For me clarity about Allan Variance was greatly increased when RoGeorge posted his explanations (
here and
here) of the conceptual reasoning behind the increase of traditional variance as tau increases. I have thought about this and think I can supply a more mathematically inclined argument for this property. The concepts behind the argument are only roughly accurate, as should become apparent during their presentation, but it does have the advantage of a bit more rigour than RoGeorge's metaphorical description. It also supports a result that I have not yet seen mentioned.
Before presenting the mathematics, it is prudent to make sure everyone is on the same page in regards to terminology. I have read a lot of papers about Allan Variance and have noted the terminology is not quite consistent.
Figure 1 shows a sine wave generated by an oscillator. Periods of this sine wave are measured from one rising edge to the next. For the purposes of this presentation, at each rising edge the preceding period is determined to be either longer or shorter than the nominal period of an ideal oscillator. So, period 1 may be shorter than the nominal period (and hence averaged over the period, the oscillator has a greater frequency than the nominal ideal frequency). Or period 1 may be longer than the nominal period (and therefore when averaged over the period, the oscillator has a smaller frequency than the ideal oscillator). While it would be more accurate to include a third possibility, i.e., period 1 has the same frequency as the nominal oscillator, that would overly complicate the model and would not provide any significant clarity to the narrative.
Figure 1 -
Figure 2 shows mulitple periods of the oscillator. An averaging interval defined by the value tau measures whether the oscillator is of higher or lower frequency than a nominal ideal oscillator. For the purposes of the following mathematics, tau is always a multiple of the nominal ideal oscillator period. The averaging measurement (conceptually) occurs by noting the "polarity" of each oscillator period. Here "polarity" means whether for that period the oscillator was of greater frequency (G) or lower frequency (L) than the nominal ideal oscillator. There are m oscillator periods in each averaging interval.
Figure 2 -
Figure 3 shows how each averaging interval is utilized to create a total measurement. In particular n intervals of length tau are statistically analyzed to produce the measurement. The sample time (ST) is the value tau times n.
Figure 3 -
Consider the situation in Figure 1. Each period produces a result - either G or L. These results are analyzed over the averaging interval. If the probability of obtaining G is p, then the probability of obtaining a
n L is 1-p. For simplicity it is assumed that p=1-p=.5.
For measuring oscillator stability the statistic of interest is not how many Gs or Ls appear in an averaging interval, but the difference between these values. The process represented by an averaging interval is well-known and is called a bernoulli trial. The expected value of the difference between the number of Gs and Ls is presented
here, specifically: 2mp - m = m(2p-1) = 0. [Note: the referenced web page uses n as the number of trials, whereas here that value is m. The value n is used here to represent the number of averaging intervals. Also, the problem solved there is stated in terms of successes and failures. The logic is exactly the same. Simply substitute L for success and G for failure.]
The variance of the difference between the two random variables in a Bernoulli trial (see above reference) is: 4mp
2 = m. Notice (!) that the variance depends on m. So, as the value of tau increases, so does the variance.
This has an interesting side-effect. The sample time equals tau * n. So, the traditional variance does not diverge as sample time increases. It diverges as the averaging time increases. Given the capabilities of computers in the 1960s and 1970s, when Allan Variance was developed, it was necessary to increase tau in order to obtain long-term measures of clock stability. Today, computers are much more powerful. So, it would be interesting to determine the sample_time/tau ratio above which an analyst would be forced to increase tau in order to obtain practical clock evaluation results. This would, of course, depend on the computer available. However, I would guess most desktop systems these days could analyze a very long data set in a practical amount of time.