Bounds on the Minimum Number of Beamformers for Integrated Sensing and Communications
Bibliographic record
Abstract
Consider a downlink integrated sensing and communications system where a base station employs linear beam-forming to estimate an unknown vector of$L$real parameters, while communicating with$K$users. What is the minimum number of beamformers needed to simultaneously perform both tasks? This paper first shows that the minimum number of beamforming vectors is bounded by$K+\sqrt{L(L+1)}/2$, when the sensing performance is measured in terms of the Cramér-Rao bound involving an$L$x$L$Fisher information matrix, and the communications performance is measured in terms of signal-to-noise-and-interference ratios. This bound can be tightened and also generalized by recognizing that when the sensing metric is a function of$M$quadratic terms involving the beamformers, the minimum number of beamformers is at most$K+\sqrt{M}$, where$M$can be less than$L(L+1)/2$. In particular, for the task of estimating the complex path loss and the angle-of-arrival (AoA) of$N$targets (with a total of$L=3N$parameters), due to the interdependencies in estimating these parameters through the beamformers, we show that the total number of beamformers needed is at most$K+\sqrt{3.5N^{2}+0.5N}$. For the sensing-only scenario with$K$= 0, the minimum number of beamformers needed is asymptotically bounded by 1.871N.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.006 | 0.054 |
| Meta-epidemiology (narrow) | 0.004 | 0.002 |
| Meta-epidemiology (broad) | 0.004 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.004 | 0.005 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.009 | 0.003 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".