On the relation between uncertainties of weighted frequency averages and the various types of Allan deviations

TL;DR

This paper explores the mathematical relationship between Allan deviations and the uncertainty in weighted frequency averages, providing methods to improve measurement accuracy especially under white phase noise conditions.

Contribution

It offers a mathematical framework for using modified Allan deviation and related two-sample deviations to accurately determine frequency measurement uncertainties.

Findings

modADEV and parADEV are advantageous for white phase noise measurements

A procedure for adaptive averaging to minimize measurement uncertainty

Discussion of theoretical aspects for real-world frequency measurement

Abstract

The power spectral density in Fourier frequency domain, and the different variants of the Allan deviation (ADEV) in dependence on the averaging time are well established tools to analyse the fluctuation properties and frequency instability of an oscillatory signal. It is often supposed that the statistical uncertainty of a measured average frequency is given by the ADEV at a well considered averaging time. However, this approach requires further mathematical justification and refinement, which has already been done regarding the original ADEV for certain noise types. Here we provide the necessary background to use the modified Allan deviation (modADEV) and other two-sample deviations to determine the uncertainty of weighted frequency averages. The type of two-sample deviation used to determine the uncertainty depends on the method used for determination of the average. We find that the modADEV, which is connected with $\Lambda$-weighted averaging, and the two sample deviation associated to a linear phase regression weighting (parADEV) are in particular advantageous for measurements, in which white phase noise is dominating. Furthermore, we derive a procedure how to minimize the uncertainty of a measurement for a typical combination of white phase and frequency noise by adaptive averaging of the data set with different weighting functions. Finally, some aspects of the theoretical considerations for real-world frequency measurement equipment are discussed.