{
  "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 $\\tr(\\Mb\\Jb_{\\rm eff}^{-1})$, 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06136",
  "author_total": 3,
  "authors": [
    {
      "name": "Zhiyong Cheng"
    },
    {
      "name": "Shengyao Chen"
    },
    {
      "name": "Di Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/crgkgn9y0a3ta.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06136",
  "fulltext_url": null,
  "id": "crgkgn9y0a3ta",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06136",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "eess.SP"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1711|Direction-of-Arrival Estimation Techniques"
    ]
  ],
  "openalex": "W7212071447",
  "pdf_url": "https://arxiv.org/pdf/2609.06136",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/crgkgn9y0a3ta.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06136",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06136"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Fundamental Limits of Adaptive Beamforming Under Finite Training",
  "title_text": "Fundamental Limits of Adaptive Beamforming Under Finite Training",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/crgkgn9y0a3ta",
  "venue": "arXiv",
  "versions": []
}