Magnus,
Correct, all the terms cancel between the end points. Note
that this is exactly equivalent to the way a traditional gated
frequency counter works -- you open the gate, wait some
sample period (maybe 1, 10, or 100 seconds) and then
close the gate. In this scenario it's clear that all the phase
information during the interval is ignored; the only points
that matter are the start and the stop.
Modern high-resolution frequency counters don't do this;
and instead they use a form of "continuous counting" and
take a massive number of short phase samples and create
a more precise average frequency out of that.
There are some excellent papers on the subject; start with
the one by Rubiola:
<
http://www.femto-st.fr/~rubiola/pdf-articles/journal/2005rsi-hi-res-freq-counters.pdf
>
There are additional papers (perhaps Bruce can locate them).
I wonder if fully overlapped frequency calculations would be
one solution to your query; similar to the advantage that the
overlapping ADEV sometimes has over back-to-back ADEV.
Related to that, I recently looked into the side-effects of using
running averages on phase or frequency data, specifically
what it does to a frequency stability plot (ADEV). See:
http://www.leapsecond.com/pages/adev-avg/
Not surprising, you get artificially low ADEV numbers when
you average in this way; the reason is that running averages,
by design, tend to smooth out (low pass filter) the raw data.
One thing you can play with is computing average frequency
using the technique that MDEV uses.
/tvb
----- Original Message -----
From: "Magnus Danielson" <[email protected]>
To: "Discussion of precise time and frequency measurement" <[email protected]>
Sent: Sunday, January 31, 2010 6:53 PM
Subject: [time-nuts] Zero dead time and average frequency estimation
Fellow time-nuts,
I keep poking around various processing algorithms trying to figure out
what they do and perform. One aspect which may be interesting to know
about is the use of zero dead time phase or frequency data and the
frequency estimation from that data. One may be compelled to
differentiate the time data into frequency data by using nearby data
samples, according to y(i) = (x(i+1)-x(i))/tau0 and then just form the
average of those. The interesting thing about that calculations is that
the x(i+1) and x(i) terms cancels except for x(1) and x(N) so
effectively only two samples of phase data is being used. This is a
simple illustration of how algorithms may provide less degrees of
freedom than one may initially assume it to have (N-1 in this case).
Similar type of cancellation occurs in linear drift estimation.
Maybe this could spark some interest in the way one estimates the
various parameters and what different estimators may do to cancel noise
of individual samples.
Cheers,
Magnus
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