NIST Technical Note 1337: Characterization of Clocks and Oscillators

206 SAMUEL R. STEINTABLE 12-2Correspondencc between Common Power-law Spectral Dcnsitics and thc Modified AllanVariancc'Noise typeWhite phasenFlicker phase[1.038 + 3 In(2,q.rl] Ih, ::-----:-'--=-=-.:.::.(21t)' r 1White frequency h o .4r 0.5Flicker frequency h_ 1 (O.936) 0.674Random-walk frequency L 1(5.42)r 0.824• Where necessary the spectral density has been assumed to be zero for frequencies greater thanthe cutoff frequency f. Thc constant n i~ the number of adjacent phase values that are averagedto produce the bandwidth reduction. The values in the last two columns arc for thc asymptoticlimit n - x.. In practice. n onIy needs to be 10 or larger before the asymptotic limit is approachedwithin a few percent. When n = I the ratio in the last column is I in all cases.phase modulation and flicker phase modulation, but for :I ~ I the conventionalAllan variance gives both an easier-to-interpret and an easier-tocalculatemeasure of stability.It is interesting to make a graph of:t versus J.1 for both the ordinary Allanvariance and the modified Allan variance, such as the one shown in Fig. 12-5.al'2'- "I "i ,1~JI-+-2-3-4 -3 -2 -, 0f-LFIG. 12-5 Relationship between a power-law spectral density whose slope on a log-log plotis " and the corresponding sample variance whose slope on a log-log plot is jJ.. The solid linedescribes the behavior of the Allan variance, while the dashed line shows the advantag.e of themodified Allan variance for white phase noise and nicker phase noise.TN-76