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

所提出的研究的目标是开发新的低复杂度波束成形算法的基础上低秩分解和集员滤波（SMF）的框架，以解决挑战#15。我们将介绍基于迭代切换和模式匹配的低秩分解的概念，以及对矩阵S_D的设计的基函数的近似。我们将制定这些分解的线性约束最小方差（LCMV）波束形成器。具体来说，目标是设计出比现有算法复杂度低一个数量级的算法。我们将开发低秩随机梯度（SG）和递归最小二乘（RLS）算法的复杂性比现有的满秩SG和RLS算法低十倍，它们至少具有相当的性能。这将是可能的，由于创新的低秩分解与基于SMF的算法相结合。所提出的低秩分解不需要复杂的特征分解或昂贵的操作。这些技术可以通过将维度从M减少到D而比满秩滤波算法简单得多。例如，对于感兴趣的场景，我们将有M=64个数组元素和秩3=D=6。然后，SMF概念将用于设计用于矩阵S_D和滤波器w_D的更新的低复杂度自适应算法。所提出的基于低秩SMF的算法的一个关键方面是利用针对矩阵S_D和滤波器w_D具有可能不同的更新比率的数据选择性更新。我们将制定LCMV波束形成问题的低秩分解使用线性代数，开发基于SMF的自适应算法和建立仿真工具来设计，测试和分析所提出的技术。其成果将是更好、更简单、更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.