[time-nuts] Some experiments and learnings with sample interpolation in phase measurement

[Erik Kaashoek]

This post is to consolidate what I learned and may be useful for others 
that are considering to use linear regression in a reciprocal counter to 
improve measurement accuracy.
The time resolution of a home build frequency/time counter is about 4ns, 
e.g. 4e-9.
Using linear regression over 100000 samples within a 1 second gate time 
the best case accuracy improvement should lead to a resolution of 4e-9 / 
square root(100000)Â  or 1.5e-11
Using two input signals with equal frequencies and thus static phase 
relation, the measured phase has an  ADEV at tau = 1 second in the order 
of 1e-11 and goes down with one decade per decade of tau. ([2] dark blue 
and pink traces)
Using two input signals with 0.002 Hz frequency difference the 
performance of the current prototype counter HW (e.g. phase pulling) can 
clearly be observed. ([1] green trace)
This, however, works only when the input signals have a frequency that 
is not harmonically related to the reference clock of the counter.
Using two input signals with 0.002 Hz frequency difference at a 
harmonically related input frequency causes the ADEV of the phase 
measurement to deteriorates with a factor 10 to about 1.5 e -10 ([2] 
light blue trace) and the dead zones, due to insufficient samples to 
observe the phase difference, are clearly visible in the linear residue 
of the phase. ([1] light blue trace)
Shifting the counters reference a couple of Hz to reduce the harmonic 
relation with the input signals brings some improvement resulting in a 
ADEV at a tau at 1 second of 2e-11 ([2] red trace) but it is not enough 
to eliminate the problem. ([1] red trace)
So, unless the counter can switch, based on the actual frequency of the 
input signals, between two sufficiently different not harmonically 
related references, using linear regression cannot guarantee substantial 
improvement in phase resolution. only an improvement in the HW time 
resolution can guarantee the improvement.
Erik.

[1] http://athome.kaashoek.com/time-nuts/Regression_phase_residue.png
[2] http://athome.kaashoek.com/time-nuts/Regression_ADEV.png