NIST Technical Note 1337: Characterization of Clocks and Oscillators

ALLAN: PRECISION CLOCKS AND OSCILLATORS651TABLE II'ClassicalClassicalTypical Noise Types Standard Standarda Name Deviation of x Deviation of y2 wllite-noise PM Q, ~-J' 7' ",( r)/../3 (constant) ",( 7) .J2(N + I )/3N1 fl icker-noise PM QI'1'- 2 -7' ",(7) Jln M/ln 2 - U,( 7) J2(N + 1/3No wllite-noise FM dor- I 70' 17.(70) J(M + 1)/6 ",( ~o)-I flicker-noise FM 0_1 T° undefined u,( ~) ,;r Ṅ :-:I- n -:-N:-:/(C":2:-("""N:----:-':-)=-In-:::2:-)- 2 random-walk FM 0_2'1' undefined u,(T) .JNj2• Note T is a general averaging time and ~o is tile initial averaging time (T = n7 0 • where n is an integer).Also note tllat tile last four entries in the founh column and the last two entries in lhe: fifth column go toinfinity as M or N go to infinity. M is the initial number of frequency difference measurements and N thenumber of phase or time difference measurements N = M + 1. If the spectral density is given by S,( f)'" haf a • thenG, = {2~ Y3Moh?G, = (2~Y (1.038 + 3 log, (hf.T) hi)a_I = 2 log. (2) h_1G_, = ;:;I (1111") 1 h_ 1•b Note this equality assumes use of modified u;(7)= a; (T).measure the phase difference or the time difference betweena pair of oscillators or clocks and we define Th =11k In other words. Th is the sample time period throughwhich the time or phase date are observed or aver:aged.Averaging n time or phase readings increases the sampletime window to nTh == T r Let 1', = 1I fr; then fr = fhln.Le., the software bandwidth is narrowed to fr. In otherwords, fr == fi,1 n decreases as we average more values;i.e., increase n (1' = n'To). One can therefore construct asecond difference composed of time deviations so-averagedand then define a modified a;' (T) == a; (T) that willremove the ambiguity through bandwidth variation:N-3n + I~j=l( 12)where N = M + 1, the number of time-deviation measurementsavailable from the data set. Now if a; (1') ­7/'0', then p.' == -ex - 1 (I ~ ex ~ 3) [10], [11]. ThusOy ( T) is typically employed as a subroutine to remove theambiguity if a ( T) - T- 1 . This is because the 11' == -exv- 1 relationship is valid as an asymptotic limit for largen and ex < 1 and is not valid in general; however, thereis evidence that U;(T) may be a better measure [12]. Specifically,for ex == 2 and I, 11' /2 equals - 3/2 and -1,respectively, providing a clean differentiation betweenwhite-noise PM and flicker-noise PM.If three or more independent oscillators or clocks areavailable along with time (phase) or frequency measure­'" See Appendix Note # 16*ments between them, then it is possible to estimate a variancefor each oscillator or clock. Often there is a referenceto which the rest are periodically measured at asampling rate 1I TO' If at each measurement the time orfrequency differences between the clocks are measured atnominally the same lime, then the time difference or frequencydifference can usually be estimated or calculatedbetween every possible pair in the set of oscillators orclocks. Given a series of measurements, variances sl canbe calculated on the time or frequency data between allpairs. It has been shown [13] that the individual clockvariances can be estimated using the following equations:wherea~ == _I- ( Esb - B)m - 2 j= I1 mB =-- ~ s2.,m - I i