NIST Technical Note 1337: Characterization of Clocks and Oscillators

12 FREQUENCY AND TIME MEASUREMENT197wherex(r) = ¢(r)/2rrvo (12-10)is the phase expressed in units of time. Alternatively, the phase could bewritten as the integral of the frequency of the oscillator:¢(r) = ¢o + {2ir[V(e) - vo] de. (12-11)However, the integral of a stationary process is generally not stationary.Thus, indiscriminate use of Eqs. (12-7) and (12-11) may violate the assumptionsof the statistical model. This contradiction is avoided when oneaccounts for the finite bandwidth of the measurement process. Although amore detailed consideration of the statistics goes beyond the scope of thistreatment, it is very important to keep in mind the assumption that lie behindthe statistical analysis of oscillators. In order to analyze the behavior of realoscillators, it is necessary to adopt a model of their performance. The modelmust be consistent with observations of the device being simulated. To makeit easier to estimate the device parameters, the models usually include certainpredictable features of the oscillator performance, such as a linear frequencydrift. A statistical analysis is useful in estimating such parameters to removetheir effect from the data. It is just these procedures for estimating thedeterministic model parameters that have proved to be the most intractable.A substantial fraction of the total noise power often occurs at Fourierfrequencies whose periods are of the same order as the data length or longer.Thus, the process of estimating parameters may bias the noise residuals byreducing the noise power at low Fourier frequencies. A general technique forminimizing this problem in the case of oscillators actually observed in thelaboratory is discussed below.It has been suggested that measurement techniques for frequency and timeconstitute a hierarchy (Allan and Daams, 1975), with the measurement of thetotal phase of the oscillator at the peak. Although more difficult to measurewith high precision than other quantities, the total phase has this statusowing to the fact that all other quantities can be derived from it.Furthermore, missing measurements produce the least deleterious effect on atime series consisting of samples of the total phase. Gaps in the data affect thecomputation of various time-dependent quantities for times equal to orshorter than the gap length, but have a negligible effect for times much longerthan the gap length. The lower levels of the hierarchy consist of the timeinterval, frequency. and frequency fluctuation. When one measures a quantitysomewhere in this hierarchy and wishes to obtain a higher quantity, it isnecessary to integrate one or more times. In this case the problem of missingdata is quite serious. For example, if frequency is measured and one wants to1N-67