有限掩码高斯过程重建在周期传感器环上的研究：掩码几何依赖性、傅里叶模式耦合与后验迹
Finite-Mask Gaussian-Process Reconstruction on a Periodic Sensor Ring: Mask-Geometry Dependence, Fourier-Mode Coupling, and Posterior Trace
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中文总结 AI 辅助
本文研究周期传感器环上任意观测掩码对高斯过程重建后验协方差的影响，揭示掩码几何形状通过傅里叶模式耦合改变归一化后验迹，并以64通道基准和麦克风阵列为例验证。
中文摘要 AI 辅助
我们分析了在等间距传感器构成的有限周期环上，任意观测掩码如何影响高斯过程重建中的后验协方差、傅里叶模式耦合以及归一化后验迹。在旋转平稳先验和齐次独立测量噪声的假设下，完全观测为傅里叶模式提供了独立的标量后验公式。然而，对于任意掩码，矩阵$Q_M=F D_MF^\top$通常是非对角的；其非对角元素是实际掩码的有限傅里叶分量，并在后验精度中耦合各模式。因此，具有相同不可用通道比例的掩码，由于几何形状不同，其归一化后验迹也可能不同。一个无量纲的64通道合成基准测试说明了这种有限矩阵效应。在圆形风扇或压缩机管道固定轴向位置的等间距壁装麦克风周向阵列，提供了一个具体的机械工程解释：失效、饱和、损坏或丢失的通道构成观测掩码。该分析是在所述旋转平稳性和公共噪声假设下的有限维参考计算，并非对非均匀或不等间距仪器的运行管道的性能声明。
英文摘要
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.
