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Random numbers with exacty zero autocorrelation

Started by jeroen_trip September 4, 2006
Hi all,

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??

Many thanks,

Jeroen


jeroen_trip said the following on 04/09/2006 14:43:
> Hi all, > > 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??
Firstly, how are you defining autocorrelation on such a small sequence? Are you taking edge effects into account? Secondly, why is it important that the autocorrelation for a given trial be exactly zero (I assume you actually mean a delta function)? Usually, all we care about is that the *expected* autocorrelation is zero, i.e. the "average" of the autocorrelations of all trials of the experiment. -- Oli
You may consider using maximum length sequences instead.

Oli Filth wrote:
> jeroen_trip said the following on 04/09/2006 14:43: >> Hi all, >> >> 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?? > > Firstly, how are you defining autocorrelation on such a small sequence? > Are you taking edge effects into account? > > Secondly, why is it important that the autocorrelation for a given trial > be exactly zero (I assume you actually mean a delta function)? Usually, > all we care about is that the *expected* autocorrelation is zero, i.e. > the "average" of the autocorrelations of all trials of the experiment. > >
-- For emailing me, please exchange abuse by a . l o d w i g

jeroen_trip wrote:
> Hi all, > > I'm trying to simulate a small sample, say 15 numbers, of white noise.
Any sequence of random numbers can be considered as a white noise.
> due to chance, the autocorrelation is not exactly 0.
Certainly. It will never be 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??
If you need short sequences with good autocorrelation properties, you may want to look at Barker codes. Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
Andre said the following on 04/09/2006 15:54:
> Oli Filth wrote: >> jeroen_trip said the following on 04/09/2006 14:43: >>> Hi all, >>> >>> 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?? >> >> Firstly, how are you defining autocorrelation on such a small >> sequence? Are you taking edge effects into account? >> >> Secondly, why is it important that the autocorrelation for a given >> trial be exactly zero (I assume you actually mean a delta function)? >> Usually, all we care about is that the *expected* autocorrelation is >> zero, i.e. the "average" of the autocorrelations of all trials of the >> experiment. >> > You may consider using maximum length sequences instead.
Maximum length sequences don't have zero autocorrelation anywhere. -- Oli
>Firstly, how are you defining autocorrelation on such a small sequence? > Are you taking edge effects into account? > >Secondly, why is it important that the autocorrelation for a given trial
>be exactly zero (I assume you actually mean a delta function)? Usually,
>all we care about is that the *expected* autocorrelation is zero, i.e. >the "average" of the autocorrelations of all trials of the experiment. > > >-- >Oli >
Hi Oli, I'm defining autocorrelation simply as average (xt-mu)*(xt-1-mu). I'm using it for simulation purposes. If I can get the autocorrelation to be exactly 0 I can use fewer simulations. Jeroen
Yes, but knowing it is exactly -1 everywhere except at t=0 might help as 
well. The OP just wants to "get rid" of it, which could mean adding 1 is 
acceptable for him to do.

Oli Filth wrote:
> Andre said the following on 04/09/2006 15:54: >> Oli Filth wrote: >>> jeroen_trip said the following on 04/09/2006 14:43: >>>> Hi all, >>>> >>>> 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?? >>> >>> Firstly, how are you defining autocorrelation on such a small >>> sequence? Are you taking edge effects into account? >>> >>> Secondly, why is it important that the autocorrelation for a given >>> trial be exactly zero (I assume you actually mean a delta function)? >>> Usually, all we care about is that the *expected* autocorrelation is >>> zero, i.e. the "average" of the autocorrelations of all trials of the >>> experiment. >>> >> You may consider using maximum length sequences instead. > > > Maximum length sequences don't have zero autocorrelation anywhere. > >
jeroen_trip wrote:
> Hi all, > > 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??
Randomness is a tricky concept. If you could devised an RNG that always produces zero autocorrelation instead of random variation, you would know it was far from random. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
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. but that's using *circular* autocorrelation (that's how the edge effects are dealt with, by wrapping it around. if the OP wants to take a look at the math surrounding the MLS, there's a little primer (or tutorial, i dunno which), at http://www.dspguru.com/info/tutor/mls.htm . Vladimir Vassilevsky wrote:
> > Any sequence of random numbers can be considered as a white noise.
so a sequence of random numbers we call "pink noise" can be considered as a white noise?
> > due to chance, the autocorrelation is not exactly 0. > > Certainly. It will never be 0.
a pseudo-random number sequence can conceivably be constructed to do that. perhaps there is an old Bell Systems Journal that spells out how. i wouldn't be so certain as you, Vladimir. r b-j
robert bristow-johnson wrote:
> 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.
another thing that i forgot to point out is that the MLS is hardly a random number since the sequence is all -1 and +1 (plus the small bias mentioned above). if you look at the p.d.f. function, it hardly looks random having two dirac deltas at +/- 1.