nominal angular frequency (in radians per second) is given by The total phase can be separated into a perfectly cyclic component , along with a fluctuating component : Time error The time-error function x(t) is the difference between expected nominal time and actual normal time: For measured values a time-error series TE(t) is defined from the reference time function T(t) as Frequency function The frequency function is the frequency over time, defined as Fractional frequency The fractional frequency y(t) is the normalized difference between the frequency and the nominal frequency : Average fractional frequency The average fractional frequency is defined as where the average is taken over observation time τ, the y(t) is the fractional-frequency error at time t, and τ is the observation time. Since y(t) is the derivative of x(t), we can without loss of generality rewrite it as Estimators This definition is based on the statistical expected value, integrating over infinite time. The real-world situation does not allow for such time-series, in which case a statistical estimator needs to be used in its place. A number of different estimators will be presented and discussed. Conventions Fixed τ estimators A first simple estimator would be to directly translate the definition into or for the time series: These formulas, however, only provide the calculation for the τ = τ0 case. To calculate for a different value of τ, a new time-series needs to be provided. Non-overlapped variable τ estimators Taking the time-series and skipping past n − 1 samples, a new (shorter) time-series would occur with τ0 as the time between the adjacent samples, for which the Allan variance could be calculated with the simple estimators. These could be modified to introduce the new variable n such that no new time-series would have to be generated, but rather the original time series could be reused for various values of n. The estimators become with , and for the time series: with . These estimators have a significant drawback in that they will drop a significant amount of sample data, as only 1/n of the available samples is being used. Overlapped variable τ estimators A technique presented by J. J. Snyder provided an improved tool, as measurements were overlapped in n overlapped series out of the original series. The overlapping Allan variance estimator was introduced by Howe, Allan and Barnes. This can be shown to be equivalent to averaging the