Flexible Null Broadening Robust Beamforming Based on JADE

In order to flexibly and completely suppress dynamic interference, a flexible and robust beamforming based on JADE is proposed in this paper. In addition, it is insensitive to the gain–phase errors of the array. Firstly, the actual steering vector with gain–phase errors is separated from the received snapshot data by the joint approximate diagonalization of eigenmatrix (JADE) algorithm. Secondly, the direction of arrival (DOA) of interference can be estimated from the separated actual steering vector by the correlation coefficient method. Thus, the actual interference steering vector with gain–phase errors can be selected by the correlation coefficient with the nominal steering vector constructed by the estimated DOA. Then, the interference covariance matrix can be reconstructed by the actual interference steering vector, and the interference power estimated by the Capon power spectral. Finally, according to the prior information of the interference, only the dynamic interference covariance matrix is tapered by the novel covariance matrix trap (CMT), which can flexibly broaden and deepen the null. Simulation results show that the depth of the proposed beamformer is more than 10 dB deeper than that of the traditional algorithm in the non-stationary interference. In addition, it can save at least 2 degrees of freedom compared to the traditional method.


Introduction
Adaptive beamforming as an effective anti-jamming technology is widely used in navigation, radar, mobile communications and sonar systems [1][2][3][4][5][6]. The adaptive weight vector obtained by a traditional minimum variance distortion-less response (MVDR) adaptive beamformer can only produce a narrow null in the beam pattern. However, the narrow null cannot mitigate the interference in a high-dynamic environment. A high-dynamic environment generally refers to the rapid movement of the interference, the high-speed movement of the receiver or the vibration of the receiver platform, which will cause the interference to move outside of the null, and the interference cannot be effectively suppressed [7]. Under high-dynamic conditions, the direction of arrival (DOA) of interference may vary in multiple resolution units, and the beamforming weight cannot be updated in real time [8]. This problem is induced by the interference steering vector mismatch caused by a high-dynamic environment.
Two ways are applied to tackle the interference steering mismatch in traditional beamforming. An effective strategy is that the minimum power distortion-less response beamformer is designed by feature analysis technology. This method formulates a quadratic constraint problem to constrain the output power in the predetermined interference angle area to be lower than a fixed value [9]. However, the disadvantage of this method is more complex. In order to address the complex problem of quadratic constraints, a set of linear constraints is proposed to replace the original quadratic constraints, and the minimum linear constraints are analyzed in [10]. In addition, based on the Newton method, a fast solution to the optimization quadratic constraints relaxed to linear constraints is proposed in [11,12]. Ors [13] proposed a null broadening beamforming based on a first-and secondorder iterative, and explained that the second-order requires fewer iterations than the first-order. Another method is the traditional null broadening technology, which was first proposed by Mailloux and Zatman [14,15]. Mailloux constructs a new interference-plusnoise covariance matrix by setting N equal intensity discrete virtual interferences near a real interference location, which achieves null broadening. Zatman takes advantage of the fact that the frequency diffusion of the incident signal can be equivalent to the expansion of the spatial angle, which achieves null broadening by the virtual frequency band to modify the covariance matrix. Because the noise item is not affected, the Zatman method is better than the Mailloux method. After summarizing the Mailloux and Zatman methods, Guerci proposed the covariance matrix tapers (CMT), which achieved null broadening by performing a Hadamard product on the sampling covariance matrix and the taper matrix [16]. The wide null can be generated in the interference angle without prior information of interference. However, the null will be shallow, and degree of freedom (DOF) will be lost when the null is broadened. The loss of degrees of freedom occur when there are non-dynamic and dynamic interferences at the same time. The CMT will broaden all the interferences corresponding to the nulls. Each broadened null will consume at least two DOFs, which will waste the DOF in non-dynamic interference. In order to tackle this problem, Liu Zhiwei proposed a flexible method to broaden the null, which uses the relationship between each eigenvector and each interference to widen different widths for different interferences and reduce the loss of DOF as much as possible [17]. However, due to eigenvalue decomposition and the ranking of eigenvalues, this algorithm cannot distinguish between different interference information by the size of eigenvalues when the power of multiple interferences is similar, which will cause the disorder of the broadening position. The common points of the above literature are based on the ideal array condition.
In the presence of gain-phase errors, the interference suppression ability of the algorithm will drop sharply. Blind source separation (BSS) is a technique that can separate and recover relatively independent source signals from unknown multi-source hybrid systems [18]. Because this technique can separate each signal steering vector with array gain-phase errors from the observation data, this feature makes it suitable for solving the problem of performance degradation of the adaptive beamforming algorithm in the presence of gain-phase errors. The joint approximate diagonalization of eigen-matrices (JADE) is a blind separation technique with high separation performance, which is proposed by Cardoso [19]. In addition, Zheng Dai proposes a direction-of-arrival (DOA) estimation method based on JADE, which is robust to the array gain-phase errors [20].
In order to effectively suppress dynamic interference in the presence of gain-phase errors, this paper proposes a flexible null broadening robust beamforming algorithm. Firstly, we use the joint approximate diagonalization of eigen-matrices (JADE) algorithm to separate the signal steering vector; secondly, the correlation coefficient method is used to match the signal steering vector. Thus, we roughly estimate the interference angle and the actual interference steering vector with gain-phase errors. Then, the dynamic interference covariance matrix can be reconstructed easily. Finally, a novel CMT with the null deepening coefficient is used to taper the covariance matrix of the dynamic interference, which can flexibly control the width and depth of the null at the dynamic interference, resulting in the saving the DOFs and an enhanced anti-interference capability.

Signal Model and MVDR Beamformer
Without loss of generality, the uniform linear array is considered here to build the signal model. In this paper, the array is composed of N sensors, as shown in Figure 1.
Considering that one desired signal impinges on the array from a known direction along with L jammer signals from unknown directions, respectively, the array-receiving signal model can be expressed as where x(t) is the snapshot data received by the array, Γ is the array gain-phase errors matrix. A is the array flow matrix, s(t) is the incident signal vector, n(t) is the Gaussian noise. The array flow matrix is further expressed as where θ is the pitch angle, and [·] T denotes transpose operation.
Appl. Sci. 2022, 12, 9329 3 of 13 Moving direction Considering that one desired signal impinges on the array from a known direction along with L jammer signals from unknown directions, respectively, the array-receiving signal model can be expressed as where ( ) x t is the snapshot data received by the array,  is the array gain-phase errors matrix. A is the array flow matrix, ( ) s t is the incident signal vector, ( ) n t is the Gaussian noise. The array flow matrix is further expressed as  n (4) where  is the pitch angle, and     denotes transpose operation.
where the phase difference between sensors is 2 sin , 0 1 The array gain-phase errors matrix Γ is defined as where ρ i and φ i are the gain and phase errors of the i-th array, diag(·) denotes matrix diagonalization. The steering vector a(θ) is written as where the phase difference between sensors is Then, the actual steering vector with array gain-phase errors is Under the maximum signal-to-interference-plus-noise ratio (SINR) criterion, the optimal weights of the MVDR beamformer can be transformed into optimization problem From (8), the mathematical model of beamforming can be interpreted as minimizing the array output power while ensuring that the response at the desired direction is 1. Therefore, it can be realized that the main lobe is formed at the desired position and the null is formed at the interference position. According to the Lagrange multipliers method, the optimal weight vector w opt is obtained by where a 0 is the desired signal steering vector, and R in is the interference-plus-noise covariance matrix (INCM). The interference-plus-noise covariance matrix R in is defined by where σ 2 l is the power of the l-th interference, σ 2 n is the noise power. Furthermore, the sample covariance matrixR is defined bŷ where, K is the snapshot number.

Preprocessing Technology Based on Blind Source Separation
Blind source separation (BSS) is a new technology to find statistically independent components in multi-channel signals. It can estimate the information of the signal source from the mixed signal when both the mixing process and the signal source are unknown. As a classical algorithm of BSS, JADE is widely used in beamforming and DOA estimation. The mixing matrix separated by JADE contains the gain-phase errors, which can improve the robustness of beamforming. The mixing matrix obtained by JADÊ Taking the first element of each column as the reference element, the column vectors in the mixing matrix are normalized as

DOA and the Actual Interference Steering Vector with Gain-Phase Errors Estimated
In order to obtain the actual interference steering vector with gain-phase errors, the correlation coefficient method is used to match the steering vector in the estimated array flow matrix. The DOA of the interference is estimated first.
The correlation coefficient between x, y is defined by where [·] H denotes conjugate transpose operation, | · | denotes modular operation and · denotes the L2 norm of vector. Then, the DOA of the i-th interference can be estimated by the following formulaθ Then, the actual interference steering vector with gain-phase errors can be selected by the correlation coefficient with the nominal steering vector constructed by the estimated DOA, i.e.,â In addition, with combined (7) and (16), we can estimate the gain-phase errors where and (·) * denotes the Hadamard product and conjugate operation, respectively.

Flexible Null Broadening Method
According to the structure of CMT in reference [15], the covariance matrix taper can be expressed as Here, Q virtual interferences are added on both sides of the i-th real interference with ∆θ interval.
The novel covariance matrix taper can be written by where P q is the null deepening coefficient, and it can control the depth of the null. In addition, the deeper null with a lager P q can suppress the interference more toughly, proved by Proposition 1.

Proposition 1.
The deepening of the beamforming null is beneficial to the suppression of interference, η i 1 − σ 2 io /σ 2 i is defined as the i-th interference suppression ratio and the null deepening coefficient P q is proportional to η i .

Proof of Proposition 1. Detailed proof is shown in Appendix A.
When η i → 0 , the interference suppression ability is poor; when η i → 1 , the interference suppression performance is close to the optimal. From Proposition 1, we can see that a larger P q can make the higher interference suppression ratio η i . Therefore, the proposed novel covariance matrix taper in this paper can improve the interference suppression ability by setting a larger null deepening coefficient P q .
To avoid the waste of DOFs and to broaden the null more flexibly, the interference mismatch matrix can be obtained by From (20), the operation of the new array taper and the interference covariance matrix is equivalent to the Hadamard product of the original one and the weighted interference covariance matrix. The null deepening coefficient P q in the weighted interference covariance matrix can adjust the power of original interference, resulting in an adjustable null depth.
Then, the interference-plus-noise covariance matrix can be expressed aŝ where L d and L nd are the number of dynamic interference and non-dynamic interference, respectively, and L = L d + L nd . In addition, R i =σ 2 i a Γ θ i a H Γ θ i is the non-dynamic interference covariance matrix, and I is an N × N identity matrix. The i-th interference power is estimated by Capon spectrum [20] The noise powerσ 2 n is estimated by small eigenvalue of matrixR [21] where λ i is the i-th eigenvalue of the sample covariance matrix and λ 1 ≥ λ 2 ≥ · · · ≥ λ L ≥ · · · ≥ λ N ≈ σ 2 n . According to the prior DOA of desired signal, the actual desired signal steering vector with the gain-phase error can be obtained as Thus, substituting (21) and (24) into (9), the weight vector in this paper can be obtained as In summary, the detailed steps for implementing the proposed algorithm are given in Table 1. Table 1. The detailed steps for implementing the proposed algorithm.

Steps Operation
1 Separate the signal steering vector from the observation data using (13); 2 Match the signal steering vector by correlation coefficient method using (15,16); 3 Reconstruct and taper the interference mismatch matrix using (20); 4 Calculate the weight vector of proposed algorithm using (25).
The interference-to-noise ratio is INR1 = 30 dB and INR2 = 25 dB , respectively, and the broadening angle is 2 degrees.
Under the above-mentioned simulation conditions, the proposed algorithm is compared with CMT [16], FCMT [17], INCM-RAB [20] and optimal beamformers in pattern, output SINR, output interference power and output SINR. It is assumed that the gain-phase errors obey Gaussian distribution N 1, 0.05 2 and N 0, 0.5 • 2 , respectively. According to the simulation conditions above, the null broadening pattern is obtained as follows: Compared with the traditional CMT, the advantage of the proposed algorithm in this paper is that the algorithm can flexibly broaden the null. As shown in Figure 2, the proposed algorithm generates a narrow null at non-dynamic interference θ 2 = 50 • and a wide null at dynamic interference θ 1 = −60 • . It can save the consumption of DOFs at non-dynamic interference θ 2 = 50 • . In addition, compared with similar flexible algorithms, FCMT, the proposed algorithm, has a deeper null and a more precise widening angle at dynamic interference, resulting in a better interference suppression capability. , respectively. The broadening angle is 2 degrees. Other simulation conditions are the same as the simulation Experiment 1. The pattern of the proposed algorithm is compared when Pq is 1,10 and 100.
As shown in Figure 3, RAB is compared with the proposed algorithm under different null deepening coefficients. When the coefficient of null deepening Pq is 1, the depth of null is about −90 dB. When Pq is set to 10, the depth of the null deepens from −90 dB to −108 dB, and the null deepens about 20 dB. When Pq is 100, the null is larger than −120 dB and exceeds RAB. Therefore, the reasonable CMT can be designed by the suitable Pq, which can suppress the dynamic interference thoroughly. In general, we should set Pq = 10~100. However, we should set Pq > 100 in a strong interference environment.  It is assumed that the gain-phase errors obey Gaussian distribution N 1, 0.01 2 and N 0, 0.5 • 2 , respectively. The broadening angle is 2 degrees. Other simulation conditions are the same as the simulation Experiment 1. The pattern of the proposed algorithm is compared when Pq is 1,10 and 100. As shown in Figure 3, RAB is compared with the proposed algorithm under different null deepening coefficients. When the coefficient of null deepening Pq is 1, the depth of null is about −90 dB. When Pq is set to 10, the depth of the null deepens from −90 dB to −108 dB, and the null deepens about 20 dB. When Pq is 100, the null is larger than −120 dB and exceeds RAB. Therefore, the reasonable CMT can be designed by the suitable Pq, which can suppress the dynamic interference thoroughly. In general, we should set Pq = 10~100. However, we should set Pq > 100 in a strong interference environment. , respectively. The broadening angle is 2 degrees. Other simulation conditions are the same as the simulation Experiment 1. The pattern of the proposed algorithm is compared when Pq is 1,10 and 100.
As shown in Figure 3, RAB is compared with the proposed algorithm under different null deepening coefficients. When the coefficient of null deepening Pq is 1, the depth of null is about −90 dB. When Pq is set to 10, the depth of the null deepens from −90 dB to −108 dB, and the null deepens about 20 dB. When Pq is 100, the null is larger than −120 dB and exceeds RAB. Therefore, the reasonable CMT can be designed by the suitable Pq, which can suppress the dynamic interference thoroughly. In general, we should set Pq = 10~100. However, we should set Pq > 100 in a strong interference environment.   This experiment considers the influence of gain-phase errors, and compares the changes of output SINR and input SNR of different algorithms. The experiment considers the gain-phase errors, and compares the output SINR of different algorithms with the change of input SNR. It is assumed that the gain-phase errors obey Gaussian distribution N 1, 0.05 2 and N 0, 5 • 2 , respectively. SNR = −10 dB ∼ 30 dB. The relationship between input signal-to-noise ratio (SNR) and output signal-to-interference-plus-noise ratio (SINR) of each algorithm is statistically compared in Figure 4 with 200 Monte Carlo experiments.
This experiment considers the influence of gain-phase errors, and compares the changes of output SINR and input SNR of different algorithms. The experiment considers the gain-phase errors, and compares the output SINR of different algorithms with the change of input SNR. It is assumed that the gain-phase errors obey Gaussian distribution , respectively. SNR 10dB 30dB  ~. The relationship between input signal-to-noise ratio (SNR) and output signal-to-interference-plus-noise ratio (SINR) of each algorithm is statistically compared in Figure 4 with 200 Monte Carlo experiments. From Figure 4, we can see that the output SINR is close to the ideal case, which shows that the proposed algorithm is robust to the gain-phase errors compared with CMT and INCM-RAB. The FCMT algorithm uses the eigenvalue decomposition method to perform better than CMT and INCM-RAB in the case of low SNR. The steering vector separated by JADE contains the information of gain-phase errors, which can eliminate the influence of gain-phase errors on the proposed algorithm. Thus, the proposed algorithm can suppress dynamic interference more thoroughly with the gain-phase errors.  Assuming SNR 0dB  and the broadened null is 4  , we carry out 200 Monte Carlo experiments to simulate and analyze the variation trend of the output SINR and the output interference power when the interference deviation angle   is from 2   to 2  .
We can see that the output SINR of INCM-RAB will decrease with the increase of the interference angle deviation from Figure 5, which indicates that the narrow null cannot suppress the dynamic interference. When the interference mismatch angle is less than the broadening angle, the broadening algorithm can effectively suppress dynamic interference. Compared with the similar CMT and FCMT, the proposed algorithm in this paper is closer to the optimal situation, because the designed matrix trap is more accurate. From Figure 4, we can see that the output SINR is close to the ideal case, which shows that the proposed algorithm is robust to the gain-phase errors compared with CMT and INCM-RAB. The FCMT algorithm uses the eigenvalue decomposition method to perform better than CMT and INCM-RAB in the case of low SNR. The steering vector separated by JADE contains the information of gain-phase errors, which can eliminate the influence of gain-phase errors on the proposed algorithm. Thus, the proposed algorithm can suppress dynamic interference more thoroughly with the gain-phase errors.

Experiment 4:
Anti-interference ability verification-Relationship between interference mismatch angle and output SINR.
Assuming SNR = 0 dB and the broadened null is 4 • , we carry out 200 Monte Carlo experiments to simulate and analyze the variation trend of the output SINR and the output interference power when the interference deviation angle ∆θ is from −2 • to 2 • .
We can see that the output SINR of INCM-RAB will decrease with the increase of the interference angle deviation from Figure 5, which indicates that the narrow null cannot suppress the dynamic interference. When the interference mismatch angle is less than the broadening angle, the broadening algorithm can effectively suppress dynamic interference. Compared with the similar CMT and FCMT, the proposed algorithm in this paper is closer to the optimal situation, because the designed matrix trap is more accurate.

Experiment 5:
The consumption DOFs of each algorithm-Comparison of large eigenvalues of covariance matrix.
The simulation conditions are the same as in Experiment 1. Under the same broadening angle, the downward trend of the eigenvalue is compared, as shown in Figure 6. The simulation conditions are the same as in Experiment 1. Under the same broadening angle, the downward trend of the eigenvalue is compared, as shown in Figure 6. The number of the covariance matrix's large eigenvalues of RAB is the least equal to the number of signal sources, which does not waste DOFs. CMT, the traditional null broadening algorithm, has the largest number of large eigenvalues. This algorithm treats all interferences as dynamic interference, which will cause the waste of DOFs. In this paper, blind source separation technology is used to separate dynamic interference and nondynamic interference, and only the dynamic interference null is widened to achieve DOFs  The simulation conditions are the same as in Experiment 1. Under the same broadening angle, the downward trend of the eigenvalue is compared, as shown in Figure 6. The number of the covariance matrix's large eigenvalues of RAB is the least equal to the number of signal sources, which does not waste DOFs. CMT, the traditional null broadening algorithm, has the largest number of large eigenvalues. This algorithm treats all interferences as dynamic interference, which will cause the waste of DOFs. In this paper, blind source separation technology is used to separate dynamic interference and nondynamic interference, and only the dynamic interference null is widened to achieve DOFs The number of the covariance matrix's large eigenvalues of RAB is the least equal to the number of signal sources, which does not waste DOFs. CMT, the traditional null broadening algorithm, has the largest number of large eigenvalues. This algorithm treats all interferences as dynamic interference, which will cause the waste of DOFs. In this paper, blind source separation technology is used to separate dynamic interference and non-dynamic interference, and only the dynamic interference null is widened to achieve DOFs saving. However, FCMT uses the eigenvalue decomposition of the sample covariance matrix to distinguish the interference state, which will fail when the interference power is equal. Therefore, the algorithm in this paper is more universal.

Conclusions
This paper obtains the mixing matrix by blind source separation technology, and the DOA of interference is obtained by the correlation method. Then, we can further estimate the array gain-phase errors, so as to improve the robustness of beamforming in a high dynamic environment. Furthermore, a new CMT is proposed, which can flexibly control the broadening angle and null depth of each interference by trapping the covariance matrix of each interference, respectively. The simulation results show that when the array has the gain-phase errors, the difference between the algorithm output SINR and the ideal value is less than 1 dB; compared with the traditional null broadening beamforming, when there is a non-dynamic interference, it saves 2-3 DOFs. The disadvantage of this algorithm is that in the case of signal coherence, it is unable to separate signal steering vectors by blind source separation technology, resulting in a serious performance degradation. Therefore, the issue will be further studied in the future.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
The i-th interference rejection ratio η i is defined as The ratio of the interference output powerσ 2 io to the interference input powerσ 2 i can be written asσ Taking the incidence of an interference as an example,R in can be expressed aŝ Introducing the null deepening coefficient P q , the above formula can be reduced tô R in = P qσ