[time-nuts] Tbolt issues

[Gregory Maxwell]

On Fri, Sep 2, 2016 at 12:51 AM, Charles Steinmetz
<csteinmetz at yandex.com> wrote:
> Now shorten the observation time to 20nS.  We see 1/5 of a complete cycle
> (72 degrees, 0.4 pi radians) of the wave.  No matter which particular 72
> degrees we see, we simply don't have enough information to reliably deduce

I do not see why you argue that.

For the purpose of discussion, lets assume you have a noiseless signal
which is stationary in frequency and amplitude over 20nS starting at
the zero crossing. Given these strong priors (single tone, constant
frequency which is not higher than one half cycle in our 20nS window,
constant amplitude, noiseless) there is exactly one frequency
consistent with any of those two observations.  If the starting phase
is unknown, I believe you need one additional observation to end up
over-determined and have an unambiguous solution again.

This kind of strong prior assumption is why sinusoidal estimators and
PLLs are able to extract tones with precision far beyond what you
would expect from taking a DFT from equivalent amount of data.

In reality, there is phase noise, non-linearities, harmonics, tidal
variations, and whatnot that make these assumptions untrue... but how
far they corrupt these assumptions depends on how useless the results
are.