NIST Technical Note 1337: Characterization of Clocks and Oscillators

204 SAMUEL R. STEINTABLE 12-1Correspondence between Common Power-Law Spectral Densities and theAllan Variance"Noise type SJIl a:(rj\\'bite phase hd 2 3f.h 2 /(211)2 r2flicker phasehI![1.038 + 31n1211:1. r)] 1h,­(211)2 .2White frequency ho !holl/.)flicker frequency h.d-' 2In(2/h_ 1Random-walk frequency h. 2 F 2 !121t)2h_ 2r• Where necessary for convergence the spectral density has been assumed tobe zero for frequencies greater than the cutotrfrequency f •.This technique is most valuable when only a few terms in Eq. (12-26) arerequired to describe the observed noise and each term dominates over severaldecades of frequency. This situation often prevails. Five power-law noiseprocesses (Allan, 1966: Vessot et aI., 1966) are common with precisionoscillators:(1) random-walk frequency modulation :x = - 2(2) flicker frequency modulation ex = -1(3) white frequency modulation 0:=0(4) flicker phase modulation 11: = 1(5) white phase modulation 0:=2The spectral density of frequency is an unambiguous description of theoscillator noise. Thus, the spectrum can be used to compute the Allanvariance (Barnes er ai., 1971):O";('r) = ~ r:r: S~(j) sin4.(rrfr) df (12-27)(rrvort J o *However, Eq. (12-27) shows that the Allan variance is very sensitive to thehigh frequency dependence of the spectral density of phase, thereby necessitatinga detailed knowledge of the bandwidth-limiting elements in themeasurement setup. The integral has been computed for each of the powerlawnoise processes, and the results are summarized in Table 11-1 (Barnes etaI., 1971). For:x in the range -2 ::::; ex ::::; 0, the Allan variance is proportionalto Til, where f.i = - ex - 1. When the log of the Allan variance is plotted as afunction of the log of the averaging time, the graph also consists of straightlinesegments with integer slopes. However, Table 12-1 also shows that even if.. See Appendix Note 1/ 8TN-74