NIST Technical Note 1337: Characterization of Clocks and Oscillators

From: Proceedings of the 15th Annual PITI Meeting, 1983.THE MEASUREMENT OF LINEAR FREQUENCY DRIFT IN OSCILLATORSJames A. BarnesAcstron, Inc., Austin, TexasABSTRACTA linear drift in frequency is an important element in moststochastic models of oscillator performance. Quartz crystaloscillators often have drifts in excess of a part in ten tothe tenth power per day. Even commercial cesium beam devicesoften show drifts of a few parts in ten to the thirteenth peryear. There are many ways to estimate the drift rates fromdata samples (e.g., regress the phase on a quadratic; regressthe frequency on a linear; compute the simple mean of thefirst difference of frequency; use Kalman filters with adrift term as one element in the state vector; and others).Although most of these estimators are unbiased, they vary inefficiency (i.e., confidence intervals). Further, the estimationof confidence intervals using the standard analysisof variance (typically associated with the specific estimationtechnique) can give amazingly optimistic results. Thesource of these problems is not an error in, say, the regressionstechniques, but rather the problems arise fromcorrelations within the residuals. That is, the oscillatormodel is often not consistent with constraints on the analysistechnique or, in other words, some specific analysistechniques are often inappropriate for the task at hand.The appropriateness of a specific analysis technique is criticallydependent on the oscillator model and can often bechecked with a simple "whiteness" test on the residuals.Following a brief review of linear regression techniques,the paper prOVides guidelines for appropriate drift estimationfor various oscillator models, including estimation ofrealistic confidence intervals for the drift.I. INTRODUCTIONAlmost all oscillators display a superposition of random and deterministicvariations in frequency and phase. The most typical model used is[1]:X(t)(1)*where X(t) is the time (phase) error of the oscillator (or clock) relati~e tosome standard; a, b. and Dr are constants for the particular clock; and ~(t) isthe random part. X(t) is a random variable by virtue of its dependence on ~(t).* See Appendix Note # 36551IN-264