NIST Technical Note 1337: Characterization of Clocks and Oscillators

The "Allan Variance" 1s defined as:U (1) - a(a) X[12]'{~hl-a+l- It\ -a+l] (10)(5)where the brackets "< >" denote infinite timeaverage. Using equation (4), one may write:It has been shown that typically 0 2(1)Y 1 2varies as 1 1l , and that 1.1 = -a-1 for -3 ~ a ~ +1. 'Hence, we see one of the dimensions of usefulnessof 0y2(t); 1.e., ascertaini ng the dependence on tallows an estimate of a (the power law spectraltype of noise). However, if a ~ +1, then 1.1 =-2,and the 1 dependence becomes somewhat ambiguous asto the type of noise in this region.It is interestingto note that in the region a ~ +1, cry2(1)is bandwi dth (fh) dependent; 1. e., the bandwi dthof the measurement system will affect the value ofo (1), and furthermore, one may use the bandwidthy 3dependence to determine the value of a (see alsoAppendix Ref. Z).Development of the Modified Allan VarianceOne may also write Oy2(1) in terms of ageneralized autocovariance function:whereand wherethe classical autocovariance function of x(t).(7)(8)(9)Using the Fourier transforms of general ized functions,one may determine the coefficients relatingthe power spectral density to o/(t). Ref. 1gives these relationships.It is of interest tonote that U (1) has the following approximate formxin the region a ~ + 1 (see Appendi.: R.. f. 2):Hence, one notes that by changing the reciprocalbandwidth as well as 1, one affects o/(t) insimilar ways, depending on the value of (Y, Fromthis, one should be able to deduce the value of a,since the bandwith dependence becomes stronger fora moving positive from +1, and the 1 dependencebecomes stronger as a moves negative from +1.Onecan change the bandwidth in the hardware or in thesoftware. In the past, it has typically been donein the hardware. 3 James Snyder 4 has shown that itis relatively easy to change the bandwidth in thedata processing by a clever technique and we havefollowed his lead.a newIn particular, we have chosenvariance analysis scheme which coincideswith the Allan variance at the minimum sampletime, 1 , (i.e., minimum data spacing), but whichochanges the bandwidth in the software as thesample time, 1, is changed.Each reading of the time deviation, xi' hasassociated with it an intrinsic nominal (hardware)measurement system bandwi dth, f h' Defi net = _1 . and similarly we may define a softwareh 2nf'bandwidt~, f = fhln, which is lin times narrowersthan the hardware bandwidth. This software bandwidth can be real i zed by averagi ng n adjacentx.'s·, 's1 = n1 where 1 =l/f. We have definedh' s sa modified Allan variance which allows the reciprocalsoftware bandwidth to change linearly with theJsamp 1e time, 1:where 1 = nl 'oG for n = 1.have formed2x . + x.) ] ~ (11)1+n 1 JEq. 11 clearly coincides with Eq.One can see that, in general, wea second difference of three timereadi ngs with each of the three be i ng an average,of n of the x.'s (with non-overlapping averages).As n increases, the (software) bandwidth decreasesand this bandwidth varies just as f s: fh/n.For a fi ni te data set of N I readi ngs of xi(i ~ 1 to N), we may write an estimate:471IN-255