Fundamental Limits of Adaptive Beamforming Under Finite Training
Abstract
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(MbJbeff−1), where Jbeff 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≤ N, a data-driven split one-step beamformer attains the coefficient Cθ≤ 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.
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BibTeX
@misc{cheng2026fundamental,
title = {{Fundamental Limits of Adaptive Beamforming Under Finite Training}},
author = {Zhiyong Cheng and Shengyao Chen and Di Song},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06136},
eprint = {2609.06136},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06136},
}
RIS
TY - UNPB
TI - Fundamental Limits of Adaptive Beamforming Under Finite Training
AU - Zhiyong Cheng
AU - Shengyao Chen
AU - Di Song
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06136
UR - https://arxiv.org/abs/2609.06136
AB - 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(MbJb_eff^−1), where Jb_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≤ N, a data-driven split one-step beamformer attains the coefficient C_θ≤ 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.
ER -