NIST Technical Note 1337: Characterization of Clocks and Oscillators

Since the time-domain definition for frequency stability is the Allanvariance, it behooves us, where possible, to relate other variances to theAllan variance. If we have an N-sample variance on data with dead-time T-rand we know the power-law spectral type (the value of ~), then we may writea; (r) (15)If we have an N-sample variance where each data entry is an average of Msamples with distributed dead time, then we may write(16)8. ConclusionFor some important power-law spectral density models often used incharacterizing precision oscillators (Sy(f) - f Q , Q = -2, -1, 0, +1, +2), wehave studied the effects on variances when there is dead time between thefrequency samples, and the frequency samples are averaged to increase theintegration time. Since dead time between measurements is a common problemthroughout metrology, the analysis here has broader applicability than just totime and frequency. Specifically, this kind of analysis has been used withgage blocks and standard volt cells--showing that the classical variance maybe non-convergent in some cases [9].Heretofore, the Allan variance has been shown to have some convenienttheoretical properties in relation to power-law spectra as the integration orsample time is varied (if Gy 2 (r) - r~, then Q = -~ -1, -2 < ~ 5 2). Sinceoy(r), by definition, is estimated from data with no dead time, the sample orintegration time can be unambiguously changed to investigate the r dependence.From our analysis, we have concluded that for the asymptotic limit of severalsamples being averaged with dead time present in the data, the r dependence ofthe variances is the same. The Q = -~ -1 relationship still remains valid forwhite FM noise (~= -1, Q = 0), flicker FM noise (~= 0, Q = -1), and for9TN-304