A 2004 paper had offered a theoretical proof that ideally the roots of the minimum variance distortionless response (MVDR) beamformer array polynomial lie on the unit circle (UC). However, existing MVDR methods fail to exploit this fundamental property adequately. This paper proposes a new adaptive beamforming design via UC roots optimization for uniform linear arrays (ULA). The proposed method starts with the sample matrix inversion (SMI) roots and optimizes the MVDR criterion for each root separately by splitting the $N$ -dimensional MVDR problem into multiple 1-D optimization problems. Conjugate symmetry is imposed on individual 1st-order factors of the MVDR polynomial during optimization that guarantees UC roots. The proposed UC Roots Constrained MVDR (UCRC-MVDR) method is non-iterative and has a closed-form solution. It also works well with limited number of snapshots. UCRC-MVDR is computationally efficient and parallelizable as all roots can be optimized concurrently. In extensive simulation studies, UCRC-MVDR exhibits consistently superior performance over existing MVDR approaches, and its performance is closer to the ideal clairvoyant case than existing MVDR methods.

中文翻译：

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通过对均匀线性阵列施加单位圆根约束的 MVDR 波束形成器设计

2004 年的一篇论文提供了理论证明，理想情况下，最小方差无失真响应 (MVDR) 波束成形器阵列多项式的根位于单位圆 (UC) 上。然而，现有的 MVDR 方法未能充分利用这一基本特性。本文通过统一线性阵列 (ULA) 的 UC 根优化提出了一种新的自适应波束成形设计。所提出的方法从样本矩阵求逆 (SMI) 根开始，并通过拆分每个根分别优化 MVDR 准则。$N$ 一维 MVDR 问题转化为多个一维优化问题。在保证 UC 根的优化过程中，对 MVDR 多项式的各个一阶因子施加共轭对称性。所提出的 UC Roots Constrained MVDR (UCRC-MVDR) 方法是非迭代的，并且具有封闭形式的解决方案。它也适用于数量有限的快照。UCRC-MVDR 具有计算效率和可并行性，因为所有根都可以同时优化。在广泛的模拟研究中，UCRC-MVDR 始终表现出优于现有 MVDR 方法的性能，并且其性能比现有 MVDR 方法更接近理想的透视情况。