An Efficient Sparse Bayesian Learning STAP Algorithm with Adaptive Laplace Prior

Abstract

Space-time adaptive processing (STAP) encounters severe performance degradation with insufficient training samples in inhomogeneous environments. Sparse Bayesian learning (SBL) algorithms have attracted extensive attention because of their robust and self-regularizing nature. In this study, a computationally efficient SBL STAP algorithm with adaptive Laplace prior is developed. Firstly, a hierarchical Bayesian model with adaptive Laplace prior for complex-value space-time snapshots (CALM-SBL) is formulated. Laplace prior enforces the sparsity more heavily than Gaussian, which achieves a better reconstruction of the clutter plus noise covariance matrix (CNCM). However, similar to other SBL-based algorithms, a large degree of freedom will bring a heavy burden to the real-time processing system. To overcome this drawback, an efficient localized reduced-dimension sparse recovery-based space-time adaptive processing (LRDSR-STAP) framework is proposed in this paper. By using a set of deeply weighted Doppler filters and exploiting prior knowledge of the clutter ridge, a novel localized reduced-dimension dictionary is constructed, and the computational load can be considerably reduced. Numerical experiments validate that the proposed method achieves better performance with significantly reduced computational complexity in limited snapshots scenarios. It can be found that the proposed LRDSR-CALM-STAP algorithm has the potential to be implemented in practical real-time processing systems.

1. Introduction

After nearly fifty years of development, space-time adaptive processing (STAP) has become a mature processing technique and has been widely used for moving target detection in airborne surveillance radar [1,2,3]. Generally, conventional STAP algorithms heavily depend on sufficiently independent and identically distributed (IID) training samples. However, the clutter environment is often heterogeneous, and the available IID samples can hardly meet the well-known Reed–Mallet–Brennan (RMB) criterion [4], which deteriorates the performance of adaptive processors.

In past decades, many efforts have been made to improve STAP performance with a small sample size, including data-independent reduced-dimension (RD) methods [5,6,7,8] and data-dependent reduced-rank (RR) methods [9,10,11,12,13]. Although the number of required training samples is further reduced to twice the reduced dimension in these methods, the requirement is still hard to satisfy in severely heterogeneous clutter environments. Then direct data domain (DDD) method was proposed in [14], which can work only with the cell under test (CUT) and solves the sample shortage problem. However, the cost is the loss of system aperture and performance degradation. Knowledge-aided (KA) STAP utilizes external prior knowledge and previously measured data to enhance performance [15,16]. Unfortunately, precise prior knowledge is still hard to obtain, causing performance degradation.

During the last decade, inspired by the vigorous development of compressed sensing methods, sparse recovery theory has shown great potential for solving the small sample problem and enhancing STAP performance. The sparse-recovery-based STAP (SR-STAP) algorithm was introduced to acquire a high-resolution clutter spectrum in 2006 [17]. SR-STAP algorithms [18] exploit the intrinsic sparsity of the clutter spectrum on the angle-Doppler plane and can achieve near-optimum performance even with a few samples. In [19], utilizing the symmetrical property of clutter, a more accurate estimation can be obtained with only one snapshot. The least absolute shrinkage and selection operator (LASSO) has been extensively applied to model the SR-STAP problem [20,21]. LASSO employs the ${\ell }_1$ norm penalty as the convex relaxation of the ${\ell }_0$ norm. However, LASSO is not an oracle procedure. To utilize the oracle properties [22], an adaptive LASSO was first proposed in [23]. Instead of using a common regularization parameter, a series of adaptive weights are assigned to different coefficients in the ${\ell }_1$ norm penalty. Although these SR algorithms can achieve acceptable clutter suppression performance, selecting the regularization parameter can be quite challenging in practice because the measurements are contaminated by noise [18]. Inappropriate parameter selection will result in pseudo-peaks at locations outside the clutter ridge [20,21], which may seriously deteriorate the accuracy of the sparse recovery and thus jeopardize the performance of clutter suppression and slow-moving target detection. Therefore, dealing with noise is an important issue in applying these algorithms [24,25,26,27]. Sparse Bayesian learning (SBL) provides a robust framework for tackling the sparse recovery problem with a self-regularizing nature, and all parameters can be updated automatically. Tipping proposed SBL in 2001 [28]. A Gaussian-prior-based SBL strategy with multiple measurement vectors (MMV) was first introduced to STAP, defined as the M-SBL-STAP, by Duan [29]. Wang proposed a fast-converging sparse Bayesian learning algorithm to improve the convergence speed [30]. A two-stage M-SBL STAP method is proposed in [31] to deal with the spatial signal model mismatch problem. From a Bayesian perspective, the aforementioned LASSO problem can be formulated to the maximum of a posteriori (MAP) estimation with a Laplace prior [32]. Furthermore, the distribution of the Laplace prior forms a sharper concave peak than a Gaussian; thus, the sparsity is enhanced. However, it is intractable to describe the posterior when Laplace priors are directly applied for modeling the complex signals, considering the Laplace distribution is not a conjugate prior to the Gaussian distribution. In [32], Babacan S.D. developed a Bayesian model in a hierarchical manner, which leads to the Laplace prior for real-value signals.

To the best of our knowledge, the adaptive Laplace-prior-based SBL framework for the complex-value two-dimensional STAP signal model has rarely been considered so far. While complex-value models can be equally converted into a real-value model, the real and imaginary parts can be recovered individually. However, the dimensions of the dictionary matrix and measurements are doubled, leading to a greater computational burden. Motivated by the hierarchical SBL framework and the superiority of the adaptive Laplace prior, we formulate a hierarchical Bayesian model with an adaptive Laplace prior for complex space-time signal MMV case (CALM-SBL), which is then applied to develop a novel STAP algorithm (CALM-SBL-STAP).

However, similar to the other SBL algorithms, CALM-SBL would also face a massive computation load for large-scale problems due to the covariance matrix inverse operation, hindering it from being implemented in the real-time processing of modern radars. Therefore, how to efficiently solve the SR problem is also an important research field, and there have been some studies on reducing computational complexity [30,33,34,35]. A clutter-spectrum-aided dictionary design method was proposed to reduce iteration time [33]; however, it still suffers from the extra computational load in the clutter power spectrum estimation step. Nevertheless, the computational burden of the above algorithms is still overwhelmed when the system has a large degree of freedom. Moreover, a beamspace post-Doppler SR-based reduced-dimension STAP (RDSR-STAP) method to save the computational load is introduced in [34].

In this work, aimed at improving computational efficiency, we next propose a localized reduced-dimension sparse recovery (LRDSR) processing scheme. A set of deeply-weighted sliding window Doppler filters is used to reduce the temporal dimension and localize the clutter. Then the prior knowledge of the clutter is applied to further refine the localized region. The sparse recovery problem is restricted to a localized reduced-dimension region (the dimension of the dictionary and the signal are both reduced). Hence the calculation efficiency of CALM-SBL is significantly improved with the LRDSR structure.

Specifically, our main contributions are listed as follows:

• We extend the hierarchical Bayesian model with real-value adaptive Laplace prior to the two-dimensional STAP complex-value application, which promotes sparsity to a greater extent.
• We develop a novel LRDSR processing scheme to accelerate CALM-SBL by eliminating the main bottleneck, i.e., the storage and inverse operation on the covariance matrix.
• Detailed comparative analyses of computational complexity, clutter suppression performance, and target detection performance between the proposed algorithms and other STAP methods are presented.

The rest of this paper is organized as follows. In Section 2, the signal model and SR-STAP problem formulation are established. The proposed two algorithms are presented in Section 3. In Section 4, numerical simulations and analyses are presented. A discussion of our algorithms is presented in Section 5. The conclusions are discussed in Section 6.

Notation: Matrices, vectors, and scalars are denoted by bold uppercase, bold lowercase letters, and italic letters, respectively. $ℝ$ and $ℂ$ separately denote the sets of real and complex values. $\mathcal{C}\mathcal{N}$ denotes the complex normal distribution. ${‖\hspace{0.33em}\cdot \hspace{0.33em}‖}_{\mathrm{F}}$ denotes the Frobenius norm and ${‖\hspace{0.33em}\cdot \hspace{0.33em}‖}_{2,1}$ represents ${\ell }_{2,1}$ mixed norm, which is defined as the sum of the column vector consisting of the ${\ell }_2$ norm of each row vector. The superscripts ${\left(\cdot \right)}^T$ and ${\left(\cdot \right)}^H$ represent transpose and conjugate transpose operation, respectively. The operator $\otimes$ stands for the Kronecker product and $\mathrm{Tr}\left(\cdot \right)$ is the matrix trace operation. $diag\left(\cdot \right)$ represents a diagonal matrix with entries of the argument vector on the diagonal. The expectation of a random variable is denoted by the symbol ${\rm E}\left(\cdot \right)$.$\left|\cdot \right|$ denotes the determinant operation on a square matrix.

2. Signal Model and SR-STAP Problem Formulation

2.1. Signal Model

In this research, consider an airborne pulse Doppler surveillance radar system with a side-looking uniform linear array (ULA), as depicted in Figure 1, consisting of N omnidirectional elements, and the spacing of array elements d is half wavelength. The platform moves with a constant velocity v_p, and M coherent pulses are emitted over a coherent processing interval at a fixed pulse repetition frequency (PRF) $f_r$.

Based on the Ward clutter model [36], the ground can be divided into distinct range rings, and each iso-range ring can be viewed as a superposition of a large number of discrete and mutually uncorrelated clutter patches. According to the geometry of the platform shown in Figure 1, the normalized Doppler frequency and normalized spatial frequency of the ith clutter patch can be defined as follows:

$$f_{d,\hspace{0.33em}i}=\frac{2v_p}{\lambda f_r}\cos {\phi }_i\cos {\theta }_i$$

(1)

$$f_{s,\hspace{0.33em}i}=\frac{d}{\lambda }\cos {\phi }_i\cos {\theta }_i$$

(2)

where ${\phi }_i$ and ${\theta }_i$ denote the elevation angle and azimuth angle, respectively.

Without considering the impact of range ambiguities, a clutter plus noise snapshot $x\in {ℂ}^{NM\times 1}$ from a given iso-range ring then can be modeled as

$$x={\displaystyle \sum _{i=1}^{N_c}{\alpha }_{i}s(f_{d,\hspace{0.33em}i},f_{s,\hspace{0.33em}i})}+n$$

(3)

where n is the thermal noise and is modeled as a zero-mean complex Gaussian vector with the covariance matrix ${\sigma }_{}^2{I}_{NM}$; N_c stands for the number of clutter patches; ${\alpha }_i$ and $s(f_{d,\hspace{0.33em}i},f_{s,\hspace{0.33em}i})$ denotes the complex amplitude and the space-time steering vector of ith clutter patch, which can be expressed as

$$s(f_{d,\hspace{0.33em}i},f_{s,\hspace{0.33em}i})=s_t(f_{d,\hspace{0.33em}i})\otimes s_s(f_{s,\hspace{0.33em}i})$$

(4)

$$s_t(f_{d,\hspace{0.33em}i})={\left[1,\hspace{0.33em}{e}^{j2\pi f_{d,\hspace{0.33em}i}},\hspace{0.33em}\cdots \hspace{0.33em},\hspace{0.33em}{e}^{j2\pi \left(M-1\right)f_{d,\hspace{0.33em}i}}\right]}^T$$

(5)

$$s_s(f_{s,\hspace{0.33em}i})={\left[1,\hspace{0.33em}{e}^{j2\pi f_{s,\hspace{0.33em}i}},\hspace{0.33em}\cdots \hspace{0.33em},\hspace{0.33em}{e}^{j2\pi \left(N-1\right)f_{s,\hspace{0.33em}i}}\right]}^T$$

(6)

Here $s_t(f_{d,\hspace{0.33em}i})$ and $s_s(f_{s,\hspace{0.33em}i})$ are the corresponding temporal and spatial steering vectors.

Since we suppose that the clutter patches are independent, the corresponding ideal clutter plus noise covariance matrix (CNCM) can then be written as

$$R={\displaystyle \sum _{i=1}^{N_c}E\left\{{\left|{\alpha }_i\right|}^2\right\}s(f_{d,\hspace{0.33em}i},f_{s,\hspace{0.33em}i})s^H(f_{d,\hspace{0.33em}i},f_{s,\hspace{0.33em}i})}+{\sigma }_{}^2{I}_{NM}$$

(7)

Based on the linearly-constrained minimum variance (LCMV) criterion, the optimum STAP weight vector can be obtained as

$$w=\frac{R^{-1}{s}_{target}}{s_{target}^H{R}^{-1}{s}_{target}}$$

(8)

where $s_{target}$ denotes the target space-time steering vector.

In reality, the ideal CNCM is unknown and can be estimated by the maximum likelihood method from the IID adjacent range cells, i.e., ${\widehat{R}}_{ML}=\left(1/L\right){\displaystyle \sum _{l=1}^L{x}_l{x}_l^{\mathrm{H}}}$, where $L$ is the number of available IID training samples. Due to the heterogeneous environment, it is often difficult to find adequate training samples, resulting in degraded clutter suppression performance.

2.2. SR-STAP Problem Formulation

The sparsity nature of the clutter spectra shows that only a small area is occupied by clutter on the entire space-time plane [18], which is exploited to estimate CNCM by sparse recovery techniques. For the SR-STAP algorithms [35], the continuous space-time plane is uniformly discretized into $N_s{M}_d$ grids, where $N_s={\rho }_{s}N$ and $M_d={\rho }_{d}M$ are the number of grids in the spatial and temporal domain, and ${\rho }_s,{\rho }_d>1$ are the resolution scales. Then the SR-STAP signal model of the received clutter plus noise echoes from L range cells $X=[x_1,x_2,\hspace{0.33em}\dots \hspace{0.33em},x_L]\in {ℂ}^{NM\times L}$ can be reformulated as

$$X=\Phi A+N$$

(9)

where $\Phi =\left[s(f_{d,\hspace{0.33em}1},f_{s,\hspace{0.33em}1}),\cdots ,s(f_{d,\hspace{0.33em}{M}_d},f_{s,\hspace{0.33em}{N}_s})\right]\in {ℂ}^{NM\times N_s{M}_d}$ is defined as an overcomplete dictionary matrix composed of the space-time steering vectors of $N_s{M}_d$ grids, $A=[a_1,a_2,\hspace{0.33em}\dots \hspace{0.33em},a_L]\in {ℂ}^{N_s{M}_d\times L}$ denotes the space-time profile matrix where each nonzero row stands for a potential clutter component, and $N=[n_1,n_2,\hspace{0.33em}\dots \hspace{0.33em},n_L]\in {ℂ}^{NM\times L}$ is the noise matrix.

Since $N_s{M}_d\gg NM$, it is an ill-posed problem to recover A using the ordinary least square method. Assuming the unknown profile A has row sparsity, i.e., the distribution of the non-zero elements is identical for each column of A. Then exploiting the sparsity nature of A, the sparse space-time profile can be acquired by solving the following mixed norm optimization problem [21]:

$$\underset{A}{\min }\hspace{0.33em}{‖X-\Phi A‖}_{\mathrm{F}}^2+\eta {‖A‖}_{2,1}$$

(10)

where $\eta$ is a nonnegative regularization parameter for balancing the row sparsity with data fidelity. This problem can be viewed as an ${\ell }_1$ norm penalized least-square estimator, also termed LASSO [37]. However, the convergence of LASSO cannot be guaranteed. In [23], Zou improved the LASSO by using a series of data-dependent weights $\mathsf{\eta }=[{\eta }_1,\hspace{0.33em}{\eta }_2,\hspace{0.33em}\dots ,\hspace{0.33em}{\eta }_{N_s{M}_d}]$ to penalize different coefficients in the space-time profile instead of a common parameter $\eta$, and the proposed the adaptive LASSO is given as:

$$\underset{A}{\min }\hspace{0.33em}{‖X-\Phi A‖}_{\mathrm{F}}^2+\eta {\displaystyle \sum _{i=1}^{N_s{M}_d}{\omega }_i}{‖A_{i·}‖}_2$$

(11)

where $A_{i·}$ is the ith row of the space-time profile A, and ${\omega }_i$ denotes the weight of $A_{i·}$. Once the estimated space-time profile A is obtained, the corresponding CNCM can be recovered from

$$\widehat{R}=\frac{1}{L}{\displaystyle \sum _{l=1}^L{\displaystyle \sum _{i=1}^{N_s{M}_d}{\left|A_{il}\right|}^2}}{s}_i{s}_i^H+{\sigma }^2{I}_{NM}$$

(12)

3. Proposed Algorithms

In this section, we first derive a complex-value adaptive Laplace-prior-based SBL STAP algorithm for the MMV case (i.e., CALM-SBL-STAP). Then an efficient localized reduced-dimension sparse recovery (LRDSR) processing scheme is developed to enhance the computational efficiency of CALM-SBL-STAP.

3.1. CALM-SBL-STAP Algorithm

From the Bayesian perspective, all of the unknown variables are considered random variables assigned to different distributions according to their priors. Similar to the other SBL models, we begin with the conventional assumption that the noise is modeled as white complex Gaussian with unknown power ${\sigma }^2$; thus, we have:

$$p(N|\xi )={\displaystyle \prod _{l=1}^L𝒞𝒩(N_{·l}|0,{\xi }^{-1}I)}$$

(13)

where $\xi ={\sigma }^{-2}$ is the inverse noise variance, with a Gamma prior assigned on it as follows:

$$p(\xi |c,d)=Gamma(\xi |c,d)$$

(14)

and the Gamma distribution is defined as [32]

$$Gamma(x|\alpha ,\beta )=\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}{x}^{\alpha -1}{e}^{-\beta x}$$

(15)

where $\alpha >0$ and $\beta >0$ represent the shape parameter and scale parameter, respectively, and $\mathrm{\Gamma }\left(\alpha \right)={\displaystyle {\int }_0^{\infty }{t}^{\alpha -1}{e}^{-t}dt}$ denotes the ‘gamma function’. The mean and variance of x are respectively given as $\frac{\alpha }{\beta }$ and $\frac{\alpha }{{\beta }^2}$.

Therefore, the Gaussian likelihood function of the measurements $X$ can be expressed as:

$$\begin{array}{l}p(X|A,\xi )={\displaystyle \prod _{l=1}^L𝒞𝒩(X_{·l}|\Phi A_{·l},{\xi }^{-1}{\rm I})}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}=\frac{1}{{(\pi {\xi }^{-1})}^{-NML}}\exp \left[-\xi {‖X-\Phi A‖}_F^2\right]\end{array}$$

(16)

Since under the assumption of Gaussian noise, the observed likelihood function is not conjugate to the adaptive Laplace prior, and hence it cannot be used directly. To formulate the Bayesian model with adaptive Laplace priors, a hierarchical framework will be modeled for the unknown space-time profile A. We suppose each column of the sparse space-time profile $A_{·l}$ is assigned with a zero-mean complex Gaussian prior as the first stage of the hierarchical model.

$$\begin{array}{l}p(A|\lambda )={\displaystyle \prod _{l=1}^L\mathcal{C}\mathcal{N}(A_{·l}|0,\Lambda )}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}={\pi }^{-N_s{M}_{d}L}{\left|\Lambda \right|}^{-L}\exp \left(-{\displaystyle \sum _{l=1}^L{A}_{·l}^H{\Lambda }^{-1}{A}_{·l}}\right)\end{array}$$

(17)

where $\lambda =[{\lambda }_1,\hspace{0.33em}{\lambda }_2,\hspace{0.33em}\dots ,\hspace{0.33em}{\lambda }_{N_s{M}_d}]$ is the variance for different rows in $A$, and $\Lambda$ is a diagonal matrix with $\lambda$ as the diagonal elements, i.e., $\Lambda =diag(\lambda )$. Further, in the second stage of the model, independent Gamma priors with parameter ${\eta }_i$ are placed on each ${\lambda }_i$ in $\lambda$ for the complex-value space-time signal model [38].

$$p(\lambda |\mathsf{\eta })={\displaystyle \prod _{i=1}^{N_s{M}_d}Gamma({\lambda }_i|\frac{3}{2},\frac{{\eta }_i}{4})}$$

(18)

where $\mathsf{\eta }={[{\eta }_1,\dots ,{\eta }_{N_s{M}_d}]}^T$.

Finally, based on the conjugate prior principle, a Gamma prior is assigned to $\mathsf{\eta }$

$$p(\mathsf{\eta }|a,b)={\displaystyle \prod _{i=1}^{N_s{M}_d}Gamma\left({\eta }_i\right|a,b)}$$

(19)

Figure 2 summarizes the probabilistic graphic model of the proposed complex hierarchical Bayesian model to illustrate the aforementioned hyperparameters. Combining the first two stages of the hierarchical Bayesian model (17) and (18), we can obtain the marginal probability distribution $p(A_{·l}|\mathsf{\eta })$ by integrating out $\lambda$:

$$\begin{array}{l}p(A_{·l}|\mathsf{\eta })={\displaystyle \int p(A_{·l}|\lambda )p(\lambda |\mathsf{\eta })\hspace{0.33em}}d\lambda \\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}={\displaystyle \prod _{i=1}^{N_s{M}_d}\frac{{\eta }_i}{2\pi }}\exp \left(-\sqrt{{\eta }_i}\left|A_{il}\right|\right)\end{array}$$

(20)

It can be seen from (20) that the probability density function $p(A_{·l}|\mathsf{\eta })$ is a joint Laplace distribution with zero location parameters and different scale parameters. Each coefficient in the space-time profile $A_{·l}$ is controlled by an independent hyperparameter ${\eta }_i$, corresponding to the adaptive LASSO problem in (11) with the relationship ${\omega }_i=\sqrt{{\eta }_i}$ [39].

Although the hierarchical model with an adaptive Laplace prior would be more complicated, the Laplace log-distribution has a sharper concave peak than Gaussian, allocating mass more on the axes so that the space-time profile is much preferred to close zero [28], as shown in Figure 3. Therefore, the Laplace distribution enforces the sparsity more heavily than a conventional Gaussian distribution. Meanwhile, the unimodal log distribution helps eliminate local minima. The modeling process of the proposed hierarchical Bayesian model in this article is motivated by [32,38], and we extend the real-value sparse recovery problem to STAP complex-value circumstance.

In the following, the expectation-maximization (EM) algorithm [29] is utilized for Bayesian inference to estimate the hyperparameters. The EM algorithm is an iterative framework for parameter estimation, alternating between the expectation step (E−step) and maximization step (M−step).

As widely known, based on the Bayesian theorem, the posterior probability $p(A|X,\lambda ,\mathsf{\eta },\xi )$ can be calculated by

$$p(A|X,\lambda ,\mathsf{\eta },\xi )=\frac{p(X|A,\xi )p(A|\lambda )p(\lambda |\mathsf{\eta })p(\mathsf{\eta })p(\xi )}{{\displaystyle \int p(X|A,\xi )p(A|\lambda )p(\lambda |\mathsf{\eta })p(\mathsf{\eta })p(\xi )dA}}$$

(21)

We can find that the posterior distribution $p(A|X,\lambda ,\mathsf{\eta },\beta )$ is a multivariate Gaussian distribution $𝒞𝒩(\mathsf{\mu },\sum )$ with parameters

$$\mathsf{\mu }=\Lambda {\Phi }^H{\left({\xi }^{-1}I+\Phi \Lambda {\Phi }^H\right)}^{-1}X$$

(22)

$$\Sigma =\Lambda -\Lambda {\Phi }^H{\left({\xi }^{-1}I+\Phi \Lambda {\Phi }^H\right)}^{-1}\Phi \Lambda$$

(23)

In E-step, the expectation of the logarithm of the joint distribution can be expressed as:

$$ℒ\left(\lambda ,\mathsf{\eta },\xi \right)=E_{p(A|X,\lambda ,\mathsf{\eta },\xi )}\left\{\log \left[p(X|A,\xi )p(A|\lambda )p(\lambda |\eta )p(\mathsf{\eta })p(\xi )\right]\right\}$$

(24)

To update $\lambda$ and $\mathsf{\eta }$, (24) can be simplified by omitting the irrelevant terms and according to (17)–(19), $\ell \left(\lambda ,\eta \right)$ can be obtained as

$$\begin{array}{l}ℒ(\lambda ,\mathsf{\eta })=E_{p(A|X,\lambda ,\mathsf{\eta },\xi )}\left\{\log \left[p(A|\lambda )p(\lambda |\eta )p(\mathsf{\eta })\right]\right\}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\propto E_{p(A|X,\lambda ,\mathsf{\eta },\xi )}\left\{\begin{array}{l}-L\ln \Lambda -{\displaystyle \sum _{l=1}^L{A}_{·l}^H{\Lambda }^{-1}{A}_{·l}^{}+\frac{3}{2}{\displaystyle \sum _{i=1}^{N_s{M}_d}\ln \frac{{\eta }_i}{4}}+\frac{1}{2}}{\displaystyle \sum _{i=1}^{N_s{M}_d}\ln {\lambda }_i}\\ -{\displaystyle \sum _{i=1}^{N_s{M}_d}\frac{{\eta }_i{\lambda }_i}{4}}+\left(a-1\right){\displaystyle \sum _{i=1}^{N_s{M}_d}\ln {\eta }_i}-b{\displaystyle \sum _{i=1}^{N_s{M}_d}{\eta }_i}\end{array}\right\}\end{array}$$

(25)

In M-step, we can find the updates of ${\lambda }_i^{new}$ and ${\eta }_i^{new}$ by maximizing (25) with respect to ${\lambda }_i$ and ${\eta }_i$ separately. Next, the partial derivative of $\ell \left(\lambda ,\mathsf{\eta }\right)$ with respect to ${\lambda }_i$ is:

$$\frac{\partial \ell \left(\lambda ,\mathsf{\eta }\right)}{\partial {\lambda }_i}=-\frac{L}{{\lambda }_i}+\frac{{\mathsf{\mu }}_{i·}{\mathsf{\mu }}_{i·}^H+L{\Sigma }_{ii}}{{\lambda }_i^2}-\frac{{\eta }_i}{4}+\frac{1}{2{\lambda }_i}$$

(26)

where ${\mathsf{\mu }}_{i·}$ is the ith row of $\mathsf{\mu }$, ${\Sigma }_{ii}$ is the (i, i)-th element of $\Sigma$. However, directly solving (26) results in a slow convergence speed for the update of ${\lambda }_i$, inspired by [40,41], we establish a new fixed-point update rule to speed up the convergence.

Let $W_i=-1+\frac{{\Sigma }_{ii}}{{\lambda }_i}$, then (26) can be rewritten as

$$\frac{\partial \ell \left(\lambda ,\mathsf{\eta }\right)}{\partial {\lambda }_i}=L\frac{W_i}{{\lambda }_i}+\frac{{\mathsf{\mu }}_{i·}{\mathsf{\mu }}_{i·}^H}{{\lambda }_i^2}-\frac{{\eta }_i}{4}+\frac{1}{2{\lambda }_i}$$

(27)

Analogously, the partial derivative of $\ell \left(\lambda ,\mathsf{\eta }\right)$ with respect to ${\eta }_i$ is:

$$\frac{\partial \ell \left(\lambda ,\mathsf{\eta }\right)}{\partial {\eta }_i}=\frac{3}{2{\eta }_i}-\frac{{\lambda }_i}{4}+\frac{a-1}{{\eta }_i}-b$$

(28)

Setting (27) and (28) to zero, the update rules of ${\lambda }_i$ and ${\eta }_i$ can be respectively expressed as:

$${\lambda }_i^{new}={\eta }_i^{-1}\left(1+2L{W}_i+2\sqrt{{(0.5+L{W}_i)}^2+{\mathsf{\mu }}_{i·}{\mathsf{\mu }}_{i·}^H{\eta }_i}\right)$$

(29)

$${\eta }_i^{new}=\frac{a+0.5}{b+{\lambda }_i/4}$$

(30)

The logarithm function of noise precision $\xi$ in E-step can be expressed as:

$$\begin{array}{l}\ell (\xi )=E_{p(A|X,\lambda ,\mathsf{\eta },\xi )}\left\{\log \left[p(X|A,\xi )p(\xi )\right]\right\}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\propto E_{p(A|X,\lambda ,\mathsf{\eta },\xi )}\left\{NML\log \xi -\xi {\displaystyle \sum _{l=1}^L{‖X_{·l}^{}-\Phi A_{·l}‖}_2^2}+(c-1)\ln \xi -d\xi \right\}\end{array}$$

(31)

Setting $\frac{\partial \ell (\xi )}{\partial \xi }=0$, we can find the update rule of ${\xi }^{new}$:

$${\xi }^{new}=\frac{NML+c-1}{\left|\right|X-\mathbf{\Phi }\mathsf{\mu }|{|}_F^2+L\cdot \mathrm{Tr}\left(\Sigma {\Phi }^H\Phi \right)+d}$$

(32)

It is worth noting that directly using (23) will result in a heavy computational burden. Fortunately, only the diagonal elements of $\Sigma$ are utilized during the process of updating the hyperparameters. Thus, in order to speed up the computation and save hardware storage space, we only need to calculate the diagonal elements ${\Sigma }_{ii}$ [41].

$$\left\{\begin{array}{l}{\Sigma }_{ii}={\lambda }_i-{\lambda }_i^2{U}_{i·}{\Phi }_{·i}\\ U={\Phi }^H{\left({\xi }^{-1}I+\Phi \Lambda {\Phi }^H\right)}^{-1}\end{array}\right.$$

(33)

The iteration operations alternate between E-step and M-step until a predefined stop criterion is satisfied. The pseudocode of the proposed algorithm (CALM-SBL-STAP) is shown in Algorithm 1.

Algorithm 1 Pseudocode for CALM-SBL-STAP algorithm

Input: training samples X, dictionary matrix $\Phi$. Initialize: $t=0$, ${\lambda }^{(0)}=1$, ${\eta }^{(0)}=1$, ${\xi }^{(0)}=\frac{NML}{0.1\left|\right|X|{|}_F^2}$. Set $a=b=c=d={10}^{-4}$, $\epsilon ={10}^{-3}$. Step 1: While $t\le {\mathrm{iter}}_{\max }$                   $t\leftarrow t+1$;                   Update $\mathsf{\mu }$ and ${\Sigma }_{ii}$ using (22) and (33);                   Update ${\lambda }^{(t)}$ using (29);                   Update ${\mathsf{\eta }}^{(t)}$ using (30);                   Update ${\xi }^{(t)}$ using (32);                   If $\left|\right|{\lambda }^{(t)}-{\lambda }^{(t-1)}|{|}_2/\left|\right|{\lambda }^{(t-1)}|{|}_2<\epsilon$                           break                   end              end Step 2: Estimate the CNCM $\tilde{R}$ by $\tilde{R}=(1/L)\Phi diag\left({\displaystyle \sum _{l=1}^L{\left|A_{·l}\right|}^2}\right){\Phi }^H+{\xi }^{-1}I$ Step 3: Compute the space-time adaptive weight $w$ by (8) Step 4: Give the output of the CALM-SBL-STAP is $Y=w^{\mathrm{H}}X$

3.2. LRDSR-STAP Scheme

Although various numerical experiments have proved that SBL algorithms can achieve favorable performance with only a few training samples, however, from (22) and (23), we can observe that it takes a costly O((NM)³) operations for matrix inversion at each iteration. Moreover, modern airborne radars often have hundreds or even thousands of system degrees of freedom, making them difficult to be implemented in real-time processing applications. Furthermore, conventional sparse-recovery algorithms need to search the entire space-time plane, and the overcomplete dictionary needs to be considerably dense to ensure the recovery accuracy, which also causes a heavy computational burden during each iteration.

To overcome these drawbacks, a novel localized reduced-dimension sparse-recovery-based space-time adaptive processing (LRDSR-STAP) method is proposed. Intuitively, to reduce the computational complexity, we can divide the existing SR-STAP problems into smaller problems to solve. If we divide the space-time plane into several regions along the Doppler frequency axis, the clutter in each localized area will still be sparsely distributed and can be recovered. Modern airborne pulse-Doppler radar systems can easily implement Doppler filter banks with extremely low sidelobes. Inspired by the formulations and properties of the PRI-staggered post-Doppler framework described in [6], deeply weighted Doppler filter banks can be utilized to acquire the narrow band localized clutter. Therefore, the received snapshots are firstly processed by K sliding window Doppler filters, for the mth Doppler bin,

$${\tilde{x}}_m={(B_m\otimes I_N)}^{H}x$$

(34)

where $B_m=\left[\begin{array}{ccc}{D}_m& \cdots & 0\\ & \ddots & \\ 0& \cdots & D_m\end{array}\right]$ is an $M\times K$ matrix whose columns represent the mth Doppler bin for the sliding window Doppler filters, and $D_m={[t_0\hspace{0.33em},\hspace{0.33em}{t}_1{e}^{j2\pi \frac{m}{M}}\hspace{0.33em},\hspace{0.33em}\cdots \hspace{0.33em},\hspace{0.33em}{t}_{M\text{'}-1}{e}^{j2\pi \frac{m}{M}\left(M\text{'}-1\right)}]}^T$, here $M\text{'}=M-K+1$ is the length of the sliding window, and $[t_0,\hspace{0.33em}\dots \hspace{0.33em},t_{M\text{'}-1}]$ is the temporal tapering window.

After filtering, the transformed clutter steering vector can be expressed as a $NK\times 1$ reduced-dimension space-time steering vector $\tilde{s}(f_{s,},f_d)$.

$$\begin{array}{l}{(B_m\otimes I_N)}^{H}s(f_{s,},f_d)=B_m^H{s}_t(f_d)\otimes s_s(f_s)\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}=\left(D_m^H{s}_{t,M\text{'}}(f_d)\right)s_{t,K}(f_d)\otimes s_s(f_s)\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}=\left(D_m^H{s}_{t,M\text{'}}(f_d)\right)\tilde{s}(f_{s,},f_d)\end{array}$$

(35)

where $s_{t,M\text{'}}(f_d)$ and $s_{t,K}(f_d)$ are $M\text{'}\times 1$ and $K\times 1$ temporal steering vectors respectively. Then (34) can be replaced with

$${\tilde{x}}_m=\hspace{0.33em}{\displaystyle \sum _{i=1}^{N_c}{\tilde{\alpha }}_i\tilde{s}(f_{d,\hspace{0.33em}i},f_{s,\hspace{0.33em}i})}+{(B_m\otimes I_N)}^{H}n$$

(36)

where ${\tilde{\alpha }}_i=\left(D_m^H{s}_{t,M\text{'}}(f_{d,i})\right){\alpha }_i$ denotes the filtered amplitude.

In this structure, the localized clutter subspace can be estimated by $⌊N+\beta \left(K-1\right)⌋$ steering vectors, where $⌊\cdot ⌋$ denotes rounding-up and $\beta =2v_p/(f_{r}d)$ represents the slope of clutter ridge, which is also the number of independent clutter observations [6]. Thus, it is proved that the localized reduced-dimension clutter is still low rank and can be sparsely recovered in the localized search region. At this time, only the grids within the mainlobe of the Doppler filter are required to form the dictionary and represent the localized clutter, as shown in the gray grids in Figure 4.

To further narrow the search region and improve the computational efficiency of sparse recovery, we present a knowledge-aided localized dictionary construction strategy. First, we can approximately estimate the slope of clutter ridge β using the prior knowledge provided by the inertial navigation system and radar system. A more refined localized dictionary is next assembled by the grids adjacent to the localized clutter ridge. As presented in Figure 4, for the mth Doppler bin, the grids in the dashed line are selected. Obviously, with the aid of prior knowledge, the size of the localized dictionary would be further reduced. Note that the size of the rectangular window ${\tilde{N}}_s{\tilde{M}}_d$ is determined by the mainlobe width Δf of the Doppler filters and corresponding coupled spatial frequency range, i.e., only the grids within the spatial-temporal frequency range of f_d,l∈[f_m − Δf/2, f_m + Δf/2] $(l=1,\hspace{0.33em}\dots \hspace{0.33em},M_d)$ and f_s,n∈[f_m − Δf/2, f_m + Δf/2]/β $(n=1,\hspace{0.33em}\dots \hspace{0.33em},N_s)$ are selected, where f_m is the central frequency of the mth Doppler bin. Finally, a prestored mask matrix Ξ_m is built to remove the redundant grids from the original dictionary Φ.

$${\Xi }_m=[{\xi }_{m,1},{\xi }_{m,2},\hspace{0.33em}\dots \hspace{0.33em},{\xi }_{m,{\tilde{N}}_s{\tilde{M}}_d}]$$

(37)

$${\xi }_{m,i}={[\underset{i-1}{\underbrace{0\hspace{0.33em},\hspace{0.33em}\dots \hspace{0.33em},\hspace{0.33em}0\hspace{0.33em}}},1\hspace{0.33em},\hspace{0.33em}\underset{N_s{M}_d-i}{\underbrace{0\hspace{0.33em},\hspace{0.33em}\dots \hspace{0.33em},\hspace{0.33em}0}}]}^T$$

(38)

In this way, an even smaller localized reduced-dimension dictionary ${\tilde{\Phi }}_m\in {ℂ}^{NK\times {\tilde{N}}_s{\tilde{M}}_d}$ of mth Doppler bin is generated as

$${\tilde{\Phi }}_m={\left(B_m\otimes I_N\right)}^H\Phi {\Xi }_m$$

(39)

Now the reduced-dimension signal model in the case of multiple measurements ${\tilde{X}}_m\in {ℂ}^{NK\times L}$ can be rewritten as

$${\tilde{X}}_m={\tilde{\Phi }}_m{\tilde{A}}_m+\tilde{N}$$

(40)

where ${\tilde{A}}_m\in {ℂ}^{{\tilde{N}}_s{\tilde{M}}_d\times L}$ is the localized space-time profile matrix, which can be estimated by solving the following optimization:

$$\underset{{\tilde{A}}_m}{\min }\hspace{0.33em}{‖{\tilde{X}}_m-{\tilde{\Phi }}_m{\tilde{A}}_m‖}_{\mathrm{F}}^2+\eta {\displaystyle \sum _{j=1}^{{\tilde{N}}_s{\tilde{M}}_d}{\omega }_j}{‖{\tilde{A}}_{m_{\hspace{0.33em}j,·}}‖}_2$$

(41)

Equation (41) is a standard adaptive LASSO optimization problem and can be solved by the previously proposed CALM-SBL algorithm. Obviously, this LRDSR-STAP processing scheme can scale down the CALM-SBL to a localized reduced-dimension region (the dimension of the dictionary and the signal are both reduced) hence the running speed of the CALM-SBL algorithm would be efficiently improved with the LRDSR processing scheme as a matter of course. We will refer to it as LRDSR-CALM-SBL in the following content. Figure 5 illustrates the processing block diagram of the proposed scheme. Since the size of the optimization problem decreases, the computational complexity of LRDSR-CALM-SBL in updating the hyperparameters and computing the STAP weights shrinks substantially.

4. Numerical Simulation

In this section, extensive experiments are performed to assess the performance of the proposed algorithms (referred to as CALM-SBL and LRDSR-CALM-SBL) and compared with other existing methods. Unless otherwise specified, we consider a side-looking ULA with the following parameters: $N=16$, $M=16$, $\lambda =0.23\hspace{0.33em}m$, $f_r=2434.8\hspace{0.33em}Hz$, and $v_p=140\hspace{0.33em}m/s$. The simulated data are generated by the Ward clutter model mentioned in Section 2, and each range bin has $N_c=181$ clutter patches evenly distributed in azimuth angles. The clutter to noise ratio CNR = 50 dB, and noise power is set to be unitary, i.e., ${\sigma }_{}^2=1$. The dictionary resolution scales are both set as ${\rho }_s={\rho }_d=5$, and the number of IID training samples L is 10. The prior parameters $a$, $b$, $c$, and $d$ are set to a small value: $a=b=c=d={10}^{-4}$. For fairness, we set the iteration tolerance $\epsilon ={10}^{-3}$ and the maximum iterations ${\mathrm{iter}}_{\max }=200$ for the proposed algorithms, M-SBL and M-FCSBL. To achieve the desired localized clutter spectrum, an 80-dB Chebyshev window is applied for the proposed LRDSR-CALM-SBL. Furthermore, all of the simulation results are averaged over 100 independent Monte-Carlo trails.

4.1. Computational Complexity Analysis

In the first experiment, we analyze the computational complexity of the proposed CALM-SBL and LRDSR-CALM-SBL for a single iteration and compare them with the other SR-STAP algorithms, including M-CVX [42], M-OMP [43,44], M-FOCUSS [45], M-SBL [29], and M-FCSBL [30]. The computational complexity is measured by the number of complex multiplications. For simplicity, the low-order terms of complex multiplications are omitted. The results are listed in

Table 1. Computational complexity comparison.

Algorithm	Estimation of R	Weight Calculation
M-CVX	$O\left({\left(N_s{M}_{d}L\right)}^3\right)$	$O\left({(NM)}^3\right)$
M-OMP	$O\left(NM{N}_s{M}_{d}L+r_s{}^3+NM{r}_s{}^2+2NM{r}_s\right)$	$O\left({(NM)}^3\right)$
M-FOCUSS	$O\left(NM{N}_s{M}_{d}L+{\left(NM\right)}^3+2{\left(NM\right)}^2{N}_s{M}_d\right)$	$O\left({(NM)}^3\right)$
M-SBL	$O\left({(NM)}^3+2{(NM)}^2{N}_s{N}_d+3{(N_s{M}_d)}^{2}NM+NM{N}_s{M}_{d}L\right)$	$O\left({(NM)}^3\right)$
M-FCSBL	$O\left({(NM)}^3+5{(NM)}^2{N}_s{M}_d+3N_s{M}_d+(2N_s{M}_{d}L+4N_s{M}_d+L)NM+L\right)$	$O\left({(NM)}^3\right)$
CALM-SBL	$O\left({(NM)}^3+2{(NM)}^2{N}_s{M}_d+2{(N_s{M}_d)}^{2}NM+NM{N}_s{M}_{d}L+{(N_s{M}_d)}^2\right)$	$O\left({(NM)}^3\right)$
LRDSR-CALM-SBL	$O\left({(NK)}^3+2{(NK)}^2{\tilde{N}}_s{\tilde{M}}_d+2{({\tilde{N}}_s{\tilde{M}}_d)}^{2}NK+NK{\tilde{N}}_s{\tilde{M}}_{d}L+{({\tilde{N}}_s{\tilde{M}}_d)}^2\right)$	$O\left({(NK)}^3\right)$

, where $r_s$ denotes the clutter rank.

As can be seen from Table 1, since only the diagonal elements are updated in (33), the computational complexity of CALM-SBL is moderately reduced and is lower than M-SBL. Because of the fact that ${\tilde{N}}_s{\tilde{M}}_d\ll N_s{M}_d$ and $NK\ll NM$, the computational efficiency of the CALM-SBL has been significantly improved with the help of the proposed LRDSR-STAP processing scheme. Meanwhile, LRDSR-STAP performs dimensionality reduction in the time domain, so the heavy burden of matrix inversion is greatly reduced to $O\left({(NK)}^3\right)$ during the process of calculating the space-time adaptive weight vector.

In order to present the comparison results more intuitively, the average code running time of these algorithms is examined, as shown in

Table 2. Average running time comparison.

Algorithm	Running Time
M-CVX	868.7062 s
M-OMP	0.0216 s
M-FOCUSS	256.2815 s
M-SBL	288.9466 s
M-FCSBL	40.8802 s
CALM-SBL	10.1729 s
LRDSR-CALM-SBL (K = 1)	0.0051 s
LRDSR-CALM-SBL (K = 3)	0.1319 s

. Note that all the simulations are operated on the same platform with Intel Xeon 4114 CPU @2.2GHz and 128GB RAM. It can be seen from Table 2 that the average running time of the proposed LRDSR-CALM-SBL is greatly reduced compared to the original CALM-SBL and is much lower than M-SBL and M-FCSBL. The running time of LRDSR-CALM-SBL is even close to M-OMP and is already less than M-OMP when K = 1. In the subsequent experiments, we will also find that the proposed algorithms can maintain a satisfactory clutter suppression performance on this basis. This superiority makes LRDSR-CALM-SBL more suitable for practical real-time processing applications. Furthermore, this LRDSR processing scheme enables further parallel acceleration via multi-core processors (i.e., GPU), which is beyond the scope of the current paper.

4.2. Clutter Suppression Performance

Due to the fact that LRDSR-CALM-SBL belongs to the reduced-dimension STAP algorithm, this means that it is essentially a suboptimal STAP method and will inevitably suffer from slight performance degradation compared with full-dimension STAP methods. Therefore, for the sake of fairness, we will separately compare the performance of CALM-SBL with other full-dimension STAP methods and LRDSR-CALM-SBL with other reduced-dimension STAP methods in our experiments.

Then we evaluate the clutter suppression performance via the metric of signal-to-interference-plus-noise (SINR) loss under various circumstances, which is defined as [21]:

$$SIN{R}_{loss}=\frac{{\sigma }_n^2}{NM}\frac{{\left|w^{H}s(f_d,f_s)\right|}^2}{w^{H}Rw}$$

(42)

where $w$ is the STAP weight vector, and $R$ is the clairvoyant CNCM.

Next, we first evaluate the SINR loss performance of CALM-SBL. Moreover, M-OMP, M-FOCUSS, M-SBL, M-FCSBL, and the optimum STAP are used as benchmarks. Note that M-CVX is not discussed here due to its overwhelming computational complexity. Figure 6 depicts the curves of the SINR loss versus the normalized Doppler frequency. As illustrated in Figure 6, all three SBL algorithms can similarly achieve near-optimum performance even with 10 training samples, and the proposed CALM-SBL consumes the least running time among these three algorithms.

Furthermore, in order to reveal the convergence speed of the proposed CALM-SBL, Figure 7 presents the average SINR loss with different numbers of training samples. The average SINR loss is defined as the mean of the SINR loss values for $f_d\in (-0.5,-0.05)\cup (0.05,0.5)$. It is indicated that the average SINR of all methods increases with the increment of training samples. Apparently, the proposed algorithm exhibits a fast convergence speed similar to the other two SBL algorithms and achieves a near-optimum performance with merely two training samples, attributed to the fact that the adaptive Laplace prior facilitates a sparser solution.

From the above full-dimension STAP experiments, we can observe that the proposed CALM-SBL has a satisfactory performance. However, in the current practical engineering applications, the full-dimensional STAP is still difficult to be implemented due to the limitation of processor capability, and thus the reduced-dimension processing scheme is more preferred. Therefore, in the following experiments, we will mainly focus on comparing the proposed LRDSR-CALM-SBL with other RD-STAP methods.

Figure 8 demonstrates the SINR loss performance of the proposed LRDSR-CALM-SBL algorithm with different K. It can be observed that the notch of K = 1, the simplest form, is much wider than others. This is because adaptive processing is only performed in the spatial domain at this time. With the increase of K, the available space-time adaptivity increase, and the clutter suppression performance is also improved with a narrower notch. Based on our empirical study, K = 3 generally leads to satisfactory performance. Considering the balance of performance and efficiency, we will employ K = 3 in the subsequent experiments.

In the next experiment, we compare the SINR loss performance with other RD methods, such as a state-of-the-art RDSR algorithm [34], and conventional statistic RD-STAP algorithms with diagonal loaded to the noise power level, such as the PRI-staggered post-Doppler [6] with three sliding windows, the direct data domain (DDD) [14] with $6\times 6$ subaperture, the joint-domain localized (JDL) [5] with $3\times 3$ beam-Doppler channels, and the optimum full-dimensional STAP. Furthermore, four typical cases, as presented in Figure 9, are considered: (a) the ideal case; (b) the spatial mismatch case in the presence of gain-phase error; (c) the temporal mismatch case in the presence of the intrinsic clutter motion (ICM); (d) the ideal case when β = 0.5. From Figure 9a, the curves show that the two reduced-dimension SR-based methods outperform the conventional methods and can obtain satisfying performance in a heterogeneous environment. Specifically, the RDSR exhibits a wider notch, and thus the proposed method has a superior minimum detectable velocity (MDV) performance. It can be observed from Figure 9b that all the SR-STAP methods suffer performance loss due to the model mismatch between the dictionary and clutter data. In Figure 9c, the proposed method shows a decreased performance compared to the ideal case since ICM leads to a broadened clutter spectrum, where more space-time vectors are needed to represent the clutter. In Figure 9d, when β = 0.5, all the SR methods encounter the off-grid effect and have a degraded performance.

In Figure 10, we evaluate the average SINR loss compared to a different number of training samples $L$, i.e., the convergence rate. The average SINR loss is also defined as the mean of the SINR loss values for $f_d\in (-0.5,-0.05)\cup (0.05,0.5)$. As demonstrated in Figure 10, the curve of the LRDSR-CALM-SBL has a 3.5 dB improvement over the curve achieved by RDSR. Therefore, the proposed method can obtain superior performance even with insufficient sample support and exhibits a faster convergence speed than the other considered algorithms.

Finally, we verify the target detection performance improvement via the probability of detection ($P_d$) versus the target SNR curves acquired by employing the cell-averaging CFAR (CA-CFAR) detector [46]. One hundred targets are assumed in the boresight and randomly injected into the whole Range-Doppler plane. The probability of a false alarm rate $P_{fa}$ is next set to be 10⁻³, and the related detection threshold is obtained by estimating the noise floor from the data surrounding the CUT. As illustrated in Figure 11, there exist over 1 dB SNR improvement when the $P_d$ is 0.8. In general, the proposed algorithm shows better detection performance than other compared methods.

5. Discussion

This research aims to develop an effective and efficient STAP algorithm for airborne radar with insufficient training samples in inhomogeneous environments.

Based on the quantitative and qualitative analysis from the previous Sections. A hierarchical structure is employed to build the sparse Bayesian learning model with adaptive Laplace priors. Compared with the existing Gaussian prior [29,30], it can promote the sparsity more heavily and ensure a higher recovery accuracy. It can be seen from Figure 6 that the proposed CALM-SBL is closer to the optimal STAP performance than the Gaussian-prior-based M-SBL algorithm, as summarized in

Table 3. SINR loss comparison of Figure 6.

	Algorithm	OPT	M-SBL	CALM-SBL
Normalized Doppler Frequency
0.06		−1.369 dB	−2.097 dB	−1.375 dB
0.2		−0.174 dB	−0.229 dB	−0.183 dB
0.49		−0.069 dB	−0.087 dB	−0.074 dB

.

In addition, although the average SINR loss of the CALM-SBL is not as good as that of M-SBL when there are only two training samples, as the number of samples increases, the performance will outperform that of M-SBL, as shown in

Table 4. The average SINR loss comparison of Figure 7.

	Algorithm	OPT	M-SBL	CALM-SBL
Training Sample Number
2		−0.252 dB	−0.879 dB	−1.453 dB
4		−0.252 dB	−0.367 dB	−0.299 dB
10		−0.252 dB	−0.356 dB	−0.260 dB

.

Similar to other SBL algorithms, large-scale matrix inversion operation and several large-scale matrix calculations are involved in each iteration process, resulting in high computational complexity, especially when the radar system has a high space-time degree of freedom. Benefiting from the fact that we only update the diagonal elements in the process of updating $\Sigma$, as detailed in Equation (33), the average running time of CALM-SBL is 10.1729 s, while that of M-SBL and M-FCSBL are 288.9466 s and 40.8802 s, respectively. Moreover, we next propose the LRDSR processing scheme to accelerate CALM-SBL by scaling down the CALM-SBL to a localized reduced-dimension region. After the computational complexity analysis in Table 1 and the comparison of average code running time in Table 2, we can observe that LRDSR-CALM-SBL only takes about one percent of the running time of the original CALM-SBL when K = 3. It should be noted that the LRDSR is essentially a reduced-dimension STAP processing scheme and inevitably suffers a certain performance loss [36].

Table 5. SINR loss comparison of CALM-SBL and LRDSR-CALM-SBL.

	Algorithm	OPT	CALM-SBL	LRDSR-CALM-SBL
Normalized Doppler Frequency
0.125		−0.373 dB	−0.381 dB	−4.223 dB
0.375		−0.080 dB	−0.092 dB	−3.035 dB

evaluates the SINR loss performance of CALM-SBL in Figure 6 and LRDSR-CALM-SBL in Figure 9a. As shown in Table 5, the SINR loss performance is about 3–4 dB lower. Although there is certain performance degradation, its running speed has been significantly improved, which is an acceptable trade-off in practical engineering applications.

6. Conclusions

In this work, we firstly developed a complex adaptive Laplace-prior-based SBL STAP algorithm by introducing an improved hierarchical Bayesian model to endow the hyperparameters with a complex Laplace prior, thus obtaining a superior performance than an SBL algorithms with a Gaussian prior in a sample shortage case. To decrease the computational requirements of the proposed CALM-SBL, especially in the context of a high-dimensional modern radar system, an efficient localized reduced-dimension sparse-recovery-based STAP processing scheme is then investigated. By exploiting the local sparsity and prior knowledhtte of the clutter, a localized reduced-dimension dictionary is constructed, and the computational efficiency is considerably enhanced. All of the simulation results have illustrated that the proposed methods overcome the limitation of the training samples and alleviate the influence of the computational complexity on the real-time processing system.