Airborne Radar Space–Time Adaptive Processing Algorithm Based on Dictionary and Clutter Power Spectrum Correction

Abstract

Sparse recovery space–time adaptive processing (SR-STAP) technology improves the moving target detection performance of airborne radar. However, the sparse recovery method with a fixed dictionary usually leads to an off-grid effect. This paper proposes a STAP algorithm for airborne radar based on dictionary and clutter power spectrum joint correction (DCPSJC-STAP). The algorithm first performs nonlinear regression in a non-stationary clutter environment with unknown yaw angles, and it corrects the corresponding dictionary for each snapshot by updating the clutter ridge parameters. Then, the corrected dictionary is combined with the sparse Bayesian learning algorithm to iteratively update the required hyperparameters, which are used to correct the clutter power spectrum and estimate the clutter covariance matrix. The proposed algorithm can effectively overcome the off-grid effect and improve the moving target detection performance of airborne radar in actual complex clutter environments. Simulation experiments verified the effectiveness of this algorithm in improving clutter estimation accuracy and moving target detection performance.

1. Introduction

Due to the movement of the platform and the downward-looking working mode, strong ground/sea clutter interferes with airborne radar when detecting moving targets, which usually leads to target detection failure. Space–time adaptive processing (STAP) technology can use two-dimensional joint information of space and time to achieve the effective filtering of space–time coupled clutter and target coherent accumulation, and it can improve the output signal-to-noise ratio. Therefore, it is one of the key technologies for current radar clutter suppression and moving target detection on various moving platforms [1]. Existing research shows that STAP technology can suppress clutter and enhance the capability of airborne radar in detecting targets more effectively. However, for STAP to achieve a loss of no more than 3 dB in the output signal-to-noise ratio, it requires independent and identically distributed (IID) snapshots that are at least twice the degree of freedom (DoF) of the radar system [2]. The above conditions are difficult to guarantee in actual radar detection. The reason for this is that the space–time two-dimensional freedom of the system is very high, and the rapid changes in an actual clutter environment makes the real clutter faced by airborne radar non-uniform and difficult to obtain.

In order to enhance the operational efficiency of space–time joint processing in real-world scenarios, the development of STAP technology in recent years has mainly focused on the following directions. Dimensionality reduction and rank reduction algorithms have been researched, which can reduce the required number of snapshots and computational complexity [3,4]. Algorithms suitable for non-uniform environments have been studied, such as the direct data domain method, which can reduce the impact of interference signals and achieve moving target detection under the conditions of a few snapshots or a single snapshot [5]. The knowledge-assisted STAP algorithm has been researched, which uses prior knowledge such as terrain maps to pre-filter radar echo data and select snapshots; it can overcome the influence of non-uniform clutter [6,7]. The utilization of sparse recovery in the STAP algorithm enables a reduction in the quantity of snapshots needed and enhances moving target detection performance, particularly when dealing with limited snapshot scenarios [8]. Among them, the STAP algorithm based on sparse recovery has received increasing attention due to its advantages in performance.

The sparse recovery space–time adaptive processing (SR-STAP) method is based on compressive sensing (CS) theory [9]. This technology breaks through the limitation of the traditional STAP method regarding the requirement of IID snapshots, and it is capable of retrieving the clutter power spectrum with minimal snapshots, resulting in a high-resolution outcome. The key to SR-STAP is determining how to quickly solve underdetermined linear problems with sparse regular terms in order to obtain high-precision clutter covariance matrix (CCM) estimation. This would provide a new method to overcome problems such as insufficient IID snapshots in non-uniform and non-stationary clutter environments. In recent years, researchers have combined the characteristics of airborne radar data with sparse recovery technology, and they have proposed a series of SR-STAP algorithms with better performance [10,11,12,13,14,15,16,17,18,19,20], such as the smooth domain STAP algorithm, weighted regularized STAP algorithm, homotopy STAP algorithm, iterative adaptive STAP algorithm, dictionary dimensionality reduction STAP algorithm, and STAP algorithm based on sparse Bayesian learning. These algorithms can obtain approximately optimal clutter suppression performance by using a few or even one snapshot under the condition that the dictionary matrix can represent the received clutter data accurately.

In consideration of the sparsity of both clutter and targets in the space–time domain, the spatial sparse spectrum estimation algorithm was proposed, which constructs an underdetermined optimization problem based on L1 norm sparse regularization using the dictionary matrix obtained by uniformly discretizing the space–time plane and the beam-Doppler domain data of the cell under test, and then, the global matched filter method is used to achieve the power spectrum estimation of the target and clutter [21,22]. This algorithm is a preliminary attempt in the field of SR-STAP, and there are still many problems that are yet to be solved. Subsequently, the sparse recovery STAP algorithm based on the direct data domain was proposed [23,24,25]. This method first qualitatively analyzes the sparsity of the clutter distribution from the space–time coupling relationship of clutter blocks and radar beam weighting, and then it reconstructs the space–time spectrum and STAP optimal weight vector using the FOCUSS algorithm. Finally, the traditional constant false alarm rate (CFAR) detector is used to achieve target detection. In order to improve the robustness of the SR-STAP algorithm, researchers have proposed a series of improvement methods. To cope with the grid mismatch phenomenon, a method was proposed to change the dictionary structure, which can generate new grid points that match the real clutter ridges through dictionary learning, localized search, adaptive partitioning, and other technologies [26,27,28]. Avoiding grid division, an algorithm was proposed that utilizes the low-rank, block-Toeplitz and semi-positive definite characteristics of clutter to construct a grid-free network based on atomic norm minimization (ANM) [29,30]. To eliminate array amplitude and phase errors, a joint estimation algorithm of the error and space–time sparse spectrum was proposed, which uses the alternating multiplier method to achieve an iterative solution of amplitude, phase errors, and sparse coefficients [31,32]. Regarding the influence of the mutual coupling of arrays, a STAP algorithm based on nested arrays and coprime arrays was proposed [33], and a spatial dimensionality reduction STAP algorithm based on the cyclic characteristics of mutual coupling matrices was proposed [34].

Although the relevant improved SR-STAP algorithms have improved the performance of airborne radar in suppressing clutter to a certain extent, they all have certain limitations and can usually only be corrected for a specific single error. According to existing research results, the main reasons for errors in SR-STAP methods include two aspects: Firstly, the process of sparse recovery requires a redundant dictionary. When the dictionary division is inaccurate, the clutter components are no longer located on the dictionary grid, and the mismatch between the clutter ridge and the grids leads to a decrease in sparse recovery accuracy, which is called the off-grid effect [35,36]. Secondly, the performance of the existing SR-STAP algorithm depends on the settings of relevant parameters. Under non-ideal conditions, such as errors or inherent clutter motion, algorithm performance is significantly degraded due to parameter blindness [37].

To cope with the above problems, this paper proposes a STAP algorithm for airborne radar based on dictionary and clutter power spectrum joint correction (DCPSJC-STAP). First, a nonlinear regression process is used to perform dictionary correction for each snapshot to eradicate the off-grid effect. Then, the clutter ridge parameters estimated by all snapshots are updated and iterated on the basis of the traditional dictionary to form a new dictionary. According to the Bayesian method, the estimation of sparse signals is equivalent to the maximum posterior probability estimation of the parameters under sparse priors [38,39]. Hence, this paper uses the Bayesian learning method to estimate hyperparameters iteratively, which can improve the estimation accuracy of the clutter power spectrum. The proposed algorithm not only reduces the error caused by off-grid problems but also accurately estimates the sparse recovery vector based on the corrected dictionary. Hence, the accuracy of estimating the covariance matrix for clutter, as well as the performance in suppressing clutter and detecting targets, shows significant improvement. The proposed algorithm’s effectiveness was confirmed through simulation experiments.

Notation: I_NM stands for identity matrix, ‖·‖₀ represents the number of non-zero elements in a vector, ‖·‖₂ represents the l2-norm, ⊗ represents the Kronecker product operation, [·]^T represents the transpose of the matrix, [·]^H represents the transpose conjugation of the matrix, [·]⁻¹ represents the inverse of the matrix, and E[·] represents the expected value of the matrix.

2. Signal Model

An airborne radar with a uniform linear array is discussed, and the influence of distance ambiguity is not considered. A working model of the airborne radar array is shown in Figure 1. The working wavelength of the radar is λ, the array element spacing is d, the number of array elements is N, the number of pulses transmitted within a coherent processing interval is K, and the pulse repetition interval is T_r. The angles θ and φ represent the azimuth and elevation angles of the radar beam, respectively. θ_e is the angle between the array axis and the flight direction. The carrier aircraft has height h and flies uniformly along the positive direction of the Y-axis at speed V.

The space–time snapshot data X∈C^NK×1 within a single range cell can be expressed as

$$\mathit{X}={\mathit{X}}_{\mathit{c}}+\mathit{n}$$

(1)

where X_c and n represent the space–time clutter vector and noise vector of the range cell under test, respectively.

The range cell under test can be regarded as N_c independent clutter blocks evenly divided along the azimuth angle, so X_c can be expressed as the sum of the reflection signals of N_c clutter blocks; that is,

$${\mathit{X}}_{\mathit{c}}={\sum }_{n=1}^{N_c}{\mathit{\gamma }}_{\mathit{n}}\mathit{s}(f_{d,n},f_{s,n}),$$

(2)

where γ_n, f_d,n, and f_s,n represent the complex amplitude, Doppler frequency, and spatial frequency of the n-th clutter block reflection signal, respectively, and s(f_d,n, f_s,n) is the space–time steering vector, expressed as [40]

$$\mathit{s}(f_{d,n},f_{s,n})={\mathit{S}(f}_{d,n})\mathbf{\otimes }{\mathit{S}(f}_{s,n}),$$

(3)

The time steering vector and the space steering vector are expressed as [3]

$${\mathit{S}(f}_{d,n})={[1,e^{2j\pi f_{d,n}},\dots , e^{2j\pi (K-1)f_{d,n}}]}^{\mathbf{T}},$$

(4)

$${\mathit{S}(f}_{s,n})={[1,e^{2j\pi f_{ s,n}},\dots , e^{2j\pi (N-1)f_{ s,n}}]}^{\mathbf{T}}.$$

(5)

The Doppler frequency and spatial frequency of the n-th clutter block are expressed as

$$f_{d,n}=\frac{2V{T}_r}{\lambda } \sin ({\theta }_n+{\theta }_e)\cos {\phi }_n ,$$

(6)

$$f_{s,n}= \frac{d}{\lambda } \sin {\theta }_n\cos {\phi }_n.$$

(7)

where the angles θ_n and φ_n represent the azimuth angle and elevation angle of the n-th clutter block, respectively, and θ_e is the angle between the array axis and the flight direction.

The clutter covariance matrix R_c can be expressed as

$${\mathit{R}}_{\mathit{c}}=\mathrm{E}\left[{\mathit{X}\mathit{X}}^{\mathit{H}}\right]={\sum }_{n=1}^{N_c}{{\gamma }_n}^2 \mathit{s}(f_{d,n},f_{s,n}) {\mathit{s}}^{\mathit{H}}(f_{d,n},f_{s,n}).$$

(8)

Assuming that clutter and noise are independent of each other, the clutter plus noise covariance matrix R can be expressed as

$$\mathit{R}={\mathit{R}}_{\mathit{c}}+{\mathit{R}}_{\mathit{N}}={\sum }_{n=1}^{N_c}{{\gamma }_n}^2 \mathit{s}(f_{d,n},f_{s,n}) {\mathit{s}}^{\mathbf{H}}(f_{d,n},f_{s,n}) + {{\delta }_n}^2{\mathit{I}}_{\mathit{N}\mathit{M}}.$$

(9)

According to the linearly constrained minimum variance criterion, the optimal weight vector can be expressed as [41]

$${\mathit{w}}_{\mathit{o}\mathit{p}\mathit{t}}=\frac{{\mathit{R}}^{-1}\mathit{s}}{{\mathit{s}}^{\mathbf{H}}{\mathit{R}}^{-1}\mathit{s}}$$

(10)

Usually, when affected by actual environmental factors, maximum likelihood estimation is used to calculate the clutter covariance matrix R′:

$${\mathit{R}}^{\prime }=\frac{1}{L}{\sum }_{l=1}^L{\mathit{x}}_{\mathit{l}}{\mathit{x}}_{\mathit{l}}^{\mathbf{H}},$$

(11)

where L is the number of snapshots, and x_l represents the snapshot data of the l-th range cell.

3. STAP Algorithm for Airborne Radar Based on Dictionary and Clutter Power Spectrum Joint Correction

3.1. Principle of Sparse Recovery

A sparse signal refers to a signal that is compressible or that is sparse under a certain transformation basis. Most actual signals have sparse characteristics, and the proportion of useful information in the signal is very small [42]. STAP based on sparse recovery technology aims to estimate a high-resolution space–time clutter power spectrum using fewer snapshots. This method first needs to construct a dictionary by evenly dividing the space–time plane into N_s × N_d grids along the spatial frequency axis and Doppler frequency axis, where N_s = ρ_s × N, N_d = ρ_d × K, ρ_s > 1, and ρ_d > 1, with the parameters ρ_s and ρ_d being the resolution scales along the spatial frequency axis and Doppler frequency axis, respectively. On this basis, the space–time steering dictionary Φ is formed by the collection of space–time steering vectors associated with each grid point, expressed as

$$\mathit{\Phi }=[\mathit{s}\left(f_{d,1},f_{s,1}\right),\dots ,\mathit{s}\left(f_{d,1},f_{s,N_s}\right),\mathit{s}\left(f_{d,2},f_{s,1}\right),\dots ,\mathit{s}\left(f_{d,N_d},f_{s,N_s}\right)] NK\times N_s{N}_d.$$

(12)

Therefore, the clutter snapshot can be expressed as

$$\mathit{X}={\sum }_{i=1}^{N_s}{\sum }_{j=1}^{N_d}{\gamma }_{ij}\mathit{s}(f_{d,j},f_{s,i})=\mathit{\Phi }\mathit{\Gamma },$$

(13)

where Γ is the sparse recovery vector, which can be written as

$$\mathit{\Gamma }={[{\gamma }_1,{\gamma }_2,\dots ,{\gamma }_{N_s{N}_d}]}^T.$$

(14)

The above sparse recovery vector can be acquired through the resolution of the subsequent optimization problem.

$$\mathit{min} {\left|\right|\gamma \left|\right|}_0\mathit{subject} \mathit{to} {\left|\right| \mathit{X}-\mathit{\Phi }\gamma \left|\right|}_2^2 \le \xi .$$

(15)

3.2. Dictionary Mismatch Problem

For the side-looking uniform linear array, the inclination of the clutter ridge in the angle-Doppler plane corresponds to the ratio between the spatial frequency and Doppler frequency associated with each clutter patch. When the clutter ridge slope value is 1, the clutter component can fall on uniform discrete grid points. However, in the actual environment, due to various interferences, the slope of the clutter ridge typically does not correspond to a whole number. Therefore, when the ratio of N_s and N_d is set improperly, it can easily lead to off-grid problems. Figure 2a presents a schematic diagram of the clutter distribution of the side-looking array under ideal conditions without off-grid effects. The clutter ridge and grids match exactly. Figure 2b presents a schematic diagram of the clutter distribution of the side-looking array with the off-grid effect. Most of the clutter components do not match the grid accurately.

3.3. Dictionary Correction Method

In order to solve the dictionary mismatch in SR-STAP, especially if there is an unknown yaw angle, the parameter estimation results obtained using the SR-STAP model can be used in nonlinear regression, which is performed to re-correct the Doppler frequency by estimating the yaw angle. Dictionary correction can improve the clutter suppression performance of the SR-STAP method. By combining Formulas (6) and (7), the Doppler frequency of the clutter point is expressed as a function of the yaw angle θ_e and the spatial frequency f_s,n:

$$f_{d,n}=f_{d,n}({\theta }_e,f_{s,n}).$$

(16)

Then, the estimated value θ_e′ of the yaw angle can be expressed as

$${{\theta }_e}^{\prime }\mathrm{argmin} {\sum }_{c=1}^C{\gamma }_i\left|\right| f_{d,n}-f_{d,n}({\theta }_e,f_{s,n}){\left|\right|}_2 ,$$

(17)

where C is the number of clutter points, and ${\mathit{\gamma }}_{\mathit{i}}$ represents the non-zero weight of the clutter spectrum. Then, the estimated value of the yaw angle θ_e′ can be used to update the Doppler frequency f_d,n′ corresponding to the clutter:

$${f_{d,n}}^{\prime }=f_{d,n}({{\theta }_e}^{\prime },f_{s,n}).$$

(18)

The above estimation method is used to update the Doppler frequency of each clutter point to obtain a new temporal steering vector. Then, combined with the spatial steering vector obtained using the SR-STAP model, the space–time steering vector for a particular snapshot is obtained. All snapshots are processed using the above method, and a three-dimensional dictionary matrix, Φ₁ (m,n,w), is formed, where m represents the number of dictionary rows, n represents the number of dictionary columns, and w represents the number of snapshots. At the same time, the clutter ridge parameter values f_d,n′ estimated using all snapshots are updated and iterated on the basis of the traditional dictionary to form a new dictionary, Φ₂ (f _d,n′, f_s,n).

3.4. Sparse Bayesian Learning Method

The sparse Bayesian learning algorithm was developed based on the correlation vector machine framework and has become an effective tool for sparse signal reconstruction and recovery [43,44]. Existing sparse Bayesian algorithms are similar to the L₁ regularization operation in solving underdetermined linear equations, and they obtain sparse solutions by giving random variables a certain prior probability distribution. An application of sparse Bayesian learning was incorporated in the recovery of sparse signals, specifically for single-measurement vectors (SMVs), and it was later extended to the case of multiple-measurement vectors (MMVs) [45].

In the SR-STAP method, the multiple-measurement vector can obtain stationary snapshots of several snapshots in the distance dimension under the condition of local clutter stationarity. To date, researchers have proposed a variety of MMV algorithms, such as the multiple convex optimization (M-CVX) [46], multiple orthogonal matching pursuit (M-OMP) [47], multiple focal underdetermined system solver (M-FOCUSS) [48], and subspace augmented multiple signal classification (SA-MUSIC) [49]. Although these SR algorithms based on MMVs have a significant improvement in convergence, their performance is not stable because the sparsity-based space–time filters are particularly sensitive to regularization parameters, clutter sparsity, and noise power. Hence, the existing MMV SR algorithm mainly relies on parameters, and, under conditions with unknown system errors or inherent clutter motion, performance will be degraded due to parameter blindness.

According to the Bayesian viewpoint, estimating a sparse signal is essentially equivalent to maximizing the posterior probability estimation while considering sparse priors on the relevant parameters. Although the M-FOCUSS algorithm can be characterized as a Bayesian method, its performance needs to be improved. Multiple-measurement vector sparse Bayesian learning (M-SBL) is different from the MMV SR algorithm. The global minimum of M-SBL is always the sparsest solution because, in the Bayesian framework, by introducing a sparse prior, the system is more likely to choose a solution with fewer non-zero elements to maximize the posterior probability. Hence, the system prefers to choose the sparsest solution. The advantage of the M-SBL algorithm is independent of a priori parameters [50]. Therefore, when the internal structure of the system is disturbed, the algorithm can ensure the robustness of sparse spectrum estimation in uncertain environments and can reduce the sensitivity to clutter.

The algorithm proposed in this paper uses the corrected three-dimensional dictionary Φ₁(m,n,w) and the new dictionary Φ₂ combined with M-SBL, and it utilizes snapshots to obtain the required hyperparameters iteratively. Then, the estimated sparse recovery vector and the new dictionary Φ₂ are combined to estimate the final clutter covariance matrix and the optimal filtering weight vector.

According to existing research conclusions, the conditional probability density function of the space–time snapshot X is

$$p\left(\mathit{X}|\mathit{\Gamma },{\delta }_n^2\right) ={\prod }_{l=1}^{L}CN\left({\mathit{x}}_{\mathit{l}}\right|\mathit{\Phi }{\mathit{\gamma }}_{\mathit{l}},{\delta }_n^2{\mathit{I}}_{\mathit{N}\mathit{M}}),$$

(19)

where Φ is the space–time steering dictionary; δ_n² is the noise power; and Γ is a joint sparse matrix, which can be written as

$$\mathit{\Gamma }=[{\mathit{\gamma }}_1,{\mathit{\gamma }}_{\mathbf{2}}\mathbf{,}\mathbf{\dots }\mathbf{,} {\mathit{\gamma }}_{\mathit{L}}].$$

(20)

The M-SBL algorithm assumes that the prior distribution is modulated by a hyperparameter vector that controls the prior variance of each row Γ, and the hyperparameters can be obtained from the snapshots. Assume that γ_l (l = 1, 2,…, L) are independent of each other and have the same complex Gaussian prior probability density, that is, γ_l ~ CN(0, Λ), and that Λ = diag(p), p = [p₁, p₂, …, p_NsNd] is the atomic power, which can represent the unknown variance corresponding to the grid points in the angle–Doppler plane and is used to estimate the clutter covariance matrix. Then, the prior distribution of matrix Γ is

$$\mathrm{P}(\mathit{\Gamma }\left|\mathit{\Lambda }\right) ={\prod }_{l=1}^L{CN(\mathit{\gamma }}_{\mathit{l}}|0,\mathit{\Lambda }).$$

(21)

Then, the posterior distribution of matrix Γ can be obtained by combining Formulas (18)–(20):

$$\mathrm{P}\left(\mathit{\Gamma }\right|\mathit{X},\mathit{\Lambda }, {\delta }_n^2)={\prod }_{l=1}^L\frac{p\left({\mathit{\gamma }}_{l,}{\mathit{x}}_{\mathit{l}}\right|\mathsf{\Lambda }, {\delta }_n^2)}{\int p\left({\mathit{\gamma }}_{l,}{\mathit{x}}_{\mathit{l}}\right|\mathsf{\Lambda }, {\delta }_n^2) d{\mathit{\gamma }}_l},$$

(22)

Hyperparameters p and δ_n² are iteratively updated according to the following two steps:

Step 1: The known k-th step hyperparameters Λ^(k) and ${{\delta }_n^2}^{\left(k\right)}$ are used to perform the posterior distribution of (k + 1)-th step Γ to obtain the mean matrix μ_l^(k+1) and the covariance matrix Σ^(k+1) of the (k + 1)-th step.

$$\mathrm{P}(\mathit{\Gamma }\left|\mathit{X},{\mathit{\Lambda }}^{\left(k\right)}, {{(\delta }_n^2)}^{\left(k\right)}\right)={\prod }_{l=1}^L\mathrm{C}\mathrm{N}\left({\mathit{\gamma }}_{\mathit{l}}|{{\mathit{\mu }}_{\mathit{l}}}^{\left(k+1\right)},{\mathit{\Sigma }}^{\left(k+1\right)}\right) ,$$

(23)

$${{\mathit{\mu }}_{\mathit{l}}}^{\left(k+1\right)}={\mathit{\Lambda }}^{\left(k\right)}{{\mathit{\Phi }}_1}^T{\left({\mathit{R}}_{\mathit{c}+\mathit{n}}^{\left(k\right)}\right)}^{-1}{\mathit{x}}_{\mathit{l}},l=\mathrm{1,2},\dots L ,$$

(24)

$${\mathit{\Sigma }}^{(k+1)}{=\mathit{\Lambda }}^{\left(k\right)}-{\mathit{\Lambda }}^{\left(k\right)}{{\mathit{\Phi }}_2}^T{\left({\mathit{R}}_{\mathit{c}+\mathit{n}}^{\left(k\right)}\right)}^{-1}{\mathit{\Phi }}_2{\mathsf{\Lambda }}^{\left(k\right)},$$

(25)

where Λ^(k) = diag(p^(k)), p^(k) = [ p₁^(k), p₂^(k), …, p_NsNd^(k)].

Step 2: The known posterior distribution of step Γ at the (k + 1)-th step is used to update hyperparameters p and δ_n².

$${\mathit{p}}_{\mathit{i}}^{\mathit{k}+1}=\frac{\frac{1}{ L}\sum _{l=1}^L{\left|{{\mathit{\mu }}_l}^{\left(k+1\right)}\right(i\left)\right|}^2}{1-\frac{{\mathit{\Sigma }}_{ii}^{(k+1)}}{{\mathit{p}}_{\mathit{i}}^{\mathit{k}}}} ,$$

(26)

$${{(\delta }_n^2)}^{(k+1)} = \frac{\frac{1}{L}\sum _{l=1}^L{\left|\right|{\mathit{x}}_{\mathit{l}}-{\mathit{\Phi }}_1{\mathit{\mu }}_l^{\left(k+1\right)}\left|\right|}_2^2}{N-M+\sum _{i=1}^{N_s{N}_d}/\frac{{\mathit{\Sigma }}_{\mathit{i}\mathit{i}}^{(\mathit{k}+1)}}{{\mathit{p}}_{\mathit{i}}^{\mathit{k}}}},$$

(27)

Formulas (23) and (26) use the aforementioned snapshot correction dictionary Φ₁. After obtaining the parameters Λ^(k+1) and ${{\delta }_n^2}^{(k+1)}$, the clutter plus noise covariance matrix R_c+n^(k+1) can be updated as follows:

$${\mathit{R}}_{c+n}^{(k+1)}={\mathit{\Phi }}_2{\mathit{\Lambda }}^{(k+1)}{{\mathit{\Phi }}_2}^{\mathit{T}}+{{\delta }_n^2}^{(k+1)}{\mathit{I}}_{\mathit{N}\mathit{M}}.$$

(28)

After obtaining the updated clutter covariance matrix, μL ^(k+1), Σ^(k+1), and other parameters can be iteratively updated. Then, the updated Λ is obtained, which is recorded as Λ₁. Λ₁ and Φ₂ are combined to estimate the final clutter covariance matrix R_c as

$${\mathit{R}}_{\mathit{c}}=\mathit{X}{\mathit{X}}^{\mathit{T}}={\mathit{\Phi }}_2{\mathsf{\Lambda }}_1{\mathsf{\Lambda }}_1^T{{\mathit{\Phi }}_2}^T={\mathit{\Phi }}_2{\mathsf{\Lambda }}_1^2{{\mathit{\Phi }}_2}^T.$$

(29)

The estimated value of the real clutter covariance matrix R_c1 is expressed as

$${\mathit{R}}_{\mathit{c}1}={\mathit{R}}_{\mathit{c}}/L .$$

(30)

After diagonally loading R_c1, the matrix R_cs is obtained. Then, the optimal filtering weight vector W is

$$\mathit{W}=\frac{{\mathit{R}}_{\mathit{c}\mathit{s}}^{-1}\mathit{S}}{{\mathit{S}}^{\prime }{\mathit{R}}_{\mathit{c}\mathit{s}}^{-1}\mathit{S}},$$

(31)

where S represents the steering vector of the target.

3.5. The Process of Proposed Algorithm

Based on the above principles of dictionary correction and clutter power spectrum correction, an airborne radar STAP algorithm with dictionary and clutter power spectrum joint correction is proposed, whose process is given as follows:

Step 1: The STAP model is initialized according to Formulas (1)–(11). The elevation angle φ corresponding to each snapshot is calculated in sequence, which is stored as an array with dimension w.

$$\phi \left(\mathrm{w}\right)=\arcsin \left(\frac{h}{R_t}\right),$$

(32)

where R_t is the radar working distance of the range cell.

At the same time, combined with Formulas (6) and (7), the Doppler frequency corresponding to the clutter point is expressed as a function of the yaw angle θ_e and the spatial frequency f_s,n.

Step 2: A traditional dictionary is constructed according to Formula (12).

Step 3: Sparse reconstruction is used to obtain the sparse vector Γ according to Formulas (13)–(15).

Step 4: The estimated value of the yaw angle θ_e′ is calculated according to Formula (17), and the Doppler frequency f_d,n′ of clutter is updated according to Formula (18). Finally, a new time steering vector is constructed to generate a new dictionary for each snapshot, and a three-dimensional dictionary matrix Φ₁ is formed. At the same time, the clutter ridge parameter values f_d,n′ estimated by all snapshots are updated to form a new dictionary, Φ₂(f_d,n′, f_s,n).

Step 5: The Bayesian iteration is performed according to Formulas (19)–(28). The mean matrix μL^(k+1) is corrected with the corrected three-dimensional dictionary Φ₁. The covariance matrix Σ^(k+1) is updated with the new dictionary Φ₂. Hyperparameter p is updated with the previously obtained parameters. Hyperparameter δ_n² is updated using the corrected three-dimensional dictionary Φ₁. The new dictionary Φ₂ is used to update the clutter plus noise covariance matrix R_c+n. By analogy, the value of c represents the iterations, while t denotes the upper limit for the number of iterations.

If c > t, then the iteration ends, and the process enters Step 6.

Step 6: The clutter covariance matrix R_c and the optimal filtering weight vector W are estimated based on Φ₂ and the finally updated hyperparameter p.

The process of the algorithm mentioned above is illustrated in Figure 3.

4. Simulation Experiment Analysis

This section analyzes the performance of the proposed algorithm through simulation experiments. The proposed algorithm is compared with SR-STAP [51], the SBL-STAP algorithm [38], the ANM-STAP algorithm [29], and the clutter reconstruction based on the nonlinear regression (CRNR-STAP) algorithm [52]. The power spectrum, clutter suppression performance, and moving target detection performance of these algorithms are analyzed in turn. The grid division coefficients ρ_s and ρ_d are both set to 6. All experimental outcomes undergo 100 iterations of Monte Carlo averaging experiments. The radar system parameters are shown in

Table 1. Parameters of the radar system.

Parameter	Value
Number of array elements	10
Number of pulses sent within one CPI	10
Wavelength (m)	0.3
Array element spacing (m)	0.15
Aircraft speed (m/s)	240
Aircraft altitude (m)	3000
Pulse repetition frequency (HZ)	4000
Signal-to-noise ratio (dB)	0
Clutter-to-noise ratio (dB)	60

.

4.1. Clutter Power Spectrum Analysis

The first experiment tests the proposed DCPSJC-STAP algorithm, the SR-STAP algorithm, the SBL-STAP algorithm, the ANM-STAP algorithm, and the CRNR-STAP algorithm. The expression of the clutter power spectrum is P = $\frac{1}{{\mathit{S}}^H{{\mathit{R}}_{\mathit{c}}}^{-1}\mathit{S}}$, where S is the steering vector and R_c is the clutter covariance matrix. The clutter power spectra estimated by these algorithms are compared, and they are shown in Figure 4a–e.

In Figure 4a, the clutter spectrum formed by the SR-STAP algorithm completely concentrates on the clutter ridge, but the estimated clutter energy is low, which is not conducive to subsequent clutter suppression. It can be seen in Figure 4b that the SBL-STAP algorithm has a serious broadening of the clutter spectrum due to the off-grid effect, and the target will be submerged in widely distributed clutter. In Figure 4c, the CRNR-STAP algorithm considers the azimuth weighting effect of the radar beam, performs nonlinear regression on the clutter spectrum in the azimuth, and uses a normalized beamformer response analysis, so the weight near the spatial frequency of 0 is larger, and the rest of the weight is smaller. Therefore, the clutter power spectrum estimated by the CRNR-STAP algorithm has a large error, which will lead to a significant decrease in clutter suppression performance. In Figure 4d, the clutter power spectrum formed by the ANM-STAP algorithm concentrates on the clutter ridge but also has a serious widening; the effect of suppressing clutter is not good. In Figure 4e, the clutter power spectrum formed by the proposed DCPSJC-STAP algorithm concentrates on the clutter ridge. Compared with the SBL-STAP and ANM-STAP algorithms, the broadening of the clutter ridge is significantly improved, and the clutter power value is higher, which contributes to the enhancement of clutter suppression and performance in detecting moving targets.

4.2. Analysis of Clutter Suppression Performance

The second experiment compares the clutter suppression performance of the proposed DCPSJC-STAP algorithm, the SR-STAP algorithm, the SBL-STAP algorithm, the ANM-STAP algorithm, and the CRNR-STAP algorithm. The expression of the signal-to-clutter-plus-noise ratio loss is ${\mathbf{S}\mathbf{C}\mathbf{N}\mathbf{R}}_{\mathit{l}\mathit{o}\mathit{s}\mathit{s}}=\frac{{\mathbf{S}\mathbf{C}\mathbf{N}\mathbf{R}}_{\mathit{o}\mathit{u}\mathit{t}}}{{\mathbf{S}\mathbf{C}\mathbf{N}\mathbf{R}}_{\mathit{o}\mathit{p}\mathit{t}}}$ = $\frac{{\mathit{\sigma }}_{\mathit{n}}^2{\left|{\mathit{W}}^{\mathit{H}}{\mathit{S}}_{\mathit{t}}\right|}^2}{{\mathit{N}\mathit{K}|\mathit{W}}^{\mathit{H}}\mathit{R}\mathit{W}|}$ [53], where W represents the estimated STAP weight; R represents the clutter plus noise covariance matrix; N is the number of array elements; K is the number of pulses transmitted within a coherent processing interval; σ_n² is the noise power; and S_t is the steering vector of target.

Figure 5 shows the relationship between the SCNR_loss of the five algorithms and the normalized Doppler frequency. It can be seen that the CRNR-STAP algorithm has a shallow depression and a wide notch. Similarly, the SR-STAP algorithm has a shallow depression depth in the main clutter area, and the clutter suppression performance is poor. Compared with SBL-STAP and ANM-STAP, the DCPSJC-STAP algorithm has the deepest void value and the narrowest notch in the main clutter region, so it has the best clutter suppression performance.

4.3. Analysis of the Convergence Performance

Figure 6 shows the relationship between SCNR_loss and the quantity of snapshots of different algorithms in the zero Doppler frequency position. It can be seen that the SCNR_loss of the proposed DCPSJC-STAP algorithm is the smallest, which represents its best clutter suppression performance. Moreover, the proposed DCPSJC-STAP algorithm only needs about 20 snapshots to achieve the optimal clutter suppression performance.

4.4. Moving Target Detection Performance Analysis

The third experiment compares the moving target detection performance of the proposed DCPSJC-STAP algorithm, the SR-STAP algorithm, the SBL-STAP algorithm, the ANM-STAP algorithm, and the CRNR-STAP algorithm. In the experiment, the snapshots from 100 range cells are filtered, and the target is located at the 151th range cell. The experimental results obtained after calculating the output power are shown in Figure 7. It can be seen that all five algorithms can detect the target in the 151th range cell. Among them, the target output power of the SR-STAP algorithm and the CRNR-STAP algorithm is low, and these algorithms have poor target detection performance. In comparison, the target output power of the DCPSJC-STAP algorithm, the ANM-STAP algorithm, and the SBL-STAP algorithm is much larger than that of the other two algorithms.

4.5. Clutter Power Spectrum Analysis with DCPSJC-STAP Algorithm at Different Yaw Angles

The fourth experiment compares the clutter power spectra of the proposed algorithm with different yaw angles. Figure 8a–d show the estimation results of the clutter power spectra with yaw angles of 3°, 5°, 7°, and 10°. It can be seen that, due to the movement of the radar platform, the distribution of the clutter spectrum in the space–time plane has different degrees of bending caused by the yaw angles. However, under the correction of this algorithm, the generated clutter power spectrum concentrates on the clutter ridge without further broadening, and the clutter power is relatively high, thus allowing for accurate estimations of the parameters of the clutter ridge model with different yaw angles.

5. Conclusions

Aiming to solve the off-grid problem and the parameter dependency problem caused by the airborne radar SR-STAP algorithm in a non-stationary clutter environment with an unknown yaw angle, this paper proposes an algorithm based on dictionary and clutter power spectrum joint correction. In the proposed DCPSJC-STAP algorithm, nonlinear regression is first performed on the clutter ridge parameters, the SR-STAP model parameters are combined to correct the corresponding dictionary for each snapshot, and then the estimated clutter of all snapshots is updated and iterated according to the traditional dictionary to form a new dictionary. Then, the idea of sparse Bayesian learning is used to update the iterative hyperparameters using the M-SBL algorithm. Finally, the estimated sparse recovery vector and the new dictionary are used to estimate the clutter covariance matrix and optimal weight vector. The proposed algorithm can not only take advantage of the non-parametric and lower snapshot requirements of the sparse Bayesian learning algorithm but can also improve the clutter suppression performance of airborne radar through dictionary correction. Simulation experiments demonstrate that the proposed algorithm significantly alleviates the expansion of clutter ridges, and, at the same time, the enhancement in clutter suppression and moving target detection capabilities exhibits a remarkable improvement, thus meeting the needs of practical applications.