随机傅里叶特征与自适应投影次梯度方法相结合的非线性波束形成

Nonlinear Beamforming Integrating Random Fourier Features and Adaptive Projected Subgradient Method

摘要: 针对有限阵元场景下传统线性波束形成存在的分辨率低、抗干扰弱问题，本文提出融合随机傅里叶特征映射与自适应投影次梯度法的非线性波束形成算法。该算法通过随机傅里叶特征将高斯核的无限维隐式计算转化为固定维显式特征向量，既降低空间复杂度又避免核字典膨胀；结合动态步长策略的凸集投影技术，实现权重向量实时更新，有效平衡收敛速度与稳态误差抑制。进一步将线性约束转化为投影约束矩阵，利用正交投影算子引导迭代收敛至可行解空间，精准控制波束图特性。实验结果表明，与核递归最小二乘算法、Frost算法及基于再生核希尔伯特空间的自适应投影次梯度法相比，所提方法在误码率、均方误差和主瓣增益等关键指标上展现出显著性能优势，并为通信、雷达等领域的非线性波束形成问题提供了具有工程实用价值的解决方案。

Abstract: To address the limitations of conventional linear beamforming in scenarios with limited array ele- ments—such as low spatial resolution and weak anti-interference capability—this paper proposes a nonlinear beamforming algorithm integrating Random Fourier Features (RFF) mapping with the Adaptive Projected Subgradient Method (APSM). The algorithm transforms the infinite-dimensional implicit computation of Gaussian kernels into fixed-dimensional explicit feature vectors through RFF, thereby reducing spatial complexity and mitigating kernel dictionary expansion. By integrating convex set projection techniques with dynamic step-size strategies within the APSM framework, it enables real-time updates of weight vectors while balancing convergence speed and steady-state error suppression. Furthermore, linear equality constraints are converted into projection constraint matrices for RFF, leveraging orthogonal projection operators to guide iterative convergence toward feasible solution spaces and achieve precise beam pattern control. Experimental results demonstrate that the proposed method exhibits significant performance advantages over Kernel Recursive Least Squares (KRLS), the Frost algorithm, and APSM based on Reproducing Kernel Hilbert Space (RKHS) across key metrics, including bit error rate, mean square error, and main lobe gain. This work provides an engineering-practical solution for nonlinear beamforming problems in communication and radar applications.