Higher-Order Allan Variance for Atomic Clocks of Arbitrary Order: Mathematical Foundation

TL;DR

This paper provides a rigorous mathematical foundation for higher-order Allan variance in atomic clocks, including analytical expressions and conditions for time-independence, enhancing understanding of clock stability analysis.

Contribution

It introduces a comprehensive analytical framework for higher-order Allan variance, establishing necessary and sufficient conditions for its properties in atomic clock models.

Findings

Derived a complete analytical expression for higher-order Allan variance.

Proved the necessity and sufficiency of the difference operation for Gaussianity.

Established conditions under which the higher-order Allan variance is time-independent.

Abstract

In this paper, we perform a time-domain analysis of the higher-order Allan variance for atomic clock models of arbitrary order. Adopting a standard atomic clock model where the time series of the clock reading deviation is expressed as a Wiener or integrated Wiener process, we define the higher-order Allan variance as the mean squared higher-order difference of the clock reading deviation. The main results of this paper are threefold. First, we prove that the higher-order difference operation of the clock reading deviation, which can be interpreted as a linear aggregation with binomial coefficients, is not only sufficient but also necessary for a resulting aggregated time series to be an independent and identically distributed Gaussian process. Second, we derive a complete analytical expression of the higher-order Allan variance, which consists of both time-dependent and time-independent terms. Third, we prove that the higher-order Allan variance is time-independent if and only if the order of difference operation is greater than or equal to the order of the atomic clock model.