有限训练下自适应波束成形的根本极限

中文摘要 AI 辅助

有限训练会降低自适应波束成形器的输出信干噪比（SINR），一个自然的问题是这种损失中有多少是不可避免的。本文解答了这一问题，提供了谱估计中Cramér--Rao界的波束成形对应物。一个精确恒等式将SINR损失表示为全知最小方差无失真响应（MVDR）权重误差的有界函数。它给出了在所有可测的数据相关波束成形规则（包括有偏和不规则规则）上的局部渐近极小极大下界。一阶系数为$\tr(\Mb\Jb_{\rm eff}^{-1})$，其中$\Jb_{\rm eff}$描述训练数据中的信息，$\Mb$衡量输出SINR的灵敏度。匹配构造确定了两个复高斯模型中的该系数。对于具有$K$个不同点干扰源且$2K+1\le N$的$N$传感器均匀线性阵列，数据驱动的分裂一步波束成形器在固定紧致正则参数集的每个内部场景达到系数$C_\theta\le K$。对于无限制协方差矩阵，样本矩阵求逆（SMI）通过经典的Reed--Mallett--Brennan定律达到系数$N-1$。该差异量化了有限源结构的一阶价值。几何公式和数值结果描述了干扰功率和阵列几何形状的依赖性，以及弱源边界附近的有限样本偏离。

英文摘要

Finite training reduces the output signal-to-interference-plus-noise ratio (SINR) of an adaptive beamformer, and a natural question is how much of this loss is unavoidable. This paper determines this question, providing a beamforming counterpart of the Cramér--Rao bound in spectral estimation. An exact identity expresses the SINR loss as a bounded function of the error in the clairvoyant minimum-variance distortionless-response (MVDR) weight. It yields a local asymptotic minimax lower bound over all measurable data-dependent beamforming rules, including biased and irregular rules. The first-order coefficient is $\tr(\Mb\Jb_{\rm eff}^{-1})$, where $\Jb_{\rm eff}$ describes the information in the training data and $\Mb$ measures the sensitivity of the output SINR. Matching constructions determine this coefficient in two complex-Gaussian models. For an $N$-sensor uniform linear array with $K$ distinct point interferers and $2K+1\le N$, a data-driven split one-step beamformer attains the coefficient $C_θ\le K$ at every interior scene of a fixed compact regular parameter set. For unrestricted covariance matrices, sample matrix inversion (SMI) attains the coefficient $N-1$ through the classical Reed--Mallett--Brennan law. The difference quantifies the first-order value of finite-source structure. Geometric formulas and numerical results describe the dependence on interference power and array geometry, and the finite-sample departure near a weak-source boundary.

发表机构

• School of Computer and Artificial Intelligence, Chaohu University（巢湖大学计算机与人工智能学院）
• School of Electronic and Optical Engineering, Nanjing University of Science and Technology（南京理工大学电子与光学工程学院）
• School of Physics and Electronic Engineering, Nanyang Normal University（南阳师范学院物理与电子工程学院）

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