Noise statistics across the three axes of a tri-axial velocity sensor constructed of pressure sensors

This paper analyzes the noise statistics of a tri-axial velocity sensor. This tri-axial sensing system comprises three orthogonal axes. Along each axis, a uni-axial velocity sensor is realized through two isotropic pressure sensors displaced along that axis. Hence, a tri-axial velocity sensor may be implemented with as few as four isotropic sensors: one at the Cartesian origin and one at each Cartesian axis. Each isotropic sensor's noise would contribute to its axis' effective noise. As the three axes share one isotropic sensor in common, the three axes' effective noises would be cross-correlated, but how? This is answered in this paper, through rigorous mathematics that analytically derive the statistical codifference across the three axes' effective noises. This analysis models the noises accommodatingly as α-stable distributed, which includes the special cases of the Gaussian distribution, the Cauchy distribution, and many other heavy-tailed probability distributions. This finding is then generalized from the aforementioned velocity sensors (i.e., differential sensors of the first order) to differential sensors of arbitrarily high orders.