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

Fri Feb 21 12:26:38 UTC 2020

Hi Taka,

On 2020-02-21 04:45, Taka Kamiya via time-nuts wrote:
> I was in electronics in big ways in 70s.  Then had a long break and came back to it in last few years.  Back then, if I wanted 1s resolution, the gate time had to be 1s.  So measuring ns and ps was pretty much impossible.  As I understand it, HP53132A (my main counter) takes thousands of samples (I assume t samples) to arrive at most likely real frequency.  That was something I had hard time wrapping my head around. 

It actually does two things.

First, it interpolates the occurrence of a rising edge (for start and
stop channel), so if this does not happen in perfect alignment with a
rising edge of the reference/coarse clock. Often the OCXO/Rubidium is
for 10 MHz,  but then a 90-500 MHz oscillator is locked to the
reference, and this higher clock is then used instead of the 10 MHz for
coarse-counting. Coarse-counting is counting of cycles just back in the
good old days of counters. The resolution is increased further not by
raising the counting frequency, but by measuring the time-error of the
trigger channel event in relation to the coarse-counter clock-edge.
Thus, measuring 0.000-0.999 of a coarse-counting cycle. In practice it
becomes hard to design for that, as the shorter end has problem is gate
delay times to be well decided, so one add one or two coarse cycles to
do 1.000-1.999 or 2.000-2.999 cycles, but these extra cycles is only for
the interpolator design, so once the fractional cycle is known the other
can be ignored.

Just to give you an idea of what different counters do, here is from the
top of my head some numbers:

HP5370A: Ref 10 MHz, Coarse 200 MHz, Interpolation gain 256, time
resolution < 20 ps
HP5328A: Ref 10 MHz, Coarse 10 MHz, Interpolation gain 1, time
resolution 100 ns
HP5328A with Option 040-042 and HP5328B: Ref 10 MHz, Coarse 100 MHZ,
Interpolation gain 1 (TI-average has other interpolation means), time
resolution 10 ns or for TI-avg 10 ps (claimed)
HP5335A: Ref 10 MHz, Coarse 10 MHz, Interpolation gain 200, time
resolution 1 ns
HP5372A: Ref 10 MHz, Coarse 500 MHz, Interpolation gain 10, time
resolution 200 ps
HP53132A: Ref 10 MHz, Coarse 100 MHz, Interpolation gain 1000, time
resolution 100 ps
SR620: Ref 10 MHz, Coarse 90 MHz, Interpolation gain 512?, time
resolution < 25 ps (don't recall details)
PM6863: Ref 10 MHz, Coarse 500 MHz, Interpolation gain 1, time
resolution 2 ns
CNT-90: Ref 10 MHz, Coarse 100 MHz, Interpolation gain 512, time
resolution 100 ps (claimed)
CNT-91: Ref 10 MHz, Coarse 100 MHz, Interpolation gain 512, time
resolutions 50 ps (claimed)
SIA3000: Ref 100 MHz, Coarse, 100 MHz, Interpolation gain 50000, time
resolution 200 fs

As I write claimed above, the actual performance can be better, but the
spec on the sheet did not overstate it more. While all the numbers may
not be 100% correct, I think they help to illustrate the relationships
very well. As you calculate the length of the coarse counter period from
it's frequency, and then divide with the interpolation gain, which is by
how many steps the period is interpolated, the raw time resolutions pops
out.

Interpolation methods differs, but typically first an error signal is
generated and then it is stored into a capacitor which is then measured
with some slower technique. The 5335A use a very simple technique where
the discharge of the capacitor is done with a much lower current than
the charging, so now the discharge time can be measured using the coarse
clock. This is called pulse-stretching. Today the far most common
technique is to use an ADC to digitize the voltage.

The 5328 counters have a unique interpolation technique by
phase-modulating the reference clock with noise, effectively shifting
the reference transitions around and that way interpolate over time a
higher resolution. It works better than claimed.

Remember that this single-shot resolution is reduced by the trigger
jitter as well as unstability of reference oscillator. In practice the
trigger jitter or resolution dominates as a 1/tau limit as you look at
the Allan Deviation, to fix that you need to buy a better counter or
signal condition for better trigger.

The second trick used in 53132 for measuring frequency is averaging. It
uses an average technique originally from optical frequency measures to
accumulate data into blocks and then subtract the time-stamps of two
subsequent blocks. This is the same as average the output of a number of
overlapping frequency estimations.

This has advantages as white noise is supressed with a steeper slope,
and the associated deviation is the modified Allan Deviation MDEV.

>  
>
> I understand most of what you said, but I've never taken statistics, so I am guessing on some part.  I can see how adev goes down as tau gets longer.  Basically, averaging is taking place.  But I am still not sure why at some point, it goes back up.  I understand noise will start to take effect, but the same noise has been there all along while adev was going down.  Then, why is this inflection point where sign of slope suddenly changes? 

OK, so the trouble is that rather than only white noise as classical
statistics deal with, we have at least 4 noise types, with different
frequency slopes. As we try to analyze this with standard deviation, the
standard deviation estimator (RMS estimator) does not converge, is
simply keeps producing noise even if we add more values. To put that in
another way, we do not gain more knowledge by doing more measurements.
The classical white noise is what is called white phase modulation
noise, we then have flicker phase noise, white frequency noise and
flicker frequency noise. All these noise-types is to be expected
according to the David Leeson model, and it is due to those that we need
to use more advanced statistics as introduced by David Allan.

The White Phase Modulation has a flat frequency response in phase noise
spectrum, 1/tau in ADEV.
The Flicker Phase Modulation has 1/sqrt(f) respone in phase noise
spectrum, 1/tau in ADEV.
The White Frequency Modulation has 1/f respone in phase noise spectrum,
1/sqrt(tau) in ADEV.
The Flicker Frequency Modulation has 1/sqrt(f^3) response in phase noise
spectrum, flat in ADEV.

In addition to this, linear frequency drift creates a slope that scales
with drift and tau, so that is an upper limit. Thermal sensitivity tends
to lay ontop as well, so does other disturbances.

Depending on details of oscillators and their sensitivity to thermal
noise, their effective minimum shifts around.

>  
>
> Also, to reach adev(tau=10), it takes longer than 10 seconds.  Manual for TimeLab basically says more samples are taken than just 10, but does not elaborate further.  Say it takes 50 seconds to get there, and say that's the lowest point of adev, does that mean it is the best to set gate time to 10 second or 50 second?  (or even, take whatever gate time and repeat the measurement until accumulated gate time equals tau?

The Allan Deviation takes a number of estimates to produce values, but
remember these are stability values for a certain observationtime for
frequency, not the frequency measure itself.

Cheers,
Magnus

>
> --------------------------------------- 
> (Mr.) Taka Kamiya
> KB4EMF / ex JF2DKG
>  
>
>     On Thursday, February 20, 2020, 7:54:22 PM EST, Magnus Danielson <magnus at rubidium.se> wrote:  
>  
>  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
>
>
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