[time-nuts] TIC Characterization

Mon Dec 31 09:49:12 UTC 2018

> Thanks Tom for your quick and extensive reply. 
> Indeed I confused Time Interval with Phase Difference….... 
...
> By the way, it also seems that HDEV at Tau=1 is  2/sqrt(3) * Sigma = 1.15 SigmaTIC
>

I don't believe that HDEV result. For many large runs of simulated normalized white phase noise input I get:
    adev(1) = 1.732, mdev(1) = 1.732, tdev(1) = 1.000, hdev(1) = 1.825

See: https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables
or google for topics like:
    sums of independent random variables
    normal sum distribution
    linear combinations of normal random variables

Then look at both calc_adev() and calc_hdev() in http://leapsecond.com/tools/adev_lib.c
Assuming you have white phase noise input with mean=0, stdev=1 and using tau=1, then,

1) The key lines for ADEV are:
    v = data[i + 2*tau] - 2 * data[i + tau] + data[i];
    sum /= 2.0;
So you would expect sqrt( (1 + 4 + 1) / 2 ) = sqrt(6/2) = 1.7321

2) The key lines for HDEV are:
    v = data[i + 3*tau] - 3 * data[i + 2*tau] + 3 * data[i + tau] - data[i];
    sum /= 6.0;
So you would expect sqrt( (1 + 9 + 9 + 1) / 6 ) = sqrt(20/6) = 1.8257

3) MDEV is the same as ADEV for tau = 1 so that's why it is also 1.7321.

4) TDEV is MDEV / sqrt(3) so that why it gives 1.0000.

For testing counter resolution TDEV is often more instructive than ADEV:

a) TDEV doesn't have that misleading, distracting, prolonged -1 slope that ADEV plots often show.
b) TDEV has the nice property of reporting 1 when given 1 RMS data.
c) The units for TDEV are seconds, which is appropriate for a TIC (time interval counter).
d) The units for ADEV are fractional frequency, which is natural for frequency standards.
e) Unless there are design flaws, instrument drift, or environmental issues, TDEV should be flat from tau 1 to forever.
f) Subtle, unwanted variations are much easier to observe in a flat line (TDEV) than a -1 line (ADEV).

/tvb

----- Original Message ----- 
From: "Club-Internet Clemgill" <clemgill at club-internet.fr>
To: "Tom Van Baak" <tvb at leapsecond.com>; "Discussion of precise time and frequency measurement" <time-nuts at lists.febo.com>
Sent: Sunday, December 30, 2018 1:07 AM
Subject: Re: [time-nuts] TIC Characterization

Thanks Tom for your quick and extensive reply. 
Indeed I confused Time Interval with Phase Difference….... 

Corrected calc: 
4/ [(Xi+2 - Xi+1) - (Xi+1 - Xi)]^2 = [(To + Ti+2) - 2(To + Ti+1) + (To + Ti)]^2 = [Ti+2 - 2Ti+1 + Ti]^2 
=  (Ti+2)^2 + 4(Ti+1)^2 + (Ti)^2 + 2(-2Ti+2*Ti+1 + Ti+2*Ti - 2Ti+1*Ti)

5/ <(Ti+1)^2> # <(Ti+1)^2> #  < (Ti)^2> for large samples and  
<(Ti+a * Ti+b)> = 0 as Ti+1 and Ti are independent
Then AVAR =  (1/2Tau^2) * 6 < (Ti)^2>  = (1/Tau^2) * 6  * SigmaTIC^2

6/ Hence ADEV = sqrt(3) * SigmaTIC / Tau

So ADEV(Tau=1)  = 1.73 * SigmaTIC (indeed…)

By the way, it also seems that HDEV at Tau=1 is  2/sqrt(3) * Sigma = 1.15 SigmaTIC

Best, 
Gilles.

> Le 30 déc. 2018 à 07:14, Tom Van Baak <tvb at LeapSecond.com> a écrit :
> 
> Hi Gilles,
> 
> Correct, the log-log slope will be -1.
> 
> But I'm not sure about your ADEV and SigmaTIC claim.
> 
> Assume the 53132A has 150 ps RMS resolution. The standard deviation is also 150 ps. The Allan deviation at tau=1 would be 1.73 * 150 ps/s or 2.60e-10.
> 
> Look at calc_adev() in http://leapsecond.com/tools/adev_lib.c and note the three data[] terms. With multiple uncorrelated terms you simply sum the variances. There are three terms so that's 3 * stdev. When you convert AVAR to ADEV the 3 becomes sqrt(3), or 1.73. Make sense?
> 
> For extra credit, note that MDEV at tau=1 is the same as ADEV. However, TDEV at tau=1 is 1.50e-10, the same as stdev. In the same file, see that the sqrt(3) factor is removed in calc_tdev().
> 
> ----
> 
> The best and largest pile of ADEV documentation is:
> 
> "information about frequency stability analysis"
> http://www.wriley.com/Freq%20Stab%20Analy%20Links.htm
> 
> There is also a wikipedia page:
> 
> https://en.wikipedia.org/wiki/Allan_variance
> 
> For simpler introductions see:
> 
> "Analysis of Time Domain Data"
> https://tf.nist.gov/phase/Properties/four.htm
> 
> "Clock Performance and Performance Measures"
> https://tycho.usno.navy.mil/mclocks2.html
> 
> "Fundamentals of Time and Frequency"
> https://tf.nist.gov/general/pdf/1498.pdf
> 
> ----
> 
> If you want to play with ADEV check out Stable32 [1] or TimeLab [2]. Both are highly recommended and are also free. For questions like yours the Stable32 noise generator feature is very useful to explore the shape(s) of ADEV for given noise types. It was used to create:
> 
> "Exploring Allan Deviation"
> http://leapsecond.com/pages/allan/Exploring_Allan_Deviation_v2.pdf
> 
> /tvb
> 
> [1] http://www.stable32.com/
> [2] http://www.ke5fx.com/timelab/readme.htm
> 
> 
> ----- Original Message ----- 
> From: "Club-Internet Clemgill" <clemgill at club-internet.fr>
> To: "Discussion of precise time and frequency measurement" <time-nuts at lists.febo.com>
> Sent: Saturday, December 29, 2018 2:28 PM
> Subject: [time-nuts] TIC Characterization
> 
> 
>> 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).
>> I found some references on the web, but no one with the associated maths.
>> 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
>> 
>> 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.
>> 
>> Gilles. 
>> 
>> 
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> 
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