Finite-Mask Gaussian-Process Reconstruction on a Periodic Sensor Ring: Mask-Geometry Dependence, Fourier-Mode Coupling, and Posterior Trace
Abstract
We analyze how an arbitrary observation mask on a finite periodic ring of equally spaced sensors affects the posterior covariance, Fourier-mode coupling, and normalized posterior trace in Gaussian-process reconstruction. Under a rotationally stationary prior and homogeneous independent measurement noise, complete observation gives independent scalar posterior formulas for the Fourier modes. For an arbitrary mask, however, the matrix QM=F DMF∗ is generally non-diagonal; its off-diagonal entries are finite Fourier components of the realized mask and couple modes in the posterior precision. Consequently, masks with the same unavailable-channel fraction can have different normalized posterior traces because their geometries differ. A dimensionless 64-channel synthetic benchmark illustrates this finite-matrix effect. A circumferential array of equally spaced wall-mounted microphones at a fixed axial station of a circular fan or compressor duct provides one concrete mechanical-engineering interpretation: failed, saturated, corrupted, or dropped-out channels form the observation mask. The analysis is a finite-dimensional reference calculation under the stated rotational-stationarity and common-noise assumptions, not a performance claim for a nonuniform or unequally instrumented operating duct.
Versions
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BibTeX
@misc{tsuzurugi2026finitemask,
title = {{Finite-Mask Gaussian-Process Reconstruction on a Periodic Sensor Ring: Mask-Geometry Dependence, Fourier-Mode Coupling, and Posterior Trace}},
author = {Jun Tsuzurugi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14007},
eprint = {2609.14007},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14007},
}
RIS
TY - UNPB
TI - Finite-Mask Gaussian-Process Reconstruction on a Periodic Sensor Ring: Mask-Geometry Dependence, Fourier-Mode Coupling, and Posterior Trace
AU - Jun Tsuzurugi
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.14007
UR - https://arxiv.org/abs/2609.14007
AB - We analyze how an arbitrary observation mask on a finite periodic ring of equally spaced sensors affects the posterior covariance, Fourier-mode coupling, and normalized posterior trace in Gaussian-process reconstruction. Under a rotationally stationary prior and homogeneous independent measurement noise, complete observation gives independent scalar posterior formulas for the Fourier modes. For an arbitrary mask, however, the matrix Q_M=F D_MF^∗ is generally non-diagonal; its off-diagonal entries are finite Fourier components of the realized mask and couple modes in the posterior precision. Consequently, masks with the same unavailable-channel fraction can have different normalized posterior traces because their geometries differ. A dimensionless 64-channel synthetic benchmark illustrates this finite-matrix effect. A circumferential array of equally spaced wall-mounted microphones at a fixed axial station of a circular fan or compressor duct provides one concrete mechanical-engineering interpretation: failed, saturated, corrupted, or dropped-out channels form the observation mask. The analysis is a finite-dimensional reference calculation under the stated rotational-stationarity and common-noise assumptions, not a performance claim for a nonuniform or unequally instrumented operating duct.
ER -