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
\(Q_M=F D_MF^\ast\) 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.
