A generalised eigenvalue reweighting covariance matrix estimation algorithm for airborne STAP radar in complex environment

National Natural Science Foundation of China, Grant/Award Number: 61801383 Abstract To improve the space‐time adaptive processing (STAP) performance of airborne radar in complex environments, a generalised eigenvalue reweighting covariance matrix estimation algorithm called GERCM is proposed here. First, the interference plus noise (IPN) covariance matrix of cell under test (CUT) data is estimated by the selected target‐free training samples around the CUT with the sample covariance matrix method. Then, with the component decompositions of the selected training samples and the assumption of approximately equal subspace, the IPN covariance matrix of CUT data is reformulated by the eigenvector matrix, eigenvalue matrix, and the eigenvalue reweighting vector. Subsequently, based on the modified covariance matching estimation criterion, the eigenvalue reweighting vector is estimated by solving the redesigned convex optimisation problem with the Lagrange dual method. Finally, the STAP weight vector is calculated to process the CUT data. The proposed algorithm can obtain a relatively accurate IPN covariance matrix of CUT data by sufficiently utilising the non‐homogeneous training samples and can effectively protect the moving targets in CUT data, which can be applied to airborne radar with arbitrary array structure and antenna configuration. Simulation results and performance analyses based on the multi‐channel airborne radar measurement data demonstrate the effectiveness of the proposed GERCM algorithm.


| INTRODUCTION
For an advanced airborne surveillance radar system, ground clutter returns are spread in Doppler because of platform motion. Compared with the ground-based radar, the airborne radar faces a more severe clutter problem in detecting small and slow moving targets. Space-time adaptive processing (STAP) is a powerful two-dimensional filtering technique for airborne phased-array radar to detect these moving targets. To test the existence of a target in the cell under test (CUT), the optimal STAP filter is calculated using the optimal interference plus noise (IPN) covariance matrix [1], which is usually estimated by the target-free training samples that are independent and identically distributed (i.i.d.) with the clutter signal in CUT data. According to the famous RMB criterion [2], the number of required target-free i.i. d. training samples is at least twice the system degrees of freedom (DoF) for obtaining less than 3dB average performance loss compared with the optimal filter. However, as a result of complex terrains, different land covers, moving targets, and geometric configuration, the sample requirements for effectively training the STAP filter cannot be met for airborne STAP radar working in a heterogeneous environment. This seriously degrades the clutter suppression performance of airborne STAP radar.
To improve the clutter suppression performance of airborne radar in a heterogeneous environment, STAP researchers proposed many different algorithms. Earlier approaches include the auxiliary channel receiver (ACR) [3], the multi-stage wiener filter (MWF) [4,5], and the knowledge-aided (KA) algorithm [6,7].
Although these algorithms can reduce the training requirements of an STAP filter, they suffer from several drawbacks. For example, it is hard for the ACR algorithm to determine the appropriate DoF for different range cells in an actual environment. Additionally, the traditional ACR algorithm cannot solve the clutter suppression problem of airborne STAP radar with non-side-looking configuration. In a heterogeneous environment, it is difficult for the MWF algorithms to select the suitable clutter rank. If the selected rank is lower than the best one, then the clutter component in CUT data cannot be effectively cancelled. On the contrary, if the selected rank is higher than the best one, then the noise will be enhanced. These will greatly reduce the clutter suppression performance of the MWF algorithms. In essence, KA algorithms are the clutter suppression algorithms that are based on the parameterised model or priori knowledge. This leads the performances of KA algorithms to be heavily dependent on the model precision and the accuracy of the priori knowledge. When the model mismatch happens or the priori knowledge is not accurate, the KA algorithms have poor clutter suppression performance. We note that the ACR, MWF, and KA algorithms only partially reduced the sample requirements for training the STAP weight vector. In terms of the increasingly complex work environment confronted by airborne radar, it is very important to further reduce the training sample requirements. To this end, many STAP algorithms with lower sample requirements have been developed. These developments include direct data domain (D3) [8,9], and sparse recovery (SR) [10,11]. These mentioned algorithms can reduce the training requirements to a single or several snapshot data but they also have some shortcomings. As the space-time sliding window processing of CUT data and the target protection is based on the direct cancellation in the element-pulse domain, D3 algorithms have an inherent problem of aperture loss and high minimum detectable velocity (MDV). In recent years, many STAP algorithms based on the clutter sparse recovery (SR) technique have been proposed successively. Utilising the sparseness of ground clutter in the angle-Doppler plane, STAP algorithms based on the SR technique (SR-STAP) can be usually summarised as a two-step procedure: first, using a certain SR algorithm to obtain the clutter space-time profile of CUT data; second, constructing the IPN covariance matrix with the obtained clutter profile and calculating the STAP weight vector to process the CUT data. Here, some typical SR-STAP algorithms are developed, such as iterative adaptive algorithm (IAA) [12], fast converging sparse Bayesian learning (FCSBL) [13], robust knowledge-aided sparse recovery (RKASR) [14], and parameter-searched orthogonal matching pursuit (PSOMP) [15]. Conventional SR-STAP algorithms can drastically decrease the training sample requirements, but they usually cannot thoroughly solve the off-grid problem even if grid mismatch correction is done [15] or a gridless technique [16] is used. As all these SR-STAP algorithms are fully space-time algorithms, they often have huge computational workload when the space-time DoF of airborne STAP radar is high. According to the latest research finding presented in [17], SR-STAP algorithms cannot accurately estimate the IPN covariance matrix using only CUT data. So, a feasible approach is to estimate the IPN covariance matrix with the sufficient employment of the non-homogeneous training samples in a complex environment, such as the sample reweighted (SRW) algorithms [18,19]. In a partially heterogeneous environment, SRW algorithms can achieve lower clutter residual than in the traditional sample covariance matrix (SCM) method, but they cannot adequately suppress the strong ground clutter in the complex environment. In other words, SRW algorithms still have the problem of a lack of fully qualified training samples. To increase the number of qualified training samples in the complex environment, the persymmetric extend factor approach (PerEFA) [20] is proposed. As the PerEFA algorithm can increase the additional i.i.d. training samples, it can obtain better clutter suppression performance than the SRW algorithms. Nevertheless, the PerEFA algorithm relies heavily on the array structure, and it is sensitive to the cross-term matrix when the antenna array has an array error or the local oscillator system has phase noise, which results in the poor performance of the PerEFA algorithm.
To overcome the problems mentioned above, a generalised eigenvalue reweighting covariance matrix estimation algorithm called GERCM, which can sufficiently utilise the nonhomogeneous training samples and reduce the influences of the cross-term matrix in the PerEFA algorithm, is proposed for airborne STAP radar working in complex environments. The proposed GERCM algorithm mainly contains the following key steps: first, estimating the initial IPN covariance matrix of CUT data by using the selected target-free training samples with the SCM method; second, taking the eigenvalue decomposition to the initially estimated IPN covariance matrix of CUT data and reformulating the IPN covariance matrix of CUT data with the eigenvector matrix, eigenvalue matrix, and the eigenvalue reweighting vector; third, estimating the eigenvalue reweighting vector by solving the redesigned convex optimisation problem with the Lagrange dual method; and finally, calculating the STAP weight vector combined with the obtained eigenvector matrix, eigenvalue matrix, and the eigenvalue reweighting vector to process the CUT data.
The main contributions of this work are listed as follows: (1) The cross-term matrix problem of the traditional PerEFA algorithm is qualitatively analysed and a new covariance matrix estimation algorithm for airborne STAP radar working in complex environments is proposed in this work. (2) By the modified covariance matching estimation criterion, a redesigned convex optimisation problem which can be solved by the Lagrange dual method is formulated to estimate the eigenvalue reweighting vector, and we derived the analytical solution of the estimated eigenvalue reweighting vector. The remaining sections are organised as follows: Section 2 describes the problem formation of airborne pulsed-Doppler radar. In Section 3, analyses of the disadvantage of the Per-EFA algorithm are provided first. Then, we introduce the key theory of the proposed GERCM algorithm. Finally, we summarise the performance advantages and computational complexity of the proposed GERCM algorithm. Simulation results and performance analyses are given in section 4. Conclusions are drawn in section 5.
Notations: ð·Þ T , ð·Þ H , and ð·Þ −1 are the transpose, conjugatetranspose, and matrix inverse operators, respectively; ‖ · ‖ 2 and ‖ · ‖ F are the 2-norm of a vector and Frobenius norm of a matrix, respectively; diagð·Þ and ≻ are the diagonalisation and element-wise greater-than operators, respectively; ℝ K�L and C K�L are the sets of real and complex matrices of dimension K � L, respectively; I K , 0 K , and 1 K are the identity matrix of order K, K -dimensional all-zero and all-one column vectors, respectively; Reð·Þ and Imð·Þ are the real and imaginary parts of a complex vector, respectively; ⊗ and ⊙ are the Kronecker and Khatri-Rao product operators, respectively; Spanð·Þ denotes the generated space spanned by a matrix;

| PROBLEM FORMATION
The platform geometry of the airborne radar is presented in Figure 1. Without loss of generality, a pulsed-Doppler radar system consisting of a uniform linear array of N elements and spacing d is under consideration. This radar system with wavelength λ r moves along the Y axis at an altitude H and constant velocity V p . M pulses are transmitted at a constant pulse repetition frequency (PRF) f r in one coherent processing interval (CPI). According to the platform geometry shown in Figure 1, the normalised spatial frequency f s and normalised Doppler frequency f d of the clutter patch S at a certain range cell can be expressed as where φ and θ are the elevation and azimuth angles, respectively, of the clutter patch S. α is the is installation angle between antenna axis and flight direction. The spatial steering vector v s and temporal steering vector v t of the clutter patch S can be defined as Considering the range-ambiguous clutter, the received clutter data that come from the rth range cell can be expressed as where a k; r; u , v s k; r; u , v t k; r; u , and v k; r; u are the complex amplitude, spatial steering vector, temporal steering vector, and the spatial-temporal steering vector, respectively, of the kth clutter patch at the uth ambiguous range cell,. Nu and Nc are the number of range-ambiguous and independent clutter patches, respectively.
To correctly decide whether the target exists or not [21,22], the core of the STAP algorithm is to estimate the optimal IPN covariance matrix of the rth range cell, that is, R Opt r . Assuming that the clutter and noise components are mutually independent, R Opt r can be given by where p k; r; u denotes the clutter patch power of the kth clutter patch at the uth ambiguous range cell, and σ 2 n is the white noise power.
In light of the linearly constrained minimum variance criterion, the well-known optimal STAP weight vector of the lth Doppler bin at the rth range cell can be written as F I G U R E 1 Platform geometry of airborne radar XIAO ET AL.
where s T; l represents the target spatial-temporal steering vector of the lth Doppler bin. Since R Opt r is unknown in advance, it is usually estimated by the selected target-free homogeneous training samples around the CUT. However, due to the limitations of many practical factors in a complex clutter environment, it is impossible to obtain sufficient homogeneous training samples for effectively estimating the optimal IPN covariance matrix R Opt r . Thus, the STAP algorithm, which can sufficiently utilise the non-homogeneous training samples, is urgently needed for future airborne radar.

| Shortcoming analyses of the traditional PerEFA algorithm
To sufficiently utilise the training samples in a complex environment, based on the persymmetric structure of the IPN covariance matrix of CUT data the Q s selected target-free training samples are used by the PerEFA algorithm [20] to estimate the IPN covariance matrix of CUT data, that is, where X ¼ ½x 1 ; x 2 ; ⋯; x Q_s � ∈ C D 0 �Q_s denotes the original training sample matrix and x q ∈ C D 0 �1 ðq ¼ 1; 2; ⋯; Q_sÞ is the selected qth training sample. D 0 ¼ 3N is chosen as the reduced-dimension system DoF given in Ref. [20].
are the spatial, temporal, and spatialtemporal transformed training sample matrices, respectively. J s , J t , and J st are the spatial, temporal, and spatial-temporal permutation matrices, respectively, which can be seen in Ref. [20]. T d is the reduced-dimension matrix of the traditional EFA algorithm [1]. In a complex environment, if the selected Q_s training samples are homogeneous with the clutter signal in the CUT, the number of i.i.d. training samples additionally increased by the PerEFA algorithm is 3Q_s, which leads to the super performance of the PerEFA algorithm. However, the PerEFA algorithm can only be applied to the clutter suppression problem of airborne STAP radar with uniform sampling in the space-time dimension, ideally calibrated antenna array and an extremely stable local oscillator system. When the airborne STAP radar is not uniformly sampling in space-time dimension, the antenna array has an array error or the local oscillator system has phase noise, all the increased 3Q_s training samples will have different statistical properties with the clutter component in CUT data. Thus, the PerEFA algorithm has the perturbation problem of the cross-term matrix. To analyse the cross-term matrix's problem, we only consider the influences of the array error and phase noise here. Then, the increased spatial training sample matrixX s , temporal training sample matrixX t , and the spatial-temporal training sample matrixX st can be expressed asX where T ta ¼ diagðe p Þ ⊗ diagðe a Þ is the space-time taper matrix, e a ∈ C N�1 and e p ∈ C M�1 are the array errors modelled in [23] and the phase noise introduced in [24], respectively. Then, the IPN covariance matrix of CUT data estimated by the PerEFA algorithm can be expressed as Because of the complex environment confronted by the airborne STAP radar, the non-homogeneous samples are inevitably selected to estimate the IPN covariance matrix of CUT data by a certain non-homogeneous detector, such as the generalised inner product (GIP) criterion. Then, the practically qth training samplex q ∈ C D 0 �1 ðq ¼ 1; 2; ⋯; Q_sÞ can be decomposed intox wherex Ho q is the homogeneous component with the same statistical property as the clutter component in CUT data,x He q is the heterogeneous component with statistical properties that are different from the clutter component in CUT data, andñ q is the noise data. Then, the cross-term matrix problem of the PerEFA algorithm can be qualitatively illustrated by the following expression:

| Key theory of the GERCM algorithm
For the sake of convenient description, we first introduce some definitions and analyses, which can be used to derive the proposed GERCM algorithm.
Considering the array error and phase noise, the practically received clutter data come from the lth and ðl þ 1Þth range cells, respectively, can be expressed as As there is a correlation between v k; l; u and v k; lþ1; u [19] and T ta is not related with the range cells, the practical clutter subspace of the lth range cell,Φ c l , is approximately equal to the clutter subspace of the ðl þ 1Þth range cell,Φ c lþ1 , that is As T d is a full column rank matrix in the context of conventional EFA algorithm, we havẽ whereΘ c l andΘ c lþ1 are the reduced-dimension clutter subspaces of the lth and ðl þ 1Þth range cells, respectively. Definẽ C q ¼ ½C c; q ;C n; q � ðq ¼ 1; 2; ⋯; Q_sÞ as the basis matrix of clutter plus noise data in the training samplẽ x q ðq ¼ 1; 2; ⋯; Q_sÞ, whereC c; q andC n; q are the corresponding basis matrices of the clutter and noise subspaces, respectively. As the heterogeneous componentx He q also falls in the clutter subspace ofx q ,x q can be represented as wherez Ho q ∈ C ρ c �1 ,z He q ∈ C ρ c �1 , andz n q ∈ ℂ ðD 0 −ρ c Þ�1 are the representation coefficients of thex Ho q ,x He q , andñ q components, respectively. ρ c denotes the effective clutter rank [1]. As the practical radar antenna has an array error and the local oscillator system has phase noise, the persymmetric structure of the IPN covariance matrix of CUT data is not on hold. Thus, the training sample matricesX s ,X t , andX st cannot be used to estimate the IPN covariance matrix of CUT data. Keeping these facts in mind, the initially estimated IPN covariance matrix of the reduced-dimension CUT datax cut ∈ C D 0 �1 by the SCM method, that is,R SCM cut , can be represented by the underlying form XIAO ET AL.
AsZ and ΔZ are diagonal matrices, we setZ ¼ diagð½z 1 ;z 2 ⋯;z D 0 �Þ and ΔZ ¼ diagð½Δz 1 ; Δz 2 ⋯; Δz D 0 �Þ, wherez i ≥ 0 ði ¼ 1; 2; ⋯; D 0 Þ and Δz i ði ¼ 1; 2; ⋯; D 0 Þ are the ith diagonal entry of matrixZ and ΔZ , respectively. Based on the aforementioned definitions and analyses, a generalised eigenvalue reweighting covariance matrix estimation algorithm is proposed to estimate the optimal IPN covariance matrix of CUT datax cut ∈ £ D 0 �1 , that is,R GERCM cut whereũ ¼ ½ũ 1 ;ũ 2 ; ⋯;ũ D 0 � is the positive eigenvalue reweighting vector which needs to be estimated, and u i ði ¼ 1; 2; ⋯; D 0 Þ is the ith entry ofũ. To estimateũ, the underlying covariance matching estimation criterion is introduced as [26] f ¼ Substitute the expression ofR GERCM cut into Equation (25). Then, Equation (25) can be recast as wherec i is the ith eigenvector inC. Then, we can estimate the eigenvalue reweighting vectorũ by the undermentioned optimisation problem where the first inequality constrains that the modified clutterto-noise ratio (CNR) is greater than the unmodified CNR [27,28], and the second inequality constrains that the entries of the eigenvalue reweighting vectorũ are positive real numbers. As the objective function in Equation (27) is convex [29] and the feasible sets of Equation (27) are convex sets, Equation (27) is a convex optimisation problem, which can be iteratively solved by the canonical primal-dual interior-point algorithm [29]. Nevertheless, we find thatũ is implied in the cost function of Equation (27). This will incur some inconveniences in solving Equation (27) with the help of the primal-dual interiorpoint algorithm. Additionally, the clutter rank ρ c is difficult to determine for the airborne STAP radar working in the complex environment [30]. To obtain the close-form solution ofũ, an alternative convex problem is redesigned to estimate the eigenvalue reweighting vectorũ. First, to avoid the matrix inverse and operations of taking the real and imaginary parts in Equation (27), we replace the cost function in Equation (27) with the following simplified form: After the cost function in Equation (27) is replaced by Equation (28), Equation (27) can be rewritten as Next, to eliminate the obstacle of determining the reduceddimension clutter rank ρ c of the IPN covariance matrix of CUT data in the complex environment, the first inequality constraint in Equation (30) can be converted into Asũ is a positive real vector, utilising the Cauchy-Schwartz inequality [25] and the property ofz i ≥ 0 ði ¼ 1; 2; ⋯; D 0 Þ we have Based on the new inequalities Equations (31) and (32), the first inequality constraint in Equation (30) can be replaced by the underlying two inequality constraints, respectively, where Equations (33a) and (33b) are mainly used to guarantee the estimated accuracy of the IPN covariance matrix of CUT data and prevent the target self-nulling of the proposed GERCM algorithm. Definer de as a column vector composed of the diagonal entries inZ with the descend order andC de as the corresponding eigenvector matrix. Then, Equation (30) can be recast as the equivalent form with the above information, the augmented Lagrange function of Equation (34) is given by are the Lagrange multipliers [29]. Define g d ¼ ½α d ; β d ; t d � as the dual variable. Then, the augmented Lagrange dual function of Equation (34) can be expressed as where and domðũÞ denotes the definition domain ofũ determined by the constraints in Equation (34). According to the conclusion presented in [29], the dual optimisation problem of Equation (34) can be represented as Solving the dual problem Equation (39), the optimal dual variable, g Opt d , can be given as As the inequality constraints in Equation (34) are affine, the Slater criterion is satisfied, which means that Equation (34) has strong duality [29]. This indicates that the eigenvalue reweighting vectorũ can be estimated by the underlying problem Solving Equation (41) by the minimisation method of unconstrained least squares problem, the estimatedũ can be represented by ð38bÞ After the eigenvector matrixC, eigenvalue matrixZ , and the eigenvector reweighting vectorũ Δ are obtained, the filtered output of the CUT datax cut at the lth Doppler bin, that is, Y cut ðlÞ, can be written as wherew GERCM cut ðlÞ is the STAP weight vector of the lth Doppler bin calculated by the proposed GERCM algorithm.

| Advantage analysis of the GERCM algorithm
Compared with the existing PerEFA algorithm, a relatively accurate IPN covariance matrix of CUT data can be estimated by the proposed GERCM algorithm by sufficiently utilising the non-homogeneous training samples, which leads to the better performance of airborne STAP radar working in complex environments. As the proposed GERCM algorithm is independent of the priori information on the array structure and antenna configuration, theoretically, the proposed GERCM algorithm can be applied to the clutter suppression problem of airborne radar with arbitrary array structure and antenna configuration. Since the reformulated IPN covariance matrix of CUT data based on the component decompositions of selected target-free training samples and the assumption of approximately equal subspace has nothing to do with the preprocessing of CUT data, the proposed GERCM algorithm derived under the framework of element-space post-Doppler can be generalised to other frameworks, such as beam-space post-Doppler. In addition, as the proposed GERCM algorithm is a sample-dependent statistical STAP algorithm, it also can suppress the rangeambiguous clutter.

| Complexity analysis of the GERCM algorithm
To analyse the computational complexity of the proposed GERCM algorithm, the computational complexity [31] is measured in terms of the number of complex multiplications in this work. For the reduced-dimension system DoF D 0 , the complexity of the eigenvalue decomposition is Oð23D 0 3 Þ; the complexity of obtaining the estimated eigenvector reweighting vectorũ Δ is Oð23D 0 3 þ 8ðD 0 2 þ 2MND 0 ÞÞ; and the complexity of the weight vector calculation for clutter suppression is Oð23D 0 3 þ 8D 0 2 Þ. Thus, the total complexity of the proposed GERCM algorithm is Oð69D 0 3 þ 16ðD 0 2 þ MND 0 ÞÞ. Implementation steps of the proposed GERCM algorithm are summarised in Table 1.

| NUMERICAL EXPERIMENTS
Based on the MCARM data from flight 5, acquisition 151, in this section we assess the algorithm performance of the proposed GERCM algorithm by comparing with other existing STAP algorithms. The main parameters of the MCARM data are listed in Table 2. As shown in Figure 2, this simulation scene contains rich terrains (such as river, lake, mountain, island, and so on) and different land covers (such as forest, road, man-made building, and so on) [32]. These factors will result in the heterogeneity of the echo data received by the airborne STAP radar. As we can see from the processing result of conventional pulse-Doppler (PD) given in Figure 3, the echo data coming from different range cells show the obvious power heterogeneity under the influences of antenna pattern, slant range, complex terrains and different land covers. Clearly, simulation results based on the measured MCARM data can sufficiently illustrate the clutter suppression performance of the proposed GERCM algorithm in complex environments. Here, the returns from only the top row of the MCARM array, consisting of 11 chosen channels from 22 available channels, are used for the following numerical experiments. Meanwhile, 66 target-free training samples are selected from 72 initial training samples by the sliding

GERCM cut
Step 1: Estimating the initial IPN covariance matrix of CUT data by the SCM method with the selected target-free training samples to obtain the matrixR SCM cut Step 2: Taking the eigenvalue decomposition to matrixR SCM cut for obtaining the eigenvector matrixC and eigenvalue matrixZ .
Step 4: Reconstructing the final IPN covariance matrix of CUT data with the obtainedC,Z , andũ Δ components by (24).
window GIP method, discarding two guard cells to prevent target self-nulling in all listed STAP algorithms.
To demonstrate the effectiveness of the proposed GERCM algorithm, the following comparative simulations including range-Doppler processing, averaged clutter power residual, target protection, and detection performance are taken into account to conduct the corresponding numerical experiments. As the radar antenna array inevitably has an array error and the optimal IPN covariance matrix of CUT data is unknown in advance, we do not conduct the comparative simulations individually in the case of an array error and plot the improvement factor curves in the underlying part of this section.

| Comparison simulations of range-Doppler processing
To demonstrate the clutter suppression performance of the proposed GERCM algorithm, we plot the range-Doppler processing results of the listed six STAP algorithms in Figure 4, which are the extended factored approach (EFA) [1], generalised eigenvalue reweighting covariance matrix extended factored approach (GERCM-EFA), PRI-staggered (PRIS) [1], generalised eigenvalue reweighting covariance matrix PRI-staggered (GERCM-PRIS) approach, persymmetric extended factored approach (PerEFA) [20], and the generalised eigenvalue reweighting covariance matrix persymmetric extended factored approach (GERCM-PerEFA). Here, 3 Doppler bins are chosen for the EFA and PRIS algorithms. As the phase noise can be ignored in a CPI [24], with the increased 66 target-free training samples obtained by the temporal transform, a total of 132 training samples are used in the context of PerEFA algorithm. Compared with the processing results shown in Figures. 4(a)

| Comparison simulations of averaged clutter power residual
To compare the averaged clutter suppression performance of the STAP algorithms listed in Figure 4, utilising the residual data taken from 1th to 557th range cells and 1th to 128th Doppler bins processed by these aforementioned algorithms, the averaged clutter power residual (ACPR) curves obtained by the average of range-dimension the average of Doppler-dimension are depicted in Figure 5 and Figure 6, respectively. Figure 5 shows that the GERCM-EFA, GERCM-PRIS, and GERCM-PerEFA algorithms can significantly outperform the EFA, PRIS, and PerEFA algorithms in the mainlobe and sidelobe Doppler regions, which implies that the proposed GERCM algorithm can effectively reduce the false alarm caused by the Doppler-spreaded ground clutter. In addition, Figure 6 shows that the GERCM-EFA, GERCM-PRIS, and GERCM-PerEFA algorithms can achieve lower ACPR values than those of the EFA, PRIS, and PerEFA algorithms at most range cells. This means that the proposed GERCM algorithm can effectively reduce the false alarm caused by the severe ground clutter coming from different range cells.

| Comparison simulations of target protection
To examine the target protection performance of the proposed GERCM algorithm, 22 moving targets are injected in the 200th and 550th range cells, respectively. In Figure 7, the target power values are plotted, where 'true value' denotes the original power values of injected moving targets. Figures 7a and 7b show that the target power values obtained from the filter-processing of the GERCM-EFA, GERCM-PRIS, and GERCM-PerEFA algorithms are almost coincided with the true power values at the shown Doppler bins, which means that the proposed GERCM algorithm can effectively protect the short-range and longrange moving targets with different size and Doppler frequencies.

| Comparison simulations of detection performance
To evaluate the detection performance of the proposed GERCM algorithm, as presented in Figure 8, we plot the probability of detection (Pd) versus input signal-to-noise ratio (SNR) curves, which are obtained by applying the cell average constant false alarm rate (CA-CFAR) detector to the filtered data processed by the EFA [1], PRIS [1], AIWCM [18], RAPR [19], and PerEFA [20] algorithms.
Here, assuming that the simulated targets are in boresight and are uniformly distributed from 101th to 300th range cells, we consider two groups of CUTs: a) 1th -45th Doppler bins which simulate the relatively fast targets moving away from airborne radar and b) 84th -128th Doppler bins which simulate the relatively fast targets moving towards airborne radar. Each group is conducted by the 9000 Monte Carlo simulations. In these simulations, the probability of false alarm (Pfa) is set to be 10 −6 . Figures 8a  and 8b show that the GERCM-EFA, GERCM-PRIS, and GERCM-PerEFA algorithms have apparent performance improvements compared to those of the EFA, PRIS, Per-EFA, AIWCM, and RAPR algorithms, which means that the proposed GERCM algorithm can help airborne STAP radar effectively detect relatively fast moving targets' fall in these regions.
To further demonstrate the detection performance of the proposed GERCM algorithm, we plot receiver operating characteristic curves in Figure 9. Here, the simulated targets are injected from 101th to 300th range cells, and the directions of arrival for all simulated targets are at the centre of the mainlobe. We also consider the two groups of CUTs: a) 46th -63th Doppler bins which simulate the relatively slow targets moving away from airborne radar and b) 66th -83th Doppler bins which simulate the relatively slow targets moving towards airborne radar. Each group is conducted by the 9000 Monte Carlo simulations. In these simulations, the SNRs of injected moving targets are set to be −36dB. Figures 9a and 9b show that the GERCM-EFA, GERCM-PRIS, and GERCM-PerEFA algorithms have higher Pd values than those of the EFA, PRIS, PerEFA, AIWCM, and RAPR algorithms at the most shown Pfa values, which indicates that the proposed GERCM algorithm can largely improve the MDV performance of relatively slow moving targets that are slightly deviated from the main beam of the radar.

| CONCLUSION
In this work, a novel generalised eigenvalue reweighting covariance matrix estimation algorithm for the clutter mitigation of airborne STAP radar working in complex environments is proposed. The proposed GERCM algorithm could obtain a relatively accurate IPN covariance matrix of CUT data by sufficiently utilising non-homogeneous training samples in a complex environment and effectively protect the moving targets with different size and Doppler frequencies in the CUT data, which could solve the clutter suppression problem of airborne radar with arbitrary array structure and antenna configuration. Although the proposed GERCM algorithm is derived in the context of elementspace post-Doppler, theoretically, it can be generalised to other frameworks, such as beam-space post-Doppler. In addition, as it is a sample-dependent statistical STAP algorithm, it also can effectively eliminate range-ambiguous clutter. Simulation results and performance analyses based on the high-fidelity MCARM data demonstrated that the proposed GERCM algorithm could effectively suppress heterogeneous clutter and greatly improve the detection performance of moving targets in complex environments. -15