DMTD Beware Counter ADEV when measuring phase

[jpb]

I've just got round to writing code to properly measure the output of my DMTD system and was puzzled (and somewhat disappointed) to find that the manually measured ADEV (of approximately 200 nsecs equivalent to 2 x 10^-12 at 0.1 secs) was coming out at ~3.5 to 3.9 x 10^{^-12} when I fed the data into STABLE32!

Having dug around in the manuals and done a bit of thinking I worked out the solution - the Counter (FCA3100) ADEV is only valid for frequency measurements not time measurements.

The counter uses the formula (ignoring the divide by 2 * (N-1))

sqrt(Sum (x_i+1-x_i)²)

STABLE32 uses the correct formula

sqrt(Sum ((x_i+1-x_i) - (x_i-x_i-1))²)
or
sqrt(Sum(x_i+1 - 2*x_i + x_i-1)²)

Assuming the noise is IID and that we sum enough terms that any cross-correlation goes to zero (zero mean) with a sd of sigma for the x_i² type terms then we only need to consider these square terms.
The counter formula amounts to

ADEV = sqrt(sigma² + sigma²) = sqrt(2) * sigma

while the STABLE32 formula comes to

ADEV = sqrt(sigma² + 2²*sigma² + sigma²) = sqrt(6)*sigma

and the ratio is sqrt(6/2) = sqrt(3) = 1.7 (roughly)

and this is what I was observing (approximately in my practical measurements.

So beware of just taking the ADEV output of the counter at face value - it will be too optimistic for time measurements.