Allan Variance - Background

Background

When investigating the stability of crystal oscillators and atomic clocks it was found that they did not have a phase noise consisting only of white noise, but also of white frequency noise and flicker frequency noise. These noise forms become a challenge for traditional statistical tools such as standard deviation as the estimator will not converge. The noise is thus said to be divergent. Early efforts in analysing the stability included both theoretical analysis and practical measurements.

An important side-consequence of having these types of noise was that, since the various methods of measurements did not agree with each other, the key aspect of repeatability of a measurement could not be achieved. This limits the possibility to compare sources and make meaningful specifications to require from suppliers. Essentially all forms of scientific and commercial uses were then limited to dedicated measurements which hopefully would capture the need for that application.

To address these problems, David Allan introduced the M-sample variance and (indirectly) the two-sample variance. While the two-sample variance did not completely allow all types of noise to be distinguished, it provided a means to meaningfully separate many noise-forms for time-series of phase or frequency measurements between two or more oscillators. Allan provided a method to convert between any M-sample variance to any N-sample variance via the common 2-sample variance, thus making all M-sample variances comparable. The conversion mechanism also proved that M-sample variance does not converge for large M, thus making them less useful. IEEE later identified the 2-sample variance as the preferred measure.

An early concern was related to time and frequency measurement instruments which had a dead time between measurements. Such a series of measurements did not form a continuous observation of the signal and thus introduced a systematic bias into the measurement. Great care was spent in estimating these biases. The introduction of zero dead time counters removed the need, but the bias analysis tools have proved useful.

Another early aspect of concern was related to how the bandwidth of the measurement instrument would influence the measurement, such that it needed to be noted. It was later found that by algorithmically changing the observation, only low values would be affected while higher values would be unaffected. The change of is done by letting it be an integer multiple of the measurement timebase .

The physics of crystal oscillators was analyzed by D. B. Leeson and the result is now referred to as Leeson's equation. The feedback in the oscillator will make the white noise and flicker noise of the feedback amplifier and crystal become the power-law noises of white frequency noise and flicker frequency noise respectively. These noise forms have the effect that the standard variance estimator does not converge when processing time error samples. This mechanics of the feedback oscillators was unknown when the work on oscillator stability started but was presented by Leeson at the same time as the statistical tools was made available by David W. Allan. For a more thorough presentation on the Leeson effect see modern phase noise literature.