Effective DoF Collapse in Sparse Near-Field Beamforming: A Block-Rank-1 Mechanism
Abstract
Sparse near-field arrays phase-cohere M widely separated sub-arrays of N elements and underpin distributed MIMO, satellite formations, and integrated sensing and communication. We show that their interference suppression over a protected region is set by coherence within a single sub-array rather than by the span of the formation: the effective rank of the region covariance can collapse from MN to M. Two independent arguments establish this. For canonical line and disk regions each diagonal covariance block is exactly a classical concentration operator, which supplies sharp per-block rank thresholds and makes fewer than one spatial degree of freedom per sub-array a conservative design rule; an orthogonal decomposition with eigenvalue interlacing gives a nonasymptotic sufficient condition that assumes nothing about the coupling between platforms. The consequences are covariance-side and beamformer-independent: the unloaded Capon functional upper-bounds worst-case signal-to-interference ratio, while its regularized counterpart scales very differently for angularly separated and for in-cone targets. An eight-satellite LEO study validates the theory, including realized worst-case SIR.
Versions
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BibTeX
@misc{he2026effective,
title = {{Effective DoF Collapse in Sparse Near-Field Beamforming: A Block-Rank-1 Mechanism}},
author = {Changhao He and Xiaojuan Zhang and Francois Chin Po Shin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15184},
eprint = {2610.15184},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15184},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Effective DoF Collapse in Sparse Near-Field Beamforming: A Block-Rank-1 Mechanism
AU - Changhao He
AU - Xiaojuan Zhang
AU - Francois Chin Po Shin
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.15184
UR - https://arxiv.org/abs/2610.15184
AB - Sparse near-field arrays phase-cohere M widely separated sub-arrays of N elements and underpin distributed MIMO, satellite formations, and integrated sensing and communication. We show that their interference suppression over a protected region is set by coherence within a single sub-array rather than by the span of the formation: the effective rank of the region covariance can collapse from MN to M. Two independent arguments establish this. For canonical line and disk regions each diagonal covariance block is exactly a classical concentration operator, which supplies sharp per-block rank thresholds and makes fewer than one spatial degree of freedom per sub-array a conservative design rule; an orthogonal decomposition with eigenvalue interlacing gives a nonasymptotic sufficient condition that assumes nothing about the coupling between platforms. The consequences are covariance-side and beamformer-independent: the unloaded Capon functional upper-bounds worst-case signal-to-interference ratio, while its regularized counterpart scales very differently for angularly separated and for in-cone targets. An eight-satellite LEO study validates the theory, including realized worst-case SIR.
ER -