NIST Technical Note 1337: Characterization of Clocks and Oscillators

12 FREQUENCY AND TIME MEASUREMENT207This graph allows one to determine power-law spectra for noninteger as wellas integer values of a. In the asymtotic limit the equation relating p and a forthe modified Allan variance isa = -p - 1 for -3 < 7. < 3. (12-30)12.1.6 Determination of the Mean Frequency and Frequency Driftof an OscillatorBefore the techniques of the previous four sections can be meaningfullyapplied to practical measurements, it is necessary to separate the deterministicand random components of the time deviation x(t). Suppose, forexample, that an oscillator has significant drift, such as might be the case for aquartz crystal oscillator. With no additional signal processing, the Allanvariance would be proportional to ,2. The variance of the Allan variancewould be very small, further demonstrating that deterministic behavior hasbeen improperly described in statistical terms and the oscillator's predictabilityis much better than the Allan variance indicated. Unfortunately, itis difficult to estimate the oscillator's deterministic behavior without introducinga bias in the noise at Fourier frequencies comparable to the inverseof the record length. In practice, it has been sufficient to consider twodeterministic terms in x(c):(12-31)The first term on the right-hand side is the synchronization error. The secondterm is due to imperfect knowledge of the mean frequency and is sometimescalled syntonization error. The quadratic term, which results from frequencydrift, is the most difficult problem for the statistical analysis because the Allanvariance is insensitive to both synchronization and syntonization errors.For white noise, the optimum estimate of the process is the mean.Therefore, a general statistical procedure that can be followed is to filter thedata until the residuals are white (Allan et aI., 1974: Barnes and Allan, 1966).For example. at shoft times the frequency fluctuations of atomic clocks areusually white. Taking the first difference of Eq. (12-31), we find that(12-32)and a linear least square fit to the frequency data yields the optimum estimateof .1\'. However, the drift in atomic clocks is generally so small that the valueobtained for D will not be statistically significant when r is small enough toTN-77