You must have read a different paper than that one. I found it (through our
library) and it says that if you have n measurements in a time period T,
the best strategy is to take n/2 measurements at the beginning of the time
and n/2 at the end to minimize the effect of the white noise phase error on the
frequency estimate. That is perfectly true, and gives an error which goes
as sqrt(4/n)delta/T rather than sqrt(12/n)(delta/T) for equally spaced
measurements (assuming large n) T is the total time interval and delta is the
std dev of each phase measurement . But it certainly does NOT say that if you
have n
measurements, just use the first and last one to estimate the slope.
If you have n measurements, the best estimate of the slope is to do a least
squares fit. If they are equally spaced, the center third do not help much
(nor do they hinder), but a least squares fit is always the best thing to
do.
"David L. Mills" <[EMAIL PROTECTED]> writes:
>Bill,
>NIST doesn't agree with you. Only the first and last are truly
>significant. Reference: Levine, J. Time synchronization over the
>Internet using an adaptive frequency locked loop. IEEE Trans. UFFC,
>46(4), 888-896, 1999.
>Dave
>Unruh wrote:
>> "David L. Mills" <[EMAIL PROTECTED]> writes:
>>
>>
>>>Bill,
>>
>>
>>>Ahem. The first point I made was that least-squares doesn't help the
>>>frequency estimate. The next point you made is that least-squares
>>>improves the phase estimate. The last point you made is that phase noise
>>
>>
>> No. The point I tried to make was the least squares improved the FREQUENCY
>> estimate by sqrt(n/6) for large n, where n is the number of points (assumed
>> equally spaced) at which the phase is measured. I am sorry that the way I
>> phrased it could have been misunderstood.
>>
>>
>> The phase is ALSO improved proportional to sqrt(n)
>> .
>> This assumes uncorrelated phase errors dominate the error budget.
>>
>>
>>
>>
>>>is not important. Our points have been made and further discussion would
>>>be boring.
>>
>>
>> Except you misunderstood my point. It may still be boring to you.
>>
>>
>>
>>>Dave
>>
>>
>>>Unruh wrote:
>>>
>>>>"David L. Mills" <[EMAIL PROTECTED]> writes:
>>>>
>>>>
>>>>
>>>>>Bill,
>>>>
>>>>
>>>>>If you need only the frequency, least-squares doesn't help a lot; all
>>>>>you need are the first and last points during the measurement interval.
>>>>
>>>>
>>>>Well, no. If you have random phase noise, a least squares fit will improve
>>>>the above estimate by roughly sqrt(n/4) where n is the number of points.
>>>>That can be significant. It is certainly true that the end points have the
>>>>most weight ( which is why the factor of 1/4). Ie, if you have 64 points,
>>>>you are better by about a factor of 4 which is not insignificant.
>>>>
>>>>
>>>>
>>>>>The NIST LOCKCLOCK and nptd FLL disciplines compute the frequency
>>>>>directly and exponentially average successive intervals. The NTP
>>>>>discipline is in fact a hybrid PLL/FLL where the PLL dominates below the
>>>>>Allan intercept and FLL above it and also when started without a
>>>>>frequency file. The trick is to separate the phase component from the
>>>>>frequency component, which requires some delicate computations. This
>>>>>allows the frequency to be accurately computed as above, yet allows a
>>>>>phase correction during the measurement interval.
>>>>
>>>>
>>>>He of course is not interested in phase corrections.
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>>Dave
>>>>
>>>>
>>>>>Unruh wrote:
>>>>>
>>>>>
>>>>>>David Woolley <[EMAIL PROTECTED]> writes:
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>>Unruh wrote:
>>>>>>
>>>>>>
>>>>>>>>I do not understand this. You seem to be measuring the offsets, not the
>>>>>>>>frequencies. The offset is irrelevant. What you want to do is to measure
>>>>>>
>>>>>>
>>>>>>>Measuring phase error to control frequency is pretty much THE standard
>>>>>>>way of doing it in modern electronics. It's called a phase locked loop
>>>>>>
>>>>>>
>>>>>>Sure. In the case of ntp you want to have zero phase error. ntp reduces
>>>>>>the
>>>>>>phase error slowly by changing the frequency. This has the advantage that
>>>>>>the frequency error also gets reduced (slowly). He wants to reduce the
>>>>>>frequency error only. He does not give a damn about the phase error
>>>>>>apparently. Thus you do NOT want to reduce the frequecy error by attacking
>>>>>>the phase error. That is a slow way of doing it. You want to estimate the
>>>>>>frequency error directly. Now in his case he is doing so by measuring the
>>>>>>phase, so you need at least two phase measurements to estimate the
>>>>>>frequency error. But you do NOT want to reduce the frequency error by
>>>>>>reducing the phase error-- far too slow.
>>>>>>
>>>>>>One way of reducing the frequency error is to use the ntp procedure but
>>>>>>applied to the frequency. But you must feed in an estimate of the frequecy
>>>>>>error. Anothr way is the chrony technique. -- collect phase points, do a
>>>>>>least squares fit to find the frequency, and then use that information to
>>>>>>drive the frequecy to zero. To reuse past data, also correct the prior
>>>>>>phase measurements by the change in frequency.
>>>>>>(t_{i-j}-=(t_{i}-t_{i-j}) df
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>>(PLL) and it is getting difficult to find any piece of electrnics that
>>>>>>>doesn't include one these days. E.g. the typical digitally tuned radio
>>>>>>
>>>>>>
>>>>>>A PLL is a dirt simply thing to impliment electronically. A few resistors
>>>>>>and capacitors. It however is a very simply Markovian process. There is
>>>>>>far
>>>>>>more information in the data than that, and digititally it is easy to
>>>>>>impliment far more complex feedback loops than that.
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>>or TV has a crystal oscillator, which is divided down to the channel
>>>>>>>spacing or a sub-multiple, and a configurable divider on the local
>>>>>>>oscillator divides that down to the same frequency. The resulting two
>>>>>>>signals are then phase locked, by measuring the phase error on each
>>>>>>>cycle, low pass filtering it, and using it to control the local
>>>>>>>oscillator frequency, resulting in their matching in frequency, and
>>>>>>>having some constant phase error.
>>>>>>
>>>>>>
>>>>>>>>the offset twice, and ask if the difference is constant or not. Ie, th
>>>>>>>>eoffset does not correspond to being off by 5Hz.
>>>>>>
>>>>>>
>>>>>>>ntpd only uses this method on a cold start, to get the initial coarse
>>>>>>>calibration. Typical electronic implementations don't use it at all,
>>>>>>>but either do a frequency sweep or simply open up the low pass filter,
>>>>>>>to get initial lock.
>>>>>>
>>>>>>
>>>>>>And? You are claiming that that is efficient or easy? I would claim the
>>>>>>latter. And his requirements are NOT ntp's requirements. He does not care
>>>>>>about the phase errors. He is onlyconcerned about the frequency errors.
>>>>>>driving the frequency errors to zero by driving the phase errors to zero
>>>>>>is
>>>>>>not a very efficient technique-- unless of course you want the phase
>>>>>>errors
>>>>>>to be zero( as ntp does, and he does not).
>>>>>>
>>>>>>
>>>>>>
>>>>>>
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