Tom,
Tom Van Baak wrote:
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.
There is a technical merit to take samples in between even if they
cancel... you avoid counter overflow, but you can do better, much better.
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.
Yes, and the main point for creating this little thread is to make
people aware that how you process your data do make a difference. It can
make a huge difference in fact. The effective two-point frequency
calculation only use two sample point to estimate the frequency and thus
also use the systematic value and noise of one sample to cancel the
noise of the first sample. For 1/f power noises this is an effect that
can even becomes larger (for 1/f^3 noise) with time as it is
non-convergent, so by looking at it briefly you don't realize it is a
noisy number.
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).
In particular, there is one paper that corrects some mistakes of
Rubiola, Australian if I remember correctly.
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.
A very simple extension leads to that. Consider the frequency estimate of
x(i+m) - x(i)
y(i,m) = -------------
m*tau0
And averaging over those:
N-m
---
1 \
y(m) = --- > y(i,m)
N-m /
---
i=1
You get 2*m samples in each end and the maximum m is of course N/2. This
can be written in another form as N/2 number of m=N/2 overlapping
frequency estimates. This is equivalent to averaging the first and
second half, subtract the first from the second and divide by m*tau0.
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.
Indeed. It creates a bias on the measurement that needs to be taken out
for a valid measurement.
One thing you can play with is computing average frequency
using the technique that MDEV uses.
Which is inspired by an article by Snyder that details how to perform
one such overlapping estimation in the counter core for better
convergence in noisy data. The MDEV is a separate basic measurement than
ADEV. So one needs to be careful not to mix results freely.
Cheers,
Magnus
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