Theo1: characterization of very long-term frequency stability

Abstract

Theo1 is the first new species of variance that addresses a particularly difficult measurement problem, namely, obtaining reliable estimation of frequency stability for sample periods that are long compared to the length of a data run. Theo1 has statistical properties that are like the Allan variance (Avar), but Theo1 also has two advantages over other estimators of frequency stability: (1) it can evaluate frequency stability at a sample period (τ) of 3/4 the length of a data run, and (2) it presently attains the highest equivalent degrees of freedom (edf) of any estimator of frequency stability including Total-var and overlapping-Avar. Theo1 is unbiased relative to Avar for WHFM noise. Theo1 is biased slightly low with FLFM and RWFM, and we present a formula for a hybrid statistic (TheoH) made up of a combination of Theo1 and Avar in which bias is automatically removed. We explain the sampling function used in Theo1 and show that its frequency response is nearly ideal for extracting power-law noise processes of the types encountered with precision oscillators and clocks. We present results which, for a given data run, show how Theo1 anticipates the levels of frequency stability that are determined by Avar when given a longer data run from the same set of clocks.

ThêoH Bias-removal Method

“…The unique data sampling of Theo1 [6], [7], [8] causes bias with respect to zero-dead-time sampling of σ 2 y (τ ). Theo1(m, τ 0 ) relative to the Allan variance can be expressed as a ratio of σ 2 y (τ ) to Theo1(m, τ 0 ) given by a normalized bias as:…”

“…An ad hoc method of determining noise type and removing bias previously reported in ref. [6] has since been found to have its bias slightly dependent on m [7]. The most accurate dependence is given by the function…”

“…The transform to S y (f) from Theo1 variance follows log-log straight lines, similar to the Allan variance [5]. The levels of the Theo1 autocovariance (ACV) functions are close to those of the Allan ACV functions, and for white FM (WHFM) ACV's are identical by definition [6], [7], [8]. The ratio of the translations to h α 's using Theo1 vs. Allan differ as the ratio 1.5 α for τ 0 τ .…”

Theo1 confidence intervals

“…Compared to Adev, Totdev has less skew in its confidence intervals [5,7]. Presently, Theo1 has the highest confidence in estimating long-term frequency stability [9,10]. In the study here, we found that Theo1's distribution function is both narrower and more symmetric than that of chi-square.…”

“…τ s has the same meaning as the traditional sampling interval τ in theσ 2 y (τ ), the traditional estimator of the Allan variance. Let us start with the equation for Theo1, the estimator of Theo1 [9,10]:…”