NIST Technical Note 1337: Characterization of Clocks and Oscillators

202 SAMUEL R. STEIN~1­!10 2 k.­t10- 1 ' I I I ","1 I 1 I II !III !! 11I!!11 1~ 1~ 1~NUMBER OF SAMPLES NFIG. 12-4 lli-sample varIance versus Allan vanance. The two-sample variance converges forthe important types of noise observed in frequency standards but the ratio of the traditionalvariance to the two-sample variance is an increasing function of sample size for flicker frequencynoise and random-walk frequenq noise.The dependence of the classical variance on the number of samples is shownin Fig. 12-4 for the case of no dead time. The quantity plotted is the ratio ofthe N-sample variance to the Allan variance. Note that O";(r) has the samevalue as the classical variance for the white-noise frequency modulation.However, the classical variance grows without bound for flicker-frequencyand random-walk-frequency noises.One may combine Eqs. (12-20) and (12-22) to obtain an equation for O"ir)in terms of the time-difference or time-deviation measurements:u;(r) =