FPGA Central

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*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?



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*am, Neu <ikarosi...@hotmail.com> wrote:
> On Aug 12, 7:55*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?
>
> 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....0b55a?hl=en&q=


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....hread/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*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.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

On Aug 14, 9:38*am, Jerry Avins <j...@ieee.org> responded:


>
> > 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?


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

[email protected] wrote:
> On Aug 14, 9:38 am, Jerry Avins <j...@ieee.org> responded:
>
>
>>> 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?
>
>
> 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 given distribution D, the probability of an event e occurring
between p1 and p2 is
p2 ( )
Integral(p(D)dp)
e=p1 ( )

The probability of an event occurring at exactly p3 is
p3 ( )
Integral(p(D)dp)
e=p3 ( )
Surely, nobody here needs help evaluating that!

> 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.....).

I don't think we're disagreeing. You put it more rigorously.

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

On 08/12/2009 01:55 PM, karl bezzoto wrote:
> Hello,
> Per definition a white noise signal has a zero mean value. yet by

Is "per definition" or is it a consequence of
the definition of white noise?

bye,

--

piergiorgio

On Aug 12, 7:55*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?

Well, you should be, since they defined white noise with the
mathematical magic of delta functions and generalized functions,
not anything to do with reality.
So, the people who actually understand engineering mostly these
days work on
Electronic Books, Miniature External Hard Disks, Laser Disk
Libraries, Flat Screen Software Debuggers,
Multiplexed Fiber Optics Signal and Conrol Systems, C++, USB, XML,
All-In-One Printers,
HDTV, Home Broadband, Cyber Batteries, Pv Cell Energy, Distributed
Processing Software
Holograms, Cell Phones, Microcomputers, mp3, mpeg, PGP, Desktop
Publishing,
On-Line Banking, On-Line Shopping, Blue Ray, HDTV, On-Line
Publishing, Atomic Clock Wristwatches,
Light Sticks, Self-Assembling Robots, and Self-Replicating Machines
and let just the
Quantum Mechanics, GM, and GE cranks worry about what power means.

karl bezzoto <[email protected]> writes:

> Hello,
> Per definition a white noise signal has a zero mean value.

Really? By whos definition? Neither of two definitions I have
found require the mean to be zero. I examine this question in
some detail here:

http://www.digitalsignallabs.com/white.pdf

--
Randy Yates % "Remember the good old 1980's, when
Digital Signal Labs % things were so uncomplicated?"
mailto://[email protected] % 'Ticket To The Moon'
http://www.digitalsignallabs.com % *Time*, Electric Light Orchestra

Randy Yates wrote:
> karl bezzoto <[email protected]> writes:
>
>> Hello,
>> Per definition a white noise signal has a zero mean value.
>
> Really? By whos definition? Neither of two definitions I have
> found require the mean to be zero. I examine this question in
> some detail here:
>
> http://www.digitalsignallabs.com/white.pdf
>

Can't you say : " If white noise has a nonzero mean, then
white noise is a signal" ( has nonzero information-content ) ,
leading to a contradiction?

--
Les Cargill

Randy Yates <[email protected]> writes:

> karl bezzoto <[email protected]> writes:
>
>> Hello,
>> Per definition a white noise signal has a zero mean value.
>
> Really? By whos definition? Neither of two definitions I have
> found require the mean to be zero. I examine this question in
> some detail here:
>
> http://www.digitalsignallabs.com/white.pdf

OK, I'm back-pedaling on the Brown conclusion in this paper and
have "published" revision PA2.

The Brown definition (constant PSD) DOES imply zero-mean, as
several of you here have already noted. However, I still
maintain that the Papoulis definition does not.

PS: You may have to press "refresh" in your browsers or
clear the cache to get the new version (PA2).
--
Randy Yates % "She's sweet on Wagner-I think she'd die for Beethoven.
Digital Signal Labs % She love the way Puccini lays down a tune, and
mailto://[email protected] % Verdi's always creepin' from her room."
http://www.digitalsignallabs.com % "Rockaria", *A New World Record*, ELO

Les Cargill <[email protected]> writes:

> Randy Yates wrote:
>> karl bezzoto <[email protected]> writes:
>>
>>> Hello,
>>> Per definition a white noise signal has a zero mean value.
>>
>> Really? By whos definition? Neither of two definitions I have found
>> require the mean to be zero. I examine this question in
>> some detail here:
>>
>> http://www.digitalsignallabs.com/white.pdf
>>
>
> Can't you say : " If white noise has a nonzero mean, then
> white noise is a signal" ( has nonzero information-content ) ,
> leading to a contradiction?

Hi Les,

First of all, note that I reversed my conclusion in the Brown definition
(constant PSD) of this paper to conclude such a definition does imply a
zero mean.

What I think you're hinting at is the concept of randomness. How random
is "random"? Obviously a plain DC value isn't random at all. But what
about a DC value with some noise on it? Is that a little random? A lot
random? COMPLETELY (purely) random?

I believe the answer lies in the autocorrelation function (which of
course is uniquely defined by the PSD as well). If the process is
_purely_ random, then there is absolutely no correlation
sample-to-sample. That's what makes it white.

Papoulis' definition is somewhat odd since he decided to utilize
not the autocorrelation function but the autocovariance function,
in which case the mean doesn't matter.

Perhaps I'm saying the same thing over and over... or hopefully this
helps.
--
Randy Yates % "She's sweet on Wagner-I think she'd die for Beethoven.
Digital Signal Labs % She love the way Puccini lays down a tune, and
mailto://[email protected] % Verdi's always creepin' from her room."
http://www.digitalsignallabs.com % "Rockaria", *A New World Record*, ELO

Randy Yates wrote:
> Les Cargill <[email protected]> writes:
>
>> Randy Yates wrote:
>>> karl bezzoto <[email protected]> writes:
>>>
>>>> Hello,
>>>> Per definition a white noise signal has a zero mean value.
>>> Really? By whos definition? Neither of two definitions I have found
>>> require the mean to be zero. I examine this question in
>>> some detail here:
>>>
>>> http://www.digitalsignallabs.com/white.pdf
>>>
>> Can't you say : " If white noise has a nonzero mean, then
>> white noise is a signal" ( has nonzero information-content ) ,
>> leading to a contradiction?
>
> Hi Les,
>
> First of all, note that I reversed my conclusion in the Brown definition
> (constant PSD) of this paper to conclude such a definition does imply a
> zero mean.
>

I saw that. *Kewl*. I always wondered that.

> What I think you're hinting at is the concept of randomness. How random
> is "random"? Obviously a plain DC value isn't random at all. But what
> about a DC value with some noise on it? Is that a little random? A lot
> random? COMPLETELY (purely) random?
>

Well, think CDMA. If there exists a transform T such that a ..
"variable" DC offset may be decoded from something that is otherwise
indistinguishable from noise ( we're assuming the transform is
secret ), then it's not purely random. Obviously, the sequences
in CDMA are *pseudo*random.


> I believe the answer lies in the autocorrelation function (which of
> course is uniquely defined by the PSD as well). If the process is
> _purely_ random, then there is absolutely no correlation
> sample-to-sample. That's what makes it white.
>

Yes, that was very nice.

> Papoulis' definition is somewhat odd since he decided to utilize
> not the autocorrelation function but the autocovariance function,
> in which case the mean doesn't matter.
>

His is generative, and may or may not be useful as a description.

> Perhaps I'm saying the same thing over and over... or hopefully this
> helps.

--
Les Cargill

On Aug 15, 9:49*pm, Randy Yates <ya...@ieee.org> wrote:
> Randy Yates <ya...@ieee.org> writes:
> > karl bezzoto <karl.bezz...@googlemail.com> writes:
>
> >> Hello,
> >> Per definition a white noise signal has a zero mean value.
>
> > Really? By whos definition? Neither of two definitions I have
> > found require the mean to be zero. I examine this question in
> > some detail here:
>
> > *http://www.digitalsignallabs.com/white.pdf
>
> OK, I'm back-pedaling on the Brown conclusion in this paper and
> have "published" revision PA2.
>
> The Brown definition (constant PSD) DOES imply zero-mean, as
> several of you here have already noted. However, I still
> maintain that the Papoulis definition does not.
>
> PS: You may have to press "refresh" in your browsers or
> clear the cache to get the new version (PA2).
> --
> Randy Yates * * * * * * * * * * *% "She's sweet on Wagner-I think she'd die for Beethoven.
> Digital Signal Labs * * * * * * *% *She love the way Puccini lays down a tune, and
> mailto://ya...@ieee.org * * * * *% *Verdi's always creepin' from her room."http://www.digitalsignallabs.com% "Rockaria", *A New World Record*, ELO *

In the Papoulis book he later states from that point on he assumes the
white noise signals have zero mean. This suggests that his definition
of white noise allows it to have a non-zero mean.

Cheers,
Dave

Dave <[email protected]> writes:
> [...]
> In the Papoulis book he later states from that point on he assumes the
> white noise signals have zero mean. This suggests that his definition
> of white noise allows it to have a non-zero mean.

Really? I didn't know that. Nice to know - thanks Dave.

--Randy

--
Randy Yates % "I met someone who looks alot like you,
Digital Signal Labs % she does the things you do,
mailto://[email protected] % but she is an IBM."
http://www.digitalsignallabs.com % 'Yours Truly, 2095', *Time*, ELO