# [time-nuts] Question about noisetypes and ADEV

Magnus Danielson magnus at rubidium.dyndns.org
Sat Oct 27 17:33:18 EDT 2018

```Hej Attila,

On 10/26/18 11:50 AM, Attila Kinali wrote:
> Hej Ole,
>
> On Fri, 26 Oct 2018 11:34:41 +0200
> Ole Petter Ronningen <opronningen at gmail.com> wrote:
>
>> I'm simulating some noise to try to improve my somewhat sketchy
>> understanding of what goes on with the various noise types as shown on an
>> ADEV plot. Nothing fancy, ~3600 points of gaussian random numbers between 0
>> and 1 in excel, imported into Timelab as phase data, scaled to ns.
>>
>> I mostly get what I expect; "pure" random noise, gives the expected slope
>> for W/F PM, -1. Integrating the same random data gives the expected slope
>> for W FM -1/2. Integrating the same random data yet again gives a slope
>> of +1/2, again as expected for RW FM.
>
> I see two issues here: If your random numbers are indeed between 0 and 1,
> as you write, then they are uniformly distributed, and not normally
> distributed. This will give you a slight bias when integrating.

Yes and no. It will not be relevant for ADEV values, but it will be
relevant for the ADEV confidence intervals.

> The other issue is the integration itself. Because you do a nummerical
> integration at discrete time steps, you will get a slight offset.
> This is something I stumbled over as well. I probably need to sit
> down once, and figure out how big that offset should be and compare
> it to what the numerical integration shows. The people here who are
> more knowledgable than me about nummerical computation should be able to
> give you a better answer.

It's very simple. Integration in continuous time and in discrete time
has somewhat different properties. I'm surprised that a stupid boy like
me needs to point out to professional when they fluked on the
difference, in peer-reviewed articles even. If you compare continuous
time integration values with discrete time values you can expect a
difference, as the devil is in the details.

Now, to make it clear, ADEV is actually a discrete time measure, which
we can approximate continuous time processes and behaviors with.

Numerical integration is really not as such a bias-generating thing. You
can end up with a scale error, so due care needs to be taken, but that
ends up being fairly obvious once you think about things.

Cheers,
Magnus

```