[time-nuts] Some long-term data
Magnus Danielson
cfmd at bredband.net
Mon Dec 25 09:50:06 EST 2006
From: Magnus Danielson <cfmd at bredband.net>
Subject: Re: [time-nuts] Some long-term data
Date: Mon, 25 Dec 2006 04:04:09 +0100 (CET)
Message-ID: <20061225.040409.-1066394447.cfmd at bredband.net>
> From: Magnus Danielson <cfmd at bredband.net>
> Subject: Re: [time-nuts] Some long-term data
> Date: Mon, 25 Dec 2006 03:43:16 +0100 (CET)
> Message-ID: <20061225.034316.-1476063739.cfmd at bredband.net>
>
> > Now, I made a quick little program that process the data, here is the result:
>
> Ehum... no... forget those plots. I just realized a little error in my
> drift(tau) estimator. I had a divide by tau too little, which will make the tau
> rise. As I said, the devil is in the details.
Brown paperbag on the head? Yes please!
I realized this morning that I had probably made yeat another error, which I
now have done the homework on and found that yes, I did. Infact, I had two
errors which canceled each other for the ADEV corrected form but not for the
drift estimate which was correct. Mumble.
When ADEV(tau) is
---------------------------------------
/ N-2n |
/ ---
/ 1 \ 2
ADEV(n*tau0) = \ / --------------- > (x - 2 x + x )
\ / 2n²tau²(N - 2n) / i+2n i+n n
\ / 0 ---
\/ i=1
The drift(tau) is
N-2n
---
1 \
drift(n*tau0) = -------------- > (x - 2 x + x )
n²tau²(N - 2n) / i+2n i+n n
0 ---
i=1
This can be concluded by letting x(t) be set to the linear assignment
delta v D 2
x(t) = TE + ------- t + - t
0 vn 2
in the inner second difference
sd(i,n) = x - 2 x + x
i+2n i+n n
and the result becomes
2 2
Dn tau
0
with that in hand it is simple enought to validate the rest of the drift(tau)
formula as given. Doing the same to the ADEV calculation gives
-
/1
ADEV (tau) = D n*tau \ / -
drift 0 \/ 2
Thus, for longer taus, even a small drift rate can bite us since it is now
multiplied with tau.
To compensate this I chose to subtract the Dn²tau0² part from the second
difference prior to squaring. This is equalent to preprocess the x samples
by removing the drift, however I always use drift(tau) which is the effective
average drift as it will contaminate the measure at a certain tau.
My corrected ADEV becomes:
----------------------------------------------------------
/ N-2n |
/ ---
/ 1 \ 2
ADEV(n*tau0) = \ / --------------- > (x - 2 x + x - drift(tau)n²tau²)
\ / 2n²tau²(N - 2n) / i+2n i+n n
\ / 0 ---
\/ i=1
The resulting measures becomes:
magnus at heaven:~/febo$ ./process cs1-gps.dat
Processing cs1-gps.dat
f = -1.371312943e-13
d = -2.464638917e-19
tau ADEV drift ADEV drift corr
600 6.669673909e-12 -2.464638917e-19 6.669673908e-12
1200 3.834415574e-12 -8.108122133e-20 3.834415573e-12
3000 2.262840286e-12 -2.392890882e-19 2.262840229e-12
6000 1.321830189e-12 -3.349535739e-20 1.321830182e-12
12000 6.871654621e-13 +3.846473719e-20 6.871653846e-13
30000 3.721752717e-13 +2.056358286e-20 3.721750161e-13
60000 2.376494277e-13 -1.810085678e-20 2.376481869e-13
120000 1.702980952e-13 +2.320652509e-21 1.702979813e-13
300000 1.549264955e-13 +3.629179727e-21 1.549245826e-13
600000 1.401676798e-13 -5.790456326e-21 1.401461493e-13
1200000 6.799762269e-14 +1.948183423e-20 6.595761175e-14
3000000 1.010049262e-13 +4.150533462e-20 4.949623724e-14
6000000 1.550028203e-13 +3.651598399e-20 4.936266040e-15
magnus at heaven:~/febo$ ./process cs2-gps.dat
Processing cs2-gps.dat
f = 5.853529458e-13
d = 1.816049729e-19
tau ADEV drift ADEV drift corr
600 7.508211487e-12 +1.816049729e-19 7.508211486e-12
1200 4.527916555e-12 +9.729746560e-20 4.527916554e-12
3000 2.722308557e-12 +4.515867825e-20 2.722308555e-12
6000 1.721060210e-12 -7.880496874e-20 1.721060178e-12
12000 1.035443547e-12 +9.560956213e-20 1.035443229e-12
30000 6.178807446e-13 +1.132639019e-19 6.178760731e-13
60000 4.117968150e-13 +1.218708103e-19 4.117643529e-13
120000 2.667708446e-13 +8.332110720e-20 2.666771423e-13
300000 1.831015144e-13 +7.249210859e-20 1.824546103e-13
600000 1.607481137e-13 +6.187329751e-20 1.585902304e-13
1200000 9.695302954e-14 +6.246924523e-20 8.117978311e-14
3000000 1.780291901e-13 +7.356014755e-20 8.569984631e-14
6000000 3.223233857e-13 +7.588775895e-20 1.520624005e-14
magnus at heaven:~/febo$ ./process rb1-gps.dat
Processing rb1-gps.dat
f = 1.081017978e-14
d = -4.280688646e-19
tau ADEV drift ADEV drift corr
600 6.497785589e-12 -4.280688646e-19 6.497785586e-12
1200 3.673171557e-12 -5.189198165e-20 3.673171556e-12
3000 2.152493532e-12 -3.462165333e-19 2.152493406e-12
6000 1.229699919e-12 -1.347603914e-19 1.229699786e-12
12000 5.983794472e-13 -2.793891384e-20 5.983794003e-13
30000 3.088139889e-13 -1.001855674e-19 3.088066758e-13
60000 2.375139899e-13 -1.183043196e-19 2.374609500e-13
120000 2.408230821e-13 -9.341989437e-20 2.406925850e-13
300000 2.703779963e-13 -5.825801115e-20 2.700954111e-13
600000 2.336053294e-13 -5.052904535e-20 2.326195966e-13
1200000 1.439223207e-13 -3.937365625e-20 1.399908188e-13
3000000 1.847594665e-13 -6.727980125e-20 1.173306781e-13
6000000 2.953067943e-13 -6.935111621e-20 2.517361716e-14
magnus at heaven:~/febo$
Sigh. Well, there you have it. For longer taus, even small drift rates will
make a difference. As long as D*tau << ADEV(tau) you don't need to worry.
By the way, the modified Allan deviation is equally sensitive to drift rate.
Cheers,
Magnus
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