[time-nuts] TIC Characterization

[Magnus Danielson]

Hi Gilles,

On 12/29/18 11:28 PM, Club-Internet Clemgill wrote:
> Hi, 
> Looking to testing my HP53132A in TIC mode. 
> I considered the Time Interval measurement technique: 
> The start channel is connected to a 1 PPS signal, and to the stop channel though a coax cable (constant delay line).

Fair enough setup. This is a static test setup which works as long as
you do not lock the counter up to a 10 MHz of the same source as the
PPS, and for all maters not accurate enough, so it's best for the test
for it to be free-running. When you lock it up, you get a very static
behavior of the systematic noise of quantization resolution, and you
will be hitting essentially the same bin all the time, and well, you are
not that lucky on real-life signals since the phase relationship glides
ever so slightly that you want to make sure you do that. So, either you
use the time-base offset to cause the quantization of the counter glide
relative to the PPS reference or you use an offset oscillator for your
signal, both achieve the same goal. The difference lies in wither you
have both start and stop channels glide, as for internal reference
offset, or you have only the stop channel glide, as you do with an
offset oscillator but have time-base and start channel being
synchronous. The jitter for the later one is expected lower, because it
will have the start-channel banging the same bin more or less each time
since the time-base of the counter, steering the phase of the
quantization is synchronous to the start-channel, thus essentially
removing the noise of the start-channel.

While you get an ADEV slope of -1 and it looks like white phase
modulation noise, the counters resolution is a very systematic noise and
you should not forget that, rather, you can use this fact in your tests
to learn more about it. You will find that it is not perfectly linear
slope either, so for an average performance you want to average over the
full set of phase-relationships between time-base and start/stop channels.

> I found some references on the web, but no one with the associated maths.

The counter resolution and slope is somewhat of a white spot. It is
"known" but not very well researched. I did one presentation on it with
associated paper, but I need to redo that one because it does not
present it properly.

> So I tried the following :
> 
> 1/ AVAR  =  (1/2*Tau^2) * < [(Xi+2 - Xi+1) - (Xi+1 - Xi)]^2 >
> with (Xi+1 - Xi) = phase difference = time interval 
> 
> 2/ Phase difference = To + Ti 
> where To is the constant delay between start and stop (coax line)
> and Ti is the counter's resolution at time i
> 
> 3/ Assuming that Ti is a Central Gaussian distribution then:
> mean = < Ti > = 0 and variance = < Ti ^2> = SigmaTIC^2

It will not be completely true, but a dominant feature.
Turns out that the quantization staircase is a very systematic property,
but then offset by the white phase modulation and flicker phase
modulation that you can expect. However, the staircase quantization will
dominate for these short taus and it is only for longer taus you go into
the flicker part.

> 4/ [(Xi+2 - Xi+1) - (Xi+1 - Xi)]^2 = [(To + Ti+1) - (To + Ti)]^2 = (Ti+1 - Ti)^2 
> =  (Ti+1)^2 + (Ti)^2 - 2(Ti+1 * Ti)
> 
> 5/ <(Ti+1)^2> #  < (Ti)^2> for large samples and 
> <2(Ti+1 * Ti)> = 0 because Ti+1 and Ti are independent
> Then AVAR =  (1/2Tau^2) * 2< (Ti)^2>  = (1/Tau^2) * SigmaTIC^2
> 
> 6/ Hence ADEV = SigmaTIC / Tau
> 
> So ADEV (log log) is a straight line with -1 slope
> And ADEV(Tau=1) provides the standard deviation = SigmaTIC  of the Time Interval Counter's resolution 
> 
> Is this right ? 
> Thanks to point me at related articles or web pages if you know any.

You do indeed get an ADEV -1 slope for the counter quantization, I've
done essentially the same analysis.

I've then done a paper showing how noise and quantization interacts and
somewhat shifts this around in, ehm, interesting ways. Unfortunately the
paper as presented was not all that good, but I should do work on that,
because there is some further insights to present more thoroughly as
well as making the real point go through better.

I have only seen an Agilent app-note which addresses some of this, but
then with the focus on frequency measurements. Others seems to have
treated the subject as a fact of life and moved on.

So, thank you for reminding me about this property, it is indeed
somewhat of a white spot.

Cheers,
Magnus