[time-nuts] Question about noisetypes and ADEV

[Magnus Danielson]

Hi Attila,

On 10/29/18 2:59 PM, Attila Kinali wrote:
> 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.

There is aspects of noise which is more or less important depending on
what you do. As we leave WPM it is no longer gaussian anyway. For ADEV
and friends the shape of the PDF isn't as important as for other things.
The slope of the frequency range is however the important one. It is
only when we do the confidence intervals where the Gaussian shape
becomes relevant for the Chi-square bounds, but those are usually not
precise enough that even rough Gaussian shape is relevant. Even for
noises with none-Gaussian properties, the Chi-square seems to be valid
enough.

For other measures, like bit-error simulations, proper Gaussian shape is
much more important, but only to a certain point. For higher BER values,
the details of the outer part of the shape isn't all that important,
it's only as you push into lower BER numbers you need to care.

> 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.

Actually, for many simulations you do not need better "shape".
There is some simulations where shape comes in, but others where it has
little to no consequence.

> 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).

Getting suitable PRNG polynomials isn't all that hard, if the length of
the "random" sequence is of concern compared to the length of the
sequence used. It's a solved problem.

Never the less, thanks for the many references. Will read up on them
eventually.

Cheers,
Magnus

> 
> 			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
>