A simple question: Based on the classical definition of "White Noise"
(a random process which has zero autocorrelation, and a purely flat PSD
from -INF to +INF), measurement of white noise to determine these
properties (rxx, PSD, mean, variance, skewness and kurtosis) poses a
paradox:

When we measure "pure white noise", we ise a band-limited measuring
device (such as a spectrum analyzer), whose pass band is obviously less
than the infinite bandwidth of the pure white noise. Upon measurement,
the measuring device LPFs the noise to pink noise, which in turn
introduces correlation between neighboring noise samples, leaving the
output of the actual measuring black box as patently "non-white".
Hence, what we measure is "assumed" to be white noise, but this
assumption may never be experimentally verified as fact, since the very
act of measurement converts it to some sort of colored noise.

Then how may the existence of pure white noise be verified, if it ever
exists in nature, or is it just another mathematical contraption to
simplify signal modeling?

(PS: This problem is similiar to the paradox of the position and
momentum of elementary particles, aka Heisenberg's principle)

Please comment where I am right or wrong in this direction of
thinking....

Partho

Ben wrote:
> A simple question: Based on the classical definition of "White Noise"
> (a random process which has zero autocorrelation, and a purely flat PSD
> from -INF to +INF), measurement of white noise to determine these
> properties (rxx, PSD, mean, variance, skewness and kurtosis) poses a
> paradox:
>
> When we measure "pure white noise", we ise a band-limited measuring
> device (such as a spectrum analyzer), whose pass band is obviously less
> than the infinite bandwidth of the pure white noise. Upon measurement,
> the measuring device LPFs the noise to pink noise, which in turn
> introduces correlation between neighboring noise samples, leaving the
> output of the actual measuring black box as patently "non-white".
> Hence, what we measure is "assumed" to be white noise, but this
> assumption may never be experimentally verified as fact, since the very
> act of measurement converts it to some sort of colored noise.
>
> Then how may the existence of pure white noise be verified, if it ever
> exists in nature, or is it just another mathematical contraption to
> simplify signal modeling?
>
> (PS: This problem is similiar to the paradox of the position and
> momentum of elementary particles, aka Heisenberg's principle)
>
> Please comment where I am right or wrong in this direction of
> thinking....

I think you are right all along:

- Perfect white noise must, if it exists, have a constant PDF
  on the interval <-inf, inf>
- Hence it has infinite power
- Hence its existence is questionable, at best.
- White noise is a useful *model* for analysis of *bandlimited *
  systems
- There s no such thing as a measurement system with a
  perfect impulse response

Nah, I agree with you on all your points.

Rune

Rune Allnor wrote:
> Ben wrote:
>> A simple question: Based on the classical definition of "White Noise"
>> (a random process which has zero autocorrelation, and a purely flat PSD
>> from -INF to +INF), measurement of white noise to determine these
>> properties (rxx, PSD, mean, variance, skewness and kurtosis) poses a
>> paradox:
>>
>> When we measure "pure white noise", we ise a band-limited measuring
>> device (such as a spectrum analyzer), whose pass band is obviously less
>> than the infinite bandwidth of the pure white noise. Upon measurement,
>> the measuring device LPFs the noise to pink noise, which in turn
>> introduces correlation between neighboring noise samples, leaving the
>> output of the actual measuring black box as patently "non-white".
>> Hence, what we measure is "assumed" to be white noise, but this
>> assumption may never be experimentally verified as fact, since the very
>> act of measurement converts it to some sort of colored noise.
>>
>> Then how may the existence of pure white noise be verified, if it ever
>> exists in nature, or is it just another mathematical contraption to
>> simplify signal modeling?
>>
>> (PS: This problem is similiar to the paradox of the position and
>> momentum of elementary particles, aka Heisenberg's principle)
>>
>> Please comment where I am right or wrong in this direction of
>> thinking....
> 
> I think you are right all along:
> 
> - Perfect white noise must, if it exists, have a constant PDF
>   on the interval <-inf, inf>
> - Hence it has infinite power
> - Hence its existence is questionable, at best.
> - White noise is a useful *model* for analysis of *bandlimited *
>   systems
> - There s no such thing as a measurement system with a
>   perfect impulse response
> 
> Nah, I agree with you on all your points.

Even the model is oversimplified. The noise power of a resistor is given 
a kTR = v^2/R, so the open-circuit noise voltage v = R*sqrt(kT). All 
well and good I suppose, and when the resistor is shorted the current is 
v/R = sqrt(kT). Consider a loop consisting of two resistors (series? 
parallel?). Are the currents the same in each? Kirchoff's laws say yes, 
but that's not reasonable. Proof: make each "resistor" one of the 
windings (rotor, stator) of a series motor. The motor's torque is 
proportional to the square if the winding current, hence unidirectional. 
If Kirchoff's current law applies, we have a perpetual motion machine.

Oh well, another one of my "inventions" shot down! :-)

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

Ben wrote:
noise.
>
> Then how may the existence of pure white noise be verified, if it ever
> exists in nature, or is it just another mathematical contraption to
> simplify signal modeling?
>
>>
> Partho


Hello Partho,

Even thermal noise has an upper limit. At room temperature a resistor
appears to have all frequencies at equal power, but a roll off starts
to occur in the tens of gigahertz.

As you stated measuring devices have frequency response limitations, so
some noise sources as measured by these devices will be
indestinguishable from mathematically ideal white noise. So this
modelling is often okay as long as some assumptions are met. 

Clay

Jerry Avins wrote:

>
> Even the model is oversimplified. The noise power of a resistor is given
> a kTR = v^2/R, so the open-circuit noise voltage v = R*sqrt(kT). All
> well and good I suppose, and when the resistor is shorted the current is
> v/R = sqrt(kT). Consider a loop consisting of two resistors (series?
> parallel?). Are the currents the same in each? Kirchoff's laws say yes,
> but that's not reasonable. Proof: make each "resistor" one of the
> windings (rotor, stator) of a series motor. The motor's torque is
> proportional to the square if the winding current, hence unidirectional.
> If Kirchoff's current law applies, we have a perpetual motion machine.
>
> Oh well, another one of my "inventions" shot down! :-)
>
> Jerry
> --

Jerry,

The derivation for the Johnson Nyquist formula assumes thermal dynamic
equilibrium and zero average current flow. If your noise source is
performing work your temperatures are unequal. Plus Kirchoff's "laws"
don't apply to statistical fluctuations.
Clay

Clay wrote:

> The derivation for the Johnson Nyquist formula assumes thermal dynamic
> equilibrium and zero average current flow. If your noise source is
> performing work your temperatures are unequal. Plus Kirchoff's "laws"
> don't apply to statistical fluctuations.

Clay,

Your last sentence was the point I tried to make obliquely. As for zero 
average current flow, that is true also of any AC device.

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

Jerry Avins wrote:
> Clay wrote:
>
> > The derivation for the Johnson Nyquist formula assumes thermal dynamic
> > equilibrium and zero average current flow. If your noise source is
> > performing work your temperatures are unequal.

OK, and the "obvious" consequence is that with an electric connection
between terminals that are kept at two different temperatures...

> > Plus Kirchoff's "laws"
> > don't apply to statistical fluctuations.
>
> Clay,
>
> Your last sentence was the point I tried to make obliquely. As for zero
> average current flow, that is true also of any AC device.

Does Kirchoff's laws distinguish between deterministic
and stichastic systems? That was new to me.

Rune

Rune Allnor wrote:

> Ben wrote:
> 
>>A simple question: Based on the classical definition of "White Noise"
>>(a random process which has zero autocorrelation, and a purely flat PSD
>>from -INF to +INF), measurement of white noise to determine these
>>properties (rxx, PSD, mean, variance, skewness and kurtosis) poses a
>>paradox:
>>
>>When we measure "pure white noise", we ise a band-limited measuring

-- snip --
>>
>>Please comment where I am right or wrong in this direction of
>>thinking....
> 
> 
> I think you are right all along:
> 
> - Perfect white noise must, if it exists, have a constant PDF
>   on the interval <-inf, inf>
> - Hence it has infinite power

-- snip --

AFAIK this argument was one of the paradoxes in then-current theory that 
motivated Plank's theory of black body radiation, and started off the 
study of quantum mechanics.  Unfortunately I absolutely can't remember 
where I saw this.

-- 

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

Tim Wescott wrote:
...
> AFAIK this argument was one of the paradoxes in then-current theory that
> motivated Plank's theory of black body radiation, and started off the
> study of quantum mechanics.  Unfortunately I absolutely can't remember
> where I saw this.

It's probably one of those things you can't simultaneously know and
remember where you read it :-).

Andor wrote:
> Tim Wescott wrote:
> ...
> > AFAIK this argument was one of the paradoxes in then-current theory that
> > motivated Plank's theory of black body radiation, and started off the
> > study of quantum mechanics.  Unfortunately I absolutely can't remember
> > where I saw this.
>
> It's probably one of those things you can't simultaneously know and
> remember where you read it :-).

Well I know I have read, but I can't remember what. What I know, I
may or may not have read, I don't remember. Read into this what
you want, I know I must remember to stay at sea for another...???

Rune