[time-nuts] theoretical Allan Variance question

Michael Wouters michaeljwouters at gmail.com
Sat Oct 29 21:59:09 EDT 2016

Dear Stuart

In a perfect world, your TI measurements would have a uniform
probability distribution function with amplitude 0.5 ns and mean value
0 ns. At least, this is the kind of PDF you would assume for
"resolution error". For this distribution, ADEV is 0.5 ns.

I don't know the HP5334B, perhaps its effective resolution is a bit
poorer than 0.5 ns, which I assume is the displayed resolution ?

BTW, one way to remove instability of the DUTs from this kind of test
is to reference the counter to 5/10 MHz and then do your TI
measurement with the same 5/10 MHz input to each channel of the


On Sun, Oct 30, 2016 at 10:38 AM, Stewart Cobb <stewart.cobb at gmail.com> wrote:
> What's the expected value of ADEV at tau = 1 s for time-interval
> measurements quantized at 1 ns?
> This question can probably be answered from pure theory (by someone more
> mathematical than me), but it arises from a very practical situation. I
> have several HP5334B counters comparing PPS pulses from various devices.
> The HP5334B readout is quantized at 1 ns, and the spec sheet (IIRC) also
> gives the instrument accuracy as 1 ns.
> The devices under test are relatively stable. Their PPS pulses are all
> within a few microseconds of each other but uncorrelated.  They are stable
> enough that the dominant error source on the ADEV plot out to several
> hundred seconds is the 1 ns quantization of the counter. The plots all
> start near 1 ns and follow a -1 slope down to the point where the
> individual device characteristics start to dominate the counter
> quantization error.
> One might expect that the actual ADEV value in this situation would be
> exactly 1 ns at tau = 1 second.  Values of 0.5 ns or sqrt(2)/2 ns might not
> be surprising. My actual measured value is about 0.65 ns, which does not
> seem to have an obvious explanation.  This brings to mind various questions:
> What is the theoretical ADEV value of a perfect time-interval measurement
> quantized at 1 ns? What's the effect of an imperfect measurement
> (instrument errors)? Can one use this technique in reverse to sort
> instruments by their error contributions, or to tune up an instrument
> calibration?
> I'd be grateful for answers to any of these questions.
> BTW, thanks to whichever time-nuts recommended the HP5334B, back in the
> archives; they're perfect for what I'm doing. And thanks to fellow time-nut
> Rick Karlquist for his part in designing them.
> Cheers!
> --Stu
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