[time-nuts] Allan variance by sine-wave fitting

[Ralph Devoe]

Here's a short reply to the comments of Bob, Attila, Magnus, and others.
Thanks for reading the paper carefully. I appreciate it. Some of the
comments are quite interesting, other seem off the mark. Let's start with
an interesting one:

The issue I intended to raise, but which I'm not sure I stated clearly
enough, is a conjecture: Is least-square fitting as efficient as any of the
other direct-digital or SDR techniques? Is the resolution of any
direct-digital system limited by (a) the effective number of bits of the
ADC and (b) the number of samples averaged? Thanks to Attila for reminding
me of the Sherman and Joerdens paper, which I have not read carefully
before. In their appendix Eq. A6 they derive a result which may or may not
be related to Eq. 6 in my paper. If the conjecture is true then the SDR
technique must be viewed as one several equivalent algorithms for
estimating phase. Note that the time deviation for a single ADC channel in
the Sherman and Joerdens paper in Fig. 3c is about the same as my value.
This suggests that the conjecture is true.

Other criticisms seem off the mark:

Several people raised the question of the filter factor of the least-square
fit.  First, if there is a filtering bias due to the fit, it would be the
same for signal and reference channels and should cancel. Second, even if
there is a bias, it would have to fluctuate from second to second to cause
a frequency error. Third, the Monte Carlo results show no bias. The output
of the Monte Carlo system is the difference between the fit result and the
known MC input. Any fitting bias would show up in the difference, but there
is none.

Attila says that I exaggerate the difficulty of programming an FPGA. Not
so. At work we give experts 1-6 months for a new FPGA design. We recently
ported some code from a Spartan 3 to a Spartan 6. Months of debugging
followed. FPGA's will always be faster and more computationally efficient
than Python, but Python is fast enough. The motivation for this experiment
was to use a high-level language (Python) and preexisting firmware and
software (Digilent) so that the device could be set up and reconfigured
easily, leaving more time to think about the important issues.

Attila has about a dozen criticisms of the theory section, mostly that it
is not rigorous enough and there are many assumptions. But it is not
intended to be rigorous. This is primarily an experimental paper and the
purpose of the theory is to give a simple physical picture of the
surprizingly good results. It does that, and the experimental results
support the conjecture above.
The limitations of the theory are discussed in detail on p. 6 where it is
called "... a convenient approximation.." Despite this the theory agrees
with the Monte Carlo over most of parameter space, and where it does not is
discussed in the text.

About units: I'm a physicist and normally use c.g.s units for
electromagnetic calculations. The paper was submitted to Rev. Sci. Instr.
which is an APS journal. The APS has no restrictions on units at all.
Obviously for clarity I should put them in SI units when possible.

Ralph
KM6IYN