Robust Adaptive Beamforming for Sparse Arrays

FAN Xuhui; WANG Yuyi; WANG Anyi; XU Yanhong; CUI Can

doi:10.11999/JEIT250952

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FAN Xuhui, WANG Yuyi, WANG Anyi, XU Yanhong, CUI Can. Robust Adaptive Beamforming for Sparse Arrays[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT250952

Citation:

FAN Xuhui, WANG Yuyi, WANG Anyi, XU Yanhong, CUI Can. Robust Adaptive Beamforming for Sparse Arrays[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT250952

FAN Xuhui, WANG Yuyi, WANG Anyi, XU Yanhong, CUI Can. Robust Adaptive Beamforming for Sparse Arrays[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT250952

Citation:

FAN Xuhui, WANG Yuyi, WANG Anyi, XU Yanhong, CUI Can. Robust Adaptive Beamforming for Sparse Arrays[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT250952

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Robust Adaptive Beamforming for Sparse Arrays

doi: 10.11999/JEIT250952 cstr: 32379.14.JEIT250952

School of Communication and Information Engineering, Xi’an University of Science and Technology, Xi’an, 710054, China

Funds: National Natural Science Foundation of China 62471384, National Natural Science Foundation of China 62301414, National Natural Science Foundation of China 62301415

Accepted Date: 2025-12-04
Rev Recd Date: 2025-12-04

Available Online: 2025-12-09

Abstract

Abstract

Objective The rapid advancement of modern communication technologies (e.g., 5G networks and IoT applications) has led to increased complexity in signal processing for wireless communication and radar systems. Adaptive beamforming techniques have found extensive applications in these areas owing to their effectiveness in extracting the signal of interest amidst interference and noise. Traditional robust adaptive beamforming methods can effectively handle steering vector mismatch. Such mismatches may arise from environmental non-stationarity, direction-of-arrival estimation errors, imperfect array calibration, antenna deformation, and local scattering effects. However, they ignore the potential benefits of the sparse arrays, which can significantly reduce hardware complexity and system cost. Moreover, they frequently fail to suppress sidelobe levels (SLL) in environments with interference source, limiting their practical utility in complex electromagnetic scenarios. To overcome these limitations, this paper proposes a robust adaptive beamforming algorithm that achieves both the sparse arrays and low SLL constraints. Methods Unlike conventional sparse approaches that place the l₀ norm penalty in the objective function, the proposed method introduces the l₀ norm into the constraint. This formulation ensures that the optimized array configuration satisfies the pre-specified number of active sensors, thereby avoiding the uncertainty caused by adjusting sparse weights in multi-objective optimization models. In addition to the sparsity constraint, a SLL suppression constraint is also introduced. This design imposes an upper bound on the array response in interference and clutter directions, thereby effectively suppressing undesired signals. By integrating these constraints into the optimization framework, the proposed method achieves a robust Minimum Variance Distortionless Response (MVDR) beamforming that exhibits sparsity, adaptivity, and robustness. To address the nonconvexity of the formulated optimization problem, a convex relaxation strategy is adopted to transform the non-convex constrain into a convex one. Therefore, this paper proposes robust adaptive beamforming methods that generates a sparse weight solution from a uniform linear array (ULA). It is worth noting that although the proposed method is derived from a ULA, obtaining a sparse weight solution provides several practical benefits. By assigning zero weights to certain sensors, the method effectively reduces the number of active elements, lowering hardware cost and computational complexity, while still maintaining desirable beamforming performance. The main contribution of this paper lies in proposing a unified framework that enables collaborative optimization of robustness, beam performance, SLL, and array sparsity. Results and Discussions A series of simulation experiments were conducted to evaluate the performance of the proposed sparse robust beamforming algorithm under various scenarios, including multiple interference environments, steering vector mismatch, angle-of-arrival (AOA) mismatch, low signal-to-noise ratio (SNR) conditions, and complex electromagnetic environments based on practical antenna arrays. Simulation results demonstrate that the proposed algorithm can maintain stable mainlobe gain in the desired signal direction while forming deep nulls in the interference directions. First, in the presence of steering vector mismatch, conventional MVDR beamformers often suffer from reduced mainlobe gain or even beam pointing deviations, which severely compromise the reception of the desired signal. In contrast, the proposed algorithm is capable of maintaining a stable and distortionless mainlobe direction under mismatch conditions, thereby ensuring high gain in the desired signal direction (Fig. 2(a), Fig. 3(a)). Second, by introducing a sidelobe constraint, the proposed algorithm effectively suppresses clutter and achieves significantly lower peak sidelobe levels compared with other approaches (Fig. 2(b)). Third, under low-SNR conditions, the algorithm demonstrates strong noise resistance. Even in severely noise-contaminated scenarios, it is able to maintain effective interference suppression and achieve high output Signal-to-Interference-plus-Noise Ratio (SINR). This indicates that the method has good adaptability in weak target detection and in cluttered environments. Moreover, the optimized sparse array configuration achieves beamforming performance close to that of a ULA despite activating only a subset of sensors (Fig. 2). Finally, experimental validation based on real antenna arrays further confirmed the effectiveness of the proposed method (Fig. 3). The algorithm maintains stable performance and is still able to achieve high gain in the desired direction even in the presence of AOA estimation mismatches (Fig. 4). In summary, experimental results demonstrate that the proposed algorithm achieves significant improvements in robustness and hardware efficiency. Furthermore, it exhibits reliable performance and effectiveness in complex electromagnetic environments. Conclusions This paper proposes a robust adaptive beamforming algorithm for sparse arrays. The core innovation lies in establishing a joint optimization model that incorporates array sparsity, steering vector mismatch robustness, and low SLL constraints into a unified framework. Compared with methods such as MVDR^[9] (which primarily focuses on interference suppression), CMR^[12] (which achieves robustness), or NA-CS^[30] (which only achieves array sparsity), the proposed method achieves a balanced across multiple dimensions. Simulation results demonstrate that, in complex scenarios involving steering vector errors, AOA estimation mismatches, and low SNR conditions, this method can maintain satisfactory beamforming performance with lower hardware costs, exhibiting stronger practical engineering value and application potential.
- Adaptive beamforming,
- sparse arrays,
- robust constraints,
- convex optimization

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References(36)

References

[1]	LUO Jie, ZHANG Shurui, HAN Yubing, et al. Low complexity robust subband adaptive beamforming with frequency-angle coupling[J]. Digital Signal Processing, 2025, 164: 105288. doi: 10.1016/j.dsp.2025.105288.
[2]	BUI V P, VAN CHIEN T, LAGUNAS E, et al. Demand-based adaptive multi-beam placement and resource allocation in non-GSO satellite systems[J]. IEEE Transactions on Vehicular Technology, 2025: 1-13. doi: 10.1109/TVT.2025.3610334. (查阅网上资料,未找到本条文献卷期信息,请确认).
[3]	SOMASUNDARAM S D. Linearly constrained robust Capon beamforming[J]. IEEE Transactions on Signal Processing, 2012, 60(11): 5845–5856. doi: 10.1109/TSP.2012.2212889.
[4]	巩朋成, 陈伟, 柯航, 等. 导向矢量失配条件下多约束鲁棒波束形成算法[J]. 信号处理, 2024, 40(10): 1855–1865. doi: 10.12466/xhcl.2024.10.010. GONG Pengcheng, CHEN Wei, KE Hang, et al. Multiple constraints robust beamforming algorithm under steering vector mismatch conditions[J]. Journal of Signal Processing, 2024, 40(10): 1855–1865. doi: 10.12466/xhcl.2024.10.010.
[5]	王兆彬, 巩朋成, 邓薇, 等. 联合协方差矩阵重构和ADMM的鲁棒波束形成[J]. 哈尔滨工业大学学报, 2023, 55(4): 64–71. doi: 10.11918/202107104. WANG Zhaobin, GONG Pengcheng, DENG Wei, et al. Robust beamforming by joint covariance matrix reconstruction and ADMM[J]. Journal of Harbin Institute of Technology, 2023, 55(4): 64–71. doi: 10.11918/202107104.
[6]	巩朋成, 刘永康, 吴云韬, 等. 基于多近似一致性ADMM的阵列方向图合成[J/OL]. https://link.cnki.net/urlid/32.1353.TN.20230113.1313.001, 2023. GONG Pengcheng, LIU Yongkang, WU Yuntao, et al. Array pattern synthesis based on multi-approximate Consistent-ADMM[J/OL]. https://link.cnki.net/urlid/32.1353.TN.20230113.1313.001, 2023.
[7]	LIU Yujin, WANG Zedan, and SUN Xuekai. Connections between least-squares migration, capon beamforming, and phase correction for seismic resolution enhancement[J]. IEEE Geoscience and Remote Sensing Letters, 2025, 22: 3002505. doi: 10.1109/LGRS.2025.3575170.
[8]	COX H, ZESKIND R, and OWEN M. Robust adaptive beamforming[J]. IEEE Transactions on Acoustics, Speech, and Signal Processing, 1987, 35(10): 1365–1376. doi: 10.1109/TASSP.1987.1165054.
[9]	CAPON J. High-resolution frequency-wavenumber spectrum analysis[J]. Proceedings of the IEEE, 1969, 57(8): 1408–1418. doi: 10.1109/PROC.1969.7278.
[10]	HUANG Yongwei, ZHOU Mingkang, and 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. doi: 10.1109/TSP.2019.2918997.
[11]	ALI M, HAMEED K, and NATHWANI K. Enhancing traditional underwater DoA estimation techniques using convolutional autoencoder-based covariance matrix reconstruction[J]. IEEE Sensors Letters, 2025, 9(1): 7000304. doi: 10.1109/LSENS.2024.3516859.
[12]	LUO Tao, CHEN Peng, CAO Zhenxin, et al. URGLQ: An efficient covariance matrix reconstruction method for robust adaptive beamforming[J]. IEEE Transactions on Aerospace and Electronic Systems, 2023, 59(5): 5634–5645. doi: 10.1109/TAES.2023.3263386.
[13]	YANG Yue, XU Xu, YANG Huichao, et al. Robust adaptive beamforming via covariance matrix reconstruction with diagonal loading on interference sources covariance matrix[J]. Digital Signal Processing, 2023, 136: 103977. doi: 10.1016/j.dsp.2023.103977.
[14]	SU Leqiang, LIU Hongwei, XU Xu, et al. Robust adaptive beamforming algorithm for multipath coherent reception based on iterative adaptive approach and covariance matrix reconstruction[J]. Digital Signal Processing, 2026, 168: 105532. doi: 10.1016/j.dsp.2025.105532.
[15]	MOHAMMADZADEH S, NASCIMENTO V H, DE LAMARE R C, et al. Maximum entropy-based interference-plus-noise covariance matrix reconstruction for robust adaptive beamforming[J]. IEEE Signal Processing Letters, 2020, 27: 845–849. doi: 10.1109/LSP.2020.2994527.
[16]	THOMAS J K, SCHARF L L, and TUFTS D W. The probability of a subspace swap in the SVD[J]. IEEE Transactions on Signal Processing, 1995, 43(3): 730–736. doi: 10.1109/78.370627.
[17]	MOLU M M, XIAO P, KHALILY M, et al. Low-complexity and robust hybrid beamforming design for multi-antenna communication systems[J]. IEEE Transactions on Wireless Communications, 2018, 17(3): 1445–1459. doi: 10.1109/TWC.2017.2778258.
[18]	SUN Linlin, QIN Yaolu, ZHUANG Zhihong, et al. A robust secure hybrid analog and digital receive beamforming scheme for efficient interference reduction[J]. IEEE Access, 2019, 7: 22227–22234. doi: 10.1109/ACCESS.2019.2899154.
[19]	CHAHROUR H, RAJAN S, DANSEREAU R, et al. Hybrid beamforming for interference mitigation in MIMO radar[C]. 2018 IEEE Radar Conference, Oklahoma City, USA, 2018: 1005–1009. doi: 10.1109/RADAR.2018.8378698.
[20]	AL KASSIR H, ZAHARIS Z D, LAZARIDIS P I, et al. A review of the state of the art and future challenges of deep learning-based beamforming[J]. IEEE Access, 2022, 10: 80869–80882. doi: 10.1109/ACCESS.2022.3195299.
[21]	LIU Fulai, QIN Hao, SUN Ziyuan, et al. WS-CACNN algorithm for robust adaptive beamforming[J]. Physical Communication, 2025, 71: 102666. doi: 10.1016/j.phycom.2025.102666.
[22]	SINGMAN M P and NARAYANAN R M. Applying machine learning to adaptive array signal processing weight generation[J]. IEEE Transactions on Aerospace and Electronic Systems, 2024, 60(4): 4952–4962. doi: 10.1109/TAES.2024.3382620.
[23]	YE Sicong, XIAO Ming, KWAN M W, et al. Extremely large aperture array (ELAA) communications: Foundations, research advances and challenges[J]. IEEE Open Journal of the Communications Society, 2024, 5: 7075–7120. doi: 10.1109/OJCOMS.2024.3486172.
[24]	SHI Wanlu, VOROBYOV S A, and LI Yingsong. ULA fitting for sparse array design[J]. IEEE Transactions on Signal Processing, 2021, 69: 6431–6447. doi: 10.1109/TSP.2021.3125609.
[25]	WANG Xiangrong, GRECO M S, and GINI F. Adaptive sparse array beamformer design by regularized complementary antenna switching[J]. IEEE Transactions on Signal Processing, 2021, 69: 2302–2315. doi: 10.1109/TSP.2021.3064183.
[26]	NOSRATI H and ABOUTANIOS E. Online antenna selection for adaptive beamforming in MIMO radar[C]. 2020 IEEE Radar Conference, Florence, Italy, 2020: 1–6. doi: 10.1109/RadarConf2043947.2020.9266672.
[27]	LI Hongtao, RAN Longyao, HE Cheng, et al. Adaptive beamforming with sidelobe level control for multiband sparse linear array[J]. Remote Sensing, 2023, 15(20): 4929. doi: 10.3390/rs15204929.
[28]	ZHANG Xuan and WANG Xiangrong. Sparse adaptive beamformer design with a good quiescent beampattern[C]. 2019 IEEE International Conference on Signal, Information and Data Processing, Chongqing, China, 2019: 1–5. doi: 10.1109/ICSIDP47821.2019.9173179.
[29]	HUANG Jiayi, ZHANG Xuan, WANG Xiangrong, et al. Transmit sparse array beamformer design for dual-function radar communication systems[C]. 2023 IEEE International Radar Conference, Sydney, Australia, 2023: 1–6. doi: 10.1109/RADAR54928.2023.10371099.
[30]	陈力恒, 马晓川, 李璇, 等. 结合压缩感知模型的稀疏阵列波束形成方法[J]. 信号处理, 2020, 36(4): 475–485. doi: 10.16798/j.issn.1003-0530.2020.04.001. CHEN Liheng, MA Xiaochuan, LI Xuan, et al. Sparse array beamforming method combined with compressed sensing model[J]. Journal of Signal Processing, 2020, 36(4): 475–485. doi: 10.16798/j.issn.1003-0530.2020.04.001.
[31]	赵友瑜, 罗桂宁, 王姣, 等. 基于改进平滑L0范数的压缩感知重构算法[J]. 电脑知识与技术, 2024, 20(12): 35–38, 41. doi: 10.14004/j.cnki.ckt.2024.0596. ZHAO Youyu, LUO Guining, WANG Jiao, et al. Compressed sensing reconstruction algorithm based on improved smooth L0norm[J]. Computer Knowledge and Technology, 2024, 20(12): 35–38, 41. doi: 10.14004/j.cnki.ckt.2024.0596. (查阅网上资料,未找到本条文献英文翻译信息,请确认).
[32]	OLIVERI G, BEKELE E T, ROBOL F, et al. Sparsening conformal arrays through a versatile BCS-based method[J]. IEEE Transactions on Antennas and Propagation, 2014, 62(4): 1681–1689. doi: 10.1109/TAP.2013.2287894.
[33]	EKSIOGLU E M and TANC A K. RLS algorithm with convex regularization[J]. IEEE Signal Processing Letters, 2011, 18(8): 470–473. doi: 10.1109/LSP.2011.2159373.
[34]	VOROBYOV S A, GERSHMAN A B, and LUO Zhiquan. 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. doi: 10.1109/TSP.2002.806865.
[35]	THAT V, MUY S, and LEE J R. Multi-UAV-aided power-up and data collection: Multi-agent DQL with genetic algorithm approach[J]. Expert Systems with Applications, 2026, 297: 129401. doi: 10.1016/j.eswa.2025.129401.
[36]	WANG Kunyu, SO A M C, CHANG T H, et al. Outage constrained robust transmit optimization for multiuser MISO downlinks: Tractable approximations by conic optimization[J]. IEEE Transactions on Signal Processing, 2014, 62(21): 5690–5705. doi: 10.1109/TSP.2014.2354312.