NIST Technical Note 1337: Characterization of Clocks and Oscillators

More specifically let us consider a sample function of time as indicated infigure 1. A measurement consists of averaging yet) over the interval r. Thenext measurement begins at a time T after the beginning of the previousmeasurement interval. There is no logical reason why T must be as large as ror larger--if T < r, then the second measurement begins before the first iscompleted, which is unusual but possible. When T = r, there is no dead timebetween measurements.The accepted definition of the Allan variance is the expected value of a twosamplevariance with no dead time between successive measurements.symbols, the Allan variance is given byIn(2)where there is no dead time between the two sample averages for the Allanvariance and the E['] denotes the expectation operator.3. The Bias Function B 1(N,r,p)Define N to be the number of sample averages of yet) used in eq (1) toestimate a sample variance (N = 2 for an Allan variance). Also define r to bethe ratio of T to r (r = 1 when there is no dead time between measurements).The parameter ~ is related to the exponent of the power law of the PSD of theprocess yet). If a is the exponent in the power-law spectrum for yet), thenthe Allan variance varies as r raised to the ~ power, where a and ~ arerelated as shown in figure 2 [2-4]. We can use estimates of ~ to infer 0, thespectral type. The ambiguity in a for ~ = -2 has been resolved by using amodified a;(r) [5-7].Often data cannot be taken without dead time between sample averages, and itis useful to consider other than two-sample variances.bias function Bl(N,r,~) by the ratio,We will define thea 2 (N,T,r)a 2 (2,T,r)(3)4TN-299