# Is a signal containing random numbers White Noise?

Started by January 11, 2007
```robert bristow-johnson wrote:

> [...]
> 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).

About two or three years ago, I brought up this issue in comp.dsp
(prompted by a lecture on white noise in a Probability and Random

I have to say, the discussion ended, and I was utterly unable to
reconcile what, to the present day, I perceive as an unreconcileable

I mean, putting aside practical aspects, and the *true reality* of
the universe (in which the mathematical *continuos* model of things
does not *truly* represent the reality, etc. etc....  Let me
emphasize again that I'm putting all that aside).

The thing is, a signal limited to the range [-1,1] *can* be white
noise, in the mathematical definition sense.

My "intuitive" interpretation of white noise comes from the fact
that any two samples of the signal --- no matter how arbitrarily
close they are --- are uncorrelated.  Thus, te picture in my head
of white noise is a signal that, even being continuos-time by nature,
is simply a set of spread *dots*  (well, a plot of such signal would
be an infinite set of dots).  That is, from one point in time, it
changes infinitely fast (with a slew-rate that is infinity) to a
different value, taken at random independently from the present
value (well, not necessarily independently --- but uncorrelated
at least)

The thing is:  you *define* a signal as having values restricted
to [-1,1], but with the property that *any two* samples of the
signal (emphasis on the "no matter how arbitrarily close they
are" part) are uncorrelated random variables, and then, by definition
(well, by Wiener's Theorem, I believe), since the PSD is the Fourier
transform of the Autocorrelation, and since the autocorrelation
*is* a delta, then the spectrum of such signal *is flat* from
-oo to +oo.

At some point in time, I was convinced that the units of the PSD
(despite the *Power* in the acronym) was really *energy* per
Hertz --- however, it was brought to my attention in this
newsgroup that that is not the case.  Which left me with two
unreconcileable properties (unreconcileable because they're both
*in the mathematical/theoretical domain* --- not being able to
reconclie something from the real world with something theoretical,
well, that's easy;  that happens all the time, if we are strict
enough).

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.

Carlos
--
```
```"Chris Barrett" <chrisbarret@0123456789abcdefghijk113322.none> wrote in
message news:Nwxph.1353\$qA7.1219@newsfe15.lga...
> 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.

Take the autocorrelation. If it's an impulse then it's white - though it may
not necessarily be Guassian white etc (this has been explained).

F.

--
Posted via a free Usenet account from http://www.teranews.com

```
```"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.

--
Posted via a free Usenet account from http://www.teranews.com

```
```On Fri, 12 Jan 2007 00:42:10 -0000, Fitlike Min <Fitlike@naeoption.com>
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.
>
> I don't think a flat spectrum telsl you if it's random or not - you need
> autocorrelation.

Autocorrelation function and PSD are a Fourier transform pair.

--
Oli
```
```Carlos Moreno wrote:
>
> The thing is, a signal limited to the range [-1,1] *can* be white
> noise, in the mathematical definition sense.

sure, in *continuous time*.  it can't be gaussian, but p.d.f. and
spectrum are not related properties of a random process.

> My "intuitive" interpretation of white noise comes from the fact
> that any two samples of the signal --- no matter how arbitrarily
> close they are --- are uncorrelated.  Thus, te picture in my head
> of white noise is a signal that, even being continuos-time by nature,
> is simply a set of spread *dots*  (well, a plot of such signal would
> be an infinite set of dots).  That is, from one point in time, it
> changes infinitely fast (with a slew-rate that is infinity) to a
> different value, taken at random independently from the present
> value (well, not necessarily independently --- but uncorrelated
> at least)

that might be accurate.  it's continuous-time, but not necessarily
continuous.

> The thing is:  you *define* a signal as having values restricted
> to [-1,1], but with the property that *any two* samples of the
> signal (emphasis on the "no matter how arbitrarily close they
> are" part) are uncorrelated random variables, and then, by definition
> (well, by Wiener's Theorem, I believe), since the PSD is the Fourier
> transform of the Autocorrelation, and since the autocorrelation
> *is* a delta, then the spectrum of such signal *is flat* from
> -oo to +oo.

but you can't represent that as samples in a discrete-time context
because what you are sampling is not bandlimited.  now imagine running
this ideal white noise into an ideal brickwall filter with cutoff
frequency of 1/(2T).  now you have a bandlimited signal that is a
random process, but, although the spectrum is flat from -1/(2T) to
+1/(2T), it's no longer white.  but that hypothetical continuous-time
signal can be sampled and the samples are sufficient to describe it
since Nyquist is happy with it. it also has a finite power that is
proportional to the sampling frequency, 1/T.

it also has (before sampling) an autocorrelation function that is not
the dirac delta (which is the autocorrelation of the impossible true
white noise).  that autocorrelation function is the inverse Fourier
Transform of the A*rect(fT) function that is your power spectrum (i
think most textbooks say A = nu/2), the (A/T)*sinc(t/T) function.  now
(A/T)*sinc(0/T) = A/T is the finite power of the bandlimited function
and is also the mean square of the voltage at any fixed time.  if the
DC is zero, that mean square is also the variance which is the AC power
of the bandlimited random signal.  but the other important thing we get
from the autocorrelation is that x(t) is not correlated to x(t+nT) for
any non-zero integer n (becauase the autocorrelation function is zero).
that means, since you sample this hypothetical random signal at
integer multiples of T, none of your samples are correlated to each
other.  "not correlated" is a weaker condition than "independent", but
at least you can say that samples that are generated from a good random
number generator and are independent are also uncorrelated and then can
be thunk of as being sampled from a hypothetical random process that
has flat spectrum from -1/(2T) to 1/(2T) and is zero outside of that
range (so Nyquist is satisfied).

> At some point in time, I was convinced that the units of the PSD
> (despite the *Power* in the acronym) was really *energy* per
> Hertz --- however, it was brought to my attention in this
> newsgroup that that is not the case.

not for a finite power signal.  random noise are power signals just
like sinusoids.  but an exponential pulse is a finite energy signal.
you gotta deal with these two classes just a little slightly
differently.  my A. Bruce Carlson Communication Systems text does a
good job spelling that out.

>  Which left me with two
> unreconcileable properties (unreconcileable because they're both
> *in the mathematical/theoretical domain* --- not being able to
> reconclie something from the real world with something theoretical,
> well, that's easy;  that happens all the time, if we are strict
> enough).
>
> 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. has infinite power.

> W.N. can not exist because it simply can't --- no signal *in our
> reality* can vary with *truly infinite* rate of change;

with infinite power you can cause an infinite rate of change.  you can
even conceptually do that with finite power, such as a square wave or
sawtooth.

>  W.N. is
> *necessarily* a theoretical construct --- albeit one very useful,
> like Dirac's delta, or complex numbers.

W.N. and dirac delta are nastier theoretical constructs (that are the
basis for brusing fights, just like religion) than complex numbers.
complex numbers are nice, well-behaved little folks that everybody
approves of.  W.N. and delta(t) are the James Dean and Sean Penn
characters.  nasty, but intriguing and sorta likable.

r b-j

```
```Fitlike Min wrote:
> "Chris Barrett" <chrisbarret@0123456789abcdefghijk113322.none> wrote in
> message news:Nwxph.1353\$qA7.1219@newsfe15.lga...
> > 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.
>
> Take the autocorrelation. If it's an impulse then it's white -

ain't no continuous-time autocorrellation for discrete-time samples.
but there *is* a discrete-time autocorrelation that we would hope would
be the "Kronecker delta".  but there ain't no meaning to a dirac-delta
to these discrete samples.

if the discrete-time samples were applied to a bandlimited
reconstruction filter, the output would not be white, but it would be
flat up to Nyquist (at least if they were good random numbers).

r b-j

```
```Fitlike Min wrote:

...

> I don't think a flat spectrum telsl you if it's random or not - you need
> autocorrelation.

Well, a flat spectrum will tell you if it's white, assuming it's noise.
Neither spectral analysis nor autocorrelation can distinguish noise from
impulses.

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.

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?
```
```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.
> > =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
>
> I don't think a flat spectrum telsl you if it's random or not - you need
> autocorrelation.

Yeah, a flat spectrum can be a white noise or a frequency sweep or a
delta function or anything else matter of fact (you could easily
transform your favourite pop song so that it would have a perfect flat
spectrum and it still won't sound like noise). And I don't see how you
could tell a white noise from either a frequency sweep or a delta
function by autocorrelation because both will result in a delta
function due to their flat spectrum.

If you want to make it simple and make sure that your signal is a white
noise just make a spectrogram of it :-)

```
```Fitlike Min wrote:
> "Chris Barrett" <chrisbarret@0123456789abcdefghijk113322.none> wrote in
> message news:Nwxph.1353\$qA7.1219@newsfe15.lga...
> > 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.
>
> Take the autocorrelation. If it's an impulse then it's white - though it may
> not necessarily be Guassian white etc (this has been explained).

I think it will only tell you whether or not the signal has a flat
spectrum, so while it may be a white noise it may be anything else that
has a flat spectrum.

```