. Furthermore, there are scatterers which are widely separated to provide some performance guarantee.

The target scene is considered to be a 2D range-azimuth space as illustrated in Figure 1. RCI discretizes the continuous target scene and generates a number of grid points, and the scatterers are assumed to be located at the grid-cell centers. Thus, the scene is discretized with azimuth bins, range bins, and associated bin sizes and . Hence, the number of grid-cells is . The th scatterer is located at the grid-cell center , where and are reference values in the respective domains and the pair represents the th grid-cell in the discretized scene. As the scatterers possess nonzero scattering coefficient which is proportional to the radar cross section (RCS), the associated means that there is no scatterer at the th grid-cell center. Denote by the index set of scatterer location.