Allan Variance Professor Hans Schuessler 689 Modern Atomic Physics Aysenur Bicer

History of Allan Variance (1966) D.W. Allan proposed M- sample variance..Determine the stability of atomic clocks. Describe the statistics of atomic frequency standards (1993) Werle et. Al detailed gave description of analysis of optical spectrometer

Figure 1.is a simulated plot of the time fluctuations, x(t) between a pair of oscillators and of the corresponding fractional frequencies calculated from the time fluctuations each averaged over a sample time t. At the bottom are the equations for the standard deviation (left) and for the time-domain measure of frequency stability as recommended by the IEEE (right) Fractional frequency If the time difference or the time fluctuations are available then the frequency or the fractional frequency fluctuations can be calculated from one period of sampling to the next over the data length

EXAMPLE : Find the two-sample (Allan) variance, s y 2 (t ), of the following sequence of fractional frequency fluctuation values y k, each value averaged over one second.

Allan Variance for CRDS Allan variance can be use to characterize the slow drift of a near-IR distributed feedback laser-based continuous wave cavity ring-down spectroscopy (CW-CRDS) system. In CRDS the accurate measurement of decay time constant can be limited by slow drift of setup. Allan variance can be used to analyze the stability of instrument. For a N time- series data xi the Allan variance is given by:

k is the subgroup size m+1 is the number of subgroups The integration time T equals to k/f, where f is sample rate. When white noise is dominant in the system (uncorrelated decays), Allan variance is proportional to 1/T and averaging data can improve the signal to noise ratio. When the drift appears Allan variance will become larger. The longest T during which the instrument can be regarded stable is determined by the drift of the system. The minimum of Allan variance gives the smallest detectable change during the longest integration time period.

Figure 2. Allan variance plot of the error in the absorbance of 13CH4 as a function of the number of ring-downs.

References IV. ANALYSIS OF TIME DOMAIN DATA Y. Chen,*,† Kevin.†K. Lehmann,‡ J. Kessler,§ B. Sherwood Lollar, ∇ G. Lacrampe Couloume, ∇ and T. C. Onstott Measurement of the 13C/12C of Atmospheric CH4 Using Near-Infrared (NIR) Cavity Ring-Down Spectroscopy Long-term stability in continuous wave cavity ringdown spectroscopy experiments Haifeng Huang and Kevin K. Lehmann*

How to calculate Allan Variance A time series of data is recorded successively and has N points: x i, i = 1...N The time interval between two successive data points is Δt. For an average size of data points of p, this N data series can be divided into m subgroups successively, with m equal to the largest integer less than or equal to N/p. For each subgroup, we have Each pair of successive A n (p) gives one sample of the Allan variance corresponding to the averaging size of p

The time averaged Allan variance of averaging size of p is defined as Evaluation of t requires ~ N-p/2+2N/p floating point operations ~ N(p max +2ϒ+2ln (p max +1))- p max (p max +1)/4 for p = 1...p max, with γ = γ the Euler–Mascheroni constant

If we assume the system is ergodic, this time averaged Allan variance t is equal to the ensemble averaged Allan variance When p= 1, the Allan variance is close to the short-term ensemble variance of the data series. If we have data without drift and uncorrelated Gaussian noise of variance σ 2, then the ensemble mean value for the Allan variance of length p is, e = σ 2 / p thus giving the straight line with a slope of −1 on a log-log plot. For calculation from a finite data series, there will be statistical fluctuations in the calculated values of t given by the variance

The slope of the Allan plot is −1 for p ≪ p min and +1 for p ≫ p min. The variance of t can be calculated in this case as

Figure.2 Allan plots of two algorithms. The modified algorithm (thinner curves) generates smoother Allan plots. For clarity, the Allan plots have been separated for (A) k1, (B) k2, and (C) k1 − k2