[time-nuts] Question about noisetypes and ADEV

Attila Kinali attila at kinali.ch
Mon Oct 29 09:59:08 EDT 2018


Moin,

I'm bunching a few mails together, to not clutter the mailinglist too much

On Sat, 27 Oct 2018 23:25:30 +0200
Magnus Danielson <magnus at rubidium.dyndns.org> wrote:

> The integration is very important aspect, as a number of assumptions
> becomes embedded into it, such as the f_H frequency which is the Nyquist
> frequency for counters, so sampling interval is also a relevant
> parameter for expected level.

An important thing to note here is that Gaussian white noise is,
as it is defined, non-continuous (by any continuity measure).
Ie if you take two samples, no matter how close they are time-wise,
their difference in value can be arbitrary large. If you are integrating
over (time-continuous) Gaussian white noise, you have to argue
carefully, why this integral is defined (meaning why calculating it
leads to a single, well defined value). In our case, it's usually
enough to assume that there is a finite cut-off frequency at which
point the signal falls off with at least 1/f^2 (or >=40dB/dec) to
ensure 1) continuity and 2) convergence of the integral.

For more details, see a textbook on Ito-calculus, e.g. [1]

On Sat, 27 Oct 2018 23:43:33 +0200
Magnus Danielson <magnus at rubidium.dyndns.org> wrote:

> A simple trick to transform uniform distribution to normal distribution
> like shape is to take 12 samples and add them together. A special trick
> is to take them pair-wise and subtract them and then add 6 differences,
> to avoid DC bias of typical uniform distribution generation (as typical
> pseudo-noise generators does not have all 0 state in them). The result
> of this subtract-add trick is a normal distribution like thing with the
> standard distribution of 1. More or fewer sample-pairs can be added if
> the product is scaled appropriately.
> 
> The Box-Mueller algorithm is another way to convert uniform distribution
> to normal distribution.

Please, plase, please, do not use "just 12 samples and add them together"
as an approach for generating normal distributed values! Even if it will
get you something that looks like a normal distribution, it's quite far
from it. It is also a very slow method and uses up a lot of randomnes.

Box-Müller is a usable alternative, though I would recommend using
the Ziggurat Method[2], which is very fast and leads to a very good
approximation. When I replaced the "take 30 samples and add them" of
François Vernotte's Sigmatheta package[3] and used the Ziggurat Method,
combined with xorshift1024*[4] for random number generation, I got
a total speed up of a factor of more than 2 (including the FFT and
everything)[5] (yes, I know that xorshift1024* does have some problems
in the quality of random numbers generated, but they shouldn't be
relevant for the application at hand).


			Attila Kinali

[1] "Stochastic Differential Equations", by Bernt Øksendal, 2013 (6th ed)

[2] "The Ziggurat Method for Generating Random Variables", 
by Marsaglia and Tsang, 2000
http://dx.doi.org/10.18637/jss.v005.i08

[3] https://theta.obs-besancon.fr/spip.php?article103&lang=fr

[4] http://xoshiro.di.unimi.it/ 
or more specifically: http://xoroshiro.di.unimi.it/xorshift1024star.c

[5] https://git.kinali.ch/attila/sigmatheta
-- 
It is upon moral qualities that a society is ultimately founded. All 
the prosperity and technological sophistication in the world is of no 
use without that foundation.
                 -- Miss Matheson, The Diamond Age, Neal Stephenson



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