[time-nuts] Frequency standards for different tau in Allen Dev measurement

[Magnus Danielson]

Hi Taka,

On 2020-02-20 19:40, Taka Kamiya via time-nuts wrote:
> I have a question concerning frequency standard and their Allen deviation.  (to measure Allen Dev in frequency mode using TimeLab)
>
> It is commonly said that for shorter tau measurement, I'd need OCXO because it's short tau jitter is superior to just about anything else.  Also, it is said that for longer tau measurement, I'd need something like Rb or Cs which has superior stability over longer term.
Seems reasonably correct.
> Here's the question part.  A frequency counter that measures DUT basically puts out a reading every second during the measurement.  When TimeLab is well into 1000s or so, it is still reading every second; it does not change the gate time to say, 1000s.
> That being the case, why this consensus of what time source to use for what tau?
> I recall reading on TICC, in time interval mode, anything that's reasonably good is good enough.  I'm aware TI mode and Freq mode is entirely different, but it is the same in fact that measurement is made for very short time span AT A TIME.
> I'm still trying to wrap my small head around this.  

OK.

I can understand that this is confusing. You are not alone being
confused about it, so don't worry.

As you measure frequency, you "count" a number of cycles over some time,
hence the name frequency counter. The number of periods (sometimes
called events) over the observation time (also known as time-base or
tau) can be used to estimate frequency like this:

f = events / time

while it is practical that average period time becomes

t = time / events

In modern counters (that is starting from early 70thies) we can
interpolate time to achieve better time-resolution for the integer
number of events.

This is all nice and dandy, but now consider that the start and stop
events is rather represented by time-stamps in some clock x, such that
for the measurements we have

time = x_stop - x_start

This does not really change anything for the measurements, but it helps
to bridge over to the measurement of Allan deviation for multiple tau.
It turns out that trying to build a standard deviation for the estimated
frequency becomes hard, so that is why a more indirect method had to be
applied, but the Allan deviation fills the role of the standard
deviation for the frequency estimation of two phase-samples being the
time-base time tau inbetween. As we now combine the counters noise-floor
with that of the reference, the Allan deviation plots provide a slopes
of different directions due to different noises. At the lowest point on
the curve, is where the least deviation of frequency measurement occurs.
Due to the characteristics of a crystal oscillator to that of the
rubidium, cesium or hydrogen maser, the lowest point occurs at different
taus, and provide different values. Lowest value is better, so there is
where I should select the time-base for my frequency measurement. So,
this may be at 10 s, 100 s or 1000 s, which means that the frequency
measurement should be using start and stop measurements with that
distance. OK, fine. So what about TimeLab in all this. Well, as we
measure with a TIC we collect a bunch of phase-samples at some base
rate, such as 10 Hz or whatever. TimeLab and other tools can then use
this to calculate Allan Deviation for a number of different taus simply
by using three samples, these being tau in between and algoritmically do
that for different taus. One then collects a number of such measurements
to form an average, the more, the better confidence interval we can but
on the Allan Deviation estimation, but it does not improve our frequency
estimation, just our estimation of uncertainty for that frequency
estimation for that tau. Once you have that Allan Deviation plot, you
can establish the lowest point and then only need two phase samples to
estimate frequency.

So, the measurement per second thing is more collection of data rather
than frequency estimation in itself.

Cheers,
Magnus