Angular Super-Resolution of Multi-Channel APAR in Interference Environments

Abstract

Aiming to resolve azimuth-dense targets in interference environments, the radar needs to have the ability of single snap echo angular super-resolution with anti-interference. To solve the problem, the angular super-resolution algorithm based on single snap echo while anti-interference with blocking matrix method is studied for active phased array radar (APAR) in this paper. Since the super-resolution ability of the conventional MUSIC algorithm and iterative adaptive algorithm (IAA) algorithm are limited in single snap echo, the iterative re-weighted least squares (IRLS) algorithm under p-norm constraint is proposed. Further, the near main lobe interference suppression ability is enhanced by the adaptive diagonal loading method. The performance differences of IAA and IRLS algorithms for single-target detection and double-target angular super-resolution are analyzed in detail by numerical simulation on three scenes of no interference, side-lobe interference, and near-main-lobe interference. The simulation results show that the proposed algorithm can effectively solve the problem of target angle estimation and super-resolution based on single sample echo in an interference environment, including near main-lobe interference.

1. Introduction

With electromagnetic countermeasures becoming increasingly fierce, the marine search radar is facing a complicated interference environment. Meanwhile, there may be multiple target echoes in the same range gate when the targets are densely distributed. Achieving angular super-resolution with suppressing the interference is important to improve the radar’s ability to detect dense targets under interference. With the development of active phased array radar (APAR), multi-channel array radar is widely used. By introducing a degree of spatial freedom, the radar theoretically has the ability of side-lobe interference suppression and angle super-resolution of the main lobe target. Since the power of the interference echo is much greater than target, and the sample of the target echo is limited, it is necessary to solve the problem of correlation signal separation with few samples under strong interference.

The research on APAR angular super-resolution started from the direction of Arriva (DOA) estimation. The DOA estimation mainly adopts the theory correlated to the signal subspace, whose representative algorithms are multiple signal classification (MUSIC) estimation based on signal noise subspace by Schmidt [1] and estimation signal parameters via rotational invariance techniques (ESPRIT) by Ryo et al. [2]. Angular super-resolution should estimate the DOAs of highly correlated signals, which is difficult to be distinguished by conventional DOA estimation algorithms. In order to overcome the negative effects of highly correlation signals, Zhang X et al. [3] proposed a decorrelation DOA estimation algorithm. Among the existing DOA algorithms, the spatial smoothing method based on a uniform linear array proposed by Evens et al. [4] is representative. The algorithm realizes DOA estimation by subspace method based on the mean smoothing correlation matrix of the overlapping subarray. On this basis, the forward backward spatial smoothing (FBSS) method is investigated in [5,6], which achieves DOA resolution of strong correlation signal by applying a bidirectional smoothing process on relative matrices to obtain the full rank matrix. A fast DOA estimation algorithm is put forward in [7], using a single snapshot to build a pseudo-covariance matrix, which can estimate the direction of the echo signal without establishing the correlation matrices of the array output. Meanwhile, the algorithm can be certainly applied to the correlation signal. In recent years, the latest algorithms [8,9,10] based on super-resolution DOA estimation have also been proposed. Although the above research results have super-resolution capacity on highly correlation signals, when they are used directly on angular super-resolution in an interference environment, there are some shortcomings: (a) the target echo is easily covered, in strong interference; (b) the super-resolution DOA estimation usually has the demand that the number of echo samples is more than two times the number of channels. In fact, most of the target echoes have only few range gates and cannot meet the processing requirements.

In recent years, with the development of artificial intelligence, the method of DOA estimation using neural networks has been proposed. A feature-to-feature learning DOA estimation method based on self-step optimization learning is proposed in [11] to solve the problem of DOA estimation under the condition of low elevation angle. Deep convolution neural network (DCNN) is used to learn the mapping relationship between hyper-complete spatial spectrums and target angle to estimate the DOAs in [12]. The regularized switching network method applied to cognitive beamforming improves the detection performance of APAR in [13]. In addition, for different scenarios and radar systems, the parameter estimation methods are studied based on machine learning in [14,15,16,17,18]. The DOA estimation technique by the neural network method can achieve excellent performance only when the network is matched with a radar system, jamming, and target environment. So, it is difficult for neural network technology to solve all the problems in this paper under different interference environments.

The APAR spatial degree of freedom can be preserved to achieve angular super resolution after anti-interference with the blocking matrix method. Then it is only necessary to find an efficient method to separate the strong correlation signal with single snap echo. The iterative adaptive approach (IAA) can realize angular super-resolution under a single snapshot based on the minimum noise constraint [19]. It is applied on inverse convolution angular super-resolution of mechanical scanning radar [20], and has satisfactory results, reaching approximately one tenth of beam width angular resolution at high SNR. The IAA algorithm is applied to mechanical scanning radar angle super-resolution based on the principle that the target multi-pulse echo is the convolution of the antenna pattern. Similarly for APAR, the target multi-channel echo is the convolution of the spatial steering vector.

Referring to the results above, to solve the problem of multi-channel APAR in resolving azimuth-dense targets under an interference environment, an angular super-resolution algorithm is studied alongside a blocking matrix to suppress interference in this paper. The performance of IAA algorithm is not very good, especially in the condition of near main-lobe interference. To overcome the shortcoming of IAA algorithm, the iterative reweighted least squares (IRLS) algorithm under p-norm constraint is proposed and improves the near-main-lobe interference suppression ability by adaptive diagonal loading. Finally, under the three conditions of no interference, side-lobe interference, and near-main-lobe interference, the performance differences of IAA and IRLS algorithms for single-target detection and double-target super-resolution are analyzed in detail by numerical simulation. Simulation results show that the diagonal loading IRLS algorithm proposed can effectively solve the problem of angular super-resolution under near-main-lobe interference.

2. Anti-Interference Angular Super-Resolution Signal Model

The adaptive digital beam forming (ADBF) technique can suppress interference in the spatial domain. However, after ADBF, the spatial degrees of freedom are lost, and the angle super-resolution cannot be further performed. By introducing the blocking matrix into multi-channel anti-interference, the interference is suppressed by the blocking matrix, and the space degrees of freedom of multi-channel are reserved to further achieve angular super-resolution. This section builds the angular super-resolution model of point targets with anti-interference by the blocking matrix method.

In the interference environment, the subarray echo of point targets for multi-channel APAR is $S_{sub}$

$${\mathit{S}}_{sub}={\displaystyle \sum _i{\mathit{S}}_j(i){\mathit{T}}_m({\theta }_0)\times \mathit{A}({\theta }_j{}^i)}+{\displaystyle \sum _k{h}_k{\mathit{T}}_m({\theta }_0)\times \mathit{A}({\theta }_t{}^k)}$$

(1)

where ${\mathit{T}}_{\mathbf{m}}({\theta }_0)=\mathit{T}\times diag(\mathit{w}\odot \mathit{A}{({\theta }_0)}^H)$, $\odot$ denotes the Hardmand product of vectors. While $\mathit{T}$ is a sub-matrix formation matrix, a matrix of M (the number of subarrays) × N (the number of antenna elements), $\mathit{w}$ is the amplitude weighting of antenna elements. $\mathit{A}(\theta )$ is the antenna element space direction vector on $\theta$. ${\mathit{S}}_j(i)$ is the echo signal of the ith interference, ${\theta }_j{}^i$ is the direction of the ith interference. $h_k$ is the complex signal echo of the kth target, ${\theta }_t{}^k$ is the direction of the kth target, ${(·)}^H$ is conjugate transpose operator. While ${\mathit{A}}_{sub}(\theta )={\mathit{T}}_m({\theta }_0)\times \mathit{A}(\theta )$, the equation above can be written as:

$${\mathit{S}}_{sub}={\displaystyle \sum _i{\mathit{A}}_{sub}({\theta }_j{}^i)\times {\mathit{S}}_j(i)}+{\displaystyle \sum _k{h}_k{\mathit{A}}_{sub}({\theta }_t{}^k)}$$

(2)

Applying the conventional interference samples selection methods to extract interference samples, the covariance matrix ${\mathit{R}}_x$ is calculated. Then performing the eigenvalue and eigenvector decomposition of the ${\mathit{R}}_x$:

$$[\mathit{V},\mathit{D}]=\mathrm{EVD}({\mathit{R}}_x)$$

(3)

where $\mathrm{EVD}(•)$ is eigenvalue and eigenvector decomposition, $\mathit{V}=\left[\begin{array}{ccc}{v}_1{}^T& \dots v_m{}^T& \dots v_M{}^T\end{array}\right]$, $\mathit{D}={\left[\begin{array}{ccc}{\lambda }_1& \dots {\lambda }_m& \dots {\lambda }_M\end{array}\right]}^T$, $v_m$ is the mth eigenvector, ${\lambda }_m$ is the eigenvalue of the mth eigenvector, and ${(·)}^T$ is the transpose operator. Extracting the eigenvectors whose eigenvalues are greater than $\eta P_n$ to reconstruct the eigenvectors of interference ${\mathit{V}}_j$:

$${\mathit{V}}_j={\left[\begin{array}{ccc}{v}_{j1}& \dots v_{ji}& \dots v_{jI}\end{array}\right]}^T$$

(4)

where $P_n$ is noise power, $\eta$ is the interference decision floating threshold suggested $\eta =100$. According to the blocking matrix theory, the interference suppression blocking matrix $\mathit{\Phi }$ is

$$\mathit{\Phi }={\mathit{I}}_M-{\mathit{V}}_j\times {({\mathit{V}}_j\times {\mathit{V}}_j{}^H)}^{-1}{\mathit{V}}_j{}^H$$

(5)

The blocking matrix $\mathit{\Phi }$ satisfies the following equation:

$$\mathit{\Phi }{\mathit{V}}_j=[{\mathit{I}}_M-{\mathit{V}}_j\times {({\mathit{V}}_j\times {\mathit{V}}_j{}^H)}^{-1}{\mathit{V}}_j{}^H]{\mathit{V}}_j=0$$

(6)

When there is no interference, the blocking matrix $\mathit{\Phi }={\mathit{I}}_M$. Since the interference signal ${\mathit{A}}_{sub}({\theta }_j{}^i)$ belongs to the interference space ${\mathit{V}}_j$, we can get that $\mathit{\Phi }{\mathit{A}}_{sub}({\theta }_j{}^i)=0$. Multiply both sides of the Equation (2) by $\mathit{\Phi }$, then

$$\begin{array}{l}\mathit{\Phi }{\mathit{S}}_{sub}={\displaystyle \sum _i{\mathit{S}}_j(i)\mathit{\Phi }{\mathit{A}}_{sub}({\theta }_j{}^i)}+{\displaystyle \sum _k{h}_k\mathit{\Phi }{\mathit{A}}_{sub}({\theta }_t{}^k)}\\ \hspace{1em}\hspace{1em} ={\displaystyle \sum _k{h}_k\mathit{\Phi }{\mathit{A}}_{sub}({\theta }_t{}^k)}\end{array}$$

(7)

Equation (7) is the angular estimation equation to be solved after suppressing the interference. Since the calculation $A_{sub}$ is complicated and the number of solutions is unknown, it is difficult to solve the equation analytically. So, numerical solution is usually applied by uniformly dispersing the main lobe into L angles to get a set of basis vectors $\mathit{X}$:

$$\mathit{X}=[\begin{array}{ccc}\mathit{\Phi }{\mathit{A}}_{sub}({\theta }_1)\dots & \mathit{\Phi }{\mathit{A}}_{sub}({\theta }_l)\dots & \mathit{\Phi }{\mathit{A}}_{sub}({\theta }_L)\end{array}]$$

(8)

Let $\mathit{y}=\mathit{\Phi }{\mathit{S}}_{sub}$, and the question above can be transformed into finding the optimal solution of the following equation:

$$\min {\Vert \mathit{y}-{\mathit{X}}^H\mathit{H}\Vert }_2$$

(9)

$\mathit{H}$ is the signal intensity in each direction and is the unknown that needs to be solved. Different from the general optimization problem, the space direction vector of density targets echo is highly dependent, so it is a strong correlation optimization problem.

3. Iterative Adaptive Algorithm

The iterative adaptive algorithm (IAA) is a super-resolution algorithm based on the weighted minimum residual noise constraint. It has good performance in solving the angular super-resolution problem of real-beam deconvolution for machine-scanned radar and can achieve an angle resolution of one tenth beam width at high SNR. To solve the question in this paper, assume that the initial solution is ${\mathit{H}}_0$, so the residual noise covariance matrix is

$${\mathit{Q}}_k=\mathit{R}-h_k{\mathit{X}}_k{\mathit{X}}_k{}^H$$

(10)

where $\mathit{R}=\mathit{X}\mathit{P}{\mathit{X}}^H$, $\mathit{P}=diag(\mathit{H}\odot \mathit{H})$, $diag(\mathit{B})$ denotes a diagonal matrix of $\mathit{B}$. Defining ${\Vert \mathit{A}\Vert }_{Q_k{}^{-1}}={\mathit{A}}^H{\mathit{Q}}_k{}^{-1}\mathit{A}$, the power of residual noise is

$${\displaystyle \sum {\Vert \mathit{y}-h_k{\mathit{X}}_k\Vert }_{Q_k{}^{-1}}}$$

(11)

Using $\min ({\displaystyle \sum {\Vert \mathit{y}-h_k{\mathit{X}}_k\Vert }_{Q_k{}^{-1}}})$ as the optimization function, we can obtain

$$h_k=\frac{{\mathit{X}}_k{}^H{\mathit{Q}}_k{}^{-1}\mathit{y}}{{\mathit{X}}_k{}^H{\mathit{Q}}_k{}^{-1}{\mathit{X}}_k}$$

(12)

According to the matrix inverse lemma:

$${(\mathit{A}+k{\mathit{B}}^H\mathit{B})}^{-1}={\mathit{A}}^{-1}-k\frac{{\mathit{A}}^{-1}{\mathit{B}}^H\mathit{B}{\mathit{A}}^{-1}}{I+k\mathit{B}{\mathit{A}}^{-1}{\mathit{B}}^H}=\frac{{\mathit{A}}^{-1}}{I+k\mathit{B}{\mathit{A}}^{-1}{\mathit{B}}^H}$$

(13)

Then:

$${\mathit{Q}}_k{}^{-1}=\frac{{\mathit{R}}^{-1}}{I-p_k{\mathit{X}}_k{}^H{\mathit{R}}^{-1}{\mathit{X}}_k}$$

(14)

Substituting Equation (14) into Equation (12), then

$$h_k=\frac{{\mathit{X}}_k{}^H{\mathit{R}}^{-1}\mathit{y}}{{\mathit{X}}_k{}^H{\mathit{R}}^{-1}{\mathit{X}}_k}$$

(15)

After the above transformation, the calculation of ${\mathit{Q}}_k{}^{-1}$ can be avoided, and it is only necessary to calculate ${\mathit{R}}^{-1}$, which is independent of k. Therefore, a cycle only needs to calculate once, greatly reducing the amount of computation. It should be pointed out that the matrix $\mathit{X}$ is combined of the space direction vectors in the main-lobe subarray of multi-channel APAR and is highly dependent. $\mathit{R}$ is approximately singular in theory, and diagonal loading can ensure solution convergence and improve efficiency. Generally, the unit array diagonal loading method is adopted and the matrix $\mathit{R}$ can be turned into

$${\mathit{R}}_{dl}=\mathit{R}+I_L\times {\gamma }_0$$

(16)

It is suggested that ${\gamma }_0=0.3P_n$. The loading coefficient should not be too large, because the unit matrix is equivalent to the uniform noise in the full feature space, which will reduce the SNR and lead to the performance degradation of the algorithm. For single echo n = 1, then $p=\Vert \mathit{X}\mathit{y}\Vert$, when n > 1, $p_k={\displaystyle \sum h_{k(n)}{}^2/N}$, while $h_{k(n)}$ is the intermedia variable of the nth iteration, and the real solution can be gradually approximated by multiple iterations. The algorithm iteration determinates when

$$\begin{array}{ccc}{N}_A>N_{\max }& or& ep<0.01\end{array}$$

(17)

where $N_A$ is the number of iterations, $N_{\max }$ is the maximum number of iterations, $ep=\Vert {\mathit{H}}_k-{\mathit{H}}_{k-1}\Vert /\Vert {\mathit{H}}_{k-1}\Vert$ and represents the relative residual of the results of two iterations. The flow chart of the diagonal loading IAA is shown in

Table 1. The flow chart of diagonal loading IAA algorithm.

Step 0: Initialize ${\mathit{H}}_0={\mathit{X}}^H\mathit{y}$

Step 1: Calculate $\mathit{P}=diag(\Vert \mathit{H}\Vert )$

Step 2: Calculate ${\mathit{R}}_{dl}=\mathit{X}\mathit{P}{\mathit{X}}^H+{\mathit{I}}_L\times {\gamma }_0$

Step 3: Calculate $h_k={\mathit{X}}_k{}^H{\mathit{R}}^{-1}\mathit{y}/({\mathit{X}}_k{}^H{\mathit{R}}^{-1}{\mathit{X}}_k)$

Step 4: Determine whether $\mathit{H}$ converges. If not, repeat Step 1 to Step 3

.

Research shows that the super-resolution performance of IAA is limited for the problem in this paper. Especially in the near main-lobe interference, the IAA algorithm is affected by the interference, and the target detection and resolution have greatly deteriorated. Considering the IAA theory, the reasons are as follows. The blocking matrix is the orthogonal projection matrix of the interference signal space. Although the interference signal is suppressed, the interference characteristics space is orthogonal to the remaining signal space. IAA sets the minimum residual noise as constraint, without norm constraint, so it cannot avoid the algorithm to converge to the direction of near main-lobe interference. To solve the above problem, other solutions are needed to enhance the angular super-resolution ability of multi-channel APAR.

4. Diagonal Loading IRLS Angular Super-Resolution Algorithm

Returning to Equation (9), without other constraints, the solution of the least square method is:

$$\mathit{H}=({\mathit{X}}^H\mathit{X}{)}^{-1}({\mathit{X}}^H\mathit{y})$$

(18)

While this paper is the optimization solution of strong dependence, constraints need to be loaded to improve the resolution ability. Theoretically, the optimal solution is the least square solution under the constraint of 0-norm, which is the optimization estimation under the least constraint in signal sources. Considering the universality, the minimum constraint of p-norm is added to solve the following problem:

$$\begin{array}{ccc}\min ({\Vert \mathit{H}\Vert }_p)& st& \mathit{y}={\mathit{X}}^H\mathit{H}\end{array}$$

(19)

where p ≥ 0. This paper uses the iteration reweighted least squares method to solve the optimization solution under any p-norm constraints. The results of the last calculation are weighted on $\mathit{H}$, and the weighted coefficient is $w_n=h_{n-1}{}^{(p/2-1)}$, so the problem is transformed into a 2-norm optimization:

$$\begin{array}{ccc}\min ({\Vert \mathit{W}\odot \mathit{H}\Vert }_2)& s.t.& \mathit{y}={\mathit{X}}^H\mathit{H}\end{array}$$

(20)

The solution of Equation (20) is:

$$\mathit{H}=\mathit{Q}{\mathit{X}}^H{(\mathit{X}\mathit{Q}{\mathit{X}}^H)}^{-1}\mathit{y}$$

(21)

where $\mathit{Q}=diag({\Vert {\mathit{H}}_{n-1}\Vert }^{\frac{2-p}{2}})$. The iterations determine when

$$\begin{array}{ccc}{N}_A>N_{\max }& or& ep<0.01\end{array}$$

(22)

The p-norm has a great influence on the resolution. According to Equation (21), when p = 2, the matrix $\mathit{Q}$ is the identity matrix, the solution of Equation (21) is the standard solution under the 2-norm constraint, and the resolution is unimproved. In theory, the smaller the p is, the better the resolution is, and the resolution performance is the best under the 0-norm constraint.

The IAA algorithm cannot work in the angular estimation and super-resolution of the targets under near main-lobe interference, neither does the IRLS. According to the IRLS theory, ${\mathit{X}}^{\mathit{H}}\mathit{X}$ in Equation (18) and $\mathit{X}\mathit{Q}{\mathit{X}}^{\mathit{H}}$ in Equation (21) are matrices of the reconstructed signal space. Considering diagonal loading on these matrices, the interference space can accommodate feature compensation, and there is no interference signal in the echo, so theoretically the interference in the near main lobe can be completely suppressed under the p-norm constraint. Besides, $\mathit{X}$ is combined with the space direction vectors in the main-lobe subarray of multi-channel APAR and is highly dependent. $\mathit{X}\mathit{Q}{X}^H$ is approximately singular in theory, and diagonal loading on $\mathit{X}\mathit{Q}{\mathit{X}}^H$ can ensure the solution convergence and improve efficiency.

In Equation (18), since the initial solution is unknown, using the conventional identity matrix diagonal loading, the initial solution is

$${\mathit{H}}_0={[{\mathit{X}}^H\mathit{X}+{\gamma }_0\times \mathit{I}]}^{-1}({\mathit{X}}^H\mathit{y})$$

(23)

The diagonal loading coefficient is suggested to be ${\gamma }_0=0.3P_n$. Research shows that during the iteration process, the IRLS algorithm can accelerate the iteration when the adaptive diagonal loading is committed on the last results. The matrix after adaptive diagonal loading is

$${\mathit{T}}_{dl}=\mathit{X}\mathit{Q}{\mathit{X}}^H+\gamma \times diag(\mathit{X}\mathit{Q}{\mathit{X}}^H)$$

(24)

where $\gamma$ is the diagonal loading coefficient, the smaller the $\gamma$ is, the higher the limit resolution can be. It is suggested that $\gamma =\min (\begin{array}{cc}10/SNR& 0.3\end{array})$, so $\gamma$ decreases with SNR, ensuring the resolution ability at high SNR. Substituting Equation (24) into Equation (21), the iteration solution is

$$\mathit{H}=\mathit{Q}{\mathit{X}}^{\mathit{H}}{\mathit{T}}_{dl}{}^{-1}\mathit{y}$$

(25)

Research shows that compared with identity matrix loading, adaptive diagonal loading can accelerate convergence, but the final results are the same.

Table 2. The flow chart of diagonal loading IRLS algorithm.

Step 0: Initialize ${\mathit{H}}_0={[{\mathit{X}}^H\mathit{X}+{\gamma }_0\times \mathit{I}]}^{-1}({\mathit{X}}^H\mathit{y})$

Step 1: Calculate $Q=diag({\Vert {\mathit{H}}_{n-1}\Vert }^{\frac{2-p}{2}})$

Step 2: Calculate ${\mathit{T}}_{dl}=\mathit{X}\mathit{Q}{\mathit{X}}^H+\gamma \times diag(\mathit{X}\mathit{Q}{\mathit{X}}^H)$

Step 3: Calculate ${\mathit{H}}_n=\mathit{Q}{\mathit{X}}^{\mathit{H}}{\mathit{T}}_{dl}{}^{-1}\mathit{y}$

Step 4: Determine whether h converges. If not, repeat Step 1 to Step 3

shows the flow chart of the adaptive diagonal loading IRLS.

Compared with Table 1 and Table 2, the flow of IAA and IRLS algorithms are basically the same, so the computation of one time iteration is basically the same. The computation of the two algorithms mainly depends on the number of iteration convergence.

5. Numerical Simulation Analysis

In this section, some necessary and important results are simulated firstly. Then, under the three conditions of no interference, side-lobe interference, and near-main-lobe interference, the performance difference of IAA and IRLS algorithms on the detection of single and double targets is analyzed by numerical simulation. The simulation parameters are shown in

Table 3. Simulation parameters.

radar parameters	antenna parameters	the number of antenna elements	64, 0.015 m apart
		the number of subarrays	16 divided evenly
		3 dB beam width	1.6°
		beam direction	0° norm to the array
		sum pattern weighted	−40 dB linear Taylor weighted
		frequency	10 GHz
	signal processing parameters	the number of fast snapshots	1000
		detection threshold	11.5 (false alarm rate 1 × 10−5)
		maximum number of iterations	20 (IAA), 20 (IRLS)
interference parameters		the number of interference	1
		interference-to-signal ratio (INR)	60 dB
		interference type	full range gate narrowband noise interference
target parameters		target type	point target
		target distance	the 800th range gate
the number of Monte Carlo simulation			1000

.

5.1. Performance Comparison of IRLS under Different P-Norm Constraints

To compare the performance of IRLS under different p-norm, Figure 1 shows the results under p-norm constraint when $p=1$ and $p=0$. The angular difference between the two targets is 0.5°, and SNR = 30 dB.

The 0th update corresponds to the initial solution without constraint in Equation (18), Figure 1a shows the simulation results under p = 1 constraint and Figure 1b shows that under p = 0. From the results, under the p = 1 constraint, the update is converged after 3 times but the two targets cannot be distinguished. While under the p = 0 constraint, the two targets are clearly distinguished after the 6th update and more obvious with the iteration process. The algorithm is converged after the 18th iteration and has only two peaks in the target direction. The angles of the two targets can be distinguished and accurately estimated. The simulation results are consistent with the initial theoretical analysis, verifying the algorithm’s correctness. The following IRLS simulation is under $p=0$.

5.2. Influence of Diagonal Loading on the Performance of IAA and IRLS

To analyze the influence of diagonal loading on the super-resolution performance of IAA and IRLS, Figure 2 shows the results before and after diagonal loading on IAA and IRLS without interference. The two target angles are 0.5° apart, and SNR = 30 dB.

In Figure 2, “NoDL” denotes the simulation results without diagonal loading, and “DL” denotes the simulation results under adaptive diagonal loading. Figure 2a shows the comparison of IAA performance before and after diagonal loading, and Figure 2b shows the comparison of IRLS performance before and after diagonal loading. From the results, the IAA and IRLS algorithm can avoid overfitting with diagonal loading, improve the convergence efficiency, and reduce the false alarm rate.

Figure 3 shows the comparison results of IAA and IRLS before and after diagonal loading under near main-lobe interference. The target parameters are the same as in Figure 2, with additional near main-lobe interference coming from the direction of 2°. As shown in Figure 2, with diagonal loading, the IRLS can suppress the near main-lobe interference, while the IAA is affected greatly, which is consistent with the analysis above. In summary, the performance of IAA and IRLS is significantly improved by diagonal loading, so the following IAA and IRLS simulations are both diagonal loaded.

5.3. Angular Super-Resolution Performance of Different Algorithms

In order to compare the performance differences between the conventional spectrum estimation MUSIC, IAA, and IRL, Figure 4 shows the comparison results without interference at a target of 1° apart. Results show that MUSIC cannot distinguish targets, while IAA and IRLS can detect and distinguish the targets efficiently. For the single sample point target, it does not meet the requirements of MUSIC algorithm, such that it is difficult for the MUSIC algorithm to distinguish two targets of 1° apart (about 0.6 times the beam-width) in a non-interference environment.

A large number of samples can enhance echo orthogonality, which makes it very important for spectral estimation algorithms to improve resolution, so that the performance under the condition of a few samples deteriorates seriously. The following simulation analysis will no longer compare the MUSIC algorithm in detail.

5.4. Comparison of Simulation without Interference

In this subsection, we consider two cases, including single target detection and double target angle resolution. In the case of single target, target detection performance and angle estimation accuracy are analyzed with different algorithms. In the case of dual targets, the probability of angle resolution is mainly analyzed.

• Case 1: single target

SNR is from 10 dB to 30 dB. The single target environment mainly focuses on the target detection probability and the angular measurement accuracy of the different algorithm.

Figure 5 shows the comparison results of IAA and IRLS on target detection probability (PD) and angular measurement accuracy. The detection probability is based on the detection threshold with a false alarm probability equal to 1 × 10⁻⁵. The angle measurement accuracy is evaluated by root mean square error (RMSE). Figure 5a shows that for the single target detection without interference, IAA performs better, with 10% more PD at SNR = 10 dB. The detection probability of both algorithms is more than 65% at SNR = 10 dB and is basically the same with the SNR increasing to 20 dB, both more than 98%. Figure 5b shows that the IRLS performs better in angular estimation accuracy generally. The angular estimation accuracy of both algorithms is more than 0.12° at SNR = 10 dB, about one fifteenth of beam width. With the SNR increasing, the accuracy improves a lot and can be better than 0.02° at SNR = 30 dB.

• Case 2: double-targets

The accuracy of super-resolution is mainly focused under the condition of double-targets. In this paper, the probability of angle super-resolution (PAR) is used to analyze the performance of the algorithms. PAR is the ratio of the correct super-resolutions numbers to the total simulation numbers, which can be computed through the following processing. After each simulation, the results are obtained by target detection and used to correlate with the real value by the global nearest neighbor association method. The association threshold is set to $\min (\begin{array}{cc}{\phi }_{3db}/\sqrt{SNR}& dAz/2\end{array})$, where ${\phi }_{3db}$ is antenna 3 dB beam width and $dAz$ is angular of the two target interval. If the two targets are detected at the same time, it is considered to be a correct super-resolution.

Figure 6 shows the accuracy comparison results of IAA and IRLS on super-resolution without interference. The angular interval between the targets is 0.4°, 0.6°, and 0.8°. The numerical simulation results show that the angular estimation accuracy of the two algorithms decreases with the targets getting closer and improves with the SNR increasing.

The IAA algorithm can distinguish the double targets with a 0.6° angular interval, and the accuracy can reach 80% at an SNR of about 40 dB. With the improvement of SNR, the super-resolution accuracy can reach 80% with about SNR = 48 dB for two targets with an angular interval of 0.4°. The IRLS algorithm can distinguish the double targets with a 0.6° angular interval, and the accuracy can reach 80% at an SNR of about 32 dB. With the improvement of SNR, the super-resolution accuracy can reach 80% at about SNR = 38 dB for two targets with an angular interval of 0.4°. So, the angular super-resolution performance of IRLS algorithm is better than IAA without interference.

5.5. Comparison of Simulation with Interferences

In this section, we mainly analyze the influence of interference on super-resolution performance. Simulation analysis with side-lobe and main-lobe interference is provided respectively. Figure 7a is the PAR comparison of side-lobe interference, and Figure 7b is of near main-lobe interference.

For the convenience of comparison, the needed SNRs are shown in

Table 4. The needed SNRs of IAA and IRLS methods.

	Environment	Method	Angular Interval
			0.4°	0.6°	0.8°
SNR/dB (PAR = 80%)	No interference	IAA	48	40	32
		IRLS	41	30	22
	Side-lobe interference	IAA	49	40	32
		IRLS	41	30	22
	Near main-lobe interference	IAA	>60	58	56
		IRLS	45	34	27

, when PAR = 80% on different conditions.

As shown in Table 4, on the condition of side-lobe interference, the performance of IRLS is the same as that without interference, and so is IAA. This shows that the side-lobe interference does not affect the super-resolution and angular estimation accuracy after blocking matrix suppression.

On the condition of near-main-lobe interference, the IAA algorithm cannot distinguish double targets of 0.8° apart, when SNR is lower than 40 dB. The performance of the IRLS algorithm also decreases, and the required SNR is increased by 4~5 dB to achieve the same PAR. This shows that the near main-lobe interference has serious impact on the IAA method and slight impact on the IRLS method, for super-resolution of double targets after blocking matrix suppression.

In order to further analyze the influence of near main lobe interference on super-resolution, Figure 8 shows the simulation results of angular super-resolution under different SNR and INR. It is shown that the diagonally loaded IRLS algorithm can completely suppress the interference, and the IAA algorithm can distinguish the target correctly only under extremely high SNR while there exists a spike in the direction of interferences.

In principle, the interference characteristics space is orthogonal to the remaining signal space after anti-interference with the blocking matrix method. So, the signal source in the interference direction will not have a great effect on the residual noise. IAA sets the minimum residual noise as constraint and cannot avoid the algorithm to converge in the direction of near main-lobe interference. IRLS algorithm has p-norm constraint, which can suppress unnecessary signal sources. So, there is no spike in the direction of interference.

6. Conclusions

To solve the angular super-resolution of multi-channel APAR in interference, an angular super-resolution model based on single snap echo with the blocking matrix method is proposed. Because of strong interference and few target samples, the MUSIC algorithm is almost ineffective. The algorithms of IAA based on minimum noise constraint and IRLS with p-norm constraint are applied to solve the problem. On three conditions of no interference, side-lobe interference, and near-main-lobe interference, the single-target detection and double-targets super-resolution performance of different algorithms are simulated and discussed. Numerical simulation results show that the IAA algorithm is slightly better than IRLS in single target detection, while the IRLS algorithm is better than IAA for angular super-resolution. The super-resolution accuracy of IRLS can reach 80% at SNR = 30 dB for two targets with an angular interval of 0.6° approximately one third of the beam width. The research results in this paper can effectively solve the angular super-resolution problem of single snapshot echo for multi-channel APAR in strong interference, including near main lobe interferences.