AllanVarianz

AllanVarianz, known in English as the Allan variance, is a statistical measure used to quantify the stability of frequency sources such as atomic clocks, quartz oscillators, and frequency synthesizers. Named after David W. Allan, it was introduced in 1966 to provide a robust description of short- to medium-term frequency fluctuations that are not captured well by conventional variance.

Definition and computation: Let y_k be the average fractional frequency deviation measured over a sampling interval of length tau. The Allan variance is defined as sigma_y^2(tau) = (1/2) E[(y_{k+1} - y_k)^2], where E denotes the expectation. In practical data analysis with a finite record of N samples, the sample Allan variance is computed as sigma_y^2(tau) = (1/(2(M-1))) sum_{k=1}^{M-1} (y_{k+1} - y_k)^2, where M is the number of consecutive intervals. The Allan deviation is the square root, sigma_y(tau), which is a standard (dimensionless) measure of stability.

Variants and interpretation: Several variants exist, including the overlapping Allan variance, which uses overlapping data blocks to improve statistical confidence, and the Modified Allan variance, which offers enhanced discrimination among noise types. The Hadamard variance is another related measure. By examining how sigma_y(tau) changes with the averaging time tau on a log-log plot, different noise processes can be identified, such as white or flicker phase noise and various forms of frequency noise. The Allan variance is widely used to assess and compare the performance of clocks, oscillators, and frequency references in metrology, telecommunications, and precision instrumentation.