[time-nuts] ADEV noise bandwidth (was LPRO Rubidium Performance)

John Miles jmiles at pop.net
Mon Feb 13 01:19:32 UTC 2012

> The ADEV difference of about 6 db at 1ms tau can be explained by the fact
> that if I apply a 500 Hz LP  filter to my 9600 sps raw data, the same
> used on the 5120A's  1K sps data, then even our 1ms ADEV answers
> become very close.
> I have found that using a 1/2 zero tau BW filter like the 5120A does can
> falsely lower its tau zero ADEV answer by 3 to 6 dB.
> The 5120A's use of a 1/2 tau zero LP cutoff filter is why the 5120A ADEV
> answers are generally not the same at Tau zero when sampled at different
> tau
> zero rates.

At least some of that effect comes down to the improvement in the instrument
floor at lower filter bandwidths, but I agree that not all of it does.  An
example that bothers me is the BVA-vs-H-maser plot at
http://www.leapsecond.com/pages/adev-bw/ compared to the BVA J1 vs J2 plot
right below it.  The latter is mostly determined by the instrument floor,
since J1 and J2 are driven by the same source through two buffers that are
presumably very phase-stable.

Here, the black and red traces are taken at 0.5 Hz and 5 Hz respectively.
The maser-vs-BVA run is at least 10x noisier than the J1-vs-J2 run.  The
divergence in the traces near t=1 second that we see in J1-vs-J2 can be
explained by saying that the instrument floor is lower, but that is not true
of the similar divergence in the BVA-vs-maser plot.  The instrument floor
should have almost no discernible effect on the BVA-vs-maser traces.  

In my own code, I was originally plotting ADEV traces from TimePod
prototypes at taus all the way down to the sampling interval, which gives
attenuated results (or "droop") below the LPF cutoff.   The approach I'm
currently using is to set the ENBW filter cutoff at 0.25x the final phase
data rate, or 0.5*Nyquist, to allow plenty of sampling margin.  "Tau zero"
for plotting purposes is then clipped to 1/(2 * ENBW).  This yields an
artifact-free trace near t0, at least for white noise.

Blue trace is clipped at t0 = 1/ (2 * ENBW):
Blue trace is unclipped, as done earlier: http://www.ke5fx.com/unclipped.png

Compare the blue trace in unclipped.png to figure 11 in Sam Stein's paper at
ities.pdf .  It's clearly the same effect, where the droop is entirely below
t=1/(2*ENBW) seconds.    

Now, I'll admit that I don't entirely follow his reasoning, or understand
why this is sufficient.  In my blue trace, if the ENBW filter's 3 dB cutoff
is 50 Hz, that means there is no good information in the data about what
happens at timescales shorter than 1/20 second.  White noise is certainly no
longer white at that point.  So how can it be valid to plot ADEV at t0
periods down to 0.01s?  Why don't we see LPF-imposed droop in the plot below
t=0.02s?  Instead, the droop begins to show up at 1/(2 * BW), just as
Stein's paper says to expect.  

It's almost as if, in using white noise to demonstrate the 1/(2*t0)
criterion in figure 11, he has forgotten what he wrote a few pages earlier
about the Allan variance of white noise being insensitive to aliasing.  I
seriously doubt he forgot any such thing... but yes, on the face of it, I
agree with you that t0 = 1/(2 * ENBW) is probably not conservative enough
for some noise types.  Certainly the unfiltered 1 ks/s residual noise trace
from the TSC 5120A (magenta) would contain aliased information near t0,
since t0 = 1/sample rate.  (The others are just boxcar-filtered copies of
the same 1 ks/s raw data.)

> The difference between our plots at 1k seconds is because the dual oven
> HP10811 reference osc I'm using is not as good as the LPRO or John's
> HP5065A reference for long term stability.
> The ADEV data can not be any better than the Reference Osc used,
> but still interesting that the 10K and 20K sec numbers from the my red
> plot nearly match John's green LPRO plot.

It might be worthwhile to put two LPROs or similar small rubidium standards
in a tightly controlled environment and benchmark them against each other
for a week or more.  E.g., put them in separately-insulated sections of the
same box.  That should suppress environmental influences, leaving only drift
caused by the Rb cell and other relevant components.  Scaling the xDEV
results by 0.707 should then give a good impression of what can be expected
from one unit when operated in a benign setting.  Repeat with other units to
add confidence as desired.

-- john

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