Relationships Between Drift Coefficient Uncertainties and Noise Levels: Application to Time Error Prediction

摘要

Oscillators are affected by drifts (linear phase drift, linear frequency drift, i.e. quadratic phase drift) and different types of noise according to the power law model of power spectral density (from f-2 to f+2 frequency noise, i.e. f-4 to f0 phase noise). Generally, for long-term instability characterization (duration greater than an hour), drift coefficients are estimated by using least squares whereas noise levels are obtained from the residuals by using variances (AVAR, MVAR, TVAR,...). However, the low frequency noises, such as random walk FM, induce very long term fluctuations which may be confused with deterministic drifts. This effect, due to the non-stationarity of these noises, depends on the low cut-off frequency which must be introduced in order to ensure power convergence for low frequencies. We calculate the standard deviation of 'artificial' drifts due to long-term random fluctuations, versus the noise levels. The first interest of these results concerns the estimation of the measurement uncertainty of drift coefficients: knowing the noise levels of an oscillator we calculate the standard deviation of the artificial drift coefficient due to these noises; thus, if a 'real' deterministic drift is identified in the signal, its coefficients are determined plus or minus the artificial drift coefficients. The standard deviation of the artificial drift coefficients may be considered as the measurement uncertainty of the deterministic drift coefficient. The second interest concerns the predictability of an oscillator affected by a deterministic drift. Thus, the knowledge of the drift coefficient uncertainties yields a criterion for quantifying the reliability of a time error prediction.