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A Low-complexity Beampattern Shaping Method for Large-scale Rectangular Phased Arrays
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摘要: 针对大规模矩形相控阵波束赋形面临的高计算复杂度瓶颈，该文提出一种基于维度解耦的波束加权向量快速设计方法，显著提升设计效率与波束调控灵活性。首先，充分利用矩形面阵的构型特性，推导方位维与俯仰维导向矢量解耦的波束形成表达式，将传统高维加权向量设计问题高效转化为两个低维加权向量的联合优化问题，从根本上降低计算复杂度。在此基础上，构建以峰值旁瓣电平最小化为代价函数、波束电平与噪声输出功率为约束条件的优化模型，开发基于近端-交替方向乘子法的迭代求解算法，并严格推导算法收敛的充分条件，保障求解稳定性与可靠性。仿真结果验证，所提方法在大幅提升计算效率的同时，不仅能依据先验信息灵活调控主瓣宽度与零陷深度，还可通过调整信噪比损失实现峰值旁瓣抑制性能的精准权衡，展现出优异的工程实用性。Abstract: To address the high computational complexity of beamforming in large-scale rectangular phased arrays, this paper proposes a low-complexity beampattern shaping method based on dimension decoupling, which markedly enhances design efficiency and beampattern adjustment flexibility. By fully exploiting the configuration characteristics of rectangular arrays, an analytical beamforming expression is derived that decouples the azimuth and elevation steering vectors. This transformation converts the traditional high-dimensional weight vector design problem into a joint optimization of two low-dimensional weight vectors, thereby substantially reducing computational complexity. On this basis, an optimization model is formulated that minimizes the peak sidelobe level under constraints on beam levels and noise output power. To solve the resultant optimization problem, an iterative algorithm based on the proximal alternating direction method of multipliers is developed, and sufficient conditions for algorithmic convergence are rigorously derived to ensure solution stability and reliability. Simulation results demonstrate that the proposed method substantially improves computational efficiency and enables flexible adjustment of mainlobe width and null depth according to prior information. Furthermore, it achieves a precise trade-off between peak sidelobe suppression and signal-to-noise ratio loss, exhibiting excellent potential for engineering applications.
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图 2 不同参数下所提算法收敛性能
Figure 2. Convergence performance of the proposed algorithm under different parameters
图 3 不同零陷电平约束下归一化波束方向图
Figure 3. Normalized beampatterns under different null level constraints
图 4 不同零陷电平约束下俯仰维波束方向图
Figure 4. Beampatterns in elevation dimension under different null level constraints
图 6 不同信噪比损失下俯仰维波束方向图
Figure 6. Beampatterns in elevation dimension under different SNR loss
图 7 所提方法与谱加权法合成的波束图
Figure 7. Beampatterns synthesized using the proposed methd and spectral weighting method
图 8 不同算法窄主瓣波束赋形：俯仰维切面
Figure 8. Beampatterns with narrow mainlobe synthesized using different algorithms: Elevation plane
图 9 不同算法宽主瓣波束赋形
Figure 9. Beampatterns with wide mainlobe synthesized using different algorithms
1 矩形相控阵波束加权向量设计
1. The design of weight vector for the rectangular phased array
输入：$ P,Q,{\varTheta _{{\text{side}}}},{\varTheta _{{\text{main}}}},{\varTheta _{{\text{null}}}},{\rho _0},{\rho _1},{\rho _2},\alpha ,\beta ,\varepsilon $ 步骤1：初始化${\boldsymbol{w}}_y^{\left( 0 \right)},{\boldsymbol{w}}_z^{\left( 0 \right)},\left\{ {{x_k}} \right\},\left\{ {{v_k}} \right\},\iota = 0$； 步骤2：$\iota = \iota + 1$； 步骤3：利用式(19)更新$\left\{ {\tilde x_k^{\left( \iota \right)}} \right\}$； 步骤4：利用二分法求得$ \bar \delta $，根据式(23)更新$ {\eta ^{\left( \iota \right)}} $； 步骤5：利用式(24)更新$ \left\{ {x_k^{\left( \iota \right)}} \right\} $，并利用式(25)更新$ \left\{ {\omega _k^{\left( \iota \right)}} \right\} $； 步骤6：分别利用式(20)、式(27)与式(28)更新$ \left\{ {\tilde v_{\bar k}^{\left( \iota \right)}} \right\} $, $ \left\{ {v_{\bar k}^{\left( \iota \right)}} \right\} $与
$ \left\{ {\varpi _{\bar k}^{\left( \iota \right)}} \right\} $；步骤7：利用二分法求得${\bar \lambda _1}$，根据式(34)更新$ {\boldsymbol{w}}_y^{\left( \iota \right)} $； 步骤8：类似步骤7，更新$ {\boldsymbol{w}}_z^{\left( \iota \right)} $； 步骤9：判断是否满足退出条件，若不满足，返回步骤2；否则，
结束迭代；输出：${\boldsymbol{w}} = {\boldsymbol{w}}_y^{\left( \iota \right)} \otimes {\boldsymbol{w}}_z^{\left( \iota \right)}$ 
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表 1 不同算法综合指标：窄主瓣波束赋形
Table 1. Comprehensive indicators of different algorithms: Beampatterns synthesis with narrow mainlobe
算法 归一化零陷 归一化PSL (dB) 计算时间(s) 所提方法 ≤-60.7 dB –25 12.8 ADMM ≤-60.2 dB –26 80.4 FOICA ≤-60 dB –27.3 22329 Case1 ≤-45.8 dB –25.7 67.5 Case2 ≤-53.6 dB –23.8 28.7 Case3 ≤-59.7 dB –19.6 4.18
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表 2 不同算法综合指标：宽主瓣波束赋形
Table 2. Comprehensive indicators of different algorithms: Beampatterns synthesis with wide mainlobe
算法 归一化零陷(dB) 归一化PSL (dB) 计算时间(s) 所提方法 ≤–61.8 –30.9 32.1 ADMM ≤–61.9 –30.3 112.1 FOICA ≤–61.4 –24.2 21698
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