STAP Based on Toeplitz Covariance Matrix Reconstruction for Airborne Radar

We consider the problem of clutter covariance matrix (CCM) estimation for space-time adaptive processing (STAP) radar in the small sample. In this paper, a fast efficient algorithm for CCM reconstruction is proposed to overcome this shortcoming for the linear structure. Particularly, we present a low-rankmatrix recovery (LRMR) question about CCMestimation based on the Toeplitz structure of CCMand the prior knowledge of the noise. +e closed-form solution is obtained by relaxing the nonconvex LRMR problem that the trace norm replaces the rank norm.+e target can then be efficiently detected by using the recovered CCM according to the STAP theorem.We also analyze the algorithm model under the linear structure in the presence of unknown mutual coupling. It is shown that our method can obtain accurateCCM in the small sample, with even higher accuracy than the traditional algorithms in the samenumber of samples. It also can reduce the coupling effect and obtain more degrees of freedom (DOF) with limited sensors and pulses by utilizing sparse linear structure (SLS) and improve angle and Doppler resolutions. Finally, numerical simulations have verified the effectiveness of the proposed method in comparison with some of the existing methods.


Introduction
Space-time adaptive processing (STAP) which has an excellent performance on clutter suppression and targets detection plays a fundamental role in radar [1][2][3]. In the practical application of STAP, the system can calculate the ideal weight vector and then obtain the optimal filter output responses based on the clutter covariance matrix (CCM) estimated with its adjacent distance cell snapshots. erefore, the training snapshots play an important role in the adaptive radar system. In practice, the system needs a large number of uniform training snapshots to obtain the reliable and accurate CCM. It is difficult for radar to meet the abovementioned condition in a heterogeneous environment, especially in a small sample [4]. e problem of sample size for STAP radar has received considerable attention recently. ese works can be classified into two categories. e first category is the failure to take advantage of the prior knowledge to reduce sample size, which can be defined as the generalized reduced sample size method. For the dimension reduction [5][6][7] and reduced rank STAP [8], the sample size is twice the reduced dimension or clutter rank, but it is still large. e direct data domain method only uses test unit data to suppress clutter at the cost of freedom loss, which is only suitable for uniform linear array and planar array [9]. Sparse recovery STAP obtains the high CCM accuracy by solving underdetermined equation [10]. It can suppress clutter very well with a small number of samples, but the existing sparse recovery STAP has some problems such as sparsity, dispersion, and base mismatch, which will affect the accuracy of the CCM. e second category approaches sample size by fully utilizing various prior knowledge. In [11], the unknown covariance matrix contained the noise was achieved by the knowledge-aided Bayesian framework and a set of training samples. A knowledge-aided model that cared for the nonhomogeneous characteristics of the disturbance with distributed MIMO radar considered in [12] solves the problems of nonhomogeneous environments and insufficient training samples. In addition to the above methods applying the prior statistical distribution, there are also algorithms employing the prior environmental data [13,14]. However, if the prior knowledge is inaccurate, the system will create inaccurate estimates for the CCM in target detecting, leading to poor performance. To avoid this, we reduce sampling size by means of the structure of CCM, including power spectral density [13,15], symmetry [16], and eigenstructure [17]. Notably, the CCM of stationary random signals is Hermitian and Toeplitz; the various estimation and approximation techniques of the Toeplitz matrix have been studied in the last ten years. However, many Toeplitz matrix estimation techniques are based on the assumption of large sample sizes [18].
In this paper, we investigate the STAP radar CCM reconstruction (TCMR-STAP) in uniform linear structure (ULS) and sparse linear structure (SLS), by taking into account the small sample constraint as well as mutual coupling effect. More specifically, the CCM restoration of ULS and SLS is proposed based on the Toeplitz structure. In particular, we first propose a low-rank matrix recovery (LRMR) problem based on Toeplitz structure CCM reconstruction which is then relaxed by replacing the rank norm with the trace norm and calculate the reconstructed CCM by a fast and feasible method. Afterwards, the CCM is used to detect the target effectively according to the STAP theory. Moreover, we reduce the influence of mutual coupling by utilizing the SLS and increase system of degrees of freedom (DOF) under the limited sensors and pulses by difference operation. Simulations demonstrate the theoretical derivations and advantages of the proposed TCMR-STAP algorithm. e remainder of this paper is organized as follows: in Section 2, we introduce the signal model with the ULS and SLS system. Section 3 gives the signal model under the mutual coupling and a fast closed CCM reconstruction method. In Section 4, simulation results and discussions prove its superiority. Finally, Section 5 gains the conclusions.
Notations: (·) T , (·) H are transpose and complex conjugate transpose operators, respectively. ⊗ , E(·), and | · | stand for the Kronecker product, the expected value, and the absolute value, respectively. I M is the M × M identity matrix, and diag(a) represents a diagonal matrix whose diagonal elements are the column vector a. Toeplitz(t) is the symmetric Toeplitz matrix consisting of the vector t. vec(·) represents the vectorization operator. tr[·] and rank[·] stand for the matrix trace and rank, respectively. ‖ · ‖ 2 denotes the l 2 norm. (·) * denotes the adjoint operator. R(·) and I(·) are the real and imaginary operators, respectively. †denotes the pseudoinversion operator.

Signal Model
2.1. e ULS System. Consider a phased array radar system that employs an N sensors uniform linear array (ULA) with intersensors spacing d and M transmitting pulses with fixed pulse repetition interval (PRI) T r during a coherent processing interval (CPI), which is called the ULS system. e velocity of the radar platform is v, and the radar wavelength is λ. e clutter at each range bin can be modeled as the superposition of N c independent clutter patches. e ith clutter patch spatial and temporal steering vectors are, respectively, given as where φ c,i and f c,i are the normalized angle and Doppler frequency of the ith clutter patch, respectively. erefore, the corresponding space-time steering vector can be calculated by e space-time clutter plus noise snapshot from a range bin without the ranger ambiguity is denoted as where a c,i represents the ith clutter patch complex amplitude and n is the Gaussian white noise vector whose power is σ 2 n . us, the clutter plus noise covariance matrix (CNCM) based on (3) can be calculated by  [2]. us, the CCM R c becomes block Toeplitz: where the spatial covariance matrix R (n) ∈ C N×N is a Toeplitz matrix determined by u (n) and v (n) , given by and the sparse linear pulse (SLP) locations are us, the spatial and temporal steering vectors of the ith respectively. Meanwhile, the space-time steering vector of the ith clutter patch can be computed by Furthermore, the space-time clutter plus noise snapshot from a range bin without the ranger ambiguity can be written as Definition 1 (difference coarray). For an array specified by an integer set A, its difference coarray D A is defined as New set D AO consists of different elements of D A . e array is a redundant array if D AO � 0, 1, . . . , N − 1 { }. Generally, the array can be classified into the ULA and SLA from the geometry. It is easy to see that the ULA itself is a redundant array, and the SLA is a redundant array only if the prerequisite mentioned previously is met.
Definition 2 (difference copulse). Similarly, a pulse location is an integer set P and its difference copulse D P is When D PO � 0, 1, . . . , M − 1 { } which is made up of the different elements of D P , the pulse is defined as a redundant pulse. Here, we can divide the pulse into the uniform linear pulse (ULP) and SLP. Normally the ULP is always a redundant pulse, but the SLP needs to meet is paper only considers the redundancy structure.
According to the above theory, further analysis indicates respectively, where the nth rows of the array selection matrix Γ A ∈ 0, 1 { } N S ×N are all zeros except for a single 1 in the A n th position, and Γ P ∈ 0, 1 { } M P ×M means the pulse selection matrix where the element of the mth row is 1 at the P m th position and the remaining elements are all 0. us, the CNCM of the SLS can be estimated by where Γ � Γ A ⊗ Γ P and R cs � ΓR c Γ H . e previous equation degenerates into the CNCM of the ULS if Γ � I. In other words, the ULS is one of the special forms of the SLS.

The Proposed Method
In this section, we describe in detail the design of the proposed TCMR-STAP algorithm with Toeplitz covariance matrix reconstruction. Whether the ULS or SLS, we can reconstruct the CCM quickly and precisely. In order to fully reflect the value of the TCMR-STAP algorithm, we consider the influence of the mutual coupling among sensors.

Mutual Coupling.
In the practical radar system, the received signal has changed owing to the electromagnetic coupling among the sensors. According to the electromagnetic coupling principle, the larger the intersensors spacing is, the smaller the coupling effect is. If the intersensors spacing is greater than a few wavelengths, the effect of coupling can be ignored. erefore, we can describe the mutual coupling effect utilizing the weight function w(n) of an array. w(n) of an array A is defined as the number of element pairs generating the coarray index n, namely, [19] e first three weight functions of the ULA from (16) are It is clear that the SLA have much smaller weight functions compared to the ULA. is means that the mutual coupling effect of SLA is relatively small.
Supposing that the mutual coupling coefficient is zero when the distance between two sensors is larger than B d [20], B is a positive integer. en, the mutual coupling matrix (MCM) of the ULA is formulated as ere are also more sophisticated mutual coupling models [21]. In theory, all of the above models can be applied to the SLA to be discussed in this paper and attain excellent performance. As a matter of fact, the MCM is unknown. If the coupling effect is ignored completely in the design of the STAP filter, the performance of radar detection will degrade or even be disabled.

e ULS System with Mutual Coupling.
e mutual coupling effect in the STAP radar will distort the space-time steering vector. We need to give careful consideration to the mutual coupling effect in actual radar systems. For the ULS system, the clutter plus noise data x um in the presence of mutual coupling can be modified as [20] x where indicates the corresponding spatial steering vector. erefore, by introducing the MCM C, (4) can be modified as where is the clutter space-time steering matrix considering the mutual coupling among the physical sensors; the corresponding CCM denotes R cm � V m PV H m . Here, by comparing (4) with (19), we find where Z � C ⊗ I M .

e SLS System with Mutual Coupling.
In this section, we extend the coupling effect to the SLS, which can be regarded as a subset of ULS. e magnitudes of coupling coefficients are inversely proportional to intersensors spacing. As a result, the SLS has been less affected by the mutual coupling due to its relatively sparse array. By considering the mutual coupling, (11) can be changed as where is the space-time steering vector of the ith clutter patch with mutual coupling. v sm (φ c,i ) � Cv s (φ c,i ) denotes the corresponding of the spatial steering vector. Furthermore, the CNCM of the SLS with mutual coupling has the form as follows: where with Γ � Γ A ⊗ Γ P , Ω � ZΓ. In the light of the minimum variance distortionless response criterion, the optimal STAP weight vector can be estimated by where is the target spacetime steering vector in the presence of mutual coupling. Typically, R usm is estimated from the training samples by where x u,i , i � 1, . . . , L, presents training samples matrix with mutual coupling and L is the number of training samples. Nevertheless, R usm estimated by (25) is subject to the error within the limited snapshots. In order to analyze the estimation error of the training samples in (25), first, note that the vectorization form of estimation error E � R usm − R usm obeys an asymptotic zero-mean normal distribution, that is, and therefore where W � (1/L)R T usm ⊗ R usm and χ 2 (N 2 S M 2 S ) represents the asymptotic Chi-square distribution with N 2 M 2 DOF.
It is remarkable that R usm in the above derivation can be approximately estimated by (25), and the matrix W should be determined based on R usm by Hence, the matrix W in (26) and (27) is replaced by W in real calculations.
Note that can be established with probability 1 − p (approaching 1) on the basis of the properties of the squared distribution, where η can be counted by N S M S and p. According to the theory of the low-rank matrix recovery, R c can be estimated by solving the following minimization problem: However, the previous equation is a NP-hard problem. We can use trace norm instead of rank norm to avoid the nonconvex and obtain min tr R c By introducing a Lagrangian multiplier, the previous equation can be reconstructed as which simplifies to where It is worth noting that the CVX [22] can seek the solution of the above convex optimization problem. However, it needs high computational complexity when the system dimension is relatively large. Here, we try to find a fast and effective closed-form solution. According to the Karush-Kuhn-Tucker (KKT) conditions [23], (33) is equivalent to the following equation: in which with (m − 1)N + 1, · · · , mN + 1 − q , q � 1, . . . , N, According to (34) and (36), we obtain where Z 1 and Z 2 obtained by evenly dividing Z have the same dimension and h stands for the conjugate of h. Nevertheless, (39) can be represented in the following form: which results in In this way, it is easy to calculate h from (41), which obtains the R c (u, tv) estimation. Finally, the optimal STAP filter weight vector can be described by R c (u, tv). It is worthy of note that we acquire the CCM of ULS if Γ A � Γ P � I.

Mathematical Problems in Engineering
Moreover, when C � I, the mutual coupling is not considered in the proposed algorithm model.

Simulation Results and Discussion
In this section, we present numerical simulation results to verify the theoretical derivation made above and compare the TCMR-STAP with the existing methods including the RCML-STAP [24], mDT-STAP [5], JDL-STAP [6], and SMI-STAP [2]. e parameters of the radar system are assumed as λ � 0.05m, d � λ/2, T r � 0.25ms, v � 50m/s, σ 2 n � 1, N c � 361, and clutter to noise ratio CNR � 30dB. In these examples, one target whose normalized angle and Doppler, respectively, are set to 0.1 and − 0.2 is present. All simulation results are average over 100 Monte Carlo experiments.

Performance of TCMR-STAP with ULS.
We begin by analyzing the presented approach feasibility and efficiency in small samples, set M � N � 10 in the ULS. As a comparison, the output SINR versus the number of training samples and the normalized Doppler frequency are shown for performance evaluation. Figure 1 reveals the output SINR of five methods by varying samples size from 2 to 200 in the target normalized Doppler frequency − 0.2. As intuitively expected, the steady SINR performance increases with sample number except for the SMI-STAP [2]. SMI have a dip due to numerical instabilities in the L � 100 training regime. It is seen that the mDT-STAP, JDL-STAP, and SMI-STAP are able to achieve poor performance when very small snapshots are collected, and the RCML-STAP enjoys higher precision among the above three methods. However, the performance of TCMR-STAP is the least affected by the number of training samples among the compared methods, which can exhibit remarkably good performance even when there are only several training samples and also converge much fastest than other schemes. In particular, the TCMR-STAP is able to work in limited samples scenario and solves the radar performance degradation caused by the lack of samples.
Moreover, we compare the SINR performance against the normalized Doppler frequency of the TCMR-STAP with others in L � 100, as shown in Figure 2. e plot shows that the TCMR-STAP outperforms other algorithms in the Doppler bins, and forms a deep null to cancel the main clutter field.

Performance of TCMR-STAP with SLS.
Here, we compare the SINR and space-time beampatterns in order to reflect the advantages of our method in SLS with A � P � 0, 1, 4, 7, 9 { } and L � 100. First, we evaluate the SINR performance against the normalized Doppler frequency for the proposed TCMR-STAP with SLS (SLS-TCMR-STAP) under N � M � 5, which is shown in Figure 3. e plot shows that our method outperforms other algorithms in the Doppler domain and has a deep null in the main clutter field. e SLS-TCMR-STAP, on the other hand, has approximately the same performance as the TCMR-STAP with ULS (ULS-TCMR-STAP) with N � M � 10.
To further illustrate the superiorities of TCMR-STAP, we compare the space-time beampatterns of the SMI-STAP and TCMR-STAP and get satisfactory results. Specifically, It is seen that all of the three methods can fully suppress the clutter and make a maximum peak at the target position. Nevertheless, the angle and Doppler resolutions of the SLS-TCMR-STAP are better than those of the SMI-STAP with ULS under the same N and M owing to the increased DOF. Moreover, the SLS-TCMR-STAP is notably better than the SMI-STAP with ULS (N � M � 10), as shown in Figure 5.
We then give the beampatterns in spatial and Doppler domains. Figure 6(a) shows the beampatterns in the spatial domain at the target normalized Doppler frequency while Figure 6(b) plots the beampatterns in the Doppler domain at   Mathematical Problems in Engineering the target spatial frequency. It is observed that the SLS-TCMR-STAP (N � M � 5) can obtain the same angle-Doppler resolution as the SMI-STAP with ULS (N � M � 10), but its sidelobe level is relatively better. It means that the SLS system can provide larger DOF in the virtual domain and improve the space-time resolution with fewer hardware resources and power consumption. On the other hand, it is shown the TCMR-STAP has a stronger capability of clutter suppression by means of analyzing the plots.

Performance of TCMR-STAP with Mutual Coupling.
In the experiment, we evaluate the mutual coupling of the SLS-TCMR-STAP with M � N � 5. en, we also show the mutual coupling effect on the beampatterns in Figure 7, where the mutual coupling model is based on (18) with c 1 � 0.5e (jπ/4) , c 2 � 0.25e j0.7π , c 3 � 0.5e j0.7π /3, and B � 3. e number of samples is L � 200. e SMI-STAP with ULS possess no deep notch in the main clutter region, implying it suffers from the severe mutual coupling effect, as shown in Figure 7(a). For the SLS-TCMR-STAP, since the intersensors spacing is increased, the mutual coupling can be further reduced; as a result, its weight functions can be lower than those of the ULS shown in Figure 7(b). erefore, the SLS is emphasized on the importance of the comprehensive balance between the DOF and the mutual coupling. Figure 8 shows the beampatterns in spatial and Doppler domain in the presence of mutual coupling. As shown in Figure 8, the

Conclusion
e traditional training samples mean the method is effective in estimating the CNCM, but it is difficult to obtain the CNCM accurately with limited training samples. is paper has proposed a robust STAP algorithm for airborne radar against unknown mutual coupling considering the Toeplitz covariance matrix reconstruction, which can be applied to the ULS and SLS system to solve the problem of insufficient training samples and reduce the mutual coupling effect. In particular, we first adopt the structure for CCM recovery based on the prior knowledge of noise and then relax it by using the trace norm instead of the rank norm. e closed-form solution of the problem is derived, and a feasible implementation algorithm's fast solution is provided. Moreover, under the conditions fixed in the number of sensors and pulses, the system can obtain higher DOF by applying the difference operation in the SLS and reduce the mutual coupling effect. e simulation and experiment results show that the proposed method has high estimation precision, even in small samples. Compared with the existing algorithms, the TCMR-STAP possesses much better performance of clutter suppression, especially a very small number of training samples. Compared with the existing    Mathematical Problems in Engineering algorithms for the ULS in the fixed number of sensors and pulses, the TCMR-STAP shows a greater DOF and decreases the mutual coupling. In future research, we will consider the target-like interference and assess the performance of the TCMR-STAP with the measured data.
Data Availability e datasets supporting the conclusions of this article are generated using MATLAB software by authors according to the radar parameters described in the manuscript.

Conflicts of Interest
e authors declare no conflicts of interest.