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

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 twosample 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 Λ-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.arXiv:1504.00466v3 [physics.data-an]

Combining fiber Brillouin amplification with a repeater laser station for fiber-based optical frequency dissemination over 1400 km

“…The scatter of the Λ 2000s fractional frequency offset of the FBA-RLS combination are larger than for the single-span loop [1], which may indicate the presence of additional out-of-loop processes in the cascaded configuration, due to for instance the RLSs interferometer. The observed offsets are mostly zero-compatible within 1 σ, with σ being the statistical uncertainty of the average [36]. The lower statistical uncertainties in the 2018's campaigns reflect the improvement in the instabilities already observable in Different processes may induce these residual offsets.…”

“…In case of CS-free link operation, such as demonstrated here for the combination of FBAs and RLSs, resolution at the level of such improved clock's can be achieved significantly faster by continuing with phase-averaging up to 500 s (Λ 500s counting), see purple curve in figure 6. Thus the fiber link uncertainty contribution is reduced according to the modADEV [1,36], and with just one RLS, it falls below the clocks' instability already at 500 s.…”

“…Let us further assume, the purpose of this fiber link is the comparison of optical clocks with the combination of top-performance instability of OADEV(τ)≈5×10 -17 (τ/s) −1/2 [41] and a top-performance systematic uncertainty of ≈2×10 -18 [5,7] (blue curve in figure 6). As shown in figure 6, the OADEV of a single-span link would drop below the systematic uncertainty of such clocks after ≈3500 s. These long averaging times can be shortened by either extending Λ-averaging to longer times for a better fiber phase noise rejection [1,36] or by segmenting the link. Sub-dividing the link in two individually stabilized segments will enhance the locking bandwidth from 17 to 33 Hz, and, hence, the often observed prominent phase noise contribution around 20 Hz Figure 6.…”

“…For the cascade of up-and downlink the stabilization delay halves leading to the delay-limited phase noise of S D ssl (f). The corresponding Λ 1s instabilities are obtained via integration: the spectral weight associated with the counting process[32,36]. link (subscript ssl) to the cascade of individually stabilized uplink (subscript u) and downlink (subscript d) in the case of constant fiber noise: obtained [1] over a 320 ks long time series with an instability of the drive frequency servo acousto-optic modulator (AOM) of ( We regard this drive frequency as an estimate for the free-running fiber-induced noise as it is the major contribution.…”

On temporal correlations in high-resolution frequency counting

“…The RCON measurement mode is an undocumented, raw continuous frequency measurement mode. [4]. One such method is internal averaging over a given time interval.…”