ThêoH Bias-removal Method

Abstract

A version of Thêo1 variance, called ThêoBR variance (BR for "bias-removed" relative to the Allan variance), is constructed. This relative bias correction is applied over a range of the longest recommended Allan τ values for a given data run. ThêoH deviation ('H' to indicate a hybrid combination of ThêoBR and Allan deviations) is the Allan deviation in short term and switches to the ThêoBR deviation in long term. In the presence of non-integer-power-law and mixed noise types, the approach is as effective and less cumbersome than past approaches, and requires little, if any, a priori knowledge or human judgment of data being analyzed. The substantially gained properties of Thêo1 at large τ , to 50 % beyond the longest possible τ using the Allan deviation alone, can be obtained without Allan bias. Long-term frequency stability can be obtained in essentially onethird less time. For example, a two-month stability can be obtained with three months of data, rather than the four months of data that are usually required for such a point.

A Fast Algorithm for Calculation of Thêo1

“…In some cases it is not necessary to calculate Thêo1 for every value of k, sometimes called an 'all-τ ' calculation, and it may be sufficient to only use k equal to powers of two. However, the more sophisticated statistics ThêoBr and ThêoH [7], which attempt to correct for bias in Thêo1 require the calculation of Thêo1 for all k as a first step. There is a technique called 'fast TheoBr' [8] which increases the speed of this calculation by averaging points within the initial dataset to reduce its size.…”