基于原子范数最小化的稀疏阵列稳健波束形成算法

doi:10.12382/bgxb.2023.0618

摘要/Abstract

摘要：

为提高稀疏阵列在信号模型存在失配时的波束形成性能,提出一种基于原子范数最小化(Atomic Norm Minimization, ANM)的稳健波束形成算法。构建基于ANM的降噪问题模型,根据稀疏阵列的协方差矩阵结构将其转化为等价的半定规划问题,同时推导该问题的对偶问题以提高运行效率,求解得到阵列降噪后的接收数据和协方差矩阵。根据互质阵列的结构特性证明其空间谱的无模糊性,对所得的协方差矩阵直接使用多重信号分类算法获得入射信号的波达方向。利用虚拟填充技术得到与互质阵列孔径相同的均匀线性阵列的接收数据,最终获得阵列输出。通过计算机仿真实验,验证了所提算法的可行性和准确性,较其他被测算法输出的信干噪比至少提高1.5dB。

关键词: 稳健波束形成, 稀疏阵列, 原子范数最小化, 对偶问题

Abstract:

Aiming at the performance degradation of the beamforming algorithm when some mismatches are present in the signal model. A robust beamforming algorithm based on atomic norm minimization (ANM) is proposed to improve the beamforming performance of sparse array during the mismatch of signal model. The proposed algorithm is used to construct an ANM-based noise reduction model and transform it into an equivalent semi-definite programming problem according to the covariance matrix structure of sparse array. Meanwhile, the dual problem of this model is derived to improve the computational efficiency, and the received data and covariance matrix of the array after noise reduction are obtained. The spatial spectrum is proved to be unambiguous based on the structural properties of a coprime array, and the directions of arrival of the incident signals are obtained directly by using the multiple signal classification algorithm for the resulting covariance matrix. The received data of a uniform linear array with the same aperture as the coprime array is obtained using the virtual filling technique, and the array output is ultimately obtained. Simulated results verify the feasibility and accuracy of the proposed algorithm, which improves the output signal to interference plus noise ratio by at least 1.5dB compared to the other tested algorithms.

Key words: robust beamforming, sparse array, atomic norm minimization, dual problem

中图分类号:

TN827⁺.2

吕岩, 曹菲, 金伟, 何川, 杨剑, 张辉. 基于原子范数最小化的稀疏阵列稳健波束形成算法[J]. 兵工学报, 2024, 45(8): 2737-2748.

LÜ Yan, CAO Fei, JIN Wei, HE Chuan, YANG Jian, ZHANG Hui. A Robust Beamforming Algorithm for Sparse Array Based on Atomic Norm Minimization[J]. Acta Armamentarii, 2024, 45(8): 2737-2748.

图/表 11

图1 互质阵列组成结构示意

Fig.1 The structure of coprime array

图2 等效ULA示意

Fig.2 Schematic diagram of the equivalent ULA

表1 所提算法步骤

Table 1 Steps of the proposed algorithm

步骤	内容
1	获取K快拍的阵列接收信号X,求解式(28)利用对偶性得到式(27)的最优解,记为Y^和 $Z 1 $ ;
2	利用MUSIC算法得到 $Z 1 *$ 的空间谱,获得入射信号的$\operatorname{DOA}\left\{\hat{\theta}_{1}, \hat{\theta}_{2}, \cdots, \hat{\theta}_{Q}\right\}$;
3	根据式(36)得到空间间隙位置的虚拟阵元接收数据$\hat{\boldsymbol{Y}}^{}$,并将其和Y^进行重排;
4	结合式(37)计算阵列权值矢量 $w ˜$ ,最终获得阵列的输出 $y ˜$ 。

表1 所提算法步骤

Table 1 Steps of the proposed algorithm

步骤	内容
1	获取K快拍的阵列接收信号X,求解式(28)利用对偶性得到式(27)的最优解,记为Y^和 $Z 1 $ ;
2	利用MUSIC算法得到 $Z 1 *$ 的空间谱,获得入射信号的$\operatorname{DOA}\left\{\hat{\theta}_{1}, \hat{\theta}_{2}, \cdots, \hat{\theta}_{Q}\right\}$;
3	根据式(36)得到空间间隙位置的虚拟阵元接收数据$\hat{\boldsymbol{Y}}^{}$,并将其和Y^进行重排;
4	结合式(37)计算阵列权值矢量 $w ˜$ ,最终获得阵列的输出 $y ˜$ 。

图3 MUSIC空间谱

Fig.3 MUSIC spatial spectrum

图4 DOA估计结果

Fig.4 DOA estimated results

图5 降噪性能对比

Fig.5 Comparison of noise reduction performances

图6 实验3.2的测试结果

Fig.6 Test results of Experiment 3.2

图7 实验3.3的测试结果

Fig.7 Test results of Experiment 3.3

图8 实验3.4的测试结果

Fig.8 Test results of Experiment 3.4

图9 波束图对比

Fig.9 Comparison of beam patterns

图10 不同算法的运行时间

Fig.10 Running times of different algorithms

参考文献 34

[1]	YANG J, YANG Y X, LIAO B. Robust adaptive Bayesian beamforming against stationary and nonstationary interferences[J]. Signal Processing, 2023, 212: 109122.
[2]	LIU S H, MAO Z H, ZHANG Y D, et al. Rank minimization-based Toeplitz reconstruction for DOA estimation using coprime array[J]. IEEE Communications Letters, 2021, 25(7): 2265-2269.
[3]	WANG C Z, ZHU X H, HUANG S M. Coherent receiver design and aperture expansion beamforming for frequency diverse array radar[J]. Digital Signal Processing, 2022, 128: 103631.
[4]	CAI C, LONG Y S, GHOSH S, et al. Bayesian adaptive beamformer for robust electromagnetic brain imaging of correlated sources in high spatial resolution[J]. IEEE Transactions on Medical Imaging, 2023, 42(9): 2502-2512.
[5]	ZHANG X Y, JIANG W D, HUO K, et al. Robust adaptive beamforming based on linearly modified atomic-norm minimization with target contaminated data[J]. IEEE Transactions on Signal Processing, 2020, 68: 5138-5151.
[6]	CHENG Y, ZHANG X Y, LIU T P, et al. Coprime array-adaptive beamforming via atomic-norm-based sparse recovery[J]. IET Radar, Sonar & Navigation, 2021, 15(11): 1494-1507.
[7]	HUANG Y W, ZHOU M K, VOROBYOV S A. New designs on MVDR robust adaptive beamforming based on optimal steering vector estimation[J]. IEEE Transactions on Signal Processing, 2019, 67(14): 3624-3638.
[8]	CHANG L, YEH C C. Performance of DMI and eigenspace-based beamformers[J]. IEEE Transactions on Antennas and Propagation, 1992, 40(11): 1336-1347.
[9]	VOROBYOV S A, GERSHMAN A B, LUO Z Q. Robust adaptive beamforming using worst-case performance optimization: a solution to the signal mismatch problem[J]. IEEE Transactions on Signal Processing, 2003, 51(2): 313-324.
[10]	XU J W, LIAO G S, ZHU S Q, et al. Response vector constrained robust LCMV beamforming based on semi-definite programming[J]. IEEE Transactions on Signal Processing, 2015, 63(21): 5720-5732.
[11]	MOHAMMADZADEH S, NASCIMENTO V H, de LAMARE R C, et al. Covariance matrix reconstruction based on power spectral estimation and uncertainty region for robust adaptive beamforming[J]. IEEE Transactions on Aerospace and Electronic Systems, 2023, 59(4): 3848-3858.
[12]	ZHOU C W, GU Y J, FAN X, et al. Direction-of-arrival estimation for coprime array via virtual array interpolation[J]. IEEE Transactions on Signal Processing, 2018, 66(22): 5956-5971.
[13]	YU L, WEI Y S, LIU W. Adaptive beamforming based on non-uniform linear arrays with enhanced degrees of freedom[C]//Proceedings of TENCON-2015 IEEE Region 10 Conference. Macao, China, IEEE, 2015: 1-5.
[14]	FU M, ZHENG Z, WANG W Q, et al. Coarray interpolation for DOA estimation using coprime EMVS array[J]. IEEE Signal Processing Letters, 2021, 28: 548-552.
[15]	ZHENG Z, YANG T, WANG W Q, et al. Robust adaptive beamforming via coprime coarray interpolation[J]. Signal Processing, 2020, 169: 107382.
[16]	YANG J, LIAO G S, LI J. Robust adaptive beamforming in nested array[J]. Signal Processing, 2015, 114: 143-149.
[17]	ZHOU C W, SHI Z G, GU Y J. Coprime array adaptive beamforming with enhanced degrees-of-freedom capability[C]// Proceedings of the 2017 IEEE Radar Conference. Seattle, WA, US: IEEE, 2017: 1357-1361.
[18]	ZHOU C W, GU Y J, HE S B, et al. A robust and efficient algorithm for coprime array adaptive beamforming[J]. IEEE Transactions on Vehicular Technology, 2018, 67(2): 1099-1112.
[19]	穆巍炜, 何滨兵, 齐尧, 等. 基于环形阵列的地面无人装备集群通信干扰抑制[J]. 兵工学报, 2023, 44(5): 1414-1421.
	MU W W, HE B B, QI Y, et al. Interference suppressionin ground unmanned equipment cluster communication based on circular array[J]. Acta Armamentarii, 2023, 44(5): 1414-1421. (in Chinese) doi: 10.12382/bgxb.2022.0904
[20]	TANG G G, BHASKAR B N, SHAH P, et al. Compressed sensing off the grid[J]. IEEE Transactions on Information Theory, 2013, 59(11): 7465-7490.
[21]	LI Y X, CHI Y J. Off-the-grid line spectrum denoising and estimation with multiple measurement vectors[J]. IEEE Transactions on Signal Processing, 2016, 64(5): 1257-1269.
[22]	YANG Z, XIE L H. Exact joint sparse frequency recovery via optimization methods[J]. IEEE Transactions on Signal Processing, 2016, 64(19): 5145-5157.
[23]	YANG Z, XIE L H. Enhancing sparsity and resolution via reweighted atomic norm minimization[J]. IEEE Transactions on Signal Processing, 2015, 64(4): 995-1006.
[24]	CHU Y H, WEI Z Q, YANG Z. New reweighted atomic norm minimization approach for line spectral estimation[J]. Signal Processing, 2023, 206: 108897.
[25]	WAGNER M, PARK Y, GERSTOFT P. Gridless DOA estimation and root-MUSIC for non-uniform linear arrays[J]. IEEE Transaction on Signal Processing, 2021, 69: 2144-2157.
[26]	GRANT M, BOYD S P, YE Y. CVX: MATLAB software for disciplined convex programming, version 2.1[EB/OL]. http://cvxr.com/cvx, 2017.
[27]	FENG B K, JENN D C. Grating lobe suppression for distributed digital subarrays using virtual filling[J]. IEEE Antennas and Wireless Propagation Letters, 2013, 12: 1323-1326.
[28]	LU J, YANG J, HOU B, et al. Virtual reconstruction-based robust adaptive beamforming for distributed digital subarray antennas[J]. International Journal of Antennas and Propagation, 2021, 2021: 6649439.
[29]	VAIDYANATHAN P P, PAL P. Why does direct-MUSIC on sparse-arrays work[C]//Proceedings of 2013 Asilomar Conference on Signals, Systems and Computers. Pacific Grove, CA, US: IEEE, 2013:2007-2011.
[30]	王永良, 陈辉, 彭应宁, 等. 空间谱估计理论与算法[M]. 北京: 清华大学出版社, 2004.
	WANG Y L, CHEN H, PENG Y N, et al. Theory and algorithm of spatial spectrum estimation[M]. Beijing: Tsinghua University Press, 2004. (in Chinese)
[31]	庞晓娇, 赵永波, 徐保庆, 等. 基于原子范数的MIMO雷达发射波形设计方法[J]. 电子与信息学报, 2019, 41(9):2143-2150.
	PANG X J, ZHAO Y B, XU B Q, et al. An atomic norm-based transmit waveform design method in MIMO radar[J]. Journal of Electronics & Information Technology, 2019, 41(9): 2143-2150. (in Chinese)
[32]	KE X C, ZHAO Y, HUANG L. On accurate source enumeration: a new Bayesian information criterion[J]. IEEE Transaction on Signal Processing, 2021, 69(1): 1012-1027.
[33]	URKOWITZ H. Energy detection of unknown deterministic signals[J]. Proceedings of the IEEE, 1967, 55(4): 523-531.
[34]	CHANG L, YEH C C. Effect of pointing errors on the performance of the projection beamformer[J]. IEEE Transactions on Antennas and Propagation, 1993, 41(8): 1045-1056.