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
Does perfect white noise really exist?
Started by ●June 27, 2006
Reply by ●June 27, 20062006-06-27
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
Reply by ●June 27, 20062006-06-27
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. �����������������������������������������������������������������������
Reply by ●June 27, 20062006-06-27
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? > >> > ParthoHello 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
Reply by ●June 27, 20062006-06-27
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
Reply by ●June 27, 20062006-06-27
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. �����������������������������������������������������������������������
Reply by ●June 27, 20062006-06-27
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
Reply by ●June 27, 20062006-06-27
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
Reply by ●June 27, 20062006-06-27
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 :-).
Reply by ●June 27, 20062006-06-27
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
