[time-nuts] Re: Testing frequency pulling on a DYI counter

[Magnus Danielson]

Erik,

OK, so it's not big magic really. There is an assumed time-base length. 
The actual time each such time-stamp is done shifts around a little. For 
a 10 MHz signal, the time-base shifts around within one period so 100 
ns. The actual trigger point 1 and actual trigger point 2 will on 
average be the tau_0 time-distance from each other. As we do this for a 
set of frequency estimations and average these, it averages out. This is 
the basic assumption being used in all estimations I've seen. My 
algorithm makes no different assumption than any of the others I've seen.

I have seen those processing this with more detail of actual delay, but 
that is when focusing on single-measurements. The danger there is that 
numeric precision eats you quickly.

Now, the variation you get is really a systematic play on the period 
time and the tau_0 and the phase-ramp you get out of that. This breaks 
down into other phase-ramps of diminishing frequency and amplitude, just 
as in a DDS. This systematic pattern rolls of quickly in averaging while 
random noise does not roll off as quickly. The systematic pattern can be 
"nulled" by matching average length to pattern length, as always. You 
can't really resolve this systematic noise before you know the 
relationship, rather it is a consequence of the actual rational number 
and how you choose to measure it. Random noise tends to smooth things out.

You need to compare the noise of the tau_0 "instability" with that of 
the signal and the time-interval measurement error, it's fairly small 
compared to the others together typically.

Now, the algorithm you have in that paper does not handle gaps in data. 
It assumed a continuous block. Essentially the linear ramp of phase and 
frequency needs to be unbroken or it will be producing the wrong 
results. You can handle gaped data by altering the algorithm, it will be 
a little more messy, but still maintain most of the benefits. Gaped data 
is a big thing, and valuable work has been done for ADEV by Dave Howe.

Cheers,
Magnus

On 8/7/22 22:21, Erik Kaashoek wrote:
> Magnus,
> Now you confuse me.
> Can you simplify the calculation assuming the samples are equally 
> spaced even if they are not? Can you assume the spread is noise and it 
> will sjaal out? How about gaps?
> Please help me to understand
> Erik
>
> On Sun, Aug 7, 2022, 22:08 Magnus Danielson <magnus at rubidium.se> wrote:
>
>     Erik,
>
>     They never are. It's a running assumption that everyone makes.
>
>     Cheers,
>     Magnus
>
>     On 8/7/22 13:28, Erik Kaashoek wrote:
>     > Magnus,
>     > Due to the design of the counter it is not possible to guarantee
>     all
>     > captures are at exactly tau_0 distance.
>     > Erik.
>     >
>     > On 6-8-2022 22:09, Magnus Danielson wrote:
>     >> The linear algebra trick actually jumps behind the linear algebra
>     >> using the knowledge that measurements is a sequence of tau_0
>     >> distance, and that allows significant reduction of the math into a
>     >> much more benign form. It also creates benign decimation
>     methods that
>     >> you can apply to any form of your liking.
>     >
>