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Is a signal containing random numbers White Noise?

Started by Chris Barrett January 11, 2007
On Jan 12, 8:45 am, Chris Barrett
<chrisbar...@0123456789abcdefghijk113322.none> wrote:
> Chris Barrett wrote: > > Let's say my audio is represented by a series of numbers and each number > > has a random value. Is my audio white noise? I think it is, but I'm > > having trouble proving it to my self.To go slightly off-topic: > > What is the basic (or not so basic) definition of gausian noise is? Is > it noise that has a bell shaped spectrum?
"Gaussian" doesn't describe the (frequency) spectrum. It describes the PDF (probability density function) of the individual sample values. -- Oli
Oli Charlesworth wrote:

   ...

>> What is the basic (or not so basic) definition of gausian noise is? Is >> it noise that has a bell shaped spectrum? > > "Gaussian" doesn't describe the (frequency) spectrum. It describes the > PDF (probability density function) of the individual sample values.
To put it into the terms in which the question was asked, The distribution of values -- the shape of a bar graph that bins the values -- is bell shaped. The spectrum of sampled noise depends on how each sample relates to those around it. The PDF -- probability density function -- depends on the shape of that bar graph I mentioned above. Most random number generators provide a uniform distribution: any number is as likely as any other. The sum of two uniformly distributed random numbers is not uniformly distributed, but triangular*. Honest playing dice generate a uniform distribution in the range 1 to 6. In the sum of two dice, middle-size numbers are more common than high or low ones. Jerry _____________________________________________ * The more in the sum, the closer to Gaussian the result becomes. -- 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;
On 12 Jan 2007 07:41:55 -0800, "Ikaro" <ikarosilva@hotmail.com> wrote:

>I don't think this will work. Like other have pointed out, flat >spectrum is a necessary, but not a sufficient condition for a white >noise process. > >I'd imagine you will also need to do some tests on the phase spectrum >to further determine if the process is noise or a determistic signal >(but you don't have phase information on the PSD). > >Also, what I previously meant was a statistical test for white noise >(some test done on a time series that will yield an alpha value >indicating how confident you are that the data is white noise). >..
It is impossible to check absolutely if the signal is random or deterministic. A good cryptographic pseudo random noise generator is completely deterministic. But unless you know the key, it is indistinguishable from true randomness. Some simple pseudo random number generators may fail some simple statistical tests, but mere failure does not guarantee that the source is random. Robert Scott Ypsilanti, Michigan
Oli Charlesworth wrote:
> > > Autocorrelation function and PSD are a Fourier transform pair. >
Yes this was proven and published by Einstein back in 1914 - long before Weiner or Khinchin.
I don't think this  will work. Like other have pointed out, flat
spectrum is a necessary, but not a sufficient condition for a white
noise process.

I'd imagine you will also need to do some tests on the phase spectrum
to further determine if the process is noise or a determistic signal
(but you don't have phase information on the PSD).

Also, what I previously meant was a statistical test for white noise
(some test done on a time series that will yield an alpha value
indicating how confident you are that the data is white noise).
Like a mentioned before, I don't think that t-test on the amplitude of
the bins in the magnitude spectrum will work.

-Ikaro


Jerry Avins wrote:
> Ikaro wrote: > > Hey, > > > > Like others pointed out, you can't determine if it is white simply by > > looking at the amplitude distribution. > > > > There are statistical tests to determine if a signal is random (like > > Runs Test and the Turning Points Methods). > > > > However I am not aware of any statistical test to determine if a > > sequence is white or not (Anyone here??)... > > Test for a flat spectrum. > > ... > > Jerry > -- > Engineering is the art of making what you want from things you can get. > =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=
=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF= =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF
Ikaro wrote:
> I don't think this will work. Like other have pointed out, flat > spectrum is a necessary, but not a sufficient condition for a white > noise process. > > I'd imagine you will also need to do some tests on the phase spectrum > to further determine if the process is noise or a determistic signal > (but you don't have phase information on the PSD). > > Also, what I previously meant was a statistical test for white noise > (some test done on a time series that will yield an alpha value > indicating how confident you are that the data is white noise). > Like a mentioned before, I don't think that t-test on the amplitude of > the bins in the magnitude spectrum will work. > > -Ikaro > > > Jerry Avins wrote: >> Ikaro wrote: >>> Hey, >>> >>> Like others pointed out, you can't determine if it is white simply by >>> looking at the amplitude distribution. >>> >>> There are statistical tests to determine if a signal is random (like >>> Runs Test and the Turning Points Methods). >>> >>> However I am not aware of any statistical test to determine if a >>> sequence is white or not (Anyone here??)...
>>
>> Test for a flat spectrum.
I thought we were looking to determine if a noise signal is white, not to determine if an arbitrary signal is white noise. 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;
>> "Gaussian" doesn't describe the (frequency) spectrum. It describes the >> PDF (probability density function) of the individual sample values. > >To put it into the terms in which the question was asked, The >distribution of values -- the shape of a bar graph that bins the values >-- is bell shaped. > >The spectrum of sampled noise depends on how each sample relates to >those around it. The PDF -- probability density function -- depends on >the shape of that bar graph I mentioned above. > >Jerry >_____________________________________________ >* The more in the sum, the closer to Gaussian the result becomes. >-- >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; >
So you mean that the values of the random numbers tend to be closer to the value of the mean as can be seen from the bell curve? If so, what is the mean in this case - the mean of all the random numbers generated?
doggie wrote:
>>> "Gaussian" doesn't describe the (frequency) spectrum. It describes the >>> PDF (probability density function) of the individual sample values. >> To put it into the terms in which the question was asked, The >> distribution of values -- the shape of a bar graph that bins the values >> -- is bell shaped. >> >> The spectrum of sampled noise depends on how each sample relates to >> those around it. The PDF -- probability density function -- depends on >> the shape of that bar graph I mentioned above. >> >> Jerry >> _____________________________________________ >> * The more in the sum, the closer to Gaussian the result becomes. >> -- >> 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; >> > > So you mean that the values of the random numbers tend to be closer to the > value of the mean as can be seen from the bell curve? If so, what is the > mean in this case - the mean of all the random numbers generated?
The mean depends on the data. The expected mean -- not necessarily even close to the measured mean for small sets -- depends on the way the data are generated. Many processes are "zero mean"; some of those that aren't are transformed to that by subtracting the mean from every instance. A playing die, with its possibilities being the integers from 1 to 6 has an expected mean of 3.5. Therefore, the sum of two dice has an expected mean of 7. But the lowest and highest possible sums are 2 and 12, with expectations of 1/36 each, while the expectation for 7 is 1/6. You can make a pretty good approximation of a zero-mean Gaussian random sequence by subtracting 35 from the sum of 10 dice and scaling the result to suit the problem. 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;
"Carlos Moreno" <moreno_at_mochima_dot_com@mailinator.com> wrote in message 
news:gJAph.12310$U04.181746@weber.videotron.net...
<snipped without loss of context I hope?>
> Maybe you could give it a second try at making me understand > what I perceive as a contradiction --- I mean, my point is that > white noise *does not* have infinite power (however, this is > directly contradicting by the fact that PSD is really that: > density *of power*, which would directly imply that the power > has to be infinite for white noise) --- more specifically, the > reason why white noise can not exist in practice *is not* that > it has infinite power; the reason is so much simpler than that! > W.N. can not exist because it simply can't --- no signal *in our > reality* can vary with *truly infinite* rate of change; W.N. is > *necessarily* a theoretical construct --- albeit one very useful, > like Dirac's delta, or complex numbers. >
Hi Carlos - If it doesn't matter how close together in time your sample points are then they aproach 0 seconds separation and your power , even with limited amplitude change _is_ infinite isn't it? I can't see any discrepancy other than that between what you can think of and what can actually be observed. Best of Luck - Mike