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Is a signal containing random numbers White Noise?

Started by Chris Barrett January 11, 2007
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.
Fitlike Min wrote:
> "Jerry Avins" <jya@ieee.org> wrote in message > news:PbSdneaEU72yWDvYnZ2dnUVZ_t7inZ2d@rcn.net... > >>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. >>&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095; > > > I don't think a flat spectrum telsl you if it's random or not - you need > autocorrelation. > > F. > > >
The spectrum is very jagged. In general it looks the same as the white noise I downloaded from Wikipedia. I think for a sequence N elements long, the auto-correlation is: auto-correlation = x[n] * x[N-n] The plot of this looks like a snake with a needle stuck in its side. If my text formats correctly that should look like this: | | ------==========='===========------
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.
Maybe, maybe not. It depends on the distribution. If you think "Gaussian" when you think white, then probably not. 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;
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.
If your audio is represented by a time series of random numbers, and each value is independent of all the others, then the noise is white. As Jerry mentioned, don't mistake the probability distribution (e.g. Gaussian, uniform, Poissan) with the spectral coloring. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Posting from Google? See http://cfaj.freeshell.org/google/ "Applied Control Theory for Embedded Systems" came out in April. See details at http://www.wescottdesign.com/actfes/actfes.html
Jerry Avins 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. > > Maybe, maybe not. It depends on the distribution. If you think > "Gaussian" when you think white, then probably not. > > Jerry
Hey Jerry, one of the rare times I get to correct you ;-). I think you meant that it doesn't matter what the distribution is, whether a sequence is white or not. For the poster, a white random sequence can be Gaussian, exponential, or what-have-you.
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??)...

If you really wanted to determine if you sequence is white or not, one
test that you can do (my suggestion) is to model your sequence as an
auto-regressive sequence of size say 2. If all coefficients in the
model are zero than you can assume the sequence to be originating from
a white noise source. Note that you will have to come up with numerical
intervals on the coefficient value to determine when small enough
numbers can be treated as zeros (ie, round off errors).

-Ikaro



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.
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.
You're missing specs --- but most likely, the answer is yes. Most likely in that probably (no pun intended) when you say "each number has a random value" you mean that each number is independent from the others. In that case, the noise is white. If the samples are independent, then they are uncorrelated; and the autocorrelation function is related to the PSD function by a Fourier Transform relationship. Samples uncorrelated implies that the autocorrelation function is a delta, and the Fourier transform of a delta is a constant value from -oo to +oo ==> BINGO!! It *is* white noise. Notice that the condition of independent samples is actually more than the minimum required --- it is sufficient that the samples be uncorrelated (independent is a "superset" of uncorrelated, except for Gaussian distributions). Carlos --
julius wrote:
> Jerry Avins 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. >> Maybe, maybe not. It depends on the distribution. If you think >> "Gaussian" when you think white, then probably not. >> >> Jerry > > Hey Jerry, one of the rare times I get to correct you ;-). > I think you meant that it doesn't matter what the distribution is, > whether a sequence is white or not. For the poster, a white > random sequence can be Gaussian, exponential, or what-have-you.
That contradicts nothing I _intended_ to write. Suppose q[n] are samples of white noise, where - <= q[n] < 1. Then 1/q[n] also samples of white noise, but it seems a bit strange 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;
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. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
just to be clear guys, truly "white" noise has infinite power, and i
doubt that Chris's random numbers have infinite variance.

random number generators usually output uniform PDF pseudo-random
numbers that are "virtually" independent of each other.  these numbers
are hypothetically "sampled" from some hypothetically continuous-time
random process that we hypothesize is bandlimited to the Nyquist
frequency.  call these random numbers "samples", x[n].  if you were to
reconstruct the bandlimited output with these samples:

     x(t) = SUM{ x[n] * sinc(t - nT) }
              n

since all of the sinc functions are explicitly bandlimited to Nyquist,
the sum of them is also.  then x(t) is "white" UP TO NYQUIST (if them's
are good random numbers), not from -inf to +inf.

true "white" noise doesn't exist anywhere, not because it is
conceptually impossible to make it flat over a given finite range of
frequencies, but because it can't be flat all the way to infinity (and
that's because it would be infinite power).

r b-j