# [time-nuts] ADEV slopes and measurement mode

Ole Petter Ronningen opronningen at gmail.com
Mon Apr 16 06:12:32 EDT 2018

```Hi, All

This will be a bit long, and I apologize for it. Perhaps someone else also
struggle with the same.

One of the properties of the familiar ADEV-plots are the slopes - and how
the slope identify the dominant noise type for the various portions of the
plots. My understanding is that the slopes stem from "how the noise behaves
under averaging" - White FM follows the usual white noise slope of
1/sqrt(N), or to put it another way; the standard deviation of a set of
many averages of length N will fall as sqrt(N) when N increases. If the
standard deviation of the whole non-averaged list is 1, the standard
deviation of a set of 100-point averages will be 1/10. Other noise types
does not follow the same 1/sqrt(N) law, hence will give rise to other
slopes as the number of points averaged increases as we look further to the
right of the plot.

If I have already droppped the ball, I'd appreciate a correction..

Now onto where I confuse myself (assuming I havent already): measuring
phase vs measuring frequency, and white FM versus white PM. Specifically,
using a frequency counter to either measure the frequency directly, or
setting up a time interval measurement to measure phase.

to the measurements, my initial reasoning was as follows:
* When measuring frequency, the quantization noise is added to the
frequency estimate. This, when converted to phase in the adev calculations,
will result in random walk in the phase plot - and hence a slope of -0.5.
"Two over, one down".
* When measuring the phase directly in a time interval measurement, the
noise is added to the phase record, and will not result in random walk in
phase - and hence a slope of -1. "One over, one down"

This is in fact not what happens - or kinda, it depends, see below. As
someone on the list helpfully pointed out; as the gate time of your
frequency measurements increase, the noise stays the same. So the slope is
-1. This makes perfect sense when measuring phase, and is consistent with
my reasoning above - the way to "average" a phase record is simply to drop
the intermediate datapoints.

My confusion concerns measuring frequency. I can not see why the slope is
-1, and I am confused for two reasons:

1. The gate time as such is not increasing - we are averaging multiple
frequency readings, all of them has noise in them which, in my mind, when
converted into phase should result in random walk in phase and follow the
-.5 slope.
2. Trying to get to grips with this, I also did a quick experiment - which
only increased my confusion:

I set up my 53230a to measure a BVA-8600 against my AHM, in three modes,
tau 1 second, each measurement lasting one hour:
1. Blue trace: Time interval mode, the slope on the left falls with a slope
of -1, as expected.
2. Pink trace: Frequency mode, "gap free"; the slope on the left also falls
with a slope of -1, somewhat confusing given my reasoning about random walk
in phase, but could maybe make sense somehow.
3. Green trace: Frequency mode, "RECiprocal" - *not* gap free. The slope on
the left falls with a slope of -0.5. Very confusing, given the result in
step 2. Random walk is evident in the phase plot.

Since the 53230A has some "peculiarities", I am not ruling out the counter
as the source of the difference in slopes for gap free vs not gap free
frequency measurement - although I can see how gaps in frequency leads to
random walk in the phase record and a slope of -0.5 - I just cant see how a
gap free frequency record (where every frequency estimate contains an
error) does *not* result in RW in phase.. :)

So my questions are:
1. Does gaps in the frequency record always lead to random walk in phase
and a slope of -0.5, is this a "known truth" that I missed? Or is this
likely another artifact of the 53230A?
2. Is my understanding of how the slopes arise, they show how the noise
types "behave under averaging", correct?
3. Depending on the answer to 1; What am I missing in my understanding of
white PM vs white FM - why does gap free frequency measurement not lead to
a slope of -0.5?

Thanks for any and all insight!
Ole

---------
makes sense that the slope is -1; the zerocrossings actually *counted* by
the frequency counter is not subject to noise, it is a nice integer. It is
only the start and stop interpolated cycles that are subject to
quantization noise, and as more and more frequency estimates are averaged,
the "portion" of the frequency estimate that is subject to the noise
decreases linearly. I suppose.
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