An Accurate Sparse Recovery Algorithm for Range-Angle Localization of Targets via Double-Pulse FDA-MIMO Radar

State Key Laboratory of Marine Resource Utilization in South China Sea and School of Information and Communication Engineering, Hainan University, Haikou 570228, China Key Laboratory for Ubiquitous Network and Service Software of Liaoning Province, School of Software, Dalian University of Technology, Dalian 116620, China Department of Communication Engineering, Institute of Information Science Technology, Dalian Maritime University, 116026, China


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][2][3][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][9][10] by effectively utilizing space diversity. However, MIMO radar cannot obtain the essential range estimates of targets. FDA-MIMO [11][12][13][14] radar, as a combination of frequency diverse array (FDA) radar [15][16][17] and MIMO radar, has a small fre-quency 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][23][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 subspacebased 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], l 1 -norm singular value decomposition (SVD) algorithm [29], and l 1 -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 doublepulse 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 l 1 -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.

Signal Model
As shown in Figure 1, a monostatic double-pulse FDA-MIMO radar that consists of uniform linear arrays (ULAs) with interelement spacing d = λ/2 is considered, where the transmitter has M antennas and the receiver has N antennas. The first antenna in the transmitter is treated as the reference point. Considering the linearly increasing frequency increments, the carrier frequency at the m-th transmitter antenna is where Δf denotes the frequency increment and f 1 stands for the carrier frequency of the first antenna in the transmitter, where Δf ≪ f 1 . Suppose the narrowband signal emitted by the m-th antenna is where T is the duration of the radar pulse and ϕ m ðtÞ is the m-th baseband waveform which follows that where τ represents the time delay. Assume that there are K 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 n-th antenna in the receiver and transmitted by the m-th antenna in the transmitter can be represented by where τðm, n, θ k , r k Þ represents the delay between the m-th antenna in the transmitter and the n-th antenna in the receiver, which is expressed as where r k and θ k are the range and angle of the k-th target. d t and d r are the interval between transmitter antennas and receiver antennas, respectively. c is the speed of light. The outputs of the received data after the matched filter (MF) can be expressed as [14] x l ð Þ = As l ð Þ + n l ð Þ, ð6Þ where sðlÞ = ½s 1 ðlÞ, s 2 ðlÞ, ⋯, s K ðlÞ T ∈ ℂ K×1 is a signal vector. nðlÞ represents the noise vector. A = ½ aðθ 1 , r 1 Þ, aðθ 2 , r 2 Þ, ⋯, aðθ K , r K Þ ∈ ℂ MN×K is a joint steering vector matrix, and aðθ k , r k Þ = a r ðθ k Þ ⊗ a t ðθ k , r k Þ with k = 1, 2, ⋯, K. The steering vectors of the receiver and transmitter can be defined by [13] a r θ k ð Þ = 1, e j2π d/λ ð Þ sin θ k , ⋯, e j2π d/λ where a r ðθ k Þ ∈ ℂ N×1 and a t ðθ k , r k Þ ∈ ℂ M×1 .

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 l 1 -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 X a = ½x a ð1Þ, x a ð2Þ, ⋯, x a ðLÞ ∈ ℂ MN×L . S a = ½s a ð1Þ, s a ð2Þ, ⋯, s a ðLÞ ∈ ℂ K×L is a transmit signal matrix. N a = ½n a ð1Þ, n a ð2Þ, ⋯, n a ðLÞ ∈ ℂ MN×L stands for the noise matrix.
Then, the steering vectors of the receiver and transmitter of the k-th target can be defined by We can utilize the sparse recovery method to achieve angle estimates. An overcomplete set of angles θ = ½θ 1 , θ 2 , ⋯, θ P is established by sampling the spatial domain range ½−π/2, π/2 uniformly, where P ≫ K is the number of grid points. Then, we need to reformulate the signal model (9) into the sparse signal model as where S a = ½ s a ð1Þ, s a ð2Þ, ⋯, s a ðLÞ ∈ ℂ P×L is a sparse matrix. A a = ½ a a ðθ 1 Þ, a a ðθ 2 Þ, ⋯, a a ðθ P Þ ∈ ℂ MN×P is an overcomplete dictionary, and a a ðθ p Þ = a ar ðθ p Þ ⊗ a at ðθ p Þ with p = 1, 2, ⋯, P. a at ðθ p Þ and a ar ðθ p Þ can be denoted as Then, we utilize the l 1 -SVD method to estimate the angle. The SVD result of the matrix X a can be represented by [29] where U a ∈ ℂ MN×MN and V a ∈ ℂ L×L are orthogonal matrices and Q a ∈ ℂ MN×L is a block matrix. We get a MN × K matrix X aSV , which contains nearly all the signal power, The sparsity vectors ðl 2 Þ a corresponds to the space spectrum, which can be calculated by the following constraint optimization problem: where η a 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 N aSV can be calculated by the regularization parameter η a [33] with a high probability of 99.9%. Finally, we utilize the second- Transmitting array Receiving array Targets Figure 1: Simplified diagram of a monostatic double-pulse FDA-MIMO radar.
3 Wireless Communications and Mobile Computing order cone (SOC) programming package, such as CVX, to solve the optimization problem in (15). Based ons ðl 2 Þ a , the one-dimensional spectral peak search can be established where the angle estimates correspond to K maximum peaks.

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 L snapshots can be expressed as (8).
We assume that angle estimates obtained from subsec- To get the range estimates, we utilize the sparse recovery method. An overcomplete set of ranges r = ½ r 1 , r 2 , ⋯, r W is established by sampling the spatial domain range ½0, c/2Δf uniformly, where W is the number of grid points and c/2Δf denotes the maximum unambiguous range [34]. We stack the range complete set corresponding to K angles into a large row vectorr = ½r ðθ 1 ,1Þ , r ðθ 1 ,2Þ , ⋯,r ðθ 1, WÞ ,r ðθ 2 ,1Þ , ⋯,r ðθ K ,W−1Þ ,r ðθ K ,WÞ 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 S r = ½ s r ð1Þ, s r ð2Þ, ⋯, s r ðLÞ ∈ ℂ KW×L denotes a sparse matrix.
A r = ½ a r ðθ 1 , r 1 Þ, a r ðθ 1 , r 2 Þ, ⋯, a r ðθ K , r W−1 Þ, a r ðθ K , r W Þ ∈ ℂ MN×KW is a known overcomplete dictionary, and a r ðθ k , r w Þ = a rr ðθ k Þ ⊗ a rt ðθ k , r w Þ with k = 1, 2, ⋯, K and w = 1, 2, ⋯, W. a rt ðθ k , r w Þ and a rr ðθ k Þ can be defined as Then, we utilize the l 1 -SVD method to estimate the range. The SVD result of the matrix X can be expressed as [29] where U r ∈ ℂ MN×MN and V r ∈ ℂ L×L are orthogonal matrices and Q r ∈ ℂ MN×L is a block matrix. We can get a MN × K matrix X rSV , which contains nearly all the signal power, X rSV = U r Q r D K = X r V r D K . Besides, suppose S rSV = S r V r D K and N rSV = NV r D K ; we can derive the expression for X rSV as According to the chi-squared distribution, the upper bound of the N rSV power can be calculated as the regularization parameter η r [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 oñ s ðl 2 Þ r , the one-dimensional spectral peak search can be established where the range estimates correspond to K maximum peaks.

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 o = 1. A simple grid r ðoÞ is constructed by discretizing the interval between 0 and c/2Δf to estimate the target parameters. The grid spacing is B o In the proposed algorithm, we use F 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 X and X a requires altogether Of2UðMNÞ 2 + 2 MNUL 2 g flops, and it takes OfðK 2 WÞ 3 + ðPKÞ 3 + 2ðU − 1Þ ðK 2 GÞ 3 g flops to solve the above SOC programming problem, where U denotes the number of iterations and G 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 4 Wireless Communications and Mobile Computing 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 A are θ 1 , θ 2 , ⋯, θ K , 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 KW × 1 sparse vectors ðl 2 Þ r can be obtained, where the first W elements represent the range estimates of the target with angle θ 1 , the W + 1 to 2W represent the range estimates of the target with angle θ 2 , and the last W elements represent the range estimates of the target with angle θ K . Hence, the corresponding angle can be found through the position of the element in the sparse space spectrums ðl 2 Þ r to obtain automatically paired range and angle estimates.
Remark 4. In this paper, we utilize F to improve the error between real parameters and their estimates. F can be written as F a in the angle estimates, which is defined by jkR a k 2 − kR a k 2 j, where R a = X a X H a and R a = ðA a S a ÞðA a S a Þ H . A a = ½ a a ðθ 1 Þ, a a ðθ 2 Þ, ⋯, a a ðθ K Þ is a new steering vector matrix constructed by θ. θ = ½θ 1 , θ 2 , ⋯, θ K T is the angle estimate for each iteration. F can be written as F r in the range estimates, which is defined by jkRk 2 − kR r k 2 j, where R = XX H and R r = ðA r SÞðA r SÞ H . A r = ½ að b θ 1 ,r 1 Þ, að b θ 2 ,r 2 Þ, ⋯, að b θ K , r K Þ is a new steering vector matrix constructed by b θ andr. b θ = ½θ∧ 1 , θ∧ 2 , ⋯, θ∧ K T is the angle estimate in advance, andr = ½r∧ 1 , r∧ 2 , ⋯, r∧ K T denotes the range estimate for each iteration.

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 v is the normalized Gaussian noise with zero mean and unit variance I. a new ðθ k, r k Þ ∈ ℂ 2MN×1 is the equivalent steering vector and can be presented by where Assuming that there are K targets, the Fisher information matrix (FIM) is [35] where k = 1, 2, ⋯, K.
where σ 2 represents the noise power, ε = a new ðθ k , r k Þ, and (1) The FDA-MIMO radar transmits a pulse with a zero frequency increment to obtain the received signal X a (2) The sparse vectors ðl 2 Þ a is obtained by CVX optimization of (15). Angle estimates are realized by searching K maximum values through a one-dimensional spectrum ofs ðl 2 Þ a (3) Use the method in subsection C to optimize the angle estimates in step 2. Then, the refined angle estimates can be received as ½θ 1 , θ 2 , ⋯, θ K (4) The FDA-MIMO radar transmits a pulse with a nonzero frequency increment to obtain the received signal X (5) The sparse vectors ðl 2 Þ r is obtained by CVX optimization of (20). Range estimates are realized by searching K maximum values through a one-dimensional spectrum ofs ðl 2 Þ r (6) Use the method in subsection C to optimize the range estimates in step 5. Then, the range estimates can be obtained as ½r 1 , r 2 , ⋯, r K , and get automatically paired range and angle estimates ðθ k , r k Þ, for k = 1, 2, ⋯, K Algorithm 1: An accurate sparse recovery algorithm for double-pulse FDA-MIMO radar.

Wireless Communications and Mobile Computing
Γ = I. dε/dθ k and dε/dr k can be expressed as a at θ k ð Þ, The CRLB for the range and angle can be expressed as

Numerical Simulation Results
In this section, we demonstrate the superiority of the proposed algorithm via simulation, where M = N = 8 and d = λ /2. The carrier frequency f 0 is 10 GHz, the frequency increment Δf is 0 kHz in angle estimation, and the frequency increment Δf is 1 kHz in range estimation.

Simple
Process of Target Estimation. Suppose K = 3 narrowband targets with angles θ 1 = −20 ∘ and θ 2 = θ 3 = 40 ∘ and ranges r 1 = 21000m, r 2 = 34000m, and r 3 = 54000m, respectively. We set SNR = 10 dB, and the number of snapshots is 50. Figure 2 depicts the spatial spectrum of the sparse vector s ðl 2 Þ a for angle estimates in the proposed algorithm. Since there are two targets from the same direction, it is necessary to extract the range estimates of targets. Figures 3 and 4 give the spatial spectrums of the first W elements and the last W elements ofs ðl 2 Þ r , respectively, where the first W elements of s ðl 2 Þ r represent range estimates of the target with angle θ 1 , and the last W elements ofs ðl 2 Þ r represent range estimates of the target with angles θ 2 and θ 3 . Figure 3 shows the range estimates of the target with angle θ 1 , and Figure 4 shows the range estimates of the target with angles θ 2 and θ 3 .

Detection and Estimation
Performance. In this subsection, we carry out a series of simulations under different conditions to verify the superiority of the proposed algorithm. The ESPRIT method [19] and the U-ESPRIT method [20] are compared with the proposed algorithm. We assume that the frequency increment and the carrier frequency of the above algorithms are 1 kHz and 10 GHz, respectively. Suppose that two targets are located: ð−10:75 ∘ , 21565 mÞ and ð25:68 ∘ , 44505 mÞ. The grids of the angle and range for the proposed algorithm are ½−90 ∘ : 1 ∘ : 90 ∘ and ½0 : 1 : 150 km, respectively. The root mean square errors (RMSEs) for the range and angle are defined as    According to the results in Figure 5, we can conclude that the proposed algorithm can achieve precise matching of the range and angle. The angles and ranges of the two targets can be accurately estimated with a small number of snapshots. As can be seen from Figures 6 and 7, the parameter estimates of the proposed algorithm can considerably approach the real ones. Figures 8 and 9 give the probability of successful detection (PSD) versus SNR for angle and range estimation in different algorithms, respectively, where L = 50. With regard to angle estimation, we define the successful detection if the SNR (dB)  9 Wireless Communications and Mobile Computing estimation result θ ik satisfies jθ ik − θ k j ≤ 0:1 ∘ . The detection is successful if the estimation result r ik satisfies jr ik − r k j ≤ 200 m with regard to range estimation. As can be seen from Figures 8 and 9, the PSD of the proposed algorithm is superior to those of other algorithms under the same condition.
Besides, with the increase of SNR, the proposed algorithm can achieve the PSD of 100% when SNR = 10 dB. Figures 10 and 11 give the RMSE results of the angle and range estimates versus SNR with L = 50, respectively. It is obvious that the RMSE results gradually decrease with the  increase of SNR, and in particular, the algorithm outperforms the other methods in range and angle estimation. Figures 12 and 13 depict the RMSE results of the angle and range estimates versus snapshots, respectively, where SNR = 0 dB. As can be seen from Figures 12 and 13, the RMSE results of all algorithms improve with the increase of snapshots. Besides, the proposed method can obtain more accurate range and angle estimates than the other methods with the same number of snapshots.

Conclusion
In this paper, an accurate sparse recovery algorithm based on a double-pulse FDA-MIMO radar is proposed. In the proposed algorithm, we decouple the range and angle parameters of the FDA-MIMO radar with two pulses. Grid partition will bring high computational complexity. 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. Compared with the subspace-based algorithms, the proposed algorithm performs better in simulation. Massive simulation results have certified that the proposed algorithm is prominent for parameter estimation of FDA-MIMO radar.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.