Low-Complexity Adaptive Beamforming Algorithms Based on Low-Rank Decompositions and Set-Membership Filtering

该研究的目标是开发基于低阶分解和集员滤波框架的新型低复杂度波束形成算法，以应对挑战#15。我们将基于迭代切换和模式匹配的低阶分解的概念以及基函数的近似引入到矩阵S_D的设计中。我们将利用这些分解来构造线性约束最小方差波束形成器。具体地说，目标是设计出比现有算法的复杂度低一个数量级的算法。我们将开发低阶随机梯度(SG)和递归最小二乘(RLS)算法，其复杂度比现有的满秩随机梯度(SG)和递归最小二乘(RLS)算法低十分之一，这两种算法的性能至少相当。这将是可能的，因为创新的低级分解与基于SMF的算法相结合。所提出的低阶分解不需要复杂的特征分解或昂贵的运算。通过将维度从M降低到D，这些技术可以比全秩滤波算法简单得多。例如，对于感兴趣的场景，我们将具有M=阵元和秩3=D=6。然后，SMF概念将被用于设计用于矩阵S_D和滤波器W_D的更新的低复杂度自适应算法。所提出的基于低阶SMF的算法的一个关键方面是利用对于矩阵S_D和滤波器W_D的可能不同的更新率来利用数据选择性更新。我们将使用线性代数来用低阶分解来描述LCMV波束形成问题，开发基于SMF的自适应算法并构建仿真工具来设计、测试和分析所提出的技术。其结果将是更好、更简单和实用的波束形成算法，以及高质量的出版物。

The goal of the proposed research is to develop novel low-complexity beamforming algorithms based on low-rank decompositions and the set-membership filtering (SMF) framework in order to address challenge #15. We will introduce concepts of low-rank decomposition based on iterative switching and pattern matching, and approximation of basis functions to the design of the matrix S_D. We will formulate the linearly constrained minimum variance (LCMV) beamformer with these decompositions. Specifically, the goal is to devise algorithms with an order of magnitude lower complexity than existing algorithms. We will develop low-rank stochastic gradient (SG) and recursive least squares (RLS) algorithms ten times less complex than the existing full-rank SG and RLS ones, which have at least comparable performance. This will be possible due to the combination of innovative low-rank decompositions with SMF-based algorithms. The proposed low-rank decompositions do not require complex eigen-decompositions or expensive operations. These techniques can be significantly simpler than full-rank filtering algorithms by reducing the dimensionality from M to D. For instance, for the scenario of interest we will have M=64 array elements and a rank 3=D=6. The SMF concept will then be used to design low-complexity adaptive algorithms for the updates of the matrix S_D and the filter w_D. One key aspect of the proposed low-rank SMF-based algorithms is to exploit data-selective updates with possibly different update ratios for the matrix S_D and the filter w_D. We will formulate the LCMV beamforming problem with the low-rank decompositions using linear algebra, develop SMF-based adaptive algorithms and build simulation tools to design, test and analyse the proposed techniques. The outcomes will be better, simpler and practical beamforming algorithms, and high-quality publications.