When Timing Uncertainty Limits Spectral Accuracy: Bias, Variance, and Frequency-Dependent Attenuation
Abstract
Uncertainty in sample timestamps is often summarized through a single root-mean-square jitter specification. However, its impact on spectral estimation depends on which timestamps are uncertain, the error distribution, and the statistics used to report spectral amplitude. This paper revisits the Fourier estimator evaluated at perturbed timestamps and derives an exact expression for its expected complex coe!cient in terms of the characteristic function of timing error. Under independent, identically distributed Gaussian dating errors, the coherent spectral contribution is attenuated by exp(→2ω 2 f 2 ε 2 t ), and an exact expression is obtained for the complex estimator variance conditional on the measured samples. These findings are extended to non-Gaussian and correlated errors. Numerical experiments corroborate both the coherent attenuation and variance formulas, while an illustrative common-o"set experiment establishes that attenuation of a complex ensemble average does not necessarily imply attenuation of the magnitude of individual estimates. Representative NI clock-jitter values are translated into nominal frequency thresholds, with careful separation between mathematical extrapolation and the physical bandwidth of a digitizer. The results provide practical uncertainty budgets and clarify the interpretation of spectral amplitude in nonuniform sampling.
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BibTeX
@misc{badaoui2026when,
title = {{When Timing Uncertainty Limits Spectral Accuracy: Bias, Variance, and Frequency-Dependent Attenuation}},
author = {Mohamed El Badaoui},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05845122v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - When Timing Uncertainty Limits Spectral Accuracy: Bias, Variance, and Frequency-Dependent Attenuation
AU - Mohamed El Badaoui
PY - 2026
DA - 2026/10/30
PB - HAL
UR - https://hal.science/hal-05845122v1
AB - Uncertainty in sample timestamps is often summarized through a single root-mean-square jitter specification. However, its impact on spectral estimation depends on which timestamps are uncertain, the error distribution, and the statistics used to report spectral amplitude. This paper revisits the Fourier estimator evaluated at perturbed timestamps and derives an exact expression for its expected complex coe!cient in terms of the characteristic function of timing error. Under independent, identically distributed Gaussian dating errors, the coherent spectral contribution is attenuated by exp(→2ω 2 f 2 ε 2 t ), and an exact expression is obtained for the complex estimator variance conditional on the measured samples. These findings are extended to non-Gaussian and correlated errors. Numerical experiments corroborate both the coherent attenuation and variance formulas, while an illustrative common-o"set experiment establishes that attenuation of a complex ensemble average does not necessarily imply attenuation of the magnitude of individual estimates. Representative NI clock-jitter values are translated into nominal frequency thresholds, with careful separation between mathematical extrapolation and the physical bandwidth of a digitizer. The results provide practical uncertainty budgets and clarify the interpretation of spectral amplitude in nonuniform sampling.
ER -