Fundamental Limits of Adaptive Beamforming Under Finite Training

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.