Monte Carlo analysis of gate-time-resolved uncertainty and oscillator noise in frequency measurements

Abstract

This study presents a gate-time-resolved evaluation of frequency-measurement uncertainty using the GUM analytical framework and large-scale Monte Carlo simulations. The experiment employed a time-interval counter (TIC) disciplined by a 10 MHz reference from a Cesium (Cs) clock to measure phase differences against a Rubidium (Rb) oscillator. Phase data were collected continuously over two days to analyse oscillator stability, and additional frequency datasets at 1 ms, 100 ms, and 1 s gate times, each spanning ten minutes, were examined to assess the effect of averaging time on measurement uncertainty. Uncertainty was evaluated using the GUM model and a Python-based parametric Monte Carlo simulation with one million iterations. A moving-block bootstrap (MBB) method was additionally applied to the same data to assess the influence of time-correlated noise. The results show that GUM slightly overestimated expanded uncertainty at short gate times (−8.38 % at 1 ms, −3.37 % at 100 ms, 0 % at 1 s). Kurtosis analysis revealed non-Gaussian behaviour at shorter averaging times, while Allan-variance analysis identified transitions between white, flicker, and drift noise regimes. These findings demonstrate that simulation-based approaches can capture temporal correlations more effectively, enabling realistic and automated uncertainty evaluation in time-and-frequency metrology under the Metrology 4.0 framework.