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confused about white noise definition

Started by karl bezzoto August 12, 2009
Hello,
Per definition a white noise signal has a zero mean value. yet by
definition also , it has a flat power density, i,e. every frequency
including zero has the same power which means it has a DC value, hence
its mean is not equal to zero.
I am confused. any help please?
On Aug 12, 7:55=A0am, karl bezzoto <karl.bezz...@googlemail.com> wrote:
> Hello, > Per definition a white noise signal has a zero mean value. yet by > definition also , it has a flat power density, i,e. every frequency > including zero has the same power which means it has a DC value, hence > its mean is not equal to zero. > I am confused. any help please?
Hi, There has been a very long discussion on white noise that is not zero mean on this group. I will see if I can find the orignal post and link it here.
On Aug 12, 11:11=A0am, Neu <ikarosi...@hotmail.com> wrote:
> On Aug 12, 7:55=A0am, karl bezzoto <karl.bezz...@googlemail.com> wrote: > > > Hello, > > Per definition a white noise signal has a zero mean value. yet by > > definition also , it has a flat power density, i,e. every frequency > > including zero has the same power which means it has a DC value, hence > > its mean is not equal to zero. > > I am confused. any help please? > > Hi, > > There has been a very long discussion on white noise that is not zero > mean on this group. > I will see if I can find the orignal post and link it here.
Found it: http://groups.google.com/group/comp.dsp/browse_thread/thread/177a51d4376130= 18/6b4628b9a0b0b55a?hl=3Den&q=3D Feel free to contribute :) hth
 > > Hello,
 > > Per definition a white noise signal has a zero mean value. yet by
 > > definition also , it has a flat power density, i,e. every
frequency
 > > including zero has the same power which means it has a DC value,
hence
 > > its mean is not equal to zero.
 > > I am confused. any help please?

 Found it:

 http://groups.google.com/group/comp.dsp/browse_thread/thread/177a51d4...


Also note that when you say "including zero has the same power which
means it has a DC value," it not necessarily true. The condition for
WN is that is the *second order * statistics (ie, autocorrelation) is
an impulse function, which has a flat power spectrum density.

On Aug 12, 4:55 am, karl bezzoto <karl.bezz...@googlemail.com> wrote:
> Hello, > Per definition a white noise signal has a zero mean value. yet by > definition also , it has a flat power density, i,e. every frequency > including zero has the same power which means it has a DC value, hence > its mean is not equal to zero. > I am confused. any help please?
[begin hand waving] In the infinite continuous domain the idealized white noise has a flat power density, not value. As you look at the power in a smaller and smaller interval, the power becomes smaller and smaller. At zero width, a point, the total power in the interval goes to zero for any finite density. [end hand waving] When we have only a finite set of samples, the sample set does not exactly match the characteristics of the theoretical continuous- infinite white noise. Larger sample sets usually get you closer. Dale B. Dalrymple
On 12 Aug, 16:38, dbd <d...@ieee.org> wrote:
> On Aug 12, 4:55 am, karl bezzoto <karl.bezz...@googlemail.com> wrote: > > > Hello, > > Per definition a white noise signal has a zero mean value. yet by > > definition also , it has a flat power density, i,e. every frequency > > including zero has the same power which means it has a DC value, hence > > its mean is not equal to zero. > > I am confused. any help please? > > [begin hand waving] > In the infinite continuous domain the idealized white noise has a flat > power density, not value. As you look at the power in a smaller and > smaller interval, the power becomes smaller and smaller. At zero > width, a point, the total power in the interval goes to zero for any > finite density. > [end hand waving] > > When we have only a finite set of samples, the sample set does not > exactly match the characteristics of the theoretical continuous- > infinite white noise. Larger sample sets usually get you closer. > > Dale B. Dalrymple
thanks to both of you. i'll study your answers more carefully
On 12 Aug, 13:55, karl bezzoto <karl.bezz...@googlemail.com> wrote:
> Hello, > Per definition a white noise signal has a zero mean value. yet by > definition also , it has a flat power density, i,e. every frequency > including zero has the same power which means it has a DC value, hence > its mean is not equal to zero. > I am confused. any help please?
I'd say that a non-zero mean would cause a Dirac Delta at f = 0. The zero-mean white noise would have a non-zero power *denisty* near f = 0, but the contribution to the power would vanish as the bandwidth vanishes. Rune
On Aug 12, 11:38&#2013266080;am, dbd <d...@ieee.org> wrote:
> On Aug 12, 4:55 am, karl bezzoto <karl.bezz...@googlemail.com> wrote: > > > Hello, > > Per definition a white noise signal has a zero mean value. yet by > > definition also , it has a flat power density, i,e. every frequency > > including zero has the same power which means it has a DC value, hence > > its mean is not equal to zero. > > I am confused. any help please? > > [begin hand waving] > In the infinite continuous domain the idealized white noise has a flat > power density, not value. As you look at the power in a smaller and > smaller interval, the power becomes smaller and smaller. At zero > width, a point, the total power in the interval goes to zero for any > finite density. > [end hand waving] > > When we have only a finite set of samples, the sample set does not > exactly match the characteristics of the theoretical continuous- > infinite white noise. Larger sample sets usually get you closer. > > Dale B. Dalrymple
So it is similar to statistics with a continuous variable given pdf (x) the probability at any individual point is zero? Don't you just love mathematics :) Cheers, Dave
Dave wrote:
> On Aug 12, 11:38 am, dbd <d...@ieee.org> wrote: >> On Aug 12, 4:55 am, karl bezzoto <karl.bezz...@googlemail.com> wrote: >> >>> Hello, >>> Per definition a white noise signal has a zero mean value. yet by >>> definition also , it has a flat power density, i,e. every frequency >>> including zero has the same power which means it has a DC value, hence >>> its mean is not equal to zero. >>> I am confused. any help please? >> [begin hand waving] >> In the infinite continuous domain the idealized white noise has a flat >> power density, not value. As you look at the power in a smaller and >> smaller interval, the power becomes smaller and smaller. At zero >> width, a point, the total power in the interval goes to zero for any >> finite density. >> [end hand waving] >> >> When we have only a finite set of samples, the sample set does not >> exactly match the characteristics of the theoretical continuous- >> infinite white noise. Larger sample sets usually get you closer. >> >> Dale B. Dalrymple > > So it is similar to statistics with a continuous variable given pdf > (x) the probability at any individual point is zero? > > Don't you just love mathematics :)
With an infinite number of probability points and a finite sum, what would you expect? 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;
On Aug 14, 9:38&#2013266080;am, Jerry Avins <j...@ieee.org> responded:


> > > So it is similar to statistics with a continuous variable given &#2013266080;pdf > > (x) the probability at any individual point is zero? > > > Don't you just love mathematics &#2013266080;:)
> > With an infinite number of probability points and a finite sum, what > would you expect?
A (countably) infinite number of points *can* have probabilities that sum to 1, but these points *cannot* all have *equal* probability. The sum is, of course, defined in the sense of a limit (this is the topic that causes most people to fall asleep in their calculus classes :-) ) and a sum of the form c + c + c + ... + c does not converge to a finite value as the number of terms in the sum increases. (Sums of the form c + c(1-c) + c(1-c)^2 + ... + c(1-c)^ {n-1} = 1-(1-c)^n do converge to 1 provided that c is in (0, 1] but the terms are not equal ....) For a continuous random variable, the number of possible value is *uncountably* infinite, and the notion of a *sum* of all such values is not defined in the above sense; the corresponding notion is that of an integral or area under the pdf, which we should remember stands for probability *density* function: it is measured in units of probability mass per unit length, and we don't get a probability from the pdf unless we "multiply" by a length or integrate the pdf over an interval. The "reason" that the probability that a continuous variable X equals c is 0 is that the "point" c has zero length (or width if you prefer) and so multiplying the pdf value by the length (or doing an integral if you prefer) gives 0; there is no *area* under the curve above the point of zero width. In short, a good reason for loving mathematics is its insistence that c times 0 is 0 for any real number c (and no, "infinity" is not a real number.....). Dilip Sarwate