Clock Performance and Performance Measures.


The quality of a clock need not be dependent upon its error or its rate, because these can be corrected for with calibration. It is the rate variations from interval to interval (usually the standard interval is a day) which determine the quality. If these variations are irregular then the clock's behavior can only be described statistically. If the rate changes systematically, i.e., if it increases by nearly the same amount every day then we talk about a drift of this clock. Quartz crystal clocks and (much less so) rubidium vapor cells typically have such a drift. Cesium clocks, unless there is something wrong with them, or they have been mis-adjusted, show no drift.

Regarding the performance of pocket or wrist crystal watches, the most important disturbance comes from the temperature variations to which the watch is exposed. As a rule of thumb, crystals have a temperature coefficient of about 1 part per million. That amounts to a rate change of 0.1s per day per degree temperature change. Some watches have an adjustment for the cooling when you are sleeping and your watch is presumably not being worn. Others have an adjustment for the slightly different heat exposures expected by wearers of "male-styles" and "female-styles".

There are many statistical ways to measure clock stability, but the most widely used is the Allan Variance, more prosaically known as two sample, zero dead-time variance. It is computed as follows: first compute the second differences and divide them by tau, the interval between the measurments you use. Form the squares of the second differences as divided by tau, add them, divide by 2 times the number of terms and that is the Allan variance. The Allan deviation is the square root of the Allan variance. You can compute the Allan variance for any multiple of tau, say N times tau, by using only every Nth point. Such stability measures are also often expressed in relative terms, i.e., as parts per million, etc. You can convert between these two styles by remembering that one day has 86400 s. Therefore rate change of 1 ms per day is equivalent to (1.0E-3) / 8.64E4 = 1.157 parts in 10 to the eight.

All measurements of clock performance, or clock stability, start with a set of regularly executed measurements of the clock correction, spaced by the interval tau. As an example, this table shows sample crystal clock measurements. The columns are the time of measurement expressed as the day number, the measured clock error against the USNO Master Clock, the first and second differences (which are divided by tau so that the units are ms/day), and the square of the second differences divided by tau:

 
 n  Clock Error   First Diff./tau  Second Diff./tau   Square(2nd diff/tau)
           ms          ms/d           ms/d
--------------------------------------------------------------- 
  0       325           0     
  1       350          25 
  2       377          27              2               4
  3       401          24             -3               9
  4       430          29              5              25
  5       461          31              2               4
  6       494          33              2               4
  7       529          35              2               4
  8       566          37              2               4
  9       601          35             -2               4
 10       636          35              0               0
 11       673          37              2               4
 12       710          37              0               0
 13       749          39              2               4
 14       790          41              2               4
 15       835          45              4              16
  
etc.
Our example clock would be a very good crystal clock because the rate variations as shown in the 4th column are small. Nevertheless, the rate shows a systematic increase of 20ms in 14 days, i.e., the clock has a noticeable drift. This drift is also visible as the average second difference (sum = 20, divide by 14 ---> average drift = 20/14 = 1.43ms/d/d).

The math for the Allan deviation gives 86 / 28 = 3.07, and the square root finally gives 1.75ms/day as the measure of stability. Converting to dimensionless units, our test clock, exhibits a frequency instability of 2.02 parts in 10 to the eight from day to day.

Remember: A clock error is given in units of time (s, ms, ns), whereas a rate difference is given as a relative number or as ms/day, ns/day, etc.