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

Authors: Qi Liu; Xianpeng Wang; Liangtian Wan; Mengxing Huang; Lu Sun
Journal: Wireless Communications and Mobile Computing
 (2020)

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## 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 improvedl1-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).


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## Body

## 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], l1-norm singular value decomposition (SVD) algorithm [29], and l1-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 l1-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. ·∗, ·−1, and ·T stand for conjugate, inverse, and transpose operations, respectively. ⊙ represents the Hadamard product, and ⊗ denotes the Kronecker product. ∙1 and ∙2 denote l1-norm and l2-norm, respectively. ℂP×Q denotes a P×Q complex matrix set.


## 2. Signal Model

As shown in Figure1, 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
(1)fm=f1+m−1Δf,m=1,2,⋯,M,where Δf denotes the frequency increment and f1 stands for the carrier frequency of the first antenna in the transmitter, where Δf≪f1.Figure 1
Simplified diagram of a monostatic double-pulse FDA-MIMO radar.Suppose the narrowband signal emitted by them-th antenna is
(2)smt=ϕmtej2πfmt,m=1,2,⋯,M,0≤t≤T,where T is the duration of the radar pulse and ϕmt is the m-th baseband waveform which follows that
(3)∫0Tϕmtϕn∗t−τej2πm−nΔftdt=1,m=n,τ=0,0,m≠n,∀τ,where τ represents the time delay.Assume that there areK 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
(4)xm,nt=∑k=1Kϕmt−τm,n,θk,rkej2πfmt−τm,n,θk,rk,where τm,n,θk,rk represents the delay between the m-th antenna in the transmitter and the n-th antenna in the receiver, which is expressed as
(5)τm,n,θk,rk=2rkc−m−1dtsinθkc−n−1drsinθkc,where rk and θk are the range and angle of the k-th target. dt and dr 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]
(6)xl=Asl+nl,where sl=s1l,s2l,⋯,sKlT∈ℂK×1 is a signal vector. nl represents the noise vector. A=a¯θ1,r1,a¯θ2,r2,⋯,a¯θK,rK∈ℂMN×K is a joint steering vector matrix, and a¯θk,rk=arθk⊗atθk,rk with k=1,2,⋯,K. The steering vectors of the receiver and transmitter can be defined by [13]
(7)arθk=1,ej2πd/λsinθk,⋯,ej2πd/λN−1sinθkT,atθk,rk=1,e−j4πΔf/crk,⋯,e−j4πΔf/cM−1rkT⊙1,ej2πd/λsinθk,⋯,ej2πd/λM−1sinθkT,where arθk∈ℂN×1 and atθk,rk∈ℂM×1.The output of the MF by collectingL snapshots can be described as
(8)X=AS+N,where X=x1,x2,⋯,xL∈ℂMN×L, S=s1,s2,⋯,sL∈ℂK×L, and N=n1,n2,⋯,nL∈ℂMN×L.


## 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 improvedl1-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]
(9)Xa=AaSa+Na,where Xa=xa1,xa2,⋯,xaL∈ℂMN×L. Sa=sa1,sa2,⋯,saL∈ℂK×L is a transmit signal matrix. Na=na1,na2,⋯,naL∈ℂMN×L stands for the noise matrix. Aa=aaθ1,aaθ2,⋯,aaθK is a MN×K steering vector matrix, and aaθk=aarθk⊗aatθk with k=1,2,⋯,K. Then, the steering vectors of the receiver and transmitter of the k-th target can be defined by
(10)aatθk=1,ej2πd/λsinθk,⋯,ej2πd/λM−1sinθkT,aarθk=1,ej2πd/λsinθk,⋯,ej2πd/λN−1sinθkT.We can utilize the sparse recovery method to achieve angle estimates. An overcomplete set of anglesθ¯=θ¯1,θ¯2,⋯,θ¯P is established by sampling the spatial domain range −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
(11)Xa=A¯aS¯a+Na,where S¯a=s¯a1,s¯a2,⋯,s¯aL∈ℂP×L is a sparse matrix. A¯a=a¯aθ¯1,a¯aθ¯2,⋯,a¯aθ¯P∈ℂMN×P is an overcomplete dictionary, and a¯aθ¯p=a¯arθ¯p⊗a¯atθ¯p with p=1,2,⋯,P. a¯atθ¯p and a¯arθ¯p can be denoted as
(12)a¯atθ¯p=1,ej2πd/λsinθ¯p,⋯,ej2πd/λM−1sinθ¯pT,a¯arθ¯p=1,ej2πd/λsinθ¯p,⋯,ej2πd/λN−1sinθ¯pT.Then, we utilize thel1-SVD method to estimate the angle. The SVD result of the matrix Xa can be represented by [29]
(13)Xa=UaQaVaT,where Ua∈ℂMN×MN and Va∈ℂL×L are orthogonal matrices and Qa∈ℂMN×L is a block matrix. We get a MN×K matrix XaSV, which contains nearly all the signal power, XaSV=UaQaDK=XaVaDK. DK=IK,0∈ℂL×K where IK∈ℂK×K is an identity matrix, and 0∈ℂT−K×K is a zero matrix. Moreover, suppose S¯aSV=S¯aVaDK and NaSV=NVaDK; we can derive XaSV as
(14)XaSV=A¯aS¯aSV+NaSV.s~ail2=∑k=1KsaiaSVk2, where i=1,2,⋯,P. The sparsity vector s~al2 corresponds to the space spectrum, which can be calculated by the following constraint optimization problem:
(15)mins~al21,subjecttoXaSV−A¯aS¯aSV22≤ηa2,where ηa 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 NaSV can be calculated by the regularization parameter ηa [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 s~al2, the one-dimensional spectral peak search can be established where the angle estimates correspond to K 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 collectingL snapshots can be expressed as (8).We assume that angle estimates obtained from subsectionA are θ¯=θ¯1,θ¯2,⋯,θ¯KT. To get the range estimates, we utilize the sparse recovery method. An overcomplete set of ranges r¯=r¯1,r¯2,⋯,r¯W is established by sampling the spatial domain range 0,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 vector r~=r~θ¯1,1,r~θ¯1,2,⋯,r~θ¯1,W,r~θ¯2,1,⋯,r~θ¯K,W−1,r~θ¯K,W to obtain automatically paired range and angle estimates. Then, we need to construct the signal model of (8) into a sparse signal model as
(16)X=A¯rS¯r+N,where S¯r=s¯r1,s¯r2,⋯,s¯rL∈ℂKW×L denotes a sparse matrix. A¯r=a¯rθ¯1,r¯1,a¯rθ¯1,r¯2,⋯,a¯rθ¯K,r¯W−1,a¯rθ¯K,r¯W∈ℂMN×KW is a known overcomplete dictionary, and a¯rθ¯k,r¯w=a¯rrθ¯k⊗a¯rtθ¯k,r¯w with k=1,2,⋯,K and w=1,2,⋯,W. a¯rtθ¯k,r¯w and a¯rrθ¯k can be defined as
(17)a¯rtθ¯k,r¯w=1,e−j4πΔf/cr¯w,⋯,e−j4πM−1Δf/cr¯wTe1,ej2πd/λsinθ¯k,⋯,ej2πd/λsinθ¯kT,a¯rrθ¯k=1,ej2πd/λsinθ¯k,⋯,ej2πd/λN−1sinθ¯kT.Then, we utilize thel1-SVD method to estimate the range. The SVD result of the matrix X can be expressed as [29]
(18)X=UrQrVrT,where Ur∈ℂMN×MN and Vr∈ℂL×L are orthogonal matrices and Qr∈ℂMN×L is a block matrix. We can get a MN×K matrix XrSV, which contains nearly all the signal power, XrSV=UrQrDK=XrVrDK. Besides, suppose S¯rSV=S¯rVrDK and NrSV=NVrDK; we can derive the expression for XrSV as
(19)XrSV=A¯rS¯rSV+NrSV.s~ril2=∑k=1KsrirSVk2, where i=1,2,⋯,KW. The sparsity of s~rl2 corresponds to the sparsity of the space spectrum, which can be calculated by the following constraint optimization problem:
(20)mins~rl21,subjecttoXrSV−A¯rS¯rSV22≤ηr2.According to the chi-squared distribution, the upper bound of theNrSV power can be calculated as the regularization parameter ηr [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 s~rl2, the one-dimensional spectral peak search can be established where the range estimates correspond to K 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 timeso=1. A simple grid ro is constructed by discretizing the interval between 0 and c/2Δf to estimate the target parameters. The grid spacing is Bo(2)
Use the proposed method in subsectionB to get r~o=r1,r2,⋯,rK and For=R2−Ror2. Then, set o=o+1(3)
According to the range estimates in step 2, a new gridro composed of K subgrids is constructed, where K subgrids are r~1,r~2,⋯,r~K and r~i is a grid established by sampling the spatial domain range with ri−Bo,ri+Bo uniformly, and i=1,2,⋯,K. Set grid spacing Bo=Bo−1/100(4)
Return to step 2 untilFo−1r<For, the final range estimates can be obtained as r~o−1In the proposed algorithm, we useF 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.Algorithm 1: An accurate sparse recovery algorithm for double-pulse FDA-MIMO radar.
(1) The FDA-MIMO radar transmits a pulse with a zero frequency increment to obtain the received signalXa(2) The sparse vectors~al2 is obtained by CVX optimization of (15). Angle estimates are realized by searching K maximum values through a one-dimensional spectrum of s~al2(3) Use the method in subsectionC to optimize the angle estimates in step 2. Then, the refined angle estimates can be received as θ1,θ2,⋯,θK(4) The FDA-MIMO radar transmits a pulse with a nonzero frequency increment to obtain the received signalX(5) The sparse vectors~rl2 is obtained by CVX optimization of (20). Range estimates are realized by searching K maximum values through a one-dimensional spectrum of s~rl2(6) Use the method in subsectionC to optimize the range estimates in step 5. Then, the range estimates can be obtained as r1,r2,⋯,rK, and get automatically paired range and angle estimates θk,rk, for k=1,2,⋯,KRemark 2.
The main computational complexity of the proposed algorithm is the singular value decomposition and the SOC programming problem. The singular value decomposition ofX and Xa requires altogether O2UMN2+2MNUL2 flops, and it takes OK2W3+PK3+2U−1K2G3 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 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 subsectionA 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 vector s~rl2 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 spectrum s~rl2 to obtain automatically paired range and angle estimates.Remark 4.
In this paper, we utilizeF to improve the error between real parameters and their estimates. F can be written as Fa in the angle estimates, which is defined by Ra2−Ra2, where Ra=XaXaH and Ra=AaSaAaSaH. Aa=aaθ¯1,aaθ¯2,⋯,aaθ¯K is a new steering vector matrix constructed by θ. θ=θ¯1,θ¯2,⋯,θ¯KT is the angle estimate for each iteration. F can be written as Fr in the range estimates, which is defined by R2−Rr2, where R=XXH and Rr=ArSArSH. Ar=a¯θ^1,r^1,a¯θ^2,r^2,⋯,a¯θ^K,r^K is a new steering vector matrix constructed by θ^ and r^. θ^=θ∧1,θ∧2,⋯,θ∧KT is the angle estimate in advance, and r^=r∧1,r∧2,⋯,r∧KT denotes the range estimate for each iteration.


## 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]
(9)Xa=AaSa+Na,where Xa=xa1,xa2,⋯,xaL∈ℂMN×L. Sa=sa1,sa2,⋯,saL∈ℂK×L is a transmit signal matrix. Na=na1,na2,⋯,naL∈ℂMN×L stands for the noise matrix. Aa=aaθ1,aaθ2,⋯,aaθK is a MN×K steering vector matrix, and aaθk=aarθk⊗aatθk with k=1,2,⋯,K. Then, the steering vectors of the receiver and transmitter of the k-th target can be defined by
(10)aatθk=1,ej2πd/λsinθk,⋯,ej2πd/λM−1sinθkT,aarθk=1,ej2πd/λsinθk,⋯,ej2πd/λN−1sinθkT.We can utilize the sparse recovery method to achieve angle estimates. An overcomplete set of anglesθ¯=θ¯1,θ¯2,⋯,θ¯P is established by sampling the spatial domain range −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
(11)Xa=A¯aS¯a+Na,where S¯a=s¯a1,s¯a2,⋯,s¯aL∈ℂP×L is a sparse matrix. A¯a=a¯aθ¯1,a¯aθ¯2,⋯,a¯aθ¯P∈ℂMN×P is an overcomplete dictionary, and a¯aθ¯p=a¯arθ¯p⊗a¯atθ¯p with p=1,2,⋯,P. a¯atθ¯p and a¯arθ¯p can be denoted as
(12)a¯atθ¯p=1,ej2πd/λsinθ¯p,⋯,ej2πd/λM−1sinθ¯pT,a¯arθ¯p=1,ej2πd/λsinθ¯p,⋯,ej2πd/λN−1sinθ¯pT.Then, we utilize thel1-SVD method to estimate the angle. The SVD result of the matrix Xa can be represented by [29]
(13)Xa=UaQaVaT,where Ua∈ℂMN×MN and Va∈ℂL×L are orthogonal matrices and Qa∈ℂMN×L is a block matrix. We get a MN×K matrix XaSV, which contains nearly all the signal power, XaSV=UaQaDK=XaVaDK. DK=IK,0∈ℂL×K where IK∈ℂK×K is an identity matrix, and 0∈ℂT−K×K is a zero matrix. Moreover, suppose S¯aSV=S¯aVaDK and NaSV=NVaDK; we can derive XaSV as
(14)XaSV=A¯aS¯aSV+NaSV.s~ail2=∑k=1KsaiaSVk2, where i=1,2,⋯,P. The sparsity vector s~al2 corresponds to the space spectrum, which can be calculated by the following constraint optimization problem:
(15)mins~al21,subjecttoXaSV−A¯aS¯aSV22≤ηa2,where ηa 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 NaSV can be calculated by the regularization parameter ηa [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 s~al2, the one-dimensional spectral peak search can be established where the angle estimates correspond to K 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 collectingL snapshots can be expressed as (8).We assume that angle estimates obtained from subsectionA are θ¯=θ¯1,θ¯2,⋯,θ¯KT. To get the range estimates, we utilize the sparse recovery method. An overcomplete set of ranges r¯=r¯1,r¯2,⋯,r¯W is established by sampling the spatial domain range 0,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 vector r~=r~θ¯1,1,r~θ¯1,2,⋯,r~θ¯1,W,r~θ¯2,1,⋯,r~θ¯K,W−1,r~θ¯K,W to obtain automatically paired range and angle estimates. Then, we need to construct the signal model of (8) into a sparse signal model as
(16)X=A¯rS¯r+N,where S¯r=s¯r1,s¯r2,⋯,s¯rL∈ℂKW×L denotes a sparse matrix. A¯r=a¯rθ¯1,r¯1,a¯rθ¯1,r¯2,⋯,a¯rθ¯K,r¯W−1,a¯rθ¯K,r¯W∈ℂMN×KW is a known overcomplete dictionary, and a¯rθ¯k,r¯w=a¯rrθ¯k⊗a¯rtθ¯k,r¯w with k=1,2,⋯,K and w=1,2,⋯,W. a¯rtθ¯k,r¯w and a¯rrθ¯k can be defined as
(17)a¯rtθ¯k,r¯w=1,e−j4πΔf/cr¯w,⋯,e−j4πM−1Δf/cr¯wTe1,ej2πd/λsinθ¯k,⋯,ej2πd/λsinθ¯kT,a¯rrθ¯k=1,ej2πd/λsinθ¯k,⋯,ej2πd/λN−1sinθ¯kT.Then, we utilize thel1-SVD method to estimate the range. The SVD result of the matrix X can be expressed as [29]
(18)X=UrQrVrT,where Ur∈ℂMN×MN and Vr∈ℂL×L are orthogonal matrices and Qr∈ℂMN×L is a block matrix. We can get a MN×K matrix XrSV, which contains nearly all the signal power, XrSV=UrQrDK=XrVrDK. Besides, suppose S¯rSV=S¯rVrDK and NrSV=NVrDK; we can derive the expression for XrSV as
(19)XrSV=A¯rS¯rSV+NrSV.s~ril2=∑k=1KsrirSVk2, where i=1,2,⋯,KW. The sparsity of s~rl2 corresponds to the sparsity of the space spectrum, which can be calculated by the following constraint optimization problem:
(20)mins~rl21,subjecttoXrSV−A¯rS¯rSV22≤ηr2.According to the chi-squared distribution, the upper bound of theNrSV power can be calculated as the regularization parameter ηr [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 s~rl2, the one-dimensional spectral peak search can be established where the range estimates correspond to K 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 timeso=1. A simple grid ro is constructed by discretizing the interval between 0 and c/2Δf to estimate the target parameters. The grid spacing is Bo(2)
Use the proposed method in subsectionB to get r~o=r1,r2,⋯,rK and For=R2−Ror2. Then, set o=o+1(3)
According to the range estimates in step 2, a new gridro composed of K subgrids is constructed, where K subgrids are r~1,r~2,⋯,r~K and r~i is a grid established by sampling the spatial domain range with ri−Bo,ri+Bo uniformly, and i=1,2,⋯,K. Set grid spacing Bo=Bo−1/100(4)
Return to step 2 untilFo−1r<For, the final range estimates can be obtained as r~o−1In the proposed algorithm, we useF 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.Algorithm 1: An accurate sparse recovery algorithm for double-pulse FDA-MIMO radar.
(1) The FDA-MIMO radar transmits a pulse with a zero frequency increment to obtain the received signalXa(2) The sparse vectors~al2 is obtained by CVX optimization of (15). Angle estimates are realized by searching K maximum values through a one-dimensional spectrum of s~al2(3) Use the method in subsectionC to optimize the angle estimates in step 2. Then, the refined angle estimates can be received as θ1,θ2,⋯,θK(4) The FDA-MIMO radar transmits a pulse with a nonzero frequency increment to obtain the received signalX(5) The sparse vectors~rl2 is obtained by CVX optimization of (20). Range estimates are realized by searching K maximum values through a one-dimensional spectrum of s~rl2(6) Use the method in subsectionC to optimize the range estimates in step 5. Then, the range estimates can be obtained as r1,r2,⋯,rK, and get automatically paired range and angle estimates θk,rk, for k=1,2,⋯,KRemark 2.
The main computational complexity of the proposed algorithm is the singular value decomposition and the SOC programming problem. The singular value decomposition ofX and Xa requires altogether O2UMN2+2MNUL2 flops, and it takes OK2W3+PK3+2U−1K2G3 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 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 subsectionA 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 vector s~rl2 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 spectrum s~rl2 to obtain automatically paired range and angle estimates.Remark 4.
In this paper, we utilizeF to improve the error between real parameters and their estimates. F can be written as Fa in the angle estimates, which is defined by Ra2−Ra2, where Ra=XaXaH and Ra=AaSaAaSaH. Aa=aaθ¯1,aaθ¯2,⋯,aaθ¯K is a new steering vector matrix constructed by θ. θ=θ¯1,θ¯2,⋯,θ¯KT is the angle estimate for each iteration. F can be written as Fr in the range estimates, which is defined by R2−Rr2, where R=XXH and Rr=ArSArSH. Ar=a¯θ^1,r^1,a¯θ^2,r^2,⋯,a¯θ^K,r^K is a new steering vector matrix constructed by θ^ and r^. θ^=θ∧1,θ∧2,⋯,θ∧KT is the angle estimate in advance, and r^=r∧1,r∧2,⋯,r∧KT 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
(21)xnew=anewθk,rk·s+v,where the v is the normalized Gaussian noise with zero mean and unit variance I. anewθk,rk∈ℂ2MN×1 is the equivalent steering vector and can be presented by
(22)anewθk,rk=aθk,rkΔf=0,aθk,rkT,where
(23)aθk,rkΔf=0=aarθk⊗aatθk,aθk,rk=arθk⊗atθk,rk.Assuming that there areK targets, the Fisher information matrix (FIM) is [35]
(24)J=JθkθkJθkrkJrkθkJrkrk,where k=1,2,⋯,K.
(25)Jθkθk=2Lσ2RedεdθkHΓ−1dεdθk,Jθkrk=2Lσ2RedεdθkHΓ−1dεdrk,Jrkθk=2Lσ2RedεdrkHΓ−1dεdθk,Jrkrk=2Lσ2RedεdrkHΓ−1dεdrk,where σ2 represents the noise power, ε=anewθk,rk, and Γ=I. dε/dθk and dε/drk can be expressed as
(26)dεdθk=∂aarθk∂θk⊗aarθk+aarθk⊗∂aatθk∂θk,∂arθk∂θk⊗atθk,rk+arθk⊗∂atθk,rk∂θk,dεdrk=0,arθk⊗∂atθk,rk∂rk,∂aarθkaθk=j2πdλcosθk0⋱N−1aarθk,∂aatθkaθk=j2πdλcosθk0⋱M−1aatθk,∂arθkaθk=j2πdλcosθk0⋱N−1arθk,∂atθk,rkaθk=j2πdcosθkλ0⋱M−1atθk,rk,∂atθk,rkark=−j4πΔfc0⋱M−1atθk,rk.The CRLB for the range and angle can be expressed as(27)CRLBθk=J−11,1,CRLBrk=J−12,2.


## 5. Numerical Simulation Results

In this section, we demonstrate the superiority of the proposed algorithm via simulation, whereM=N=8 and d=λ/2. The carrier frequency f0 is 10 GHz, the frequency increment Δf is 0 kHz in angle estimation, and the frequency increment Δf is 1 kHz in range estimation.


### 5.1. Simple Process of Target Estimation

SupposeK=3 narrowband targets with angles θ1=−20∘ and θ2=θ3=40∘ and ranges r1=21000m, r2=34000m, and r3=54000m, respectively. We set SNR=10dB, and the number of snapshots is 50. Figure 2 depicts the spatial spectrum of the sparse vector s~al2 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 of s~rl2, respectively, where the first W elements of s~rl2 represent range estimates of the target with angle θ1, and the last W elements of s~rl2 represent range estimates of the target with angles θ2 and θ3. Figure 3 shows the range estimates of the target with angle θ1, and Figure 4 shows the range estimates of the target with angles θ2 and θ3.Figure 2
The spatial spectrum of angle estimation.Figure 3
Range estimation of the target with angleθ1.Figure 4
Range estimation of the target with angleθ2.


### 5.2. 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: −10.75∘,21565m and 25.68∘,44505m. The grids of the angle and range for the proposed algorithm are −90∘:1∘:90∘ and 0:1:150km, respectively. The root mean square errors (RMSEs) for the range and angle are defined as
(28)RMSEθ=1ρK∑i=1ρ∑k=1Kθik−θk2,RMSEr=1ρK∑i=1p∑k=1Krik−rk2,where ρ=100 is the number of Monte Carlo experiments. θik and rik represent the estimates of θk and rk in the i-th Monte Carlo experiment.The angle-range estimation results are exhibited in Figure5 for FDA-MIMO radar using the proposed algorithm in which Monte Carlo trials are implemented 100 times. Moreover, the results of U-ESPRIT and the proposed algorithm are provided for comparison with the proposed algorithm in Figures 6 and 7, where SNR=0dB and L=50. 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.Figure 5
Range and angle estimation results using the proposed algorithm.Figure 6
Comparison of DOA estimated results of 50 independent simulations.Figure 7
Comparison of range estimated results of 50 independent simulations.Figures8 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 estimation result θik satisfies θik−θk≤0.1∘. The detection is successful if the estimation result rik satisfies rik−rk≤200m 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=10dB.Figure 8
PSD versus SNR for angle estimation.Figure 9
PSD versus SNR for range estimation.Figures10 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.Figure 10
RMSE results of angle estimation versus SNR.Figure 11
RMSE results of range estimation versus SNR.Figures12 and 13 depict the RMSE results of the angle and range estimates versus snapshots, respectively, where SNR=0dB. 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.Figure 12
RMSE results of angle estimation versus snapshots.Figure 13
RMSE results of range estimation versus snapshots.


## 5.1. Simple Process of Target Estimation

SupposeK=3 narrowband targets with angles θ1=−20∘ and θ2=θ3=40∘ and ranges r1=21000m, r2=34000m, and r3=54000m, respectively. We set SNR=10dB, and the number of snapshots is 50. Figure 2 depicts the spatial spectrum of the sparse vector s~al2 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 of s~rl2, respectively, where the first W elements of s~rl2 represent range estimates of the target with angle θ1, and the last W elements of s~rl2 represent range estimates of the target with angles θ2 and θ3. Figure 3 shows the range estimates of the target with angle θ1, and Figure 4 shows the range estimates of the target with angles θ2 and θ3.Figure 2
The spatial spectrum of angle estimation.Figure 3
Range estimation of the target with angleθ1.Figure 4
Range estimation of the target with angleθ2.


## 5.2. 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: −10.75∘,21565m and 25.68∘,44505m. The grids of the angle and range for the proposed algorithm are −90∘:1∘:90∘ and 0:1:150km, respectively. The root mean square errors (RMSEs) for the range and angle are defined as
(28)RMSEθ=1ρK∑i=1ρ∑k=1Kθik−θk2,RMSEr=1ρK∑i=1p∑k=1Krik−rk2,where ρ=100 is the number of Monte Carlo experiments. θik and rik represent the estimates of θk and rk in the i-th Monte Carlo experiment.The angle-range estimation results are exhibited in Figure5 for FDA-MIMO radar using the proposed algorithm in which Monte Carlo trials are implemented 100 times. Moreover, the results of U-ESPRIT and the proposed algorithm are provided for comparison with the proposed algorithm in Figures 6 and 7, where SNR=0dB and L=50. 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.Figure 5
Range and angle estimation results using the proposed algorithm.Figure 6
Comparison of DOA estimated results of 50 independent simulations.Figure 7
Comparison of range estimated results of 50 independent simulations.Figures8 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 estimation result θik satisfies θik−θk≤0.1∘. The detection is successful if the estimation result rik satisfies rik−rk≤200m 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=10dB.Figure 8
PSD versus SNR for angle estimation.Figure 9
PSD versus SNR for range estimation.Figures10 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.Figure 10
RMSE results of angle estimation versus SNR.Figure 11
RMSE results of range estimation versus SNR.Figures12 and 13 depict the RMSE results of the angle and range estimates versus snapshots, respectively, where SNR=0dB. 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.Figure 12
RMSE results of angle estimation versus snapshots.Figure 13
RMSE results of range estimation versus snapshots.


## 6. 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.



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*Source: 6698446-2020-12-16.xml*