# Random numbers with exacty zero autocorrelation

Started by September 4, 2006
```robert bristow-johnson said the following on 05/09/2006 17:04:
> Oli Filth wrote:
>> Andre said the following on 04/09/2006 15:54:
>>
>>> You may consider using maximum length sequences instead.
>>
>> Maximum length sequences don't have zero autocorrelation anywhere.
>
> they can be biased upward a little so that they are zero everywhere
> except the zero lag and lags that are an integer multiple of the
> period.

How so?  Surely MLS are always odd in length, so they must always
contain a slight DC offset, and hence a slight DC offset in the
autocorrelation?

>
> but that's using *circular* autocorrelation (that's how the edge
> effects are dealt with, by wrapping it around.

--
Oli
```
```Oli Filth said the following on 05/09/2006 17:33:
> robert bristow-johnson said the following on 05/09/2006 17:04:
>> Oli Filth wrote:
>>> Andre said the following on 04/09/2006 15:54:
>>>
>>>> You may consider using maximum length sequences instead.
>>>
>>> Maximum length sequences don't have zero autocorrelation anywhere.
>>
>> they can be biased upward a little so that they are zero everywhere
>> except the zero lag and lags that are an integer multiple of the
>> period.
>
> How so?  Surely MLS are always odd in length, so they must always
> contain a slight DC offset, and hence a slight DC offset in the
> autocorrelation?

Ah, of course.  You mean by subtracting the DC value from each sample
(with the proviso that the samples are no longer {+1,-1}).

--
Oli
```
```MY SUGGESTION
go for chaotic map generators.

logical map
bernoulli map
or other chaotic maps
which generate almost near to zero A.C
refere kenichi kitayama paper of AFRICON 2002

of related properties of CHAOTIC SEQUENCES

regards
particle

```
```jeroen_trip wrote:

> I'm trying to simulate a small sample, say 15 numbers, of white noise. But
> due to chance, the autocorrelation is not exactly 0. Of course the standard
> deviation is also not exacty 1 but I can fix that. Is there transformation
> to get rid of the unwanted autocorrelation??

I would say that if you did that, that they aren't random anymore.

Well, at least one of them isn't, and another one if you normalize
the standard deviation.

Zero autocorrelation and the appropriate standard deviation is only
true in the limit as the length of the sequence goes to infinity.

-- glen

```
```glen herrmannsfeldt <gah@ugcs.caltech.edu> writes:

> jeroen_trip wrote:
>
>
>> I'm trying to simulate a small sample, say 15 numbers, of white noise. But
>> due to chance, the autocorrelation is not exactly 0. Of course the standard
>> deviation is also not exacty 1 but I can fix that. Is there transformation
>> to get rid of the unwanted autocorrelation??
>
> I would say that if you did that, that they aren't random anymore.
>
> Well, at least one of them isn't, and another one if you normalize
> the standard deviation.
>
> Zero autocorrelation and the appropriate standard deviation is only
> true in the limit as the length of the sequence goes to infinity.

I agree with you, Glen (sorry, I just can't bring myself to write
"glen").

I think the key here is to realize that the probabilistic
autocorrelation function is not a time-domain function. It
is computed from the joint PDF(s) of the underlying random
variables.

The OP is attempting to apply function-based autocorrelation
in order to gain probabilistic results, and it just ain't
gonna happen.
--
%  Randy Yates                  % "Though you ride on the wheels of tomorrow,
%% Fuquay-Varina, NC            %  you still wander the fields of your
%%% 919-577-9882                %  sorrow."
%%%% <yates@ieee.org>           % '21st Century Man', *Time*, ELO
```
```Randy Yates wrote:

...

> The OP is attempting to apply function-based autocorrelation
> in order to gain probabilistic results, and it just ain't
> gonna happen.

"The generation of random numbers is too important to be left to chance
- Robert R. Coveyou" is from http://www.firstpr.com.au/dsp/rand31/ which
might be worth reading for other reasons.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```Jerry Avins wrote:

(snip)

> "The generation of random numbers is too important to be left to chance
> - Robert R. Coveyou" is from http://www.firstpr.com.au/dsp/rand31/ which
> might be worth reading for other reasons.

Somewhere in Knuth's "Art of Computer Programming" he has an example
of random number generators being left to chance.  He designs a very
complicated generator which after a short while settles on a nice short
repeating pattern.

-- glen

```