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.