A Robust Space-Time Adaptive Processing Method by Linear Programming

Abstract

The main aim of the airborne early warning (AEW) system is to search the potential targets in a large surveillance area. The underlying assumption is that the desired target signals only exist in a few range cells for space-time adaptive processing (STAP), i.e., targets (with certain look direction and Doppler) are sparsely distributed in the entire range cells and most of the range cells are target-free. By utilizing the sparsity of the target distribution, we propose a new STAP method by minimizing the l1-norm of the output magnitude. Unlike conventional STAP methods, which exclude the cell under test from the training samples to avoid target self-nulling, our method processes the cell under test (CUT) and the training samples simultaneously without sample selection. Moreover, to achieve robustness against target steering vector mismatch, we constrain the l1-modulus of the response of any steering vector within a rhombus uncertainty set to exceed unity. Additionally, based on a new definition of the l1-norm of a complex-valued vector, the original nonlinear programming problem can be transformed into a linear programming problem. On the other hand, unlike the slide window processor (SWP) whose weights need to be updated for each range cell, the adaptive weight of our method for a block of samples requires no updating. Consequently, the computational complexity of the proposed method is much lower than that of conventional STAP methods. Finally, since the CUT is used to compute the STAP weights, our method can also suppress the discrete interference. The robustness, computational effectiveness and superiority of the proposed STAP method are verified based on simulated data and the MCARM data.

1. Introduction

Space-time adaptive processing (STAP) is a classic two-dimensional adaptive filtering technique for airborne radar systems. It suppresses ground clutter in both spatial and temporal domains to improve the detection performance of slow-moving targets. By utilizing the distinct spatial and Doppler features of target echoes, ground clutter and external interference, STAP effectively extracts target signals from complex radar backgrounds [1,2,3,4,5]. Despite these advantages, traditional STAP methods still face three key limitations in practical applications. First, real airborne clutter environments are time-varying and non-stationary, making it difficult for conventional statistical STAP methods to obtain enough homogeneous training samples for reliable clutter covariance matrix (CCM) estimation [6,7]. Second, the inversion of high-dimensional CCMs requires substantial matrix calculations, leading to high computational cost that restricts real-time radar processing [8]. More importantly, to avoid target self-nulling, conventional STAP methods remove the cell under test (CUT) and other target-containing samples from the training set. This requires updating adaptive weights for each range cell and further increases computational burden [9,10]. Third, actual radar systems always suffer from imperfect target parameter knowledge. Even small errors in preset target direction or Doppler frequency can cause severe self-nulling effects and significantly degrade detection performance [1,11,12].

To overcome the above problems, various improved STAP algorithms have been developed over the past decades, which can be classified into six typical categories according to their implementation principles. (1) Reduced-dimensional STAP (RD-STAP): Typical methods such as the extended factored algorithm (EFA) and joint domain localized (JDL) algorithm reduce the dimension of the processing domain, and thus decrease the required number of training samples [1,2]. (2) Reduced-rank STAP: Techniques including principal component STAP (PC-STAP) [13,14,15] and multistage Wiener filter STAP (WMF-STAP) [16] remove redundant clutter components and reduce sample demand based on the actual clutter rank. (3) Knowledge-aided STAP (KA-STAP) [3,17]: This method uses prior geographic and radar data to assist CCM estimation, while its performance heavily relies on the accuracy of prior information, resulting in limited adaptability in unknown scenarios. (4) Non-statistical STAP [3,18]: Represented by direct data domain (D3) algorithms, such methods work without statistical training and perform well against discrete interference, but their ability to suppress uniform clutter is limited. (5) Sparse representation STAP (SR-STAP) [19,20]: Based on the space-time sparsity of clutter, SR-STAP estimates clutter statistics using few samples. However, the iterative sparse reconstruction process brings heavy computation and limits its practical use. (6) Sparsity-constrained STAP (SC-STAP) [8,21]: As a recently developed fast STAP method, it adds sparse constraints to the traditional cost function to achieve faster convergence and better clutter suppression performance.

Most existing STAP algorithms estimate the clutter-plus-noise covariance matrix using training samples from adjacent range cells, so as to reduce the target self-nulling effect. These methods assume that the training samples are target-free and share similar clutter statistical characteristics with the CUT. However, in multi-target scenarios, target echoes from neighboring range cells easily contaminate the training set, which biases the CCM estimation and reduces detection accuracy. To solve this issue, many sample selection methods have been proposed to remove non-uniform and target-contaminated training data. Nevertheless, since the CUT is excluded during training in traditional STAP frameworks, the CCM and its inverse must be recalculated for each range cell, which produces repeated calculations and greatly lowers the real-time processing efficiency [9].

In addition, conventional STAP approaches assume accurate prior knowledge of target parameters. In practical airborne radar operation, slight mismatches between preset and real target angles or Doppler frequencies will lead to filter self-nulling and reduce the output signal-to-clutter-plus-noise ratio. To mitigate such performance degradation, several robust STAP strategies have been proposed. The first method enlarges the main beam coverage by applying multiple linear constraints, but it consumes system degrees of freedom (DOFs) for clutter suppression and cannot suppress sidelobe interference, which may cause false alarms under distributed interference conditions [22]. The second method improves target parameter estimation by using the minimum-norm property of STAP filters, which relieves self-nulling but increases computational load [23]. Another group of robust STAP methods constructs different types of uncertainty regions (polyhedron, sphere, ellipsoid) for target steering vectors, and optimizes filter response to resist parameter errors [1,11,24,25]. Although these methods achieve good robustness, their complex solving processes are not suitable for real-time radar applications. Therefore, it is necessary to develop a robust and low-complexity STAP method to handle both training sample contamination and target parameter mismatch.

In practical airborne early warning scenarios, potential targets are sparsely distributed in the surveillance region, which means target echoes only appear in a small number of range cells. Motivated by this range-sparse characteristic, this paper proposes a robust STAP method based on l1-norm output amplitude regularization with fixed target gain. The introduced l1-norm constraint produces different weights for training samples: higher weights are assigned to pure clutter samples and lower weights to target-contaminated samples, which effectively reduces the self-nulling effect caused by impure training data. Meanwhile, the proposed method alleviates performance loss caused by steering vector mismatch. Unlike existing robust STAP methods that adopt quadratic uncertainty constraints, the proposed method uses a rhombic uncertainty region, which can be converted into linear constraints. Furthermore, the l1-norm optimization problem can be transformed into a standard linear programming problem, which greatly reduces computational complexity. In addition, the proposed method adopts a block processing strategy, where all range cells in one block share the same adaptive weight. This avoids repeated weight calculation for each cell and significantly improves computational efficiency for practical radar detection tasks.

The remainder of this paper is organized as follows. In Section 2, the signal model for STAP detection is given and the full-dimension STAP is briefly reviewed. Section 3 present the proposed STAP based on minimum output amplitude. In Section 4, we present the robust linear programming STAP (RLP-STAP) for target steering vector mismatch case. Simulation results are provided in Section 5 to demonstrate the effectiveness of the proposed STAP. Finally, conclusions are drawn in Section 6.

2. Target Detection Model

In STAP, target detection is a binary hypothesis test problem

$$\left\{\begin{array}{c}{H}_0:x_k=c_k+n_k\\ H_1:x_k=\alpha s+c_k+n_k\end{array}\right.$$

(1)

where $x_k\in {ℂ}^{NM\times 1}$ is the snapshot of the CUT, $c_k$ is the clutter component, $n_k\in {ℂ}^{NM\times 1}$ is the receiver noise vector, $\alpha$ is the unknown target amplitude, $s\in {ℂ}^{NM\times 1}$ is the steering vector of target, the subscript $k$ denotes the range cell index, $N$ is the number of array elements and $M$ is the number of pulses. The condition $H_0$ is the null hypothesis, or case of target absence, while $H_1$ is the alternative hypothesis indicating target presence. Under hypothesis $H_0$, $x$ is a zero mean complex Gaussian vector with distribution $x\sim CN\left(0,R_1\right)$ and $R_1$ is the covariance matrix of the clutter and noise and can be written as

$$R_k=R_C+{\sigma }^{2}I$$

(2)

where $R_C$ is the clutter matrix and ${\sigma }^2$ is the power of receiver noise. A STAP weight is required to discriminate whether the target is present in the CUT or not. Generally, the space-time processor linearly combines the elements of the data snapshot, yielding the scalar output

$$y_k=\left|w_k^H{x}_k\right|$$

(3)

where $w_k^H$ is the space-time adaptive weight to enhance the target and suppress the clutter. For the conventional STAP, the adaptive weight can be computed by solving following optimization problem

$$\begin{array}{l}\min \hspace{0.33em}{w}_k^H{R}_k{w}_k\\ s.t.\hspace{0.33em}\hspace{0.33em}{w}_k^{H}s=1\end{array}$$

(4)

where $R_k=E\left\{x_{k/H_0}^H{x}_{k/H_0}\right\}$ is the clutter and noise covariance matrix (CCM); therefore the STAP weight is given as (the scalar is omitted for simplicity)

$$w_k=R_k^{-1}s$$

(5)

which is known as the MVDR method. However, the true covariance matrix is unknown in practice and can be estimated by the secondary or training data as

$${\widehat{R}}_k={\displaystyle \frac{1}{L}}{\displaystyle \sum _{m=1}^L{x}_m^H{x}_m}={\widehat{R}}_{cn}$$

(6)

which yields the so-called sample matrix inversion (SMI) methods. To avoid target self-nulling, the training samples are assumed to be target-free and obey the independent and identical distribution (IID) with the CUT. However, some target-like signals present in the training samples will also cause target self-nulling. In this case, the covariance matrix has the following form

$$\begin{array}{ll}{\widehat{R}}_k& ={\displaystyle \frac{1}{L_{wo}}}{\displaystyle {\displaystyle \sum _{m=1}^{L_{wo}}}{x}_{m/H_0}{x}_{m/H_0}^H}+{\displaystyle \frac{1}{L_w}}{\displaystyle {\displaystyle \sum _{k=1}^{L_w}}{x}_{k/H_1}{x}_{k/H_1}^H}\\ & ={\displaystyle \frac{1}{L_{wo}}}{\displaystyle {\displaystyle \sum _{m=1}^{L_{wo}}}\left(c_m+n_m\right){\left(c_m+n_m\right)}^H}+{\displaystyle \frac{1}{L_w}}{\displaystyle {\displaystyle \sum _{k=1}^{L_w}}\left({\alpha }_{k}s+c_k+n_k\right){\left({\alpha }_{k}s+c_k+n_k\right)}^H}\\ & ={\widehat{R}}_{cn}+{\widehat{R}}_s+{\displaystyle \frac{1}{L_w}}{\displaystyle {\displaystyle \sum _{m=1}^{L_w}}\left(\alpha s\right){\left(c_k+n_k\right)}^H}+{\displaystyle \frac{1}{L_w}}{\displaystyle {\displaystyle \sum _{k=1}^{L_w}}\left(c_k+n_k\right){\alpha }_k{s}^H}\\ & ={\widehat{R}}_{cn}+{\widehat{R}}_s+{\widehat{R}}_{sc}\end{array}$$

(7)

where

$${\widehat{R}}_s={\displaystyle \frac{1}{L_w}}{\displaystyle {\displaystyle \sum _{k=1}^{L_w}}{\left|{\alpha }_k\right|}^2{s}^{H}s}={\sigma }_s^{2}s{s}^H$$

(8)

$${\widehat{R}}_{cn}={\displaystyle \frac{1}{L_{wo}+L_w}}{\displaystyle \sum _{m=1}^{L_{wo}+L_w}\left(c_m+n_m\right){\left(c_m+n_m\right)}^H}$$

(9)

$$\begin{array}{cc}{\widehat{R}}_{cs}& ={\displaystyle \frac{1}{L_w}}{\displaystyle {\displaystyle \sum _{k=1}^{L_w}}\left(c_k+n_k\right){\alpha }_k{s}^H}+{\displaystyle \frac{1}{L_w}}{\displaystyle {\displaystyle \sum _{m=1}^{L_w}}\left({\alpha }_{k}s\right){\left(c_k+n_k\right)}^H}\\ & =b{s}^H+s{b}^H\end{array}$$

(10)

$$b={\displaystyle \frac{1}{L_w}}{\displaystyle {\displaystyle \sum _{k=1}^{L_w}}{\alpha }_k\left(c_k+n_k\right)}\in {ℂ}^{NM\times 1}$$

(11)

And $L_{wo},L_w$ denote the number of samples with targets and without targets respectively. The covariance matrix estimated by samples contaminated by targets is composed of three parts: the clutter covariance matrix, the target covariance matrix and the target–clutter cross covariance matrix. By substituting Equation (7) into Equation (4), we have the following optimization

$$\begin{array}{l}\underset{w_k}{\min }{w}^H{\widehat{R}}_{cn}w+w^H{\widehat{R}}_{s}w+w^H{\widehat{R}}_{cs}w\\ s.t.\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{w}^{H}s=1\end{array}$$

(12)

By substituting the constraint $w^{H}s=1$ into the cost function, formulation (12) can be rewritten as

$$\begin{array}{l}\underset{w_k}{\min }{w}^H{\widehat{R}}_{cn}w+\left(w^{H}b+b^{H}w\right)+{\sigma }_s^2\\ s.t.\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{w}^{H}s=1\end{array}$$

(13)

Comparing Equations (13) with Equation (4), we find that the presence of the desired target in training samples imposes a regularization term $\left(w^{H}b+b^{H}w\right)$ on the adaptive weight. From Equation (10), we note that the regularization term is determined by target power. When the cost function of Equation (13) is minimized, the regularization term is minimized as well. Consequently, the target self-nulling is caused. Therefore, to avoid the target self-nulling, the samples with targets or target-like signals should be excluded from the training samples.

3. STAP Based on Minimum Output Amplitude

In this section, we propose a novel STAP method based on minimum output amplitude criterion [26]. Most of the existing STAP methods are based on minimizing the variance of the output and suffer target self-nulling when the target is present in the training data. However, in many STAP applications (e.g., AEW radar), the desired target is only present in a few range cells, which implies that the targets are sparsely distributed in the receiving samples. Considering the noise-free case, the optimal STAP weight should satisfy

$$\begin{array}{l}E\left\{\left|w_k^H{x}_{k/H_0}\right|\right\}=0\\ E\left\{\left|w_k^H{x}_{k/H_1}\right|\right\}=\left|\alpha \right|\end{array}$$

(14)

where $E\left\{\cdot \right\}$ denotes the expectation operation. Stacking L training sample output into a vector, one has the following matrix-vector formulation

$$y^{\ast }=X^{H}w$$

(15)

where $X=\left[x_1,x_2,\dots ,x_L\right]\in {ℂ}^{NM\times L}$, $y={\left[\begin{array}{cccc}{y}_1& y_2& \dots & y_L\end{array}\right]}^T$, and the superscript $\ast$ denotes the conjugate operation. As is stated above, the desired signal is only present in a few samples; therefore, the most elements of $y$ are very small corresponding to the $H_0$ hypothesis and only a few elements are relatively large corresponding to the $H_1$ hypothesis. In other words, the output vector $y$ is supposed to be a sparse vector. An often-used sparsity constraint is l1-norm minimization. Then by utilizing the sparsity of vector $y$, we propose a novel STAP method that minimizes the l1-norm of $y$ while maintaining the gain on desired signal. The $l_1$-norm penalty puts the most weight on the small element and the least weight on the large element of a vector. For the $l_1$-optimal solution, many elements of output vector $y$ are either zero or small, and the other elements are relatively large. Consequently, we have the following optimization problem

$$\begin{array}{l}\underset{w}{\min } {\left|y^{*}\right|}_1=\underset{w}{\min } {\left|X^{H}w\right|}_1=\underset{w}{\min } {\displaystyle \sum _{i=1}^L\sqrt{w^H{x}_i{x_i}^{H}w}}\\ s.t.\hspace{0.33em}{w}^H\mathrm{s}=1\end{array}$$

(16)

Since the gain on target is maintained to unity, when a target is present in a certain range cell, the corresponding element in $y$ is likely big and vice versa. The constrained optimization (16) can be transformed into an unconstrained optimization

$$\underset{w}{\min }{\displaystyle \sum _i^L\sqrt{w^H{x}_i{x_i}^{H}w}}+\lambda (w^H\mathrm{s}-1)$$

(17)

Letting the gradient of the cost function (17) with respect to $\mathbf{w}$ be equal to zero, we have the following relation

$${\displaystyle {\displaystyle \sum _i^L}\frac{x_i{x_i}^{H}w}{\sqrt{w^H{x}_i{x_i}^{H}w}}}+\lambda s=X\Lambda Xw+\lambda s=0$$

(18)

where

$$\Lambda =diag\left({\displaystyle \frac{1}{\sqrt{w^H{x}_1{x_1}^{H}w}}},\dots ,{\displaystyle \frac{1}{\sqrt{w^H{x}_L{x_L}^{H}w}}}\right)$$

(19)

We can solve Equation (18) by an iterative way as

$$w_k={\mu }_k{\left(X{\Lambda }_{k-1}X\right)}^{-1}s$$

(20)

where

$${\Lambda }_k=\left\{\begin{array}{cc}diag\left({\displaystyle \frac{1}{\sqrt{w_k^H{x}_1{x_1}^H{w}_k}}},\dots ,{\displaystyle \frac{1}{\sqrt{w_k^H{x}_L{x_L}^H{w}_k}}}\right)& k=1,2,3\\ I_{L\times L}& k=0\end{array}\right.$$

(21)

$${\mu }_k={\displaystyle \frac{1}{w_k^{H}s}}$$

(22)

Comparing with the traditional sample matrix inversion (SMI) method, the proposed method can be viewed as an iterative SMI where the training samples are iteratively weighted. Larger weights are assigned to signal-free samples, while smaller weights are assigned to samples containing the desired target, then the contribution of the samples with target present will be iteratively attenuated. Consequently the target self-nulling phenomenon is avoided or attenuated. On the other hand, the convergence rate of Equation (16) is similar to that of the SMI, i.e., about twice the system dimension. Small eigenvalues of the CCM cause significant, undetermined changes in the adaptive weights, which lead to the performance degradation and slow convergence rate. To attenuate the undetermined significant change of $w$ caused by small eigenvalues of CCM, we can use the well-known diagonal loading technique; then the problem (16) can be written as

$$\begin{array}{l}\underset{w}{\min }{\displaystyle \sum _{i=1}^L\sqrt{w^H{x}_i{x_i}^{H}w}+\lambda w^{H}w}\\ s.t.\hspace{0.33em}{w}^H\mathrm{s}=1\end{array}$$

(23)

Consequently, the iterative formulation can be rewritten as

$$w_k={\mu }_k{\left(X{\Lambda }_{k-1}X+\lambda I_{NM}\right)}^{-1}s$$

(24)

where $\lambda$ is the loading factor. From expression (24), we note that the contribution of the samples with target of interest is iteratively decreased and the contribution of the samples without target is iteratively stabilized by the loading factor. Therefore, optimization (23) is a robust l1-norm minimization STAP (RL1M-STAP). To solve problem (23), we can also use CVX, a package for specifying and solving convex programs [27]. The diagonal loading factor is usually selected empirically, typically about $10{\sigma }_n^2$ where ${\sigma }_n^2$ is the noise power in a single sensor.

4. Robust STAP by Linear Programming

The computational load of solving the l1-norm minimization is very heavy for real-time application. Fortunately, l1-norm minimization can be translated into linear programming by some relaxation. In this section, a robust STAP by linear programming (RSTAP-LP) is formulated. First, the complex output $y_k$ can be written as

$$y_k=y_{k,r}+\mathrm{j}{y}_{k,i}$$

(25)

Then

$${\left|y_k\right|}_1={\left|y_{k,r}+\mathrm{j}{y}_{k,i}\right|}_1\le {\left|y_{k,r}\right|}_1+{\left|y_{k,i}\right|}_1$$

(26)

Define ${\left|y_k\right|}_{l_1}={\left|y_{k,r}\right|}_1+{\left|y_{k,i}\right|}_1$, then ${\left|y^{*}\right|}_{l_1}={\left|y_r\right|}_1+{\left|y_i\right|}_1$ where $y=y_r+\mathrm{j}{y}_i$. According to the definition of norm [28], it can be easily verified that the operator ${\left|\cdot \right|}_{l_1}$ is still a norm. Replace ${\left|y^{*}\right|}_1$ with ${\left|y^{*}\right|}_{l_1}$ and the cost function of (9) can be translated into

$$\underset{w}{\min } {\left|y^{*}\right|}_{l_1}=\underset{w}{\min } {\left|X^{H}w\right|}_{l_1}$$

(27)

Denoting $w=w_r+\mathrm{j}{w}_i$, $X=X_r+\mathrm{j}{X}_i$, then the expanded real-valued matrix and vector are written as

$$\overline{w}=\left[\begin{array}{c}{w}_r\\ w_i\end{array}\right]\in {ℝ}^{2NM},\overline{X}=\left[\begin{array}{cc}{X}_r& -X_i\\ X_i& X_r\end{array}\right]\in {ℝ}^{2NM\times 2L}$$

(28)

We can convert the l1-norm of complex variables into that of real variables

$${\left|X^{H}w\right|}_{l_1}={\left|{\overline{X}}^H\overline{w}\right|}_1$$

(29)

Remark 1.

Since the operator ${\left|\cdot \right|}_{l_1}$is a norm, according to the equivalence of norms, there exists a positive constant $\eta ,\xi$,$\tilde{\eta },\tilde{\xi }$such that for all $y$,

$$\eta {\left|y^{*}\right|}_2\le {\left|y^{*}\right|}_{l_1}\le \xi {\left|y^{*}\right|}_2$$

(30)

Since the CUT is used to compute the adaptive weight, the small mismatches may cause performance degradation. In many practical applications, due to a variety of mismatches, the steering vector is not known exactly. The actual steering vector can be formulated as

$$a=s+e$$

(31)

where $e\in {ℂ}^N$ is the steering vector error. It is assumed that the error is in an uncertainty set. The goal of robust STAP is to ensure that the magnitude response does not attenuate in the uncertainty set. In Refs. [1,27], the uncertainty region is modeled as a sphere and ellipsoid respectively. Under the spherical or ellipsoidal model, the robust STAP can be categorized as a second-order cone programming (SOCP) or quadratically constrained quadratic program (QCQP) problem [11]. Here, we use the rhombus uncertainty set model proposed in [24], which is given by

$$\Im =\left\{e\left|{\left|e\right|}_{l_1}\le {\epsilon }_1\right.\right\}$$

(32)

where ${\epsilon }_1>0$ determines the area of the rhombus uncertainty set. The proposed robust STAP minimizes the $l_1$-norm of the output magnitude vector while requiring the $l_{\infty }$-modulus of the array response to exceed unity for all $e\in \Im$. Then we have the following optimization

$$\begin{array}{l}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\underset{w}{\min }{\left|X^{H}w\right|}_{l_1}\\ \mathrm{s}.\mathrm{t}.\hspace{0.33em}{\left|{\left(s+e\right)}^{H}w\right|}_{\infty }\ge 1,\hspace{0.33em}for\hspace{0.33em}all\hspace{0.33em}\hspace{0.33em}e\le \Im \end{array}$$

(33)

where the $l_{\infty }$-modulus of a complex number $z$ is defined as ${\left|z\right|}_{\infty }={\left|z_r+\mathrm{j}{z}_i\right|}_{\infty }=\max (\left|z_r\right|,\left|z_i\right|)$. However, due to the non-convex constraint, it is not easy to directly solve Equation (33). Fortunately, X. Jiang [24] reformulates the constraints into linear programming as

$${\overline{s}}^T\overline{w}-{\epsilon }_1{\left|\overline{w}\right|}_{\infty }\ge 1$$

(34)

where $s=s_r+\mathrm{j}{s}_i$ and $\overline{s}={\left[s_r^T,s_i^T\right]}^T$. More details about the deduction can be found in [24]. Substituting this inequality into the constraint in Equation (33) leads to the following optimization problem

$$\begin{array}{c}\underset{w}{\min }{\left|X^{H}w\right|}_{l_1}\\ \mathrm{s}.\mathrm{t}.\hspace{0.33em}{\overline{s}}^T\overline{w}-{\epsilon }_1{\left|\overline{w}\right|}_{\infty }\ge 1\end{array}$$

(35)

Furthermore, by introducing two auxiliary variables $b$ and $r$, optimization (35) can be converted into a standard LP problem as

$$\begin{array}{ll}\underset{\overline{w},b}{\min }& b^T{1}_{2L}\\ \mathrm{s}.\mathrm{t}.& -b\le {\overline{X}}^H\overline{w}\le b\\ & {\overline{s}}^T\overline{w}\ge {\epsilon }_{1}r+1\\ & -r{1}_{2NM}\le \overline{w}\le r{1}_{2NM}\\ & 0\le b\end{array}$$

(36)

In summary, we converted the robust STAP (33) to the LP problem (36). The LP is the simplest convex optimization problem and, thus, can be efficiently solved by the well-established simplex method [29,30] or interior point method [31]. Furthermore, it is known that the computational effort per iteration required by interior point methods to solve the LP problem is much less than that required to solve the SOCP problem with a similar size and structure [32,33]. Therefore, the proposed RSTAP-LP is more computationally efficient than SOCP-based STAP methods.

Remark 2.

In order to avoid target self-nulling, the conventional STAP has to exclude the cell under test and the training samples with target-like signals for the covariance matrix estimation. Therefore, the conventional STAP requires training sample selection. Furthermore, the covariance matrix estimation and its inversion need to be recomputed for each CUT for the sliding window processor (SWP), which leads to a heavy computation load. Our method can simultaneously process the CUT and the secondary data without training sample selection. Thus the adaptive weight for a block of samples requires no updating which leads to a significant reduction in computation.

Remark 3.

Since all the receiving samples are used to compute the adaptive weight, it implies an opportunity to suppress both the statistical clutter and non-statistical interference (discrete interference). By analysis of the cost function of Equation (33), we note that the output residue of all the samples is minimized. Therefore, the discrete interference and statistical clutter are suppressed, while the gain on the desired signal is maintained by the constraint. Furthermore, the target self-nulling is attenuated by minimizing the L1 norm of the output residue vector.

Remark 4.

According to Remark 1 and Remark 2, we note that the proposed algorithms have three significant advantages over the existing robust STAP methods. First, it requires no sample selection operations to avoid the target self-nulling, which significantly reduces the computational load; second, it is robust to signal steering vector mismatch by constraining the l1-modulus of the response of any steering vector within a rhombus uncertainty set to exceed unity; finally, our method can also suppress the discrete interference by using all the samples to compute the adaptive weight.

5. Experiments

In this section, both the simulated data and the MCARM data are used to verify the clutter suppression performance of the proposed STAP based on linear programming (RSTAP-LP). The performance of our method is compared with six STAP methods in terms of the output SCNR and the output clutter residue: loaded sample matrix inversion (LSMI), the eigen-canceler (EIG) and the L1-norm conventional conjugate gradient (L1-CCG) STAP [21], the extended factored space-time processing (EFA) [34], the robust Worst-Case STAP algorithm [25], and the direct data-domain D3-STAP algorithm [18]. In the following experiment, the output SCNR is defined as

$$SCNR\triangleq {\displaystyle \frac{{\left|y_s\right|}^2/L_s}{{\left|y_c\right|}^2/L_c}}$$

(37)

where $y_s=w^H{X}_s$ denotes the output signal power, $y_c=w^H{X}_c$ denotes the output clutter power, $X_s$ is the data matrix stacked by $L_s$ samples with targets, and $X_c$ is the data matrix stacked by $L_c$ samples without targets.

In this paper, we adopt two processing modes: sliding window processing (SWP) and concurrent block processing (CBP). For sliding window processing, training samples exclude the current range cell and samples containing target signals. To avoid the performance degradation of comparison algorithms resulting from unreasonable sample selection, prior information is introduced in sample selection to remove non-homogeneous samples contaminated by target-like signals and discrete interferences, and the remaining samples are used for adaptive weight calculation. In the sliding window mode, training samples are separately selected for each cell under test to derive adaptive weights, which are only applied to the corresponding cell under test. Once the adaptive filtering of one cell is finished, the identical operation is performed on the next cell under test in sequence. In contrast, concurrent block processing requires no guard cells, and all range cells participate in adaptive weight solving. Only one set of adaptive weights is calculated and applied to all cells under test in a unified manner. In other words, merely a single adaptive weight solution is required throughout the entire processing procedure when utilizing concurrent block processing.

In subsequent simulation experiments, the parameter configurations of all comparative algorithms are kept consistent without special specifications. For the LSMI algorithm, the diagonal loading factor is set to tenfold of the minimum eigenvalue. The subspace dimension of the EIG algorithm is assigned as the ideal value of N + M − 1, i.e., 23. As for the Worst-Case STAP algorithm in Reference [25], the parameter corresponding to the uncertainty set is configured to 0.02. In the L1-CCG algorithm proposed in Reference [21], the regularization parameter adopted to balance the sparsity constraint and total squared error is set to 10. With respect to the D3-STAP method presented in Reference [21], within its two-dimensional sliding window mechanism, the sliding quantities of spatial-domain window and temporal-domain window are designated as 3 and 5 respectively.

5.1. Simulated Data

The simulation parameters used to generate the raw data, including radar system configuration, platform motion, and clutter environment, are provided in

Table 1. Parameters of the simulated data.

Parameter	Value
Number of spatial elements	8
Number of temporal pulses in a CPI	16
Radar frequency	1240 MHz
Platform velocity	200 m/s
Channel spacing	λ/2
Main beam look direction	side-looking
Target normalized Doppler frequency	0.1
Pulse repetition frequency	900
Height of platform	3 km
Clutter-to-noise ratio (CNR)	50 dB

. These settings are kept identical for all compared algorithms to guarantee a controlled and unbiased performance evaluation.

In this simulation scenario, the true space-time steering vector of the target signal is perfectly known. Two synthetic targets are injected into range bins [100, 290], both located at the broadside direction (azimuth 0 degree) with a normalized Doppler frequency of −0.1, and with input signal-to-clutter-plus-noise ratios (SCNR) of −60 dB and −52 dB, respectively.

Example 1: Exactly Known Signal Space-Time Steering Vector: Figure 1 compares the output clutter power residue of different STAP algorithms, where Figure 1a presents the clutter suppression performance under the concurrent block processing (CBP) mode, and Figure 1b illustrates the corresponding performance under the sliding window processing (SWP) mode. It is observed that even in this ideal case, the conventional LSMI algorithm suffers severe performance degradation under the CBP mode when the target-of-interest signal is present in the processing block.

Consistent with the theoretical deductions presented earlier, Figure 1 verifies that the proposed RLM-STAP and RSTAP-LP algorithms effectively mitigate the target self-nulling effect without requiring additional sample selection procedures. For RLM-STAP, the self-nulling induced by target-contaminated training samples is iteratively attenuated. Since the cost function of RSTAP-LP is an approximation to that of RLM-STAP, both algorithms achieve performance comparable to the sliding window LSMI (LSMI-SWP) benchmark. Notably, however, RLM-STAP and RSTAP-LP only require a single adaptive weight computation for the entire data block.

While the benchmark reduced-dimension STAP methods exhibit lower computational complexity under this scenario, the proposed algorithm demonstrates distinct performance advantages. Meanwhile, the computational overhead of non-reduced-dimension robust STAP methods is comparable to that of the proposed approach.

Figure 2 plots the output SCNR varying with the normalized Doppler frequency for the azimuth angle 0 degree. The samples with targets are always used to compute the adaptive weight for the proposed method. It can be seen from Figure 2 that our method can effectively attenuate the target self-nulling effect and the LSMI-CBP, EIG-CBP and L1-CCG-CBP suffer from the target self-nulling effect caused by samples contaminated by targets. The performance of our method is close to that of LSMI-SWP, EIG-SWP and L1-CCG-SWP without adaptive weight updating.

Example 2: Space-time steering vector mismatch: In this simulation, the desired space-time steering vector is distorted by array element amplitude and phase errors. The actual steering vector is modeled as:

$$\tilde{s}=s\odot (1+n)$$

(38)

where $\odot$ denotes the Hadamard product, and $n$ is the complex-valued steering vector error vector. The real and imaginary parts of $n$ are assumed to be uniformly distributed over the interval $\left[0,\kappa \right]$, with $\kappa =0.1$ in this paper.

Figure 3 depicts the output clutter power residue as a function of range bin index, while Figure 4 presents the output signal-to-clutter-plus-noise ratio (SCNR) versus normalized Doppler frequency. The results confirm that the proposed methods remain effective in mitigating target self-nulling, even under this steering vector mismatch condition. It is worth noting that L1-CCG-SWP, LSMI-SWP, and EIG-SWP are inherently immune to self-nulling effects, as the cross-covariance between target and clutter components vanishes under sliding window processing. The proposed algorithms achieve performance comparable to these sliding window benchmarks, while delivering a substantial reduction in computational complexity.

Example 3: Discrete interference suppression: To further comprehensively validate the discrete interference suppression performance and practical superiority of the proposed STAP algorithms, the basic simulation configurations are consistent with the aforementioned experimental settings. Specifically, two target signals are injected into the range bins of 100 and 290, which are characterized by a normalized Doppler frequency of 0.1, with input signal-to-clutter ratios (SCRs) of −52 dB and −44 dB, respectively. Meanwhile, two discrete interferences are introduced at the range bins of 15 and 350. Unlike the target signals, these discrete interferences are set with a normalized Doppler frequency of 0.3 and an identical input SCR of −52 dB.

Figure 5 illustrates the clutter suppression performance of different STAP algorithms in the presence of discrete interferences. Among them, Figure 5a presents the performance comparison of all evaluated algorithms under the block processing framework, while Figure 5b demonstrates the corresponding results under the sliding window processing scheme. To further highlight the performance differences in competitive algorithms with favorable interference suppression capability, Figure 5c provides a refined local comparison based on the results in Figure 5a.

The quantitative and qualitative comparison results indicate that the D3-STAP, L1-CCG-STAP, and conventional LSMI algorithms, as well as the proposed algorithms, can effectively suppress discrete interferences under block processing mode. However, these benchmark approaches suffer from a substantial degradation in output signal-to-clutter-plus-noise ratio (SCNR) compared with the proposed algorithms, resulting in degraded target detection capability. In contrast, for the sliding window processing mode, samples containing target components and discrete interferences are excluded from the adaptive weight estimation process. Consequently, although the L1-CCG-STAP and LSMI algorithms can achieve a comparable target output SCNR with the proposed method, they are incapable of suppressing discrete interferences.

Figure 6 compares the space-time response characteristics of the adaptive weights derived from the LSMI algorithm and the proposed method. As observed from the simulation results, the proposed method forms distinct adaptive nulls at the discrete interference positions. The underlying mechanism is that the proposed algorithm can adaptively distinguish target signals from discrete interferences according to their distinct Doppler characteristics. Owing to the difference in Doppler frequencies between targets and discrete interferences, the discrete interferences are recognized as clutter components and further suppressed during adaptive weight solving. This demonstrates the superior interference suppression capability of the proposed algorithm in complex discrete interference scenarios.

In summary, the proposed algorithm achieves an excellent balance between target signal preservation and discrete interference suppression. It exhibits superior comprehensive performance compared with other benchmark algorithms in discrete interference scenarios, thereby possessing greater potential for practical engineering applications.

Example 4: Algorithm Parameter and Sparsity Performance Analysis: According to the cost function of the proposed algorithm, its performance is affected by multiple factors, including the relaxation parameter, the number of training samples, and target sparsity. This section further analyzes the correlation between the performance of the proposed method and the above influencing factors through numerical simulations. In this group of simulations, only a single target is considered, located at the 30th range bin, with a normalized Doppler frequency of 0.1 and an input SCNR of −55 dB. Space-time steering vector errors are considered, and other simulation settings are consistent with previous experiments.

Figure 7 shows the variation in target output SCNR with the relaxation factor. The results verify that the proposed algorithm is robust and insensitive to changes in the relaxation factor. In practical engineering applications, the optimal relaxation parameter can be determined via statistical simulation. Figure 8 illustrates the detection performance of different algorithms under varying target signal-to-noise ratio (SNR) conditions, which verifies that the proposed method achieves superior detection capability compared with the benchmark algorithms.

Figure 9 illustrates how the target detection probability varies with the number of training samples. It can be observed that the proposed algorithm fails to detect targets effectively when the number of training samples is less than the system space-time dimension. For STAP systems, the theoretical dimension is determined by the product of spatial channels and temporal pulses. Insufficient training samples fail to support the optimization solution, causing the cost function to converge abnormally and further leading to target detection failure. When the number of training samples exceeds the system space-time dimension, the detection performance improves rapidly. In addition, the false alarm probability drops sharply with the increase in training samples, and the overall algorithm performance converges steadily when the sample size reaches twice the system space-time dimension.

Figure 10 characterizes the detection performance with varying target occupancy ratios. To eliminate random errors and enhance result reliability, 200 Monte Carlo simulations are conducted, and the statistical outcomes are summarized in Figure 11. The results show that the algorithm’s sparsity assumption becomes invalid when the fraction of target-contaminated range bins exceeds 15%, which induces significant performance degradation. The algorithm performance is gradually restored with the reduction in target occupancy. When the target proportion decreases below 10%, the system performance tends to be stable. In practical radar operating environments, the proportion of target-contaminated range bins is typically less than 1%. This demonstrates that the proposed algorithm possesses a sufficient application margin and is well applicable to practical radar detection scenarios.

5.2. Computational Complexity Analysis

To quantitatively evaluate the computational efficiency of the proposed algorithm, this section presents a comparative analysis of the computational complexity between the proposed algorithm and several typical STAP methods. The computational overhead of STAP is mainly concentrated in three core stages: covariance matrix estimation, adaptive weight calculation, and clutter suppression. Let the number of spatial channels, pulses within a coherent processing interval, and range training samples be denoted as N, M, and L, respectively. For the D3-STAP algorithm, the spatial and temporal sliding window sizes are denoted as P and Q, respectively, with the dimension of the reduced space-time data defined as $D=\left(N-P\right)\left(M-Q\right)$. The number of iterations for the RLM-STAP algorithm is set to three. The comparison algorithms cover representative methods from different technical routes: LSMI and EIG are traditional STAP algorithms based on direct covariance matrix estimation; EFA achieves dimensionality reduction via space-time decomposition; D3-STAP performs data dimensionality reduction using a two-dimensional sliding window; L1-CCG and RSTAP-LP solve weights based on sparse optimization and robust estimation, respectively; and RLM-STAP realizes robust clutter suppression via iteratively reweighted minimization. The computational complexity analysis results of each typical algorithm at the aforementioned three stages are presented in

Table 2. Computational complexity analysis of typical STAP algorithms.

Algorithms	Covariance Matrix Estimation	Adaptive Weight	Clutter Suppression
LSMI	${\left(NM\right)}^{2}L$	$3{\left(NM\right)}^3+{\left(NM\right)}^{2}L$	$3\left(NM\right)LM$
EFA	${\displaystyle \frac{LNM}{2}}{\log }_2\left(M\right)+{\left(3N\right)}^{2}LM$	$3{\left(3N\right)}^{3}M+{\left(3N\right)}^{2}M$	$3NML$
D3-STAP	$O\left(LPQ{D}^2\right)$	$L\left(3{\left(D\right)}^3+{\left(D\right)}^2\right)$	$LPQD$
L1-CCG	——	——	$10{\left(NM\right)}^2$
EIG	${\left(NM\right)}^{2}L$	$3{\left(NM\right)}^3+{\left(NM\right)}^{2}L$	$3\left(N+M\right)LM$
RSTAP-LP	——	——	$O\left({\left(2NM+2L\right)}^3\right)$
RLM-STAP	${\left(NM\right)}^{2}L$	$10{\left(NM\right)}^3+{\left(NM\right)}^{2}L$	$3\left(NM\right)LM$
Worst-Case STAP	${\left(NM\right)}^{2}L$	$O\left({\left(NM\right)}^3\right)$	$3\left(NM\right)LM$

.

As illustrated in the table, the proposed algorithm achieves a computational complexity comparable to that of conventional robust STAP methods under small-sample conditions, with its computational advantages further enhanced as the number of training samples increases. Although the proposed method incurs a higher computational cost than typical dimensionality-reduced STAP algorithms (EFA, D3-STAP, EIG), it eliminates the requirement for sample selection. Specifically, the adaptive weight is calculated only once for each Doppler channel, enabling superior suppression of discrete interference. By balancing overall performance and computational overhead, the proposed algorithm exhibits prominent comprehensive advantages. Furthermore, the adopted linear programming strategy facilitates straightforward engineering implementation, which endows the method with excellent practical application value.

5.3. MCARM Data

In this section, the measured MCARM dataset is adopted to validate the performance of the proposed robust STAP algorithm. The MCARM dataset [23] was collected in 1995–1996 over the Delmarva Peninsula near the eastern coast of Maryland; the data used in this paper is selected from Acquisition 575 of Flight 5 (file RL050575). The radar system operates at L-band with a phased array consisting of 22 antenna elements arranged in a 2 × 11 configuration. Prior to experimental validation, the raw echo data is reconstructed into a standard space-time data cube for joint space-time processing. To guarantee data quality, valid array channels and stable range bins are screened to remove abnormal and noisy echo components. The dataset is calibrated and normalized at a fixed CNR of 55 dB to standardize the background noise level, and the clutter covariance matrix is estimated via sample averaging to characterize the practical statistical properties of real measured clutter. In the experiments, 11 array elements and 12 coherent pulses are utilized for STAP, with training samples extracted from the range bins of 200 to 400. Different from the benchmark algorithms that exclude target-contaminated range bins from the training set to avoid target self-nulling, the proposed method adopts intact training samples without removing target-containing range bins to fully validate its robustness. Further details concerning the MCARM dataset are provided in [35].

Figure 12 depicts the output power across different range cell indices. Two synthetic targets are injected into range cells [249, 279] with a normalized Doppler frequency of −0.15 and an SCNR of −55 dB. Consistent with the simulation settings in the previous numerical experiments, Gaussian perturbations are added to the ideal space-time steering vector to simulate practical array mismatch errors. The results show that the proposed algorithm achieves competitive detection performance compared with conventional SWP-based methods including LSMI-SWP, EIG-SWP, and L1-CCG-SWP. Furthermore, to evaluate the discrete interference suppression capability, two discrete interferences are injected into range cells [299, 349] with a normalized Doppler of −0.4. Figure 13 compares the output power of the proposed method with several benchmark algorithms using CBP and SWP strategies. It is observed that the SWP-based methods suffer severe false alarms in the presence of discrete interference. Although the existing CBP-based methods can suppress discrete interference to a certain extent, their suppression performance is significantly degraded compared with the proposed method. Overall, the experimental results in Figure 8 and Figure 9 verify that the proposed method possesses excellent capability for discrete interference suppression while effectively alleviating the target self-nulling effect.

6. Conclusions

Based on the underlying fact that the desired targets are sparsely distributed in the entire surveillance area, the output clutter residue vector is supposed to be an approximately sparse vector with many small elements and a few large elements. By utilizing the sparsity of the target distribution, a well-known sparse constraint (l1-norm minimization) is imposed on the output residue vector. Furthermore, a new definition of the l1-norm of the complex vector is introduced to transform the original optimization problem into linear programming. There are three advantages of the proposed method. First, it requires no sample selection operations to avoid the target self-nulling and significantly reduce the computational load; second, it is robust to signal steering vector mismatch by constraining the l1-modulus of the response of any steering vector within a rhombus uncertainty set to exceed unity. Finally, our method can also suppress the discrete interference by using all the samples to compute the adaptive weight. Experimental results verified the effectiveness of the proposed algorithm.