Abstract

In this paper, a sparse recovery algorithm based on a double-pulse FDA-MIMO radar is proposed to jointly extract the angle and range estimates of targets. Firstly, the angle estimates of targets are calculated by transmitting a pulse with a zero frequency increment and employing the improved -SVD method. Subsequently, the range estimates of targets are achieved by utilizing a pulse with a nonzero frequency increment. Specifically, after obtaining the angle estimates of targets, we perform dimensionality reduction processing on the overcomplete dictionary to achieve the automatically paired range and angle in range estimation. Grid partition will bring a heavy computational burden. Therefore, we adopt an iterative grid refinement method to alleviate the above limitation on parameter estimation and propose a new iteration criterion to improve the error between real parameters and their estimates to get a trade-off between the high-precision grid and the atomic correlation. Finally, the proposed algorithm is evaluated by providing the results of the Cramér-Rao lower bound (CRLB) and numerical root mean square error (RMSE).

1. Introduction

Target localization has been acting as a pivotal part in the field of array signal processing, which expects various applications in radar, navigation, and communication [1–4]. In recent years, multiple-input multiple-output (MIMO) radar [5, 6] has attracted widespread consideration in target localization due to many potential merits [7], where multiple antennas are utilized to transmit different waveforms at the same time and simultaneously receive reflected signals. Compared with the phased array radar, MIMO radar can obtain enhanced spatial resolution, improved estimation performance, and increased degrees of freedom (DOF) [8–10] by effectively utilizing space diversity. However, MIMO radar cannot obtain the essential range estimates of targets. FDA-MIMO [11–14] radar, as a combination of frequency diverse array (FDA) radar [15–17] and MIMO radar, has a small frequency increment in adjacent transmitting array antennas to achieve the joint estimation of the angle and range [18].

Nowadays, the traditional DOA estimation algorithms have been applied for the joint angle-range estimation of FDA-MIMO radar, such as the estimation of signal parameters via rotational invariance techniques (ESPRIT) [19], unitary ESPRIT (U-ESPRIT) [20], and two-dimensional multiple signal classification (2D-MUSIC) [21]. However, the aforementioned algorithms based on subspace decomposition usually require a large number of snapshots and encounter performance degradation in the case of highly correlated targets.

The compressed sensing technique has attracted extensive attention in sparse signal reconstruction to deal with the above limitation of the subspace-based algorithms. The sparse signal recovery (SSR) algorithms [22–24] mainly estimate target parameters by constructing a sparse signal model and reconstructing the spatial spectrum. In an environment with a low signal-to-noise ratio (SNR) or a small number of snapshots, the SSR algorithm outperforms the subspace-based method in parameter estimation performance [25, 26]. In the past several years, some SSR algorithms have been presented, such as the sparse Bayesian learning (SBL) algorithm [27, 28], -norm singular value decomposition (SVD) algorithm [29], and -norm sparse representation of array covariance vector (SRACV) algorithm [30]. Nevertheless, they suffer from a complex two-dimensional overcomplete dictionary, which will bring a heavy computational burden.

In this paper, a sparse recovery algorithm is proposed based on a double-pulse FDA-MIMO radar. We extend the double-pulse concept of FDA radar in [31] to FDA-MIMO radar to solve the high-complexity problem of sparse recovery algorithms and simultaneously improve the parameter estimation performance of FDA-MIMO radar. Firstly, the angle estimates of targets are calculated by utilizing a pulse with a zero frequency increment and employing the improved -SVD method. Subsequently, the range estimates of targets are achieved by transmitting a pulse with a nonzero frequency increment. Specifically, after obtaining the angle estimates of targets, we deleted the unnecessary elements in the overcomplete dictionary to reduce its dimensionality during the range estimation. Therefore, this algorithm not only decouples the angle and range of FDA-MIMO radar but also reduces the dimension of the overcomplete dictionary. Grid partition will bring the problem of the heavy computational burden. As a result, we utilize an iterative grid refinement method to overcome the adverse effects caused by the grid partition on parameter estimation. Furthermore, we propose a new iteration criterion to improve the error between real parameters and their estimates to get a trade-off between the high-precision grid and the atomic correlation, so the proposed algorithm can achieve better target localization performance with FDA-MIMO radar as compared with the subspace-based algorithm. Finally, we derive the CRLB for the target parameter of the double-pulse FDA-MIMO radar. Numerical simulation verifies the superior performance of the proposed algorithm.

Notation. Capital bold letters and lowercase bold letters represent matrices and vectors, respectively. , , and stand for conjugate, inverse, and transpose operations, respectively. represents the Hadamard product, and denotes the Kronecker product. and denote -norm and -norm, respectively. denotes a complex matrix set.

2. Signal Model

As shown in Figure 1, a monostatic double-pulse FDA-MIMO radar that consists of uniform linear arrays (ULAs) with interelement spacing is considered, where the transmitter has antennas and the receiver has antennas. The first antenna in the transmitter is treated as the reference point. Considering the linearly increasing frequency increments, the carrier frequency at the -th transmitter antenna is where denotes the frequency increment and stands for the carrier frequency of the first antenna in the transmitter, where .

Figure 1 

Simplified diagram of a monostatic double-pulse FDA-MIMO radar.

Suppose the narrowband signal emitted by the -th antenna is where is the duration of the radar pulse and is the -th baseband waveform which follows that where represents the time delay.

Assume that there are far-field targets in the far-field whose ranges are much larger than the aperture of FDA-MIMO radar. Subsequently, the signal received by the -th antenna in the receiver and transmitted by the -th antenna in the transmitter can be represented by where represents the delay between the -th antenna in the transmitter and the -th antenna in the receiver, which is expressed as where and are the range and angle of the -th target. and are the interval between transmitter antennas and receiver antennas, respectively. is the speed of light.

The outputs of the received data after the matched filter (MF) can be expressed as [14] where is a signal vector. represents the noise vector. is a joint steering vector matrix, and with . The steering vectors of the receiver and transmitter can be defined by [13] where and .

The output of the MF by collecting snapshots can be described as where , , and .

3. Range and Angle Estimation for Monostatic Double-Pulse FDA-MIMO Radar

In this section, we propose a target localization algorithm with a double-pulse FDA-MIMO radar based on iterative grid refinement to alleviate the problem that grid partition brings about, a heavy computational burden and correlation. Firstly, we decouple the range and angle parameters of the FDA-MIMO radar with two pulses. Then, the improved -SVD method is utilized to estimate the angle and range of the target.

3.1. Angle Estimation for FDA-MIMO Radar

The angle estimates of targets are calculated by transmitting a pulse with a zero frequency increment and avoiding the range parameter. According to (8), the output of the FDA-MIMO radar after MF can be reconstructed by [32] where . is a transmit signal matrix. stands for the noise matrix. is a steering vector matrix, and with . Then, the steering vectors of the receiver and transmitter of the -th target can be defined by

We can utilize the sparse recovery method to achieve angle estimates. An overcomplete set of angles is established by sampling the spatial domain range uniformly, where is the number of grid points. Then, we need to reformulate the signal model (9) into the sparse signal model as where is a sparse matrix. is an overcomplete dictionary, and with . and can be denoted as

Then, we utilize the -SVD method to estimate the angle. The SVD result of the matrix can be represented by [29] where and are orthogonal matrices and is a block matrix. We get a matrix , which contains nearly all the signal power, . where is an identity matrix, and is a zero matrix. Moreover, suppose and ; we can derive as

, where . The sparsity vector corresponds to the space spectrum, which can be calculated by the following constraint optimization problem: where denotes the regularization parameter [29] to balance the mismatch degree of the model and the sparsity. According to the chi-squared distribution, the upper bound of can be calculated by the regularization parameter [33] with a high probability of 99.9%. Finally, we utilize the second-order cone (SOC) programming package, such as CVX, to solve the optimization problem in (15). Based on , the one-dimensional spectral peak search can be established where the angle estimates correspond to maximum peaks.

3.2. Range Estimation for FDA-MIMO Radar

The range estimates of targets are calculated by transmitting a pulse with a nonzero frequency increment. The output of the MF by collecting snapshots can be expressed as (8).

We assume that angle estimates obtained from subsection are . To get the range estimates, we utilize the sparse recovery method. An overcomplete set of ranges is established by sampling the spatial domain range uniformly, where is the number of grid points and denotes the maximum unambiguous range [34]. We stack the range complete set corresponding to angles into a large row vector to obtain automatically paired range and angle estimates. Then, we need to construct the signal model of (8) into a sparse signal model as where denotes a sparse matrix. is a known overcomplete dictionary, and with and . and can be defined as

Then, we utilize the -SVD method to estimate the range. The SVD result of the matrix can be expressed as [29] where and are orthogonal matrices and is a block matrix. We can get a matrix , which contains nearly all the signal power, . Besides, suppose and ; we can derive the expression for as

, where . The sparsity of corresponds to the sparsity of the space spectrum, which can be calculated by the following constraint optimization problem:

According to the chi-squared distribution, the upper bound of the power can be calculated as the regularization parameter [33] with a high probability of 99.9%. Finally, we utilize the SOC programming package, such as CVX, to solve the optimization problem in (20). Based on , the one-dimensional spectral peak search can be established where the range estimates correspond to maximum peaks.

3.3. Grid Refinement

Since it is impossible that all parameter estimates fall on the grid points, the refining operation for the grid is required, which will bring high computational complexity and produce highly correlated atoms. To tackle the problems, we propose an improved iterative grid refinement algorithm. For example, the algorithm steps of range estimation are given as follows: (1)Set refinement times . A simple grid is constructed by discretizing the interval between 0 and to estimate the target parameters. The grid spacing is (2)Use the proposed method in subsection to get and . Then, set (3)According to the range estimates in step 2, a new grid composed of subgrids is constructed, where subgrids are and is a grid established by sampling the spatial domain range with uniformly, and . Set grid spacing (4)Return to step 2 until , the final range estimates can be obtained as

In the proposed algorithm, we use to improve the error between real parameters and their estimates to get a trade-off between the high-precision grid and the atomic correlation. The detailed steps of the proposed sparse recovery algorithm based on a double-pulse FDA-MIMO radar are summarized in Algorithm 1.

Remark 2. The main computational complexity of the proposed algorithm is the singular value decomposition and the SOC programming problem. The singular value decomposition of and requires altogether flops, and it takes flops to solve the above SOC programming problem, where denotes the number of iterations and represents the number of refinement grid points for each target. Compared with the U-ESPRIT algorithm [20], the proposed algorithm requires more computation. However, this algorithm has outstanding advantages, which can not only adapt to the scene of insufficient snapshots and high target correlation but also provide higher precision and resolution.

Remark 3. We assume that angle estimates obtained from subsection are , respectively. Therefore, we can construct a simplified overcomplete dictionary containing range and angle information via adding a sparse grid of the range dimension. By solving the SOC programming problem in (20), a sparse vector can be obtained, where the first elements represent the range estimates of the target with angle , the to represent the range estimates of the target with angle , and the last elements represent the range estimates of the target with angle . Hence, the corresponding angle can be found through the position of the element in the sparse space spectrum to obtain automatically paired range and angle estimates.

Remark 4. In this paper, we utilize to improve the error between real parameters and their estimates. can be written as in the angle estimates, which is defined by , where and . is a new steering vector matrix constructed by . is the angle estimate for each iteration. can be written as in the range estimates, which is defined by , where and . is a new steering vector matrix constructed by and . is the angle estimate in advance, and denotes the range estimate for each iteration.

4. CRLB Analysis

In this section, we derive the CRLB results with regard to the angle and range. According to (8) and (9), we can rewrite the signal model as vector where the is the normalized Gaussian noise with zero mean and unit variance . is the equivalent steering vector and can be presented by where

Assuming that there are targets, the Fisher information matrix (FIM) is [35] where . where represents the noise power, , and . and can be expressed as