NIST Technical Note 1337: Characterization of Clocks and Oscillators

12 FREQUENCY AND TIME MEASUREMENT217l/)WWO::;:Ec:lW 0Ow oU-w00::0::U­WU­COO5z100~10­...~ ...'\~"'-""~""" "'~"" ,\""'.'-\".'\ .~' ....\' :'" \'.\::..1;-1------'10-:-----=>"'--,-!100T (IN UNITS OF DATA SPACING)FIG.12-15 Number ofdegrees offreedom as a function ofaveraging time for the case of 101phase measurements: The heavy broken line is for random-walk frequency noise, the lightbroken line is for flicker frequency noise, the dotted line is for white frequency noise. the heavysolid line is for flicker phase noise, and the light solid line is for white phase noise,An expression for each individual variance is obtained by adding two jointvariances and subtracting the third:Uf = !(ut + ui - a]k)'uf = !(O}k + u~ - a!k), (12-44)uf = t(U7k + Urk - 0'5)·This method works best if the three devices are comparable in performance.Caution must be exercised since Eqs. (12-44) may give a negative sampleAllan variance despite the fact that the true Allan variance is positive definite,This is possible because the confidence interval of the estimate is sufficientlylarge to include negative variances. Such a result is an indication that theconfidence intervals of the sample Allan variances are too large and thatmore data is required.12,2 DIRECT DIGITAL MEASUREMEJ\T12.2.1 Time-Interval MeasurementsA common technique for measuring the phase difference between oscillatorshaving nearly equal nominal frequencies is the use of direct timeintervalmeasurements. In this section and those that follow, the symbols Vtaand Vl0 are used to indicate the nominal values of \'1 and v 2 , respectively. Inthe simplest form of this technique, a time-interval counter is started on someIN-87