Forums

does WGN(white Gaussian noise) imple zero mean?

Started by kiki December 25, 2004
Hi all,

I read through several books but did not get clarification on whether 
WGN(white Gaussian noise process) imply zero mean or not...

Another confusion I have is that the definition of WGN is it has flat power 
spectrum density, let's say S(f)=1, then Rx(t)=delta(t) is its 
autocorrelation function, I don't see how people say the power of this noise 
process is E((x(t))^2)=sigma_x, something like that... the power should be 
infinite, right?

Any clarifications? Thanks a lot!



"kiki" <lunaliu3@yahoo.com> wrote in message
news:cqlbr7$a4m$1@news.Stanford.EDU...
> Hi all, > > I read through several books but did not get clarification on whether > WGN(white Gaussian noise process) imply zero mean or not...
Zero mean.
> > Another confusion I have is that the definition of WGN is it has flat
power
> spectrum density, let's say S(f)=1, then Rx(t)=delta(t) is its > autocorrelation function, I don't see how people say the power of this
noise
> process is E((x(t))^2)=sigma_x, something like that... the power should be > infinite, right?
Power cannot be infinite, unless you have identical waveforms for autocorrelation peak.
> > Any clarifications? Thanks a lot! > > >
"kiki" <lunaliu3@yahoo.com> writes:

> Hi all, > > I read through several books but did not get clarification on whether > WGN(white Gaussian noise process) imply zero mean or not...
Hi Kiki, Now you've got me wondering. On one hand, I've heard the term "zero-mean additive white Gaussian noise" many times, but on the other hand, "white" implies a flat PSD, which in term implies that there is some power at DC. So I can't answer your question.
> Another confusion I have is that the definition of WGN is it has flat power > spectrum density, let's say S(f)=1, then Rx(t)=delta(t) is its > autocorrelation function, I don't see how people say the power of this noise > process is E((x(t))^2)=sigma_x,
Rxx(t) is defined to be Rxx(tau)= E[x(t)*x(t-tau)] for a real random process x(t). Then, by definition, E[x^2(t)] = E[x(t) * x(t-0)] = Rxx(0) = undefined (infinity) when Rxx(t) = delta(t). Thus you're contradicting yourself somewhat. A truly white-noise process does have infinite power (hence the Dirac delta function in the autocorrelation), but most transistors I know of burn out after a few gigawatts, so we usually speak of a band-limited white noise process, i.e., a process which has a PSD Sxx(w) = c, |w| < a, and in which case the power is finite and Rxx(0) = a*c/pi. -- % Randy Yates % "She's sweet on Wagner-I think she'd die for Beethoven. %% Fuquay-Varina, NC % She love the way Puccini lays down a tune, and %%% 919-577-9882 % Verdi's always creepin' from her room." %%%% <yates@ieee.org> % "Rockaria", *A New World Record*, ELO http://home.earthlink.net/~yatescr
kiki wrote:

> Hi all, > > I read through several books but did not get clarification on whether > WGN(white Gaussian noise process) imply zero mean or not...
OK. Think. Hmm. The definition of a white noise process is that the PSD is 1 everywhere. OK, I understand that. If I know the PSD of a function I can find the expected power between any two frequencies by just integrating the PSD over that interval. OK, I've read my books, I understand that. Now, DC means the frequency interval between 0 and 0 (technically between 0- and 0+). Integrating 1 between 0 and 0 I get -- ZERO! WOW!
> > Another confusion I have is that the definition of WGN is it has flat power > spectrum density, let's say S(f)=1, then Rx(t)=delta(t) is its > autocorrelation function, I don't see how people say the power of this noise > process is E((x(t))^2)=sigma_x, something like that... the power should be > infinite, right?
For a truly white PSD the power is infinite (see, I'm not raking you over the coals for this -- this is actually a bit of a brain twister and therefore not a blindingly obvious question). This is actually the mathematical difficulty that led Plank to his discovery of black-body radiation -- before Plank you had to just describe thermal noise in a resistor or whatnot as white noise, and shrug your shoulders when the question of infinite power came up. So for all practical purposes you should remember that white noise of any sort is a mathematical fiction, and only it to predict the responses of physical systems who's bandwidths are much lower than the bandwidth limitation of the noise you have at hand.
> > Any clarifications? Thanks a lot! > > >
-- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Tim Wescott wrote:

  ...

> So for all practical purposes you should remember that white noise of > any sort is a mathematical fiction, and only it to predict the responses > of physical systems who's bandwidths are much lower than the bandwidth > limitation of the noise you have at hand.
Man, I wish I had had you for a teacher, back when it mattered! 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;
Tim Wescott <tim@wescottnospamdesign.com> writes:

> kiki wrote: > >> Hi all, >> I read through several books but did not get clarification on >> whether WGN(white Gaussian noise process) imply zero mean or not... > > OK. Think. > > Hmm. The definition of a white noise process is that the PSD is 1 > everywhere. OK, I understand that. > > If I know the PSD of a function I can find the expected power between > any two frequencies by just integrating the PSD over that interval. > OK, I've read my books, I understand that. > > Now, DC means the frequency interval between 0 and 0 (technically > between 0- and 0+). Integrating 1 between 0 and 0 I get -- ZERO! WOW!
Hey Tim, You get the same result when integrating between 1- and 1+, or 253,392- and 253,392+, etc., and we know have power at there. What have you answered, then? -- % Randy Yates % "With time with what you've learned, %% Fuquay-Varina, NC % they'll kiss the ground you walk %%% 919-577-9882 % upon." %%%% <yates@ieee.org> % '21st Century Man', *Time*, ELO http://home.earthlink.net/~yatescr
Randy Yates wrote:

> Tim Wescott <tim@wescottnospamdesign.com> writes: > > >>kiki wrote: >> >> >>>Hi all, >>>I read through several books but did not get clarification on >>>whether WGN(white Gaussian noise process) imply zero mean or not... >> >>OK. Think. >> >>Hmm. The definition of a white noise process is that the PSD is 1 >>everywhere. OK, I understand that. >> >>If I know the PSD of a function I can find the expected power between >>any two frequencies by just integrating the PSD over that interval. >>OK, I've read my books, I understand that. >> >>Now, DC means the frequency interval between 0 and 0 (technically >>between 0- and 0+). Integrating 1 between 0 and 0 I get -- ZERO! WOW! > > > Hey Tim, > > You get the same result when integrating between 1- and 1+, or > 253,392- and 253,392+, etc., and we know have power at there. > > What have you answered, then?
Well, _I've_ answered that there's no DC content (which is another way of saying zero mean), when you take DC to it's mathematical limit (note that white noise will appear to have DC content if you only observe it for a finite amount of time, such as the time from the big bang to right now). _You've_ extended this to show that you can pick any one, zero-bandwidth, filter and find no energy there. Hopefully I've also pointed out to Kiki that he has all this information at his fingertips if he'd just collate, think, and use a pencil and paper every once in a while. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Tim Wescott <tim@wescottnospamdesign.com> writes:

> Randy Yates wrote: > >> Tim Wescott <tim@wescottnospamdesign.com> writes: >> >>>kiki wrote: >>> >>> >>>>Hi all, >>>>I read through several books but did not get clarification on >>>>whether WGN(white Gaussian noise process) imply zero mean or not... >>> >>>OK. Think. >>> >>>Hmm. The definition of a white noise process is that the PSD is 1 >>>everywhere. OK, I understand that. >>> >>>If I know the PSD of a function I can find the expected power between >>>any two frequencies by just integrating the PSD over that interval. >>>OK, I've read my books, I understand that. >>> >>>Now, DC means the frequency interval between 0 and 0 (technically >>>between 0- and 0+). Integrating 1 between 0 and 0 I get -- ZERO! WOW! >> Hey Tim, >> You get the same result when integrating between 1- and 1+, or >> 253,392- and 253,392+, etc., and we know have power at there. What >> have you answered, then? > > Well, _I've_ answered that there's no DC content (which is another way > of saying zero mean), when you take DC to it's mathematical limit > (note that white noise will appear to have DC content if you only > observe it for a finite amount of time, such as the time from the big > bang to right now). _You've_ extended this to show that you can pick > any one, zero-bandwidth, filter and find no energy there.
No, you've shown that there is no power there. There is indeed energy there since, for white noise, Sxx(w) at w = w0 is strictly greater than zero for any value of w0 (including 0), and the units of power spectral density are [joules] ([watts/Hz] == [joules]). One obtains power upon integration of the Sxx(w) (no matter how small of an integration interval is chosen) since \int_{w_0-}^{w_0+} Sxx(w) dw has units of [joules] * [1/seconds], i.e., power.
> Hopefully I've also pointed out to Kiki that he has all this > information at his fingertips if he'd just collate, think, and use a > pencil and paper every once in a while.
If you confuse me and I've had two classes in it, I can't imagine what's going on in kiki's mind. It is very possible that my mind is screwed on wrong - if you think so, show me where my thinking has gone astray. -- % Randy Yates % "With time with what you've learned, %% Fuquay-Varina, NC % they'll kiss the ground you walk %%% 919-577-9882 % upon." %%%% <yates@ieee.org> % '21st Century Man', *Time*, ELO http://home.earthlink.net/~yatescr
Randy Yates wrote:

> Tim Wescott <tim@wescottnospamdesign.com> writes: > > >>Randy Yates wrote: >> >> >>>Tim Wescott <tim@wescottnospamdesign.com> writes: >>> >>> >>>>kiki wrote: >>>> >>>> >>>> >>>>>Hi all, >>>>>I read through several books but did not get clarification on >>>>>whether WGN(white Gaussian noise process) imply zero mean or not... >>>> >>>>OK. Think. >>>> >>>>Hmm. The definition of a white noise process is that the PSD is 1 >>>>everywhere. OK, I understand that. >>>> >>>>If I know the PSD of a function I can find the expected power between >>>>any two frequencies by just integrating the PSD over that interval. >>>>OK, I've read my books, I understand that. >>>> >>>>Now, DC means the frequency interval between 0 and 0 (technically >>>>between 0- and 0+). Integrating 1 between 0 and 0 I get -- ZERO! WOW! >>> >>>Hey Tim, >>>You get the same result when integrating between 1- and 1+, or >>>253,392- and 253,392+, etc., and we know have power at there. What >>>have you answered, then? >> >>Well, _I've_ answered that there's no DC content (which is another way >>of saying zero mean), when you take DC to it's mathematical limit >>(note that white noise will appear to have DC content if you only >>observe it for a finite amount of time, such as the time from the big >>bang to right now). _You've_ extended this to show that you can pick >>any one, zero-bandwidth, filter and find no energy there. > > > No, you've shown that there is no power there. There is indeed energy > there since, for white noise, Sxx(w) at w = w0 is strictly greater > than zero for any value of w0 (including 0), and the units of power > spectral density are [joules] ([watts/Hz] == [joules]). One obtains > power upon integration of the Sxx(w) (no matter how small of an > integration interval is chosen) since \int_{w_0-}^{w_0+} Sxx(w) dw has > units of [joules] * [1/seconds], i.e., power. >
Oy -- good point. Geeze these limits-to-infinity things get tricky. There must be energy there because if you integrate a white noise process the variance of the result goes up with the integration time. But if you take the average of the white noise process (average = integral / integration time) then the variance goes _down_ with the integration time, eventually going to zero as the integration time goes to infinity. So; zero mean, infinite energy.
> >>Hopefully I've also pointed out to Kiki that he has all this >>information at his fingertips if he'd just collate, think, and use a >>pencil and paper every once in a while. > > > If you confuse me and I've had two classes in it, I can't imagine > what's going on in kiki's mind. It is very possible that my mind > is screwed on wrong - if you think so, show me where my thinking > has gone astray.
I was confused about this stuff, too, and asking questions didn't clear it up. What _did_ clear it up (for the most part; see your comment above) was thinking about it. I had the advantage that I take long bike rides, and for some reason it really worked for me to ponder these questions while riding. This is why I'm trying to get the guy pulled away from Matlab simulations. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
"kiki" <lunaliu3@yahoo.com> writes:

> I read through several books but did not get clarification on > whether WGN(white Gaussian noise process) imply zero mean or not...
For an infinity of samples, yes. But if you have just n samples, the mean you'll get will be 0 only in the mean, but have a variance of V/n where V is the variance of a single sample.
> Another confusion I have is that the definition of WGN is it has > flat power spectrum density, let's say S(f)=1, then Rx(t)=delta(t) > is its autocorrelation function, I don't see how people say the > power of this noise process is E((x(t))^2)=sigma_x, something like > that... the power should be infinite, right?
That's a problem of defining your scale factors in a manner that yields workable results. Fourier transforms for random processes are somewhat special here. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum